## 1. Homo Economicus and Expected Utility Theory



Bounded rationality has come to broadly encompass models of effective
behavior that weaken, or reject altogether, the idealized conditions
of perfect rationality assumed by models of economic man. In this
section we state what models of economic man are committed to and
their relationship to expected utility theory. In later sections we
review proposals for departing from expected utility theory.



The perfect rationality of *homo economicus* imagines a
hypothetical agent who has complete information about the options
available for choice, perfect foresight of the consequences from
choosing those options, and the wherewithal to solve an optimization
problem (typically of considerable complexity) that identifies an
option which maximizes the agent's personal utility. The meaning
of 'economic man' has evolved from John Stuart
Mill's description of a hypothetical, self-interested individual
who seeks to maximize his personal utility (1844); to Jevon's
mathematization of marginal utility to model an economic consumer
(1871); to Frank Knight's portrayal of the *slot-machine
man* of neo-classical economics (1921), which is Jevon's
*calculator man* augmented with perfect foresight and
determinately specified risk; to the modern conception of an
economically rational economic agent conceived in terms of Paul
Samuelson's *revealed preference* formulation of utility
(1947) which, together with von Neumann and Morgenstern's
axiomatization (1944), changed the focus of economic modeling from
reasoning behavior to choice behavior.



Modern economic theory begins with the observation that human beings
like some consequences better than others, even if they only assess
those consequences hypothetically. A perfectly rational person,
according to the canonical paradigm of synchronic decision making
under risk, is one whose comparative assessments of a set of
consequences satisfies the recommendation to maximize expected
utility. Yet, this recommendation to maximize expected utility
presupposes that qualitative comparative judgments of those
consequences (i.e., preferences) are structured in such a way (i.e.,
satisfy specific axioms) so as to admit a mathematical representation
that places those objects of comparison on the real number line (i.e.,
as inequalities of mathematical expectations), ordered from worst to
best. This structuring of preference through axioms to admit a
numerical representation is the subject of expected utility
theory.


### 1.1 Expected Utility Theory



We present here one such axiom system to derive expected utility
theory, a simple set of axioms for the binary relation \(\succeq\),
which represents the relation "is weakly preferred to".
The objects of comparison for this axiomatization are
*prospects*, which associate probabilities to a fixed set of
consequences, where both probabilities and consequences are known to
the agent. To illustrate, the prospect ([?]EUR10, 1/2;
EUR20, 1/2) concerns two consequences, *losing 10
Euros* and *winning 20 Euros*, each assigned the
probability one-half. A rational agent will prefer this prospect to
another with the same consequences but greater chance of losing than
winning, such as (\(-\)EUR10, 2/3; EUR20, 1/3), assuming
his aim is to maximize his financial welfare. More generally, suppose
that \(X = \{x\_1, x\_2, \ldots, x\_n\}\) is a mutually exclusive and
exhaustive set of consequences and that \(p\_i\) denotes the
probability of \(x\_i\), where each \(p\_i \geq 0\) and \(\sum\_{i}^{n}
p\_i = 1\). A prospect *P* is simply the set of
consequence-probability pairs, \(P = (x\_1, p\_1; \ x\_2, p\_2; \ldots; \
x\_n, p\_n)\). By convention, a prospect's consequence-probability
pairs are ordered by the value of each consequence, from least
favorable to most. When prospects *P*, *Q*, *R* are comparable under a
specific preference relation, \(\succeq\), and the (ordered) set of
consequences *X* is fixed, then prospects may be simply
represented by a vector of probabilities.



The *expected utility hypothesis* Bernoulli (1738) states that
rational agents ought to maximize expected utility. If your
qualitative preferences \(\succeq\) over prospects satisfy the
following three constraints, *ordering*, *continuity*,
and *independence*, then your preferences will maximize
expected utility (Neumann & Morgenstern 1944).



A1.**Ordering**.
The ordering condition states
that preferences are both *complete* and *transitive*.
For all prospects *P*, *Q*, completeness entails that either \(P
\succeq Q\), \(Q \succeq P\), or both \(Q \succeq P\) and \(Q \succeq
P\), written \(P \sim Q\). For all prospects \(P, Q, R\), transitivity
entails that if \(P \succeq Q\) and \(Q \succeq R\), then \(P \succeq
R\).
A2.**Archimedean**.
For all prospects \(P, Q,
R\) such that \(P \succeq Q\) and \(Q \succeq R\), then there exists
some \(p \in (0,1)\) such that \((P, p; \ R, (1-p)) \sim Q\), where
\((P, p; R, (1-p))\) is the *compound prospect* that yields the
prospect *P* as a consequence with probability *p* or yields
the prospect *R* with probability
 \(1-p\).[1]
A3.**Independence**.
For all prospects \(P, Q,
R\), if \(P \succeq Q\), then 

 \[(P, p; \ R, (1-p)) \succeq (Q, p; \ R, (1-p))\] for all *p*.


Specifically, if A1, A2, and A3 hold, then there is a real-valued
function \(V(\cdot)\) of the form

 

\[\label{eq:seu}
 V(P) = \sum\_i (p\_i \cdot u(x\_i))\]


where *P* is any prospect and \(u(\cdot)\) is a von Neumann and
Morgenstern utility function defined on the set of consequences
*X*, such that \(P \succeq Q\) if and only if \(V(P) \geq V(Q)\).
In other words, if your qualitative comparative judgments of prospects
at a given time satisfy A1, A2, and A3, then those qualitative
judgments are representable numerically by inequalities of functions
of the form \(V(\cdot)\), yielding a logical calculus on an interval
scale for determining the consequences of your qualitative comparative
judgments at that time.


### 1.2 Axiomatic Departures from Expected Utility Theory



It is commonplace to explore alternatives to an axiomatic system and
expected utility theory is no exception. To be clear, not all
departures from expected utility theory are candidates for modeling
bounded rationality. Nevertheless, some confusion and misguided
rhetoric over how to approach the problem of modeling bounded
rationality stems from unfamiliarity with the breadth of contemporary
statistical decision theory. Here we highlight some axiomatic
departures from expected utility theory that are motivated by bounded
rationality considerations, all framed in terms of our particular
axiomatization from
 section 1.1.


#### 1.2.1 Alternatives to A1



Weakening the ordering axiom introduces the possibility for an agent
to forgo comparing a pair of alternatives, an idea both Keynes and
Knight advocated (Keynes 1921; Knight 1921). Specifically, dropping
the completeness axiom allows an agent to be in a position to neither
prefer one option to another nor be indifferent between the two
(Koopman 1940; Aumann 1962; Fishburn 1982). Decisiveness, which the
completeness axiom encodes, is more mathematical convenience than
principle of rationality. The question, which is the question that
every proposed axiomatic system faces, is what logically follows from
a system which allows for incomplete preferences. Led by Aumann
(1962), early axiomatizations of rational incomplete preferences were
suggested by Giles (1976) and Giron & Rios (1980), and later
studied by Karni (1985), Bewley (2002), Walley (1991), Seidenfeld,
Schervish, & Kadane (1995), Ok (2002), Nau (2006),
Galaabaatar & Karni (2013) and Zaffalon & Miranda (2017). In
addition to accommodating indecision, such systems also allow for you
to reason about someone else's (possibly) complete preferences
when your information about that other agent's preferences is
incomplete.



Dropping transitivity limits extendability of elicited preferences
(Luce & Raiffa 1957), since the omission of transitivity as an
axiomatic constraint allows for cycles and preference reversals.
Although violations of transitivity have been long considered both
commonplace and a sign of human irrationality (May 1954; Tversky
1969), reassessments of the experimental evidence challenge this
received view (Mongin 2000; Regenwetter, Dana, & Davis-Stober
2011). The axioms impose synchronic consistency constraints on
preferences, whereas the experimental evidence for violations of
transitivity commonly conflate dynamic and synchronic consistency
(Regenwetter et al. 2011). Specifically, a person's preferences
at one moment in time that are inconsistent with his preferences at
another time is no evidence for that person holding logically
inconsistent preferences at a single moment in time. Arguments to
limit the scope of transitivity in normative accounts of rational
preference similarly point to diachronic or group preferences, which
likewise do not contradict the axioms (Kyburg 1978; Anand 1987;
Bar-Hillel & Margalit 1988; Schick 1986). Arguments that point to
psychological processes or algorithms that admit cycles or reversals
of preference over time also point to a misapplication of, rather than
a counter-example to, the ordering condition. Finally, for decisions
that involve explicit comparisons of options over time, violating
transitivity may be rational. For example, given the goal of
maximizing the rate of food gain, an organism's current food
options may reveal information about food availability in the near
future by indicating that a current option may soon disappear or that
a better option may soon reappear. Information about availability of
options over time can, and sometimes does, warrant non-transitive
choice behavior over time that maximizes food gain
(McNamara, Trimmer, & Houston 2014).


#### 1.2.2 Alternatives to A2



Dropping the Archimedean axiom allows for an agent to have
*lexicographic preferences* (Blume, Brandenburger, & Dekel
1991); that is, the omission of
 A2
 allows the possibility for an agent to prefer one option infinitely
more than another. One motivation for developing a non-Archimedean
version of expected utility theory is to address a gap in the
foundations of the standard subjective utility framework that prevents
a full reconciliation of *admissibility* (i.e., the principle
that one ought not select a weakly dominated option for choice) with
*full conditional preferences* (i.e., that for any event, there
is a well-defined conditional probability to represent the
agent's conditional preferences; Pedersen 2014). Specifically,
the standard subjective expected utility account cannot accommodate
conditioning on zero-probability events, which is of particular
importance to game theory (P. Hammond 1994). Non-Archimedean variants
of expected utility theory turn to techniques from nonstandard
analysis (Goldblatt 1998), full conditional probabilities
(Renyi 1955; Coletii & Scozzafava 2002; Dubins 1975; Popper
1959), and lexicographic probabilities (Halpern 2010; Brickhill &
Horsten 2016
 [Other Internet Resources]),
 and are all linked to imprecise probability theory.



Non-compensatory single-cue decision models, such as the Take-the-Best
heuristic
 (section 7.2),
 appeal to lexicographically ordered cues, and admit a numerical
representation in terms of non-Archimedean expectations
(Arlo-Costa & Pedersen 2011).


#### 1.2.3 Alternatives to A3



A1
 and
 A2
 together entail that \(V(\cdot)\) assigns a real-valued index to
prospects such that \(P \succeq Q\) if and only if \(V(P) \geq V(Q)\).
The independence axiom,
 A3,
 encodes a separability property for choice, one that ensures that
expected utilities are linear in probabilities. Motivations for
dropping the independence axiom stem from difficulties in applying
expected utility theory to describe choice behavior, including an
early observation that humans evaluate possible losses and possible
gains differently. Although expected utility theory can represent a
person who either gambles or purchases insurance, Friedman and Savage
remarked in their early critique of von Neumann and Morgenstern's
axiomatization, it cannot simultaneously do both (M. Friedman &
Savage 1948).



The principle of loss aversion (Kahneman & Tversky 1979; Rabin
2000) suggests that the subjective weight that we assign to potential
losses is larger than those we assign to potential gains. For example,
the *endowment effect* (Thaler 1980)--the observation that
people tend to view the value of a good higher when viewed as a
potential loss than when viewed as a potential gain--is supported
by neurological evidence for gains and losses being processed by
different regions of the brain (Rick 2011). However, even granting the
affective differences in how we process losses and gains, those
differences do not necessarily translate to a general
"negativity bias" (Baumeister, Bratslavsky, &
Finkenauer 2001) in choice behavior (Hochman & Yechiam 2011;
Yechiam & Hochman 2014). Yechiam and colleagues report experiments
in which participants do not exhibit loss aversion in their choices,
such as cases in which participants respond to repetitive situations
that issue losses and gains and single-case decisions involving small
stakes. That said, observations of risk aversion (Allais 1953) and
ambiguity aversion (Ellsberg 1961) have led to alternatives to
expected utility theory, all of which abandon
 A3.
 Those alternative approaches include prospect theory
 (section 2.4),
 regret theory (Bell 1982; Loomes & Sugden 1982), and
rank-dependent expected utility (Quiggin 1982).



Most models of bounded rationality do not even fit into this broad
axiomatic family just outlined. One reason is that bounded rationality
has historically emphasized the procedures, algorithms, or
psychological processes involved in making a decision, rendering a
judgment, or securing a goal
 (section 2).
 Samuelson's shift from reasoning behavior to choice behavior
abstracted away precisely these details, however, treating them as
outside the scope of rational choice theory. For Simon, that was
precisely the problem. A second reason is that bounded rationality
often focuses on adaptive behavior suited to an organism's
environment
 (section 3).
 Since ecological modeling involves goal-directed behavior mitigated
by the constitution of the organism and stable features of its
environment, focusing on (synchronically) coherent comparative
judgments is often not, directly at least, the best way to frame the
problem.



That said, one should be cautious about generalizations sometimes made
about the limited role of decision theoretic tools in the study of
bounded rationality. Decision theory--broadly construed to
include statistical decision theory (Berger 1980)--offers a
powerful mathematical toolbox even though historically, particularly
in its canonical form, it has traded in psychological myths such as
"degrees of belief" and logical omniscience
 (section 1.3).
 One benefit of studying axiomatic departures from expected utility
theory is to loosen the grip of Bayesian dogma to expand the range of
possibilities for applying a growing body of practical and powerful
mathematical methods.


### 1.3 Limits to Logical Omniscience



Most formal models of judgment and decision making entail *logical
omniscience*--complete knowledge of all that logically
follows from one's current commitments combined with any set of
options considered for choice--which is as psychologically
unrealistic as it is difficult, technically, to avoid (Stalnaker
1991). A descriptive theory that presumes or a prescriptive theory
that recommends to disbelieve a claim when the evidence is logically
inconsistent, for example, will be unworkable when the belief in
question is sufficiently complicated for all but logically omniscient
agents, even for non-omniscient agents that nevertheless have access
to unlimited computational resources (Kelly & Schulte 1995).



The problem of logical omniscience is particularly acute for expected
utility theory in general, and the theory of subjective probability in
particular. For the postulates of subjective probability imply that an
agent knows all the logical consequences of her commitments, thereby
mandating logical omniscience. This limits the applicability of the
theory, however. For example, it prohibits having uncertain judgments
about mathematical and logical statements. In an article from 1967,
"Difficulties in the theory of personal probability",
reported in Hacking 1967 and Seidenfeld, Schervish, & Kadane 2012
but misprinted in Savage 1967, Savage raises the problem of logical
omniscience for the subjective theory of probability:



> 
> 
> The analysis should be careful not to prove too much; for some
> departures from theory are inevitable, and some even laudable. For
> example, a person required to risk money on a remote digit of \(\pi\)
> would, in order to comply fully with the theory, have to compute that
> digit, though this would really be wasteful if the cost of computation
> were more than the prize involved. For the postulates of the theory
> imply that you should behave in accordance with the logical
> implication of all that you know. Is it possible to improve the theory
> in this respect, making allowances within it for the cost of thinking,
> or would that entail paradox, as I am inclined to believe but unable
> to demonstrate? (Savage 1967 excerpted from Savage's
> prepublished draft; see notes in Seidenfeld et al. 2012)
> 
> 
> 



Responses to Savage's problem include a game-theoretic treatment
proposed by I.J. Good (1983), which swaps the extensional variable
that is necessarily true for an intensional variable representing an
accomplice who knows the necessary truth but withholds enough
information from you for you to be (coherently) uncertain about what
he knows. This trick changes the subject of your uncertainty, from a
necessarily true proposition that you cannot coherently doubt to a
coherent guessing game about that truth facilitated by your
accomplice's incomplete description. Another response sticks to
the classical line that failures of logical omniscience are deviations
from the normative standard of perfect rationality but introduces an
index for incoherence to accommodate reasoning with incoherent
probability assessments (Schervish, Seidenfeld, & Kadane 2012). A
third approach, suggested by de Finetti (1970), is to restrict
possible states of affairs to observable states with a finite
verifiable procedure--which may rule out theoretical states or
any other that does not admit a verification protocol. Originally,
what de Finetti was after was a principled way to construct a
partition over possible outcomes to distinguish serious possible
outcomes of an experiment from wildly implausible but logically
possible outcomes, yielding a method for distinguishing between
genuine doubt and mere "paper doubts" (Peirce 1955). Other
proposals follow de Finetti's line by tightening the
admissibility criteria and include *epistemically possible*
events, which are events that are logically consistent with the
agent's available information; *apparently possible*
events, which include any event by default unless the agent has
determined that it is inconsistent with his information; and
*pragmatically possible* events, which only includes events
that are judged sufficiently important (Walley 1991: 2.1).



The notion of *apparently possible* refers to a procedure for
determining inconsistency, which is a form of bounded procedural
rationality
 (section 2).
 The challenges of avoiding paradox, which Savage alludes to, are
formidable. However, work on bounded fragments of Peano arithmetic
(Parikh 1971) provide coherent foundations for exploring these ideas,
which have been taken up specifically to formulate bounded-extensions
of default logic for *apparent possibility* (Wheeler 2004) and
more generally in models of *computational rationality* (Lewis,
Howes, & Singh 2014).


### 1.4 Descriptions, Prescriptions, and Normative Standards



It is commonplace to contrast how people render judgments, or make
decisions, from how they ought to do so. However, interest in
cognitive processes, mechanisms, and algorithms of boundedly rational
judgment and decision making suggests that we instead distinguish
among three aims of inquiry rather than these two. Briefly, a
*descriptive theory* aims to explain or predict what judgments
or decisions people in fact make; a *prescriptive theory* aims
to explain or recommend what judgments or decisions people ought to
make; a *normative theory* aims to specify a normative standard
to use in evaluating a judgment or decision.



To illustrate each type, consider a domain where differences between
these three lines of inquiry are especially clear: arithmetic. A
descriptive theory of arithmetic might concern the psychology of
arithmetical reasoning, a model of approximate numeracy in animals, or
an algorithm for implementing arbitrary-precision arithmetic on a
digital computer. The normative standard of full arithmetic is
Peano's axiomatization of arithmetic, which distills natural
number arithmetic down to a function for one number succeeding another
and mathematical induction. But one might also consider
Robinson's induction-free fragment of Peano arithmetic (Tarski,
Mostowski, & Robinson 1953) or axioms for some system of cardinal
arithmetic in the hierarchy for large cardinals. A prescriptive theory
for arithmetic will reference both a fixed normative standard and
relevant facts about the arithmetical capabilities of the organism or
machine performing arithmetic. A curriculum for improving the
arithmetical performance of elementary school children will differ
from one designed to improve the performance of adults. Even though
the normative standard of Peano arithmetic is the same for both
children and adults, stable psychological differences in these two
populations may warrant prescribing different approaches for improving
their arithmetic. Continuing, even though Peano's axioms are the
normative standard for full arithmetic, nobody would prescribe
Peano's axioms for the purpose of improving anyone's sums.
There is no mistaking Peano's axioms for a descriptive theory of
arithmetical reasoning, either. Even so, a descriptive theory of
arithmetic will presuppose the Peano axioms as the normative standard
for full arithmetic, even if only implicitly. In describing how people
sum two numbers, after all, one presumes that they are attempting to
sum two numbers rather than concatenate them, count out in sequence,
or send a message in code.



Finally, imagine an effective pedagogy for teaching arithmetic to
children is known and we wish to introduce children to cardinal
arithmetic. A reasonable start on a prescriptive theory for cardinal
arithmetic for children might be to adapt as much of the successful
pedagogy for full arithmetic as possible while anticipating that some
of those methods will not survive the change in normative standards
from Peano to (say) ZFC+. Some of those differences can be seen as a
direct consequence of the change from one standard to another, while
other differences may arise unexpectedly from the observed interplay
between the change in task, that is, from performing full arithmetic
to performing cardinal arithmetic, and the psychological capabilities
of children to perform each task.



To be sure, there are important differences between arithmetic and
rational behavior. The objects of arithmetic, numerals and the numbers
they refer to, are relatively clear cut, whereas the objects of
rational behavior vary even when the same theoretical machinery is
used. Return to expected utility theory as an example. An agent may be
viewed as deliberating over options with the aim to choose one that
maximizes his personal welfare, or viewed to act as if he deliberately
does so without actually doing so, or understood to do nothing of the
kind but to instead be a bit part player in the population fitness of
his kind.



Separating the question of how to choose a normative standard from
questions about how to evaluate or describe behavior is an important
tool to reduce misunderstandings that arise in discussions of bounded
rationality. Even though Peano's axioms would never be
prescribed to improve, nor proposed to describe, arithmetical
reasoning, it does not follow that the Peano axioms of arithmetic are
irrelevant to descriptive and prescriptive theories of arithmetic.
While it remains an open question whether the normative standards for
human rational behavior admit axiomatization, there should be little
doubt over the positive role that clear normative standards play in
advancing our understanding of how people render judgments, or make
decisions, and how they ought to do so.


## 2. The Emergence of Procedural Rationality



Simon thought the shift in focus from reasoning behavior to choice
behavior was a mistake. Since, in the 1950s, little was known about
the processes involved in making judgments or reaching decisions, we
were not in the position to freely abstract away all of those features
from our mathematical models. Yet, this ignorance of the psychology of
decision-making also raised the question of how to proceed. The answer
was to attend to the costs in effort from operating a procedure for
making decisions and comparing those costs to the resources available
to the organism using the procedure and, conversely, to compare how
well an organism performs in terms of accuracy
(section 8.2) with its limited
cognitive resources in order to investigate models with comparable
levels of accuracy within those resource bounds. Effectively managing
the trade-off between the costs and quality of a decision involves
another type of rationality, which Simon later called *procedural
rationality* (Simon 1976: 69).



In this section we highlight early, key contributions to modeling
procedures for boundedly rational judgment and decision-making,
including the origins of the *accuracy-effort trade-off*,
Simon's *satisficing* strategy, *improper linear
models*, and the earliest effort to systematize several features
of high-level, cognitive judgment and decision-making: *cumulative
prospect theory*.


### 2.1 Accuracy and Effort



Herbert Simon and I.J. Good were each among the first to call
attention to the cognitive demands of subjective expected utility
theory, although neither one in his early writings abandoned the
principle of expected utility as the normative standard for rational
choice. Good, for instance, referred to the recommendation to maximize
expected utility as the *ordinary principle* of rationality,
whereas Simon called the principle *objective rationality* and
considered it the central tenet of *global rationality*. The
rules of rational behavior are costly to operate in both time and
effort, Good observed, so real agents have an interest in minimizing
those costs (Good 1952: 7(i)). Efficiency dictates that one choose
from available alternatives an option that yields the largest result
given the resources available, which Simon emphasized is not
necessarily an option that yields the largest result overall (Simon
1947: 79). So reasoning judged deficient without considering the
associated costs may be found meritorious once all those costs are
accounted for--a conclusion that a range of authors soon came to
endorse, including Amos Tversky:



> 
> 
> It seems impossible to reach any definitive conclusions concerning
> human rationality in the absence of a detailed analysis of the
> sensitivity of the criterion and the cost involved in evaluating the
> alternatives. When the difficulty (or the costs) of the evaluations
> and the consistency (or the error) of the judgments are taken into
> account, a [transitivity-violating method] may prove superior.
> (Tversky 1969)
> 
> 
> 



Balancing the quality of a decision against its costs soon became a
popular conception of bounded rationality, particularly in economics
(Stigler 1961), where it remains commonplace to formulate boundedly
rational decision-making as a constrained optimization problem. On
this view boundedly rational agents are utility maximizers after all,
once all the constraints are made clear (Arrow 2004). Another reason
for the popularity of this conception of bounded rationality is its
compatibility with Milton Friedman's *as if* methodology
(M. Friedman 1953), which licenses models of behavior that ignore the
causal factors underpinning judgment and decision making. To say that
an agent behaves *as if* he is a utility maximizer is at once
to concede that he is not but that his behavior proceeds as if he
were. Similarly, to say that an agent behaves as if he is a utility
maximizer under certain constraints is to concede that he does not
solve constrained optimization problems but nevertheless behaves as if
he did.



Simon's focus on computationally efficient methods that yield
solutions that are good enough contrasts with Friedman's as if
methodology, since evaluating whether a solution is "good
enough", in Simon's terms, involves search procedures,
stopping criteria, and how information is integrated in the course of
making a decision. Simon offers several examples to motivate inquiry
into computationally efficient methods. Here is one. Applying the
game-theoretic minimax algorithm to the game of chess calls for
evaluating more chess positions than the number of molecules in the
universe (Simon 1957a: 6). Yet if the game of chess is beyond the reach
of exact computation, why should we expect everyday problems to be any
more tractable? Simon's question is to explain how human beings
manage to solve complicated problems in an uncertain world given their
meager resources. Answering Simon's question, as opposed to
applying Friedman's method to fit a constrained optimization
model to observed behavior, is to demand a model with better
predictive power concerning boundedly rational judgment and decision
making. In pressing this question of how human beings solve uncertain
inference problems, Simon opened two lines of inquiry that continue to
today, namely:


1. How do human beings actually make decisions "in the
wild"?
2. How can the standard theories of global rationality be simplified to
render them more tractable?



Simon's earliest efforts aimed to answer the second question
with, owing to the dearth of psychological knowledge at the time about
how people actually make decisions, only a layman's
"acquaintance with the gross characteristics of human
choice" (Simon 1955a: 100). His proposal was to replace the
optimization problem of maximizing expected utility with a simpler
decision criterion he called *satisficing*, and by models with
better predictive power more generally.


### 2.2 Satisficing



Satisficing is the strategy of considering the options available to
you for choice until you find one that meets or exceeds a predefined
threshold--your aspiration level--for a minimally acceptable
outcome. Although Simon originally thought of procedural rationality
as a poor approximation of global rationality, and thus viewed the
study of bounded rationality to concern "the behavior of human
beings who *satisfice* because they have not the wits to
*maximize*" (Simon 1957a: xxiv), there are a range of
applications of satisficing models to sequential choice problems,
aggregation problems, and high-dimensional optimization problems,
which are increasingly common in machine learning.



Given a specification of what will count as a good-enough outcome,
satisficing replaces the optimization objective from expected utility
theory of selecting an undominated outcome with the objective of
picking an option that meets your aspirations. The model has since
been applied to business (Bazerman & Moore 2008; Puranam,
Stieglitz, Osman, & Pillutla 2015), mate selection (Todd &
Miller 1999) and other practical sequential-choice problems, like
selecting a parking spot (Hutchinson, Fanselow, et al. 2012). Ignoring
the procedural aspects of Simon's original formulation of
satisficing, if one has a fixed aspirational level for a given
decision problem, then admissible choices from satisficing can be
captured by so-called \(\epsilon\)-efficiency methods (Loridan 1984;
White 1986).



Hybrid optimization-satisficing techniques are used in machine
learning when many metrics are available but no sound or practical
method is available for combining them into a single value. Instead,
hybrid optimization-satisficing methods select one metric to optimize
and satisfice the remainder. For example, a machine learning
classifier might optimize accuracy (i.e., maximize the proportion of
examples for which the model yields the correct output; see
 section 8.2)
 but set aspiration levels for the false positive rate, coverage, and
runtime.



Selten's *aspiration adaption theory* models decision
tasks as problems with multiple incomparable goals that resist
aggregation into a complete preference order over all alternatives
(Selten 1998). Instead, the decision-maker will have a vector of goal
variables, where those vectors are comparable by weak dominance. If
vector A and vector B are possible assignments for my goals, then A
dominates vector B if there is no goal in the sequence in which B
assigns a value that is strictly less than A, and there is some goal
for which A assigns a value strictly greater than B. Selten's
model imagines an aspiration level for each goal, which itself can be
adjusted upward or downwards depending on the set of feasible
(admissible) options. Aspiration adaption theory is a highly
procedural and local account in the tradition of Newell and
Simon's approach to human problem solving (Newell & Simon
1972), although it was not initially offered as a psychological
process model. Analogous approaches have been explored in the AI
planning literature (Bonet & Geffner 2001; Ghallab, Nau, &
Traverso 2016).


### 2.3 Proper and Improper Linear Models



Proper linear models represent another important class of optimization
models. A proper linear model is one where predictor variables are
assigned weights, which are selected so that the linear combination of
those weighted predictor variables optimally predicts a target
variable of interest. For example, linear regression is a proper
linear model that selects weights such that the squared
"distance" between the model's predicted value of
the target variable and the actual value (given in the data set) is
minimized.



Paul Meehl's review in the 1950s of psychological studies using
statistical methods versus clinical judgment cemented the statistical
turn in psychology (Meehl 1954). Meehl's review found that
studies involving the prediction of a numerical target variable from
numerical predictors is better done by a proper linear model than by
the intuitive judgment of clinicians. Concurrently, the psychologist
Kenneth Hammond formulated Brunswik's lens model
 (section 3.2)
 as a composition of proper linear models to model the differences
between clinical versus statistical predictions (K. Hammond 1955).
Proper linear models have since become a workhorse in cognitive
psychology in areas that include decision analysis (Keeney &
Raiffa 1976; Kaufmann & Wittmann 2016), causal inference
(Waldmann, Holyoak, & Fratianne 1995; Spirtes 2010), and
response-times to choice (Brown & Heathcote 2008; Turner,
Rodriguez, et al. 2016).



Robin Dawes, returning to Meehl's question about statistical
versus clinical predictions, found that even improper linear models
perform better than clinical intuition (Dawes 1979). The
distinguishing feature of improper linear models is that the weights
of a linear model are selected by some non-optimal method. For
instance, equal weights might be assigned to the predictor variables
to afford each equal weight or a unit-weight, such as 1 or [?]1, to tally features supporting a positive or negative prediction,
respectively. As an example, Dawes proposed an improper model to
predict subjective ratings of marital happiness by couples based on
the difference between their rates of lovemaking and fighting. The
results? Among the thirty happily married couples, two argued more
than they had intercourse. Yet all twelve unhappy couples fought more
frequently. And those results replicated in other laboratories
studying human sexuality in the 1970s. Both equal-weight regression
and unit-weight *tallying* have since been found to commonly
outperform proper linear models on small data sets. Although no simple
improper linear model performs well across all common benchmark
datasets, for almost every data set in the benchmark there is some
simple improper model that performs well in predictive accuracy
(Lichtenberg & Simsek 2016). This observation, and many others in
the heuristics literature, points to biases of simplified models that
can lead to better predictions when used in the right circumstances
 (section 4).



Dawes's original point was not that improper linear models
outperform proper linear models in terms of accuracy, but rather that
they are more efficient and (often) close approximations of proper
linear models. "The statistical model may integrate the
information in an optimal manner", Dawes observed, "but it
is always the individual ...who chooses variables" (Dawes
1979: 573). Moreover, Dawes argued that it takes human judgment to
know the direction of influence between predictor variables and target
variables, which includes the knowledge of how to numerically code
those variables to make this direction clear. Recent advances in
machine learning chip away at Dawes's claims about the unique
role of human judgment, and results from Gigerenzer's ABC Group
about unit-weight tallying outperforming linear regression in
out-of-sample prediction tasks with small samples is an instance of
improper linear models outperforming proper linear models (Czerlinski,
Gigerenzer, & Goldstein 1999). Nevertheless, Dawes's general
observation about the relative importance of variable selection over
variable weighting stands (Katsikopoulos, Schooler, & Hertwig
2010).


### 2.4 Cumulative Prospect Theory



If both satisficing and improper linear models are examples addressing
Simon's second question at the start of this
section--namely, how to simplify existing models to render them
both tractable and effective--then Daniel Kahneman and Amos
Tversky's *cumulative prospect theory* is among the first
models to directly incorporate knowledge about how humans actually
make decisions.



In our discussion in
 section 1.1
 about alternatives to the Independence Axiom,
 (A3),
 we mentioned several observed features of human choice behavior that
stand at odds with the prescriptions of expected utility theory.
Kahneman and Tversky developed prospect theory around four of those
observations about human decision-making (Kahneman & Tversky 1979;
Wakker 2010).


1. **Reference Dependence**. Rather than make decisions by
comparing the absolute magnitudes of welfare, as prescribed by
expected utility theory, people instead tend to value prospects by
their change in welfare with respect to a reference point. This
reference point can be a person's current state of wealth, an
aspiration level, or a hypothetical point of reference from which to
evaluate options. The intuition behind reference dependence is that
our sensory organs have evolved to detect changes in sensory stimuli
rather than store and compare absolute values of stimuli. Therefore,
the argument goes, we should expect to see the cognitive mechanisms
involved in decision-making to inherit this sensitivity to changes in
perceptual attributes values.



In prospect theory, reference dependence is reflected by utility
changing sign at the origin of the valuation curve \(v(\cdot)\) in
 Figure 1(a).
 The *x*-axis represents gains (right side) and losses (left
side) in euros, and *y*-axis plots the value placed on relative
gains and losses by a valuation function \(v(\cdot)\), which is fit to
experimental data on people's choice behavior.
2. **Loss Aversion**. People are more sensitive to losses
than gains of the same magnitude; the thrill of victory does not
measure up to the agony of defeat. So, Kahneman and Tversky
maintained, people will prefer an option that does not incur a loss to
an alternative option that yields an equivalent gain. The disparity in
how potential gains and losses are evaluated also accounts for the
endowment effect, which is the tendency for people to value a good
that they own more than a comparatively valued substitute (Thaler
1980).



In prospect theory, loss aversion appears in
 Figure 1(a)
 in the (roughly) steeper slope of \(v(\cdot)\) to the left of the
origin, representing losses relative to the subject's reference
point, than the slope of \(v(\cdot)\) for gains on the right side of
the reference point. Thus, for the same magnitude of change in reward
*x* from the reference point, the magnitude of the consequence
of gaining *x* is less than the magnitude of losing *x*.



Note that differences in affective attitudes toward, and the
neurological processes responsible for processing, losses and gains do
not necessarily translate to differences in people's choice
behavior (Yechiam & Hochman 2014). The role and scope that loss
aversion plays in judgment and decision making is less clear than was
initially assumed
 (section 1.2).
3. **Diminishing Returns for both Gains and Losses**. Given
a fixed reference point, people's sensitivity to changes in
asset values (*x* in
 Figure 1a)
 diminish the further one moves from that reference point, both in the
domain of losses and the domain of gains. This is inconsistent with
expected utility theory, even when the theory is modified to
accommodate diminishing marginal utility (M. Friedman & Savage
1948).



In prospect theory, the valuation function \(v(\cdot)\) is concave for
gains and convex for losses, representing a diminishing sensitivity to
both gains and losses. Expected utility theory can be made to
accommodate sensitivity effects, but the utility function is typically
either strictly concave or strictly convex, not both.
4. **Probability Weighting**. Finally, for known exogenous
probabilities, people do not calibrate their subjective probabilities
by direct inference (Levi 1977), but instead systematically
underweight high-probability events and overweight low-probability
events, with a cross-over point of approximately one-third
 (Figure 1b).
 Thus, changes in very small or very large probabilities have greater
impact on the evaluation of prospects than they would under expected
utility theory. People are willing to pay more to reduce the number of
bullets in the chamber of a gun from 1 to 0 than from 4 bullets to 3
in a hypothetical game of Russian roulette.



Figure 1(b)
 plots the median values for the probability weighting function
\(w(\cdot)\) that takes the exogenous probability *p* associated
with prospects, as reported in Tversky & Kahneman 1992. Roughly,
below probability values of one-third people overestimate the
probability of an outcome (consequence), and above probability
one-third people tend to underestimate the probability of an outcome
occurring. Traditionally, overweighting is thought to concern the
systematic miscalibration of people's subjective estimates of
outcomes against a known exogenous probability, *p*, serving as
the reference standard. In support of this view, miscalibration
appears to disappear when people learn a distribution through sampling
instead of learning identical statistics by description (Hertwig,
Barron, Weber, & Erev 2004). Miscalibration in this context ought
to be distinguished from overestimating or underestimating subjective
probabilities when the relevant statistics are not supplied as part of
the decision task. For example, televised images of the aftermath of
airplane crashes lead to an overestimation of the low-probability
event of commercial airplanes crashing. Even though a person's
subjective probability of the risk of a commercial airline crash would
be too high given the statistics, the mechanism responsible is
different: here the *recency* or *availability* of
images from the evening news is to blame for scaring him out of his
wits, not the sober fumbling of a statistics table. An alternative
view maintains that people understand that their weighted
probabilities are different than the exogenous probability but
nevertheless prefer to act as if the exogenous probability were so
weighted (Wakker 2010). On this view, probability weighting is not a
(mistaken) belief but a preference.



![two graphs, a and b](fig1.svg)

Figure 1:
 (a) plots the value function \(v(\cdot)\) applied to consequences of
a prospect; (b) plots the median value of the probability weighting
function \(w(\cdot)\) applied to positive prospects of the form \((x,
p; 0, 1-p)\) with probability *p*.
[An extended description of this figure is in the supplement.]





Prospect theory incorporates these components into models of human
choice under risk by first identifying a reference point that either
refers to the status quo or some other aspiration level. The
consequences of the options under consideration then are framed in
terms of deviations from this reference point. Extreme probabilities
are simplified by rounding off, which yields miscalibration of the
given, exogenous probabilities. Dominance reasoning is then applied,
where dominated alternatives are eliminated from choice, along with
additional steps to separate options without risk, probabilities
associated with a specific outcome are combined, and a version of
eliminating irrelevant alternatives is applied (Kahneman & Tversky
1979: 284-285).



Nevertheless, prospect theory comes with problems. For example, a
shift of probability from less favorable outcomes to more favorable
outcomes ought to yield a better prospect, all things considered, but
the original prospect theory violates this principle of stochastic
dominance. *Cumulative prospect theory* satisfies stochastic
dominance, however, by appealing to a rank-dependent method for
transforming probabilities (Quiggin 1982). For a review of the
differences between prospect theory and cumulative prospect theory,
along with an axiomatization of cumulative prospect theory, see
Fennema & Wakker 1997.


## 3. The Emergence of Ecological Rationality



Imagine a meadow whose plants are loaded with insects but few are in
flight. Then, this meadow is a more favorable environment for a bird
that gleans rather than hawks. In a similar fashion, a decision-making
environment might be more favorable for one decision-making strategy
than for another. Just as it would be "irrational" for a
bird to hawk rather than glean, given the choice for this meadow, so
too what may be an irrational decision strategy in one environment may
be entirely rational in another.



If procedural rationality attaches a cost to the making of a decision,
then ecological rationality locates that procedure in the world. The
questions ecological rationality ask are what features of an
environment can help or hinder decision making and how should we model
judgment or decision-making ecologies. For example, people make causal
inferences about patterns of covariation they observe--especially
children, who then perform experiments testing their causal hypotheses
(Glymour 2001). Unsurprisingly, people who draw the correct inferences
about the true causal model do better than those who infer the wrong
causal model (Meder, Mayrhofer, & Waldmann 2014). More surprising,
Meder and his colleagues found that those making correct causal
judgments do better than subjects who make no causal judgments at all.
And perhaps most surprising of all is that those with true causal
knowledge also beat the benchmark standards in the literature which
ignore causal structure entirely; the benchmarks encode, spuriously,
the assumption that the best we can do is to make no causal judgments
at all.



In this section and the next we will cover five important
contributions to the emergence of ecological rationality. In this
section, after reviewing Simon's proposal for distinguishing
between *behavioral constraints* and *environmental
structure*, we turn to three historically important contributions:
*the lens model*, *rational analysis*, and *cultural
adaptation*. Finally, in
 section 4,
 we review the *bias-variance decomposition*, which has figured
in the Fast and Frugal Heuristics literature
 (section 7.2).


### 3.1 Behavioral Constraints and Environmental Structure



Simon thought that both behavioral constraints and environmental
structure ought to figure in a theory of bounded rationality, yet he
cautioned against identifying behavioral and environmental properties
with features of an organism and features of its physical environment,
respectively:



> 
> 
> we must be prepared to accept the possibility that what we call
> "the environment" may lie, in part, within the skin of the
> biological organisms. That is, some of the constraints that must be
> taken as givens in an optimization problem may be physiological and
> psychological limitations of the organism (biologically defined)
> itself. For example, the maximum speed at which an organism can move
> establishes a boundary on the set of its available behavior
> alternatives. Similarly, limits on computational capacity may be
> important constraints entering into the definition of rational choice
> under particular circumstances. (Simon 1955a: 101)
> 
> 
> 



That said, what is classified as a behavioral constraint rather than
an environmental affordance varies across disciplines and the
theoretical tools pressed into service. For example, one computational
approach to bounded rationality, *computational rationality
theory* (Lewis et al. 2014), classifies the cost to an organism of
executing an optimal program as a behavioral constraint, classifies
limits on memory as an environmental constraint, and treats the costs
associated with searching for an optimal program to execute as
exogenous. Anderson and Schooler's study and computational
modeling of human memory (Anderson & Schooler 1991) within the
ACT-R framework, on the other hand, views the limits on memory and
search-costs as behavioral constraints which are adaptive responses to
the structure of the environment. Still another broad class of
computational approaches are found in *statistical signal
processing*, such as adaptive filters (Haykin 2013), which are
commonplace in engineering and vision (Marr 1982; Ballard & Brown
1982). Signal processing methods typically presume the sharp
distinction between device and world that Simon cautioned against,
however. Still others have challenged the distinction between
behavioral constraints and environmental structure by arguing that
there is no clear way to separate organisms from the environments they
inhabit (Gibson 1979), or by arguing that features of cognition which
appear body-bound may not be necessarily so (Clark & Chalmers
1998).



Bearing in mind the different ways the distinction between behavior
and environment have been drawn, and challenges to what precisely
follows from drawing such a distinction, ecological approaches to
rationality all endorse the thesis that the ways in which an organism
manages structural features of its environment are essential to
understanding how deliberation occurs and effective behavior arises.
In doing so theories of bounded rationality have traditionally focused
on at least some of the following features, under this rough
classification:


* **Behavioral Constraints**--may refer to bounds on
computation, such as the *cost of searching* the best algorithm
to run, an appropriate rule to apply, or a satisficing option to
choose; the *cost of executing* an optimal algorithm,
appropriate rule, or satisficing choice; and *costs of storing*
the data structure of an algorithm, the constitutive elements of a
rule, or the objects of a decision problem.
* **Ecological Structure**--may refer to
*statistical*, *topological*, or other *perceptible
invariances* of the task environment that an organism is adapted
to; or to *architectural features* or *biological
features* of the computational processes or cognitive mechanisms
responsible for effective behavior, respectively.


### 3.2 Brunswik's Lens Model



Egon Brunswik was among the first to apply probability and statistics
to the study of human perception, and was ahead of his time in
emphasizing the role ecology plays in the generalizability of
psychological findings. Brunswik thought psychology ought to aim for
statistical descriptions of adaptive behavior (Brunswik 1943). Instead
of isolating a small number of independent variables to manipulate
*systematically* to observe the effects on a dependent
variable, psychological experiments ought instead to assess how an
organism adapts to its environment. So, not only should experimental
subjects be representative of the population, as one would presume,
but the experimental situations they are subjected to ought to be
representative of the environment that the subjects inhabit (Brunswik
1955). Thus, Brunswik maintained, psychological experiments ought to
employ a *representative design* to preserve the causal
structure of an organism's natural environment. For a review of
the development of representative design and its use in the study of
judgment and decision-making, see Dhami, Hertwig, & Hoffrage
2004.



Brunswik's *lens model* is formulated around his ideas
about how behavioral and environmental conditions bear on organisms
perceiving proximal cues to draw inferences about some distal feature
of its "natural-cultural habitat" (Brunswik 1955: 198). To
illustrate, an organism may detect the color markings (distal object)
of a potential mate through contrasts in light frequencies reflecting
across its retina (proximal cues). Some proximal cues will be more
informative about the distal objects of interest than others, which
Brunswik understood as a difference in the "objective"
correlations between proximal cues and the target distal objective.
The *ecological validity* of proximal cues thus refers to their
capacity for providing the organism useful information about some
distal object within a particular environment. Assessments of
performance for an organism then amount to a comparison of the
organism's actual use of cue information to the cue's
information capacity.



Kenneth Hammond and colleagues (K. Hammond, Hursch, & Todd 1964)
formulated Brunswik's lens model as a system of linear bivariate
correlations, as depicted in
 Figure 2
 (Hogarth & Karelaia 2007). Informally,
 Figure 2
 says that the accuracy of a subject's judgment (response),
\(Y\_s\), about a numerical target criterion, \(Y\_e\), given some
informative cues (features) \(X\_1, \ldots, X\_n\), is determined by the
correlation between the subject's response and the target. More
specifically, the linear lens model imagines two large linear systems,
one for the environment, *e*, and another for the subject,
*s*, which both share a set of cues, \(X\_1, \ldots, X\_n\). Note
that cues may be associated with one another, i.e., it is possible
that \(\rho(X\_i,X\_j) \neq 0\) for indices \(i\neq j\) from 1 to
*n*.



The accuracy of the subject's judgment \(Y\_s\) about the target
criterion value \(Y\_e\) is measured by an *achievement index*,
\(r\_a\), which is computed by Pearson's correlation coefficient
\(\rho\) of \(Y\_e\) and \(Y\_s\). The subject's predicted
response \(\hat{Y}\_s\) to the cues is determined by the weights
\(\beta\_{s\_i}\) the subject assigns to each cue \(X\_i\), and the
linearity of the subject's response, \(R\_s\), measures the noise
in the system, \(\epsilon\_s\). Thus, the subject's response is
conceived to be a weighted linear sum of subject-weighted cues plus
noise. The analogue of *response linearity* in the environment
is *environmental predictability*, \(R\_e\). The environment, on
this model, is thought to be probabilistic--or
"chancy" as some say. Finally, the environment-weighted
sum of cues, \(\hat{Y}\_e\), is compared to the subject-weighted sum of
cues, \(\hat{Y}\_s\), by a *matching index*, *G*.



![a complex diagram described in full in the next link](LensModel.png)

Figure 2:
 Brunswik's Lens Model  

[An extended description of this figure is in the supplement.]





In light of this formulation of the lens model, return to
Simon's remarks concerning the classification of environmental
affordance versus behavioral constraint. The conception of the lens
model as a linear model is indebted to signal detection theory, which
was developed to improve the accuracy of early radar systems. Thus,
the model inherits from engineering a clean division between subject
and environment. However, suppose for a moment that both the
environmental mechanism producing the criterion value and the
subject's predicted response are linear. Now consider the
error-term, \(\epsilon\_s\). That term may refer to biological
constraints that are responses to adaptive pressures on the whole
organism. If so, ought \(\epsilon\_s\) be classified as an
environmental constraint rather than a behavioral constraint? The
answer will depend on what follows from the reclassification, which
will depend on the model and the goal of inquiry
 (section 8).
 If we were using the lens model to understand the ecological validity
of an organism's judgment, then reclassifying \(\epsilon\_s\) as
an environmental constraint would only introduce confusion; If instead
our focus was to distinguish between behavior that is subject to
choice and behavior that is precluded from choice, then the proposed
reclassification may herald clarity--but then we would surely
abandon the lens model for something else, or in any case would no
longer be referring to the parameter \(\epsilon\_s\) in
 Figure 2.



Finally, it should be noted that the lens model, like nearly all
linear models used to represent human judgment and decision-making,
does not scale well as a descriptive model. In multi-cue
decision-making tasks involving more than three cues, people often
turn to simplifying heuristics due to the complications involved in
performing the necessary calculations
 (section 2.1;
 see also
 section 4).
 More generally, as we remarked in
 section 2.3,
 linear models involve calculating trade-offs that are difficult for
people to perform. Lastly, the supposition that the environment is
linear is a strong modeling assumption. Quite apart from the
difficulties that arise for humans to execute the necessary
computations, it becomes theoretically more difficult to justify model
selection decisions as the number of features increases. The matching
index *G* is a goodness-of-fit measure, but goodness-of-fit tests
and residual analysis begin to lead to misleading conclusions for
models with as five or more dimensions. Modern machine learning
techniques for supervised learning get around this limitation by
focusing on analogues of the achievement index, construct predictive
hypotheses purely instrumentally, and dispense with matching
altogether (Wheeler 2017).


### 3.3 Rational Analysis



Rational analysis is a methodology applied in cognitive science and
biology to explain why a cognitive system or organism engages in a
particular behavior by appealing to the presumed goals of the
organism, the adaptive pressures of its environment, and the
organism's computational limitations. Once an organism's
goals are identified, the adaptive pressures of its environment
specified, and the computational limitations are accounted for, an
optimal solution under those conditions is derived to explain why a
behavior that is otherwise ineffective may nevertheless be effective
in achieving that goal under those conditions (Marr 1982; Anderson
1991; Oaksford & Chater 1994; Palmer 1999). Rational analyses are
typically formulated independently of the cognitive processes or
biological mechanisms that explain how an organism realizes a
behavior.



One theme to emerge from the rational analysis literature that has
influenced bounded rationality is the study of memory (Anderson &
Schooler 1991). For instance, given the statistical features of our
environment, and the sorts of goals we typically pursue, forgetting is
an advantage rather than a liability (Schooler & Hertwig 2005).
Memory traces vary in their likelihood of being used, so the memory
system will try to make readily available those memories which are
most likely to be useful. This is a rational analysis style argument,
which is a common feature of the Bayesian turn in cognitive psychology
(Oaksford & Chater 2007; Friston 2010). More generally, spacial
arrangements of objects in the environment can simplify perception,
choice, and the internal computation necessary for producing an
effective solution (Kirsch 1995). Compare this view to the discussion
of recency or availability effects distorting subjective probability
estimates in
 section 2.4.



Rational analyses separate the goal of behavior from the mechanisms
that cause behavior. Thus, when an organism's observed behavior
in an environment does not agree with the behavior prescribed by a
rational analysis for that environment, there are traditionally three
responses. One strategy is to change the specifications of the
problem, by introducing an intermediate step or changing the goal
altogether, or altering the environmental constraints, et cetera
(Anderson & Schooler 1991; Oaksford & Chater 1994). Another
strategy is to argue that mechanisms matter after all, so details of
human psychology are taken into an alternative account (Newell &
Simon 1972; Gigerenzer, Todd, et al. 1999; Todd, Gigerenzer, et al.
2012). A third option is to enrich rational analysis by incorporating
computational mechanisms directly into the model (Russell &
Subramanian 1995; Chater 2014). Lewis, Howes, and Singh, for instance,
propose to construct theories of rationality from (i) structural
features of the task environment; (ii) the bounded machine the
decision-process will run on, about which they consider four different
classes of computational resources that may be available to an agent;
and (iii) a utility function to specify the goal, numerically, so as
to supply an objective function against which to score outcomes (Lewis
et al. 2014).


### 3.4 Cultural Adaptation



So far we have considered theories and models which emphasize an
individual organism and its surrounding environment, which is
typically understood to be either the physical environment or, if
social, modeled as if it were the physical environment. And we
considered whether some features commonly understood to be behavioral
constraints ought to be instead classified as environmental
affordances.



Yet people and their responses to the world are also part of each
person's environment. Boyd and Richardson argue that human
societies ought to be viewed as an adaptive environment, which in turn
has consequences for how individual behavior is evaluated. Human
societies contain a large reservoir of information that is preserved
through generations and expanded upon, despite limited, imperfect
learning by the members of human societies. *Imitation,* which
is a common strategy in humans, including pre-verbal infants (Gergely,
Bekkering, & Kiraly 2002), is central to cultural
transmission (Boyd & Richerson 2005) and the emergence of social
norms (Bicchieri & Muldoon 2014). In our environment, only a few
individuals with an interest in improving on the folk lore are
necessary to nudge the culture to be adaptive. The main advantage that
human societies have over other groups of social animals, this
argument runs, is that cultural adaptation is much faster than genetic
adaptation (Bowles & Gintis 2011). On this view, human psychology
evolved to facilitate speedy adaptation. Natural selection did not
equip our large-brained ancestors with rigid behavior, but instead
selected for brains that allowed then to modify their behavior
adaptively in response to their environment (Barkow, Cosmides, &
Tooby 1992).



But if human psychology evolved to facility fast social learning, it
comes at the cost of human credulity. To have speedy adaptation
through imitation of social norms and human behavior, the risk is the
adoption of maladaptive norms or stupid behavior.


## 4. The Bias-Variance Trade-off



The *bias-variance trade-off* refers to a particular
decomposition of overall prediction error for an estimator into its
central tendency (bias) and dispersion (variance). Sometimes overall
error can be reduced by increasing bias in order to reduce variance,
or vice versa, effectively trading an increase in one type of error to
afford a comparatively larger reduction in the other. To give an
intuitive example, suppose your goal is to minimize your score with
respect to the following targets.



![four targets further described in the next link](fig3.svg)

Figure 3
  

[An extended description of this figure is in the supplement.]
 




Ideally, you would prefer a procedure for delivering your
"shots" that had both a low bias and low variance. Absent
that, and given the choice between a low bias and high variance
procedure versus a high bias and low variance procedure, you would
presumably prefer the latter procedure if it returned a lower overall
score than the former, which is true of the corresponding figures
above. Although a decision maker's learning algorithm ideally
should have low bias and low variance, in practice it is common that the
reduction in one type of error yields some increase in the other. In
this section we explain the conditions under which the relationship
between expected squared loss of an estimator and its bias and
variance holds and then remark on the role that the bias-variance
trade-off plays in research on bounded rationality.


### 4.1 The Bias-Variance Decomposition of Mean Squared Error



Predicting the exact volume of gelato to be consumed in Rome next
summer is more difficult than predicting that more gelato will be
consumed next summer than next winter. For although it is a foregone
conclusion that higher temperatures beget higher demand for gelato,
the precise relationship between daily temperatures in Rome and
*consumo di gelato* is far from certain. Modeling quantitative,
predictive relationships between random variables, such as the
relationship between the temperature in Rome, *X*, and volume of
Roman gelato consumption, *Y*, is the subject of *regression
analysis*.



Suppose we predict that the value of *Y* is *h*. How should
we evaluate whether this prediction is any good? Intuitively, the best
we can do is to pick an *h* that is as close to *Y* as we
can make it, one that would minimize the difference \(Y - h\). If we
are indifferent to the direction of our errors, viewing positive
errors of a particular magnitude to be no worse than negative errors
of the same magnitude, and vice versa, then a common practice is to
measure the performance of *h* by its squared difference from
*Y*, \((Y - h)^2\). (We are not always indifferent; consider the
plight of William Tell aiming at that apple.) Finally, since the
values of *Y* vary, we might be interested in the average value
of \((Y - h)^2\) by computing its expectation, \(\mathbb{E} \left[ (Y
- h)^2 \right]\). This quantity is the *mean squared error* of
*h*, 

 

\[\textrm{MSE}(h) := \mathbb{E} \left[ (Y - h)^2 \right].\] 


Now imagine our prediction of *Y* is based on some data
\(\mathcal{D}\) about the relationship between *X* and *Y*,
such as last year's daily temperatures and daily total sales of
gelato in Rome. The role that this particular dataset \(\mathcal{D}\)
plays as opposed to some other possible data set is a detail that will
figure later. For now, view our prediction of *Y* as some
function of *X*, written \(h(X)\). Here again we wish to pick an
\(h(\cdot)\) to minimize \(\mathbb{E} \left[ (Y - h(X))^2 \right]\),
but how close \(h(\cdot)\) is to *Y* will depend on the possible
values of *X*, which we can represent by the conditional
expectation 

 

\[\mathbb{E} \left[ (Y - h(X))^2 \right] := \mathbb{E} \left[ \mathbb{E} \left[ Y - h(X) \mid X\right] \right].\] 


How then should we evaluate this conditional prediction? The same as
before, only now accounting for *X*. For each possible value
*x* of *X*, the best prediction of *Y* is the
conditional mean, \(\mathbb{E}\left[ Y \mid X = x\right]\). The
*regression function* of *Y* on *X*, \(r(x)\), gives
the optimal value of *Y* for each value \(x \in X\): 



\[r(x) := \mathbb{E}\left[ Y \mid X = x\right].\]


Although the regression function represents the true population value
of *Y* given *X*, this function is usually unknown,
typically complicated, therefore often approximated by a simplified
model or learning algorithm, \(h(\cdot)\).



We might restrict candidates for \(h(X)\) to linear (or affine)
functions of *X*, for instance. Yet making predictions about the
value of *Y* with a simplified linear model, or some other
simplified model, can introduce a systematic prediction error called
*bias*. Bias results from a difference between the central
tendency of data generated by the true model, \(r(X)\) (for all \(x
\in X\)), and the central tendency of our estimator,
\(\mathbb{E}\left[h(X)\right]\), written 

 

\[\textrm{Bias}(h(X)) := r(X) - \mathbb{E}\left[h(X) \right],\]


where any non-zero difference between the pair is interpreted as a
systematically positive or systematically negative error of the
estimator, \(h(X)\).



*Variance* measures the average deviation of a random variable
from its expected value. In the current setting we are comparing the
predicted value \(h(X)\) of *Y*, with respect to some data
\(\mathcal{D}\) about the relationship between *X* and *Y*,
and the average value of \(h(X)\), \(\mathbb{E}\left[ h(X) \right]\),
which we will write 

 

\[\textrm{Var}(h(X)) = \mathbb{E}\left[( \mathbb{E}\left[ h(X) \right] - h(X))^2\right].\]


The bias-variance decomposition of mean squared error is rooted in
frequentist statistics, where the objective is to compute an estimate
\(h(X)\) of the true parameter \(r(X)\) with respect to data
\(\mathcal{D}\) about the relationship between *X* and *Y*.
Here the parameter \(r(X)\) characterizing the truth about *Y* is
assumed to be fixed and the data \(\mathcal{D}\) is treated as a
random quantity, which is exactly the reverse of Bayesian statistics.
What this means is that the data set \(\mathcal{D}\) is interpreted to
be one among many possible data sets of the same dimension generated
by the true model, the deterministic process \(r(X)\).



Following Christopher M. Bishop (2006), we may derive the
bias-variance decomposition of mean squared error of *h* as
follows. Let *h* refer to our estimate \(h(X)\) of *Y*,
*r* refer to the true value of *Y*, and \(\mathbb{E}\left[ h
\right]\) the expected value of the estimate *h*. Then, 



\[\begin{align}
 &\textrm{MSE}(h) \\
 &\quad = \mathbb{E}\left[ ( r -h)^2 \right] \\
 &\quad = \mathbb{E}\left[ \left( \left( r - \mathbb{E}\left[ h \right] \right)
 + \left( \mathbb{E}\left[ h \right] - h \right) \right )^2 \right] \\
 &\quad = \mathbb{E}\left[ \left( r - \mathbb{E}\left[ h \right] \right)^2 \right] 
 + \mathbb{E}\left(\left( \mathbb{E}\left[ h \right] - h \right )^2\right) 
 + 2 \mathbb{E}\left[ \left( \mathbb{E}\left[ h \right] - h \right) \cdot 
 \left( r - \mathbb{E}\left[ h \right] \right) \right] \\
 &\quad = \left( r - \mathbb{E}\left[h \right] \right)^2 
 + \mathbb{E}\left[ \left( \mathbb{E}\left[h \right] - h \right)^2\right] + 0\\
 &\quad = \mathrm{B}(h)^2 \ + \ \textrm{Var}(h) 
\end{align}\]


where the term \(2 \mathbb{E}\left[ \left( \mathbb{E}\left[ h \right]
- h \right) \cdot \left( r - \mathbb{E}\left[ h \right] \right)
\right]\) is zero, since 

 

\[\begin{align}
 &\mathbb{E}\left[ \left( \mathbb{E}\left[ h \right] - h \right) \cdot \left( r - \mathbb{E}\left[ h \right] \right) \right] \notag\\
 &\qquad = \left(\mathbb{E} \left[ r \cdot \mathbb{E} \left[ h \right] \right] 
 - \mathbb{E}\left[ \mathbb{E}\left[ h \right]^2 \right] - \mathbb{E}\left[ h \cdot r \right] 
 + \mathbb{E}\left[ h\cdot \mathbb{E}\left[ h \right] \right] \right) \nonumber \tag{1}\\
 &\qquad = r \cdot \mathbb{E} \left[ h \right] - \mathbb{E}\left[ h \right]^2 - r \cdot \mathbb{E} \left[ h \right] + \mathbb{E}\left[ h \right]^2 \label{eq:owl}\tag{2}\\
 &\qquad = 0. \tag{3} 
\end{align}\]


Note that the frequentist assumption that *r* is a deterministic
process is necessary for the derivation to go through; for if *r*
were a random quantity, the reduction of \(\mathbb{E} \left[ r \cdot
\mathbb{E} \left[ h \right] \right]\) to \( r \cdot \mathbb{E} \left[
h \right]\) in line (2) would be invalid.



One last detail that we have skipped over is the prediction error of
\(h(X)\) due to noise, *N*, which occurs independent of the
model/learning algorithm used. Thus, the full bias-variance
decomposition of the mean-squared error of an estimate *h* is the
sum of the bias (squared), variance, and irreducible error: 




\[\tag{4}\label{eq-puppy} \textrm{MSE}(h)\ = \ \mathrm{B}(h)^2 \ + \ \textrm{Var}(h) \ + \ N\]


### 4.2 Bounded Rationality and Bias-Variance Generalized



Intuitively, the bias-variance decomposition brings to light a
trade-off between two extreme approaches to making a prediction. At
one extreme, you might adopt as an estimator a constant function which
produces the same answer no matter what data you see. Suppose 7 is
your lucky number and your estimator's prediction, \(h(X) = 7\).
Then the variance of \(h(\cdot)\) would be zero, since its prediction
is always the same. The bias of your estimator, however, will be very
large. In other words, your lucky number 7 model will massively
*under fit* your data.



At the other extreme, suppose you aim to make your bias error zero.
This occurs just when the predicted value of *Y* and the actual
value of *Y* are identical, that is, \(h(x\_i) = y\_i\), for every
\((x\_i, y\_i)\). Since you are presumed to not know the true function
\(r(X)\) but instead only see a sample of data from the true model,
\(\mathcal{D}\), it is from this sample that you will aspire to
construct an estimator that generalizes to accurately predict examples
outside your training data \(\mathcal{D}\). Yet if you were to fit
\(h\_{\mathcal{D}}(X)\) perfectly to \(\mathcal{D}\), then the variance
of your estimator will be very high, since a different data set
\(\mathcal{D}'\) from the true model is not, by definition, identical
to \(\mathcal{D}\). How different is \(\mathcal{D}'\) to
\(\mathcal{D}\)? The variation from one data set to another among all
the possible data sets is the variance or irreducible noise of the
data generated by the true model, which may be considerable.
Therefore, in this zero-bias case your model will massively
*overfit* your data.



The bias-variance trade-off therefore concerns the question of how
complex a model ought to be to make reasonably accurate predictions on
unseen or out-of-sample examples. The problem is to strike a balance
between an under-fitting model, which erroneously ignores available
information about the true function *r*, and an overfitting
model, which erroneously includes information that is noise and
thereby gives misleading information about the true function
*r*.



One thing that human cognitive systems do very well is to generalize
from a limited number of examples. The difference between humans and
machines is particularly striking when we compare how humans learn a
complicated skill, such as driving a car, from how a machine learning
system learns the same task. As harrowing an experience it is to teach
a teenager how to drive a car, they do not need to crash into a
utility pole 10,000 times to learn that utility poles are not
traversable. What teenagers learn as children about the world through
play and observing other people drive lends to them an understanding
that utility poles are to be steered around, a piece of
commonsense that our current machine learning systems do not have but
must learn from scratch on a case-by-case basis. We, unlike our
machines, have a remarkable capacity to transfer what we learn from
one domain to another domain, a capacity fueled in part by our
curiosity (Kidd & Hayden 2015).



Viewed from the perspective of the bias-variance trade-off, the
ability to make accurate predictions from sparse data suggests that
variance is the dominant source of error but that our cognitive system
often manages to keep these errors within reasonable limits
(Gigerenzer & Brighton 2009). Indeed, Gigerenzer and Brighton make
a stronger argument, stating that "the bias-variance dilemma
shows formally why a mind can be better off with an adaptive toolbox
of biased, specialized heuristics" (Gigerenzer & Brighton
2009: 120); see also
 section 7.2.
 However, the bias-variance decomposition is a decomposition of
squared loss, which means that the decomposition above depends on how
total error (loss) is measured. There are many loss functions,
however, depending on the type of inference one is making along with
the stakes in making it. If one were to use a 0-1 loss function, for
example, where all non-zero errors are treated equally--meaning
that "a miss as good as a mile"--the decomposition
above breaks down. In fact, for 0-1 loss, bias and variance combine
multiplicatively (J. Friedman 1997)! A generalization of the
bias-variance decomposition that applies to a variety of loss
functions \(\mathrm{L}(\cdot)\), including 0-1 loss, has been offered
by (Domingos 2000), 

 

\[\mathrm{L}(h)\ = \ \mathrm{B}(h)^2 \ + \ \beta\_1\textrm{Var}(h) \ + \ \beta\_2\mathrm{N}\]


where the original bias-variance decomposition,
 Equation 4,
 appears as a special case, namely when \(\mathrm{L}(h) =
\textrm{MSE}(h)\) and \(\beta\_1 = \beta\_2 = 1\).


## 5. Better with Bounds



Our discussion of improper linear models
 (section 2.3)
 mentioned a model that often comes surprisingly close to
approximating a proper linear model, and our discussion of the
bias-variance decomposition
 (section 4.2)
 referred to conjectures about how cognitive systems might manage to
make accurate predictions with very little data . In this section we
review examples of models which deviate from the normative standards
of global rationality yet yield markedly *improved*
outcomes--sometimes even yielding results which are impossible
under the conditions of global rationality. Thus, in this section we
will survey examples from the *statistics of small samples* and
*game theory* which point to demonstrable advantages to
deviating from global rationality.


### 5.1 Homo Statisticus and Small Samples



In a review of experimental results assessing human statistical
reasoning published in the late 1960s that took stock of research
conducted after psychology's full embrace of statistical
research methods
 (section 2.3),
 Petersen and Beach argued that the normative standard of probability
theory and statistical optimization methods were "a good first
approximation for a psychological theory of inference" (Peterson
& Beach 1967: 42). Petersen and Beach's view that humans
were *intuitive statisticians* that closely approximate the
ideal standards of *homo statisticus* fit into a broader
consensus at that time about the close fit between the normative
standards of logic and intelligent behavior (Newell & Simon 1956,
1976). The assumption that human judgment and decision-making closely
approximates normative theories of probability and logic would later
be challenged by experimental results by Kahneman and Tversky, and the
biases and heuristics program more generally
 (section 7.1).



Among Kahneman and Tversky's earliest findings was that people
tend to make statistical inferences from samples that are too small,
even when given the opportunity to control the sampling procedure.
Kahneman and Tversky attributed this effect to a systematic failure of
people to appreciate the biases that attend small samples, although
Hertwig and others have offered evidence that samples drawn from a
single population are close to the known limits to working memory
(Hertwig, Barron et al. 2004).



Overconfidence can be understood as an artifact of small samples. The
*Naive Sampling Model* (Juslin, Winman, & Hansson
2007) assumes that agents base judgments on a small sample retrieved
from long-term memory at the moment a judgment is called for, even
when there are a variety of other methods available to the agent. This
model presumes that people are naive statisticians (Fiedler &
Juslin 2006) who assume, sometimes falsely, that samples are
representative of the target population of interest and that sample
properties can be used directly to yield accurate estimates of a
population. The idea is that when sample properties are uncritically
taken as estimators of population parameters a reasonably accurate
probability judgment can be made with overconfidence, even if the
samples are unbiased, accurately represented, and correctly processed
by the cognitive mechanisms of the agent. When sample sizes are
restricted, these effects are amplified.



However, sometimes effective behavior is aided by inaccurate judgments
or cognitively adaptive illusions (Howe 2011). The statistical
properties of small samples are a case in point. One feature of small
samples is that correlations are amplified, making them easier to
detect (Kareev 1995). This fact about small samples, when combined
with the known limits to human short-term memory, suggests that our
working-memory limits may be an adaptive response to our environment
that we exploit at different stages in our lives. Adult short-term
working memory is limited to seven items, plus or minus two. For
correlations of 0.5 and higher, Kareev demonstrates that sample sizes
between five and nine are most likely to yield a sample correlation
that is greater than the true correlation in the population (Kareev
2000), making those correlations nevertheless easier to detect.
Furthermore, children's short-term memories are even more
restricted than adults, thus making correlations in the environment
that much easier to detect. Of course, there is no free lunch: this
small-sample effect comes at the cost of inflating estimates of the
true correlation coefficients and admitting a higher rate of false
positives (Juslin & Olsson 2005). However, in many contexts,
including child development, the cost of error arising from
under-sampling may be more than compensated by the benefits from
simplifying choice (Hertwig & Pleskac 2008) and accelerating
learning. In the spirit of Brunswik's argument for
representative experimental design
 (section 3.2),
 a growing body of literature cautions that the bulk of experiments on
adaptive decision-making are performed in highly simplified
environments that differ in important respects from the natural world
in which human beings make decisions (Fawcett et al. 2014). In
response, Houston, MacNamara and colleagues argue, we should
incorporate more environmental complexity in our models.


### 5.2 Game Theory



Pro-social behavior, such as cooperation, is challenging to explain.
Evolutionary game theory predicts that individuals will forgo a public
good and that individual utility maximization will win over collective
cooperation. Even though this outcome is often seen in economic
experiments, in broader society cooperative behavior is pervasive
(Bowles & Gintis 2011). Why? The traditional evolutionary
explanations of human cooperation in terms of *reputation*,
*reciprocation*, and *retribution* (Trivers 1971; R.
Alexander 1987), are unsatisfactory because they do not uniquely
explain why cooperation is a stable behavior. If a group punishes
individuals for failing to perform a behavior, and the punishment
costs exceed the benefit of doing that behavior, then this behavior
will become stable regardless of its social benefits. Anti-social
norms arguably take root by precisely the same mechanisms (Bicchieri
& Muldoon 2014). Although reputation, reciprocation, and
retribution may explain how large-scale cooperation is sustained in
human societies, it does not explain how the behavior emerged (Boyd
& Richerson 2005). Furthermore, cooperation is observed in
microorganisms (Damore & Gore 2012), which suggests that much
simpler mechanisms are sufficient for the emergence of cooperative
behavior.



Whereas the 1970s saw a broader realization of the advantages of
improper models to yield results that were often good enough
 (section 2.3),
 the 1980s and 1990s witnessed a series of results involving improper
models yielding results that were strictly better than what was
prescribed by the corresponding proper model. In the early 1980s
Robert Axelrod held a tournament to empirically test which among a
collection of strategies for playing iterations of the
prisoner's dilemma performed best in a round-robin competition.
The winner was a simple reciprocal altruism strategy called
*tit-for-tat* (Rapoport & Chammah 1965), which simply
starts off each game cooperating then, on each successive round,
copies the strategy the opposing player played in the previous round.
So, if your opponent cooperated in this round, then you will cooperate
on the next round; and if your opponent defected this round, then you
will defect the next. Subsequent tournaments have shown that
tit-for-tat is remarkably robust against much more sophisticated
alternatives (Axelrod 1984). For example, even a rational utility
maximizing player playing against an opponent who only plays
tit-for-tat (i.e., will play tit-for-tat no matter whom he faces) must
adapt and play tit-for-tat--or a strategy very close to it
(Kreps, Milgrom, et al. 1982).



Since tit-for-tat is a very simple strategy, computationally, one can
begin to explore a notion of rationality that emerges in a group of
boundedly rational agents and even see evidence of those bounds
contributing to the emergence of pro-social norms. Rubinstein
(Rubinstein 1986) studied finite automata which play repeated
prisoner's dilemmas and whose aims are to maximize average
payoff while minimizing the number of states of a machine. Finite
automata capture regular languages, the lowest-level of the
Chomsky-hierarchy, thus model a type of boundedly rational agents.
Solutions are a pair of machines in which the choice of the machine is
optimal for each player at every stage of the game. In an evolutionary
interpretation of repeated games, each iteration of Rubinstein's
can be seen as successive generations of agents. This approach is in
contrast to Neyman's study of players of repeated games who can
only play mixtures of pure strategies that can be programmed on finite
automata, where the number of states that are available is an
exogenous variable whose value is fixed by the modeler. In
Neyman's model, each generation plays the entire game and thus
traits connected to reputation can arise (Neyman 1985). More
generally, although cooperation is impossible for infinitely repeated
prisoner's dilemmas, for finitely repeated prisoner's
dilemmas, a cooperative equilibrium exists for finite automata players
whose number of states is less than exponential in the number of
rounds of the game (Papadimitriou & Yannakakis 1994; Ho 1996). The
demands on memory may exceed the psychological capacities of people,
however, even for simple strategies like tit-for-tat played by a
moderately sized group of players (Stevens, Volstorf, et al. 2011).
These theoretical models showing a number of simple paths to
pro-social behavior may not, on their own, be simple enough to offer
plausible process models for cooperation.



On the heels of work on the effects of time (finite iteration versus
infinite iteration) and memory/cognitive ability (finite state
automata versus Turing machines), attention soon turned to
environmental constraints. Nowak and May looked at the spatial
distribution on a two-dimensional grid of 'cooperators'
and 'defectors' in iterated prisoner's dilemmas and
found cooperation to emerge among players without memories or
strategic foresight (Nowak & May 1992). This work led to the study
of *network topology* as a factor in social behavior (Jackson
2010), including social norms (Bicchieri 2005; J. Alexander 2007),
signaling (Skyrms 2003), and wisdom of crowd effects (Golub &
Jackson 2010). When social ties in a network follow a scale-free
distribution, the resulting diversity in the number and size of
public-goods games is found to promote cooperation, which contributes
to explaining the emergence of cooperation in communities without
mechanisms for reputation and punishment (F. Santos, M. Santos, &
Pacheco 2008).



But, perhaps the simplest case for bounded rationality are examples of
agents achieving a desirable goal without any deliberation at all.
Insects, flowers, and even bacteria exhibit evolutionary stable
strategies (Maynard Smith 1982), effectively arriving at Nash
equilibria in strategic normal form games. If we imagine two species
interacting with one another, say honey bees (*Apis mellifera*)
and a species of flower, each interaction between a bee and a flower
has some bearing on the fitness of each species, where fitness is
defined as the expected number of offspring. There is an incremental
payoff to bees and flowers, possibly negative, after each interaction,
and the payoffs are determined by the genetic endowments of bees and
flowers each. The point is that there is no choice exhibited by these
organisms nor in the models; the process itself selects the traits.
The agents have no foresight. There are no strategies that the players
themselves choose. The process is entirely mechanical. What emerges in
this setting are *evolutionary dynamics*, a form of bounded
rationality without foresight.



Of course, any improper model can misfire. A rule of thumb shared by
people the world-over is to not let other people take advantage of
them. While this rule works most of the time, it misfires in the
*ultimatum game* (Guth, Schmittberger, & Schwarze
1982). The ultimatum game is a two-player game in which one player,
endowed with a sum of money, is given the task of splitting the sum
with another player who may either accept the offer--in which
case the pot is accordingly split between the two players--or
reject, in which case both players receive nothing. People
receiving offers of 30 percent or less of the pot are often observed
to reject the offer, even when players are anonymous and therefore
would not suffer the consequences of a negative reputation signal
associated with accepting a very low offer. In such cases, one might
reasonably argue that no proposed split is worse than the status quo
of zero, so people ought to accept whatever they are offered.


### 5.3 Less is More Effects



Simon's remark that people satisfice when they haven't the
wits to maximize (Simon 1957a: xxiv) points to a common assumption,
that there is a trade-off between effort and accuracy
 (section 2.1).
 Because the rules of global rationality are expensive to operate
(Good 1952: 7(i)), people will trade a loss in accuracy for gains in
cognitive efficiency (Payne, Bettman, & Johnson 1988). The
methodology of rational analysis
 (section 3.3)
 likewise appeals to this trade-off.



The results surveyed in
 Section 5.2
 caution against blindly endorsing the accuracy-effort trade-off as
universal, a point that has been pressed in the defense of heuristics
as reasonable models for decision-making (Katsikopoulos 2010; Hogarth
2012).



Simple heuristics like *Tallying*, which is a type of improper
linear model
 (section 2.3),
 and *Take-the-best*
 (section 7.2),
 when tested against linear regression on many data sets, have been
both found to outperform linear regression on out-of-sample prediction
tasks, particularly when the training-sample size is low (Czerlinski
et al. 1999; Rieskamp & Dieckmann 2012).


## 6. Aumann's Five Arguments and One More



Aumann advanced five arguments for bounded rationality, which we
paraphrase here (1997).


1. Even in very simple decision problems, most economic agents are not
(deliberate) maximizers. People do not scan the choice set and
consciously pick a maximal element from it.
2. Even if economic agents aspired to pick a maximal element from a
choice set, performing such maximizations are typically difficult and
most people are unable to do so in practice.
3. Experiments indicate that people fail to satisfy the basic assumptions
of rational decision theory.
4. Experiments indicate that the conclusions of rational analysis
(broadly construed to include rational decision theory) do not match
observed behavior.
5. Some conclusions of rational analysis appear normatively
unreasonable.



In the previous sections we covered the origins of each of
Aumann's arguments. Here we briefly review each, highlighting
material in other sections under this context.



The first argument, that people are not deliberate maximizers, was a
working hypothesis of Simon's, who maintained that people tend
to satisfice rather than maximize
 (section 2.2).
 Kahneman and Tversky gathered evidence for the reflection effect in
estimating the value of options, which is the reason for reference
points in prospect theory
 (section 2.4)
 and analogous properties within rank-dependent utility theory more
generally (sections
 1.2
 and
 2.4).
 Gigerenzer's and Hertwig's groups at the Max Planck
Institute for Human Development both study the algorithmic structure
of simple heuristics and the adaptive psychological mechanisms which
explain their adoption and effectiveness; both of their research
programs start from the assumption that expected utility theory is not
the right basis for a descriptive theory of judgment and
decision-making (sections
 3,
 5.3, and
 7.2).



The second argument, that people are often unable to maximize even if
they aspire to, was made by Simon and Good, among others, and later by
Kahneman and Tversky. Simon's remarks about the complexity of
\(\Gamma\)-maxmin reasoning in working out the end-game moves in chess
 (section 2.2)
 is one of many examples he used over the span of his career, starting
before his seminal papers on bounded rationality in the 1950s. The
*biases and heuristics* program spurred by Tversky and
Kahneman's work in the late 1960s and 1970s
 (section 7.1)
 launched the systematic study of when and why people's
judgments deviate from the normative standards of expected utility
theory and logical consistency.



The third argument, that experiments indicate that people fail to
satisfy the basic assumptions of expected utility theory, was known
from early on and emphasized by the very authors who formulated and
refined the homo economicus hypothesis
 (section 1)
 and whose names are associated with the mathematical foundations. We
highlighted an extended quote from Savage in
 section 1.3,
 but could mention as well a discussion of the theory's
limitations by de Finetti and Savage (1962), and even a closer reading
of the canonical monographs of each, namely Savage 1954 and de Finetti
1970. A further consideration, which we discussed in
 section 1.3
 is the demand of *logical omniscience* in expected utility
theory and nearly all axiomatic variants.



The fourth argument, regarding the differences between the predictions
of rational analysis and observed behavior, we addressed in
discussions of Brunswik's notion of ecological validity
 (section 3.2)
 and the traditional responses to these observations by rational
analysis
 (section 3.3).
 The fifth argument, that some of the conclusions of rational analysis
do not agree with a reasonable normative standard, was touched on in
sections
 1.2,
 1.3, and the subject of
 section 5.



Implicit in Aumann's first four arguments is the notion that
global rationality
 (section 2)
 is a reasonable normative standard but problematic for descriptive
theories of human judgment and decision-making
 (section 8).
 Even the literature standing behind Aumann's fifth argument,
namely that there are problems with expected utility theory as a
normative standard, nevertheless typically address those shortcomings
through modifications to, or extensions of, the underlying
mathematical theory
 (section 1.2).
 This broad commitment to optimization methods, dominance reasoning,
and logical consistency as bedrock normative principles is behind
approaches that view bounded rationality as *optimization under
constraints*:



> 
> 
> Boundedly rational procedures are in fact fully optimal procedures
> when one takes account of the cost of computation in addition to the
> benefits and costs inherent in the problem as originally posed (Arrow
> 2004).
> 
> 
> 



For a majority of researchers across disciplines, bounded rationality
is identified with some form of optimization problem under
constraints.



Gerd Gigerenzer is among the most prominent and vocal critics of the
role that optimization methods and logical consistency plays in
commonplace normative standards for human rationality (Gigerenzer
& Brighton 2009), especially the role those standards play in
Kahneman and Tversky's biases and heuristics program (Kahneman
& Tversky 1996; Gigerenzer 1996). We turn to this debate next, in
 section 7.


## 7. Two Schools of Heuristics



Heuristics are simple rules of thumb for rendering a judgment or
making a decision. Some examples that we have seen thus far include
Simon's satisficing, Dawes's improper linear models,
Rapoport's tit-for-tat, imitation, and several effects observed
by Kahneman and Tversky in our discussion of prospect theory.



There are nevertheless two views on heuristics that are roughly
identified with the research traditions associated with Kahneman and
Tversky's *biases and heuristics* program and
Gigerenzer's *fast and frugal heuristics* program,
respectively. A central dispute between these two research programs is
the appropriate normative standard for judging human behavior (Vranas
2000). According to Gigerenzer, the biases and heuristics program
mistakenly classifies all biases as errors (Gigerenzer, Todd, et al.
1999; Gigerenzer & Brighton 2009) despite evidence pointing to
some biases in human psychology being adaptive. In contrast, in a rare
exchange with a critic, Kahneman and Tversky maintain that the dispute
is merely terminological (Kahneman & Tversky 1996; Gigerenzer
1996).



In this section, we briefly survey each of these two schools. Our aim is
to give a characterization of each research program rather than an
exhaustive overview.


### 7.1 Biases and Heuristics



Beginning in the 1970s, Kahneman and Tversky conducted a series of
experiments showing various ways that human participants'
responses to decision tasks deviate from answers purportedly derived
from the appropriate normative standards (sections
 2.4
 and
 5.1).
 These deviations were given names, such as *availability*
(Tversky & Kahneman 1973), *representativeness*, and
*anchoring* (Tversky & Kahneman 1974). The set of cognitive
biases now numbers into the hundreds, although some are minor variants
of other well-known effects, such as "The IKEA effect"
(Norton, Mochon, & Ariely 2012) being a version of the well-known
endowment effect
 (section 1.2).
 Nevertheless, core effects studied by the biases and heuristics
program, particularly those underpinning prospect theory
 (section 2.4),
 are entrenched in cognitive psychology (Kahneman, Slovic, &
Tversky 1982).



An example of a probability judgment task is Kahneman and
Tversky's Taxi-cab problem, which purports to show that subjects
neglect base rates.



> 
> 
> A cab was involved in a hit and run accident at night. Two cab
> companies, the Green and the Blue, operate in the city. You are given
> the following data:
> 
> 
> * 85% of the cabs in the city are Green and 15% are Blue.
> * A witness identified the cab as a Blue cab. The court tested his
> ability to identify cabs under the appropriate visibility conditions.
> When presented with a sample of cabs (half of which were Blue and half
> of which were Green) the witness made correct identifications in 80%
> of the cases and erred in 20% of the cases.
> 
> 
> 
> Question: What is the probability that the cab involved in the
> accident was Blue rather than Green? (Tversky & Kahneman 1977:
> 3-3).
> 
> 
> 



Continuing, Kahneman and Tversky report that several hundred subjects
have been given slight variations of this question and for all
versions the modal and median responses was 0.8, instead of the
correct answer of \(\bfrac{12}{29}\) (\(\approx 0.41\)). 



> 
> 
> Thus, the intuitive judgment of probability coincides with the
> credibility of the witness and ignores the relevant base-rate, i.e.,
> the relative frequency of Green and Blue cabs. (Tversky & Kahneman
> 1977: 3-3)
> 
> 
> 



Critical responses to results of this kind fall into three broad
categories. The first types of reply is to argue that the
experimenters, rather than the subjects, are in error (Cohen 1981). In
the Taxi-cab problem, arguably Bayes sides with the folk (Levi 1983)
or, alternatively, is inconclusive because the normative standard of
the experimenter and the presumed normative standard of the subject
requires a theory of witness testimony, neither of which is specified
(Birnbaum 1979). Other cognitive biases have been ensnared in the
replication crises, such as *implicit bias* (Oswald, Mitchell,
et al. 2013; Forscher, Lai et al. 2017) and *social priming*
(Doyen, Klein, et al. 2012; Kahneman 2017
 [Other Internet Resources]).



The second response is to argue that there is an important difference
between identifying a normative standard for combining probabilistic
information and applying it across a range of cases
 (section 8.2),
 and it is difficult in practice to determine that a decision-maker is
representing the task in the manner that the experimenters intend
(Koehler 1996). Observed behavior that appears to be boundedly
rational or even irrational may result from a difference between the
intended specification of a problem and the actual problem subjects
face.



For example, consider the systematic biases in people's
perception of randomness reported in some of Kahneman and
Tversky's earliest work (Kahneman & Tversky 1972). For
sequences of flips of a fair coin, people expect to see, even for
small samples, a roughly-equal number heads and tails and alternation
rates between heads and tails that are slightly higher than long-run
averages (Bar-Hillel & Wagenaar 1991). This effect is thought to
explain the *gambler's fallacy*, the false belief that a
run of heads from an i.i.d. sequence of fair coin tosses will make the
next flip more likely to land tails. Hahn and Warren argue that the
limited nature of people's experiences with random sequences is
a better explanation than to view them as cognitive deficiencies.
Specifically, people only ever experience finite sequence of outputs
from a randomizer, such as a sequence of fair coin tosses, and the
limits to their memory
 (section 5.1)
 of past outcomes in a sequence will mean that not all possible
sequences of a given length with appear to them with equal
probability. Therefore, there is a psychologically plausible
interpretation of the question, "is it more likely to see
*HHHT* than *HHHH* from flips of a fair coin?",
for which the correct answer is, "Yes" (Hahn & Warren
2009). If the gambler's fallacy boils down to a failure to
distinguish between sampling with and without replacement, Hahn and
Warren's point is that our intuitive statistical abilities
acquired through experience alone is unable to make the distinction
between these two sampling methods. Analytical reasoning is
necessary.



Consider also the risky-choice framing effect that was mentioned
briefly in
 section 2.4.
 An example is the Asian disease example,


* If program *A* is adopted, 200 people will be saved.
* If program *B* is adopted, there is a 1/3 probability that
600 people will be saved, and a 2/3 probability that no people
will be saved (Tversky & Kahneman 1981: 453).



Tversky and Kahneman report that a majority of respondents (72
percent) chose option (a), whereas a majority of respondents (78
percent) shown an equivalent reformulation of the problem in terms of
the number of people who would die rather than survive chose (b). A
meta-analysis of subsequent experiments has shown that the framing
condition accounts for most of the variance, but it also reveals no
linear combination of formally specified predictors that are used in
prospect theory, cumulative prospect theory, and Markowitz's
utility theory, suffices to capture this framing effect
(Kuhberger, Schulte-Mecklenbeck, & Perner 1999).



The point to this second line of criticism is not that people's
responses are at variance with the correct normative standard but
rather that the explanation for why they are at variance will matter
not only for assessing the rationality of people but what prescriptive
interventions ought to be taken to counter the error. It is rash to
conclude that people, rather than the peculiarities of the task or the
theoretical tools available to us at the moment, are in error.



Lastly, the third type of response is to accept the experimental
results but challenge the claim that they are generalizable. In a
controlled replication of Kahneman and Tversky's lawyer-engineer
example (Tversky & Kahneman 1977), for example, a crucial
assumption is whether the descriptions of the individuals were drawn
at random, which was tested by having subjects draw blindly from an
urn (Gigerenzer, Hell, & Blank 1988). Under these conditions,
base-rate neglect disappeared. In response to the Linda example
(Tversky & Kahneman 1983), rephrasing the example in terms of
*which alternative is more frequent* rather than *which
alternative is more probable* reduces occurrences of the
conjunction fallacy among subjects from 77% to 27% (Fiedler 1988).
More generally, a majority of people presented with the Linda example
appear to interpret 'probability' non-mathematically but
switch to a mathematical interpretation when asked for frequency
judgments (Hertwig & Gigerenzer 1999). Ralph Hertwig and
colleagues have since noted a variety of other effects involving
probability judgments to diminish or disappear when subjects are
permitted to learn the probabilities through sampling, suggesting that
people are better adapted to making a *decision by experience*
of the relevant probabilities as opposed to making a decision by their
*description* (Hertwig, Barron et al. 2004).


### 7.2 Fast and Frugal Heuristics



The Fast and Frugal school and the Biases and Heuristics school both
agree that heuristics are biased. Where they disagree, and disagree
sharply, is whether those biases are necessarily a sign of
irrationality. For the Fast and Frugal program the question is under
what environmental conditions, if any, does a particular heuristic
perform effectively. If the heuristic's structural bias is
well-suited to the task environment, then the bias of that heuristic
may be an advantage for making accurate judgments rather than a
liability
 (section 4).
 We saw this adaptive strategy before in our discussion of
Brunswik's lens model
 (section 3.2),
 although there the bias in the model was to assume that both the
environment and the subject's responses were linear. The aim of
the Fast and Frugal program is to adapt this Brunswikian strategy to a
variety of improper models.



This general goal of the Fast and Frugal program leads to a second
difference between the two schools. Because the Fast and Frugal
program aims to specify the conditions under which a heuristic will
lead to better outcomes than competing models, heuristics are treated
as algorithmic models of decision-making rather than descriptions of
errant effects; heuristics are themselves objects of study. To that
end, all heuristics in the fast and frugal tradition are conceived to
have three components: (i) a search rule, (ii) a stopping rule, and
(iii) a decision rule. For example, *Take-the-Best* (Gigerenzer
& Goldstein 1996), is a heuristic applied to binary, forced-choice
problems. Specifically, the task is to pick the correct option
according to an external criterion, such as correctly picking which of
a pair of cities has a larger population, based on cue information
that is available to the decision-maker, such as whether she has heard
of one city but not the other, whether one city is known to have a
football franchise in the professional league, et cetera. Based on
data sets, one can compute the predictive validity of different cues,
and thus derive their weights. Take-the-Best then has the following
structure: *Search rule:* Look up the cue with the highest
cue-validity; *Stopping rule:* If the pair of objects have
different cue values, that is, one is positive and the other negative,
stop the search. If the cue values are the same, continue searching
down the cue-order; *Decision rule:* Predict that the
alternative with the positive cue value has the higher
target-criterion value. If all cues fail to discriminate, that is, if
all cue values are the same, then predict the alternative randomly by
a coin flip. The bias of Take-the-Best is that it ignores relevant
cues. Another example is *tallying*, which is a type of
improper linear model
 (section 2.3).
 Tallying has the following structure for a binary, forced-choice
task: *Search rule:* Look up cues in a random order;
*Stopping rule:* After some exogenously determined *m*
\((1 < m \leq N)\) of the *N* available cues are evaluated,
stop the search; *Decision rule:* Predict that the alternative
with the higher number of positive cue values has the higher
target-criterion value. The bias in tallying is that it ignores cue
weights. One can see then how models are compared to one another by
how they process cues and their performance is evaluated with respect
to a specified criterion for success, such as the number of correct
answers to the city population task.



Because Fast and Frugal heuristics are computational models, this
leads to a third difference between the two schools. Kahneman endorses
the System I and System II theory of cognition (Stanovich & West
2000). Furthermore, Kahneman classifies heuristics as fast, intuitive,
and non-deliberative System I thinking. Gigerenzer, by contrast, does
not endorse the System I and System II hypothesis, thus rejects
classifying heuristics as, necessarily, non-deliberative cognitive
processes. Because heuristics are computational models in the Fast and
Frugal program, in principle each may be used deliberatively by a
decision-maker or used by a decision-modeler to explain or predict a
decision-maker's non-deliberative behavior. The Linear Optical
Trajectory (LOT) heuristic (McBeath, Shaffer, & Kaiser 1995) that
baseball players use intuitively, without deliberation, to catch fly
balls, and which some animals appear to use to intercept prey, is the
same heuristic that the "Miracle on the Hudson" airline
pilots used deliberatively to infer that they could not reach an
airport runway and decided instead to land their crippled plane in the
Hudson river.



Here are a list of heuristics studied in the Fast and Frugal program
(Gigerenzer, Hertwig, & Pachur 2011), along with an informal
description for each along with historical and selected contemporary
references.


1. **Imitation.** *People have a strong tendency to
imitate the successful members of their communities* (Henrich
& Gil-White 2001). 



> 
> 
> If some one man in a tribe ...invented a new snare or weapon, or
> other means of attack or defense, the plainest self-interest, without
> the assistance of much reasoning power, would prompt other members to
> imitate him. (Darwin 1871, 155)
> 
> 
> 
> 



Imitation is presumed to be fundamental to the speed of cultural
adaptation including the adoption of social norms
 (section 3.4).
2. **Preferential Attachment.** *When given the choice to
form a new connection to someone, pick the individual with the most
connections to others* (Yule 1925; Barabasi & Albert
1999; Simon 1955b).
3. **Default rules.** *If there is an applicable default
rule, and no apparent reason for you to do otherwise, follow the
rule.* (Fisher 1936; Reiter 1980; Thaler & Sustein 2008;
Wheeler 2004).
4. **Satisficing.** *Search available options and choose
the first one that exceeds your aspiration level.* (Simon 1955a;
Hutchinson et al. 2012).
5. **Tallying.** *To estimate a target criterion, rather
than estimate the weights of available cues, instead count the number
of positive instances* (Dawes 1979; Dana & Dawes
2004).
6. **One-bounce Rule (Hey's Rule B).** *Have at
least two searches for an option. Stop if a price quote is larger than
the previous quote.* The one-bounce rule plays
"winning-streaks" by continuing search while you keep
receiving a series of lower and lower quotes, but stops as soon as
your luck runs out (Hey 1982; Charness & Kuhn 2011).
7. **Tit-for-tat.** *Begin by cooperating, then respond
in kind to your opponent; If your opponent cooperates, then cooperate;
if your opponent defects, then defect* (Axelrod 1984; Rapaport,
Seale, & Colman 2015).
8. **Linear Optical Trajectory (LOT).** *To intersect
with another moving object, adjust your speed so that your angle of
gaze remains constant.* (McBeath et al. 1995; Gigerenzer
2007).
9. **Take-the-best.** *To decide which of two
alternatives has a higher value on a specific criterion, (i) first
search the cues in order of their predictive validity; (ii) next, stop
search when a cue is found which discriminates between the
alternatives; (iii) then, choose the alternative selected by the
discriminating cue. (iv) If all cues fail to discriminate between the
two alternatives, then choose an alternative by chance* (Einhorn
1970; Gigerenzer & Goldstein 1996).
10. **Recognition:** *To decide which of two alternatives
has a higher value on a specific criterion and one of the two
alternatives is recognized, choose the alternative that is
recognized* (Goldstein & Gigerenzer 2002; Davis-Stober, Dana,
& Budescu 2010; Pachur, Todd, et al. 2012).
11. **Fluency:** *To decide which of two alternatives has
a higher value on a specific criterion, if both alternatives are
recognized but one is recognized faster, choose the alternative that
is recognized faster* (Schooler & Hertwig 2005; Herzog &
Hertwig 2013).
12. **\(\frac{1}{N}\) Rule:** *For *N* feasible
options, invest resources equally across all *N* options*
(Hertwig, Davis, & Sulloway 2002; DeMiguel, Garlappi, & Uppal
2009).



There are three lines of responses to the Fast and Frugal program to
mention. Take-the-Best is an example of a *non-compensatory*
decision rule, which means that the first discriminating cue cannot be
"compensated" by the cue-information remaining down the
order. This condition, when it holds, is thought to warrant taking a
decision on the first discriminating cue and ignoring the remaining
cue-information. The computational efficiency of Take-the-Best is
supposed to come from only evaluating a few cues, which number less
than 3 on average in benchmarks tests (Czerlinski et al. 1999).
However, all of the cue validities need to be known by the
decision-maker and sorted before initiating the search. So,
Take-the-Best by design treats a portion of the necessary
computational costs to execute the heuristic as exogenous. Although
the lower-bound for sorting cues by comparison is \(O(n \log n)\),
there is little evidence to suggest that humans sort cues by the most
efficient sorting algorithms in this class. On the contrary, such
operations are precisely of the kind that qualitative probability
judgments demand
 (section 1.2).
 Furthermore, in addition to the costs of ranking cue validities,
there is the cost of acquisition and the determination that the
agent's estimates are non-compensatory. Although the exact
accounting of the cognitive effort presupposed is unknown, and argued
to be lower than critics suggest (Katsikopoulos et al. 2010),
nevertheless these necessary steps threaten to render Take-the-Best
non-compensatory in execution but not in what is necessary prior to
setting up the model to execute.



A second line of criticism concerns the cognitive plausibility of Take
the Best (Chater, Oaksford, Nakisa, & Redington 2003). Nearly all
of the empirical data on the performance characteristics of
Take-the-Best are by computer simulations, and those original
competitions pitted Take the Best against standard statistical models
(Czerlinski et al. 1999) but omitted standard machine learning
algorithms that Chater, Oaksford and colleagues found performed just
as well as Take the Best. Since these initial studies, the focus has
shifted to machine learning, and includes variants of Take-the-Best,
such as "greedy cue permutation" that performs provably
better than the original and is guaranteed to always find accurate
solutions when they exist (Schmitt & Martignon 2006). Setting
aside criticisms targeting the comparative performance advantages of
Take the Best qua decision model, others have questioned the
plausibility of using Take-the-Best as a cognitive model. For example,
Take-the-Best presumes that cue-information is processed serially, but
the speed advantages of the model translate to an advantage in human
decision-making derives only if humans process cue information on a
serial architecture. If instead people process cue information on a
parallel cognitive architecture, then the comparative speed advantages
of Take-the-Best would become moot (Chater et al. 2003).



The third line of criticism concerns whether the Fast-and-Frugal
program truly mounts a challenge to the normative standards of
optimization, dominance-reasoning, and consistency, as advertised.
Take-the-Best is an algorithm for decision-making that does not
comport with the axioms of expected utility theory. For one thing, its
lexicographic structure violates the Archimedean axiom
 (section 1.1, A2).
 For another, it is presumed to violate the transitivity condition of
the Ordering axiom
 (A1).
 Further still, the "less-is-more" effects appear to
violate Good's principle (Good 1967), a central pillar of
Bayesian decision theory, which recommends to delay making a terminal
decision between alternative options if the opportunity arises to
acquire free information. In other words, according canonical
Bayesianism, free advice is a bore but no one ought to turn down free
information (Pedersen & Wheeler 2014). If noncompensatory decision
rules like Take-the-Best violate Good's principle, then perhaps
the whole Bayesian machinery ought to go (Gigerenzer & Brighton
2009).



But these points merely tell us that attempts to formulate
Take-the-Best in terms of an ordering of prospects on a real-valued
index won't do, not that ordering and numerical indices have all
got to go. As we saw in
 section 1.1,
 there is a long and sizable literature on lexicographic probabilities
and non-standard analysis, including early work specifically
addressing non-compensatory nonlinear models (Einhorn 1970). Second,
Gigerenzer argues that "cognitive algorithms...need to meet
more important constraints than internal consistency"
(Gigerenzer & Goldstein 1996), which includes transitivity, and
elsewhere advocates abandoning coherence as a normative standard
(Arkes, Gigerenzer, & Hertwig 2016). However, Take-the-Best
presupposes that cues are ordered by cue validity, which naturally
entails transitivity, otherwise Take-The-Best could neither be
coherently specified nor effectively executed. More generally, the
Fast and Frugal school's commitment to formulating heuristics
algorithmically and implementing them as computational models commits
them to the normative standards of optimization, dominance reasoning,
and logical consistency.



Finally, Good's principle states that a decision-maker facing a
single-person decision-problem cannot be worse (in expectation) from
receiving free information. Exceptions are known in game theory
(Osborne 2003: 283), however, that involve asymmetric information
among two or more decision-makers. But there is also an exception for
single-person decision-problems involving indeterminate or imprecise
probabilities (Pedersen & Wheeler 2015). The point is that
Good's principle is not a fundamental principle of probabilistic
methods, but instead is a specific result that holds for the canonical
theory of single-person decision-making with determinate
probabilities.


## 8. Appraising Human Rationality



The rules of logic, the axioms of probability, the principles of
utility theory--humans flout them all, and do so as a matter of
course. But are we irrational to do so? That depends on what being
rational amounts to. For a Bayesian, any qualitative comparative
judgment that does not abide by the axioms of probability is, by
definition, irrational. For a baker, any recipe for bread that is
equal parts salt and flour is irrational, even if coherent. Yet
Bayesians do not war with bakers. Why? Because bakers are satisfied
with the term 'inedible' and do not aspire to commandeer
'irrational'.



The two schools of heuristics
 (section 7)
 reach sharply different conclusions about human rationality. Yet,
unlike bakers, their disagreement involves the meaning of
'rationality' and how we ought to appraise human judgment
and decision making. The "rationality wars" are not the
result of "rhetorical flourishes" concealing a broad
consensus (Samuels, Stich, & Bishop 2002), but substantive
disagreements
 (section 7.2)
 that are obscured by ambiguous use of terms like
'rationality'.



In this section we first distinguish seven different notions of
rationality, highlighting the differences in aim, scope, standards of
assessment, and differences in the objects of evaluation. We then turn
to consider two importantly different normative standards
used in bounded rationality, followed by an example, the
*perception-cognition gap*, illustrating how slight variations
of classical experimental designs in the biases and heuristics
literature change both the results and the normative standards used to
evaluate those results.


### 8.1 Rationality



While Aristotle is credited with saying that humans are rational,
Bertrand Russell later confessed to searching a lifetime in vain for
evidence in Aristotle's favor. Yet 'rationality' is
what Marvin Minsky called a suitcase word, a term that needs to be
unpacked *before* getting anywhere.



One meaning, central to decision theory, is *coherence*, which
is merely the requirement that your commitments not be self-defeating.
The subjective Bayesian representation of rational preference over
options as inequalities in subjective expected utility delivers
coherence by applying a dominance principle to (suitably structured)
preferences. A closely related application of dominance reasoning is
the minimization of expected loss (or maximization of expected gain in
economics) according to a suitable loss function, which may even be
asymmetric (Elliott, Komunjer, & Timmermann 2005) or applied to
radically restricted agents, such as finite automata (Rubinstein
1986). Coherence and dominance reasoning underpin expected utility
theory
 (section 1.1),
 too.



A second meaning of rationality refers to an interpretive stance or
disposition that we take to understand the beliefs, desires, and
actions of another person (Dennett 1971) or to understand anything
they might say in a shared language (Davidson 1974). On this view,
rationality refers to a bundle of assumptions we grant to another
person in order to understand their behavior, including speech. When
we offer a reason-giving explanation for another person's
behavior, we take such a stance. If I say "*the driver
laughed because she made a joke*" you would not get far in
understanding me without granting to me, and even this imaginary
driver and woman, a lot. So, in contrast to the lofty normative
standards of coherence that few if any mortals meet, the standards of
rationality associated with an interpretive stance are met by
practically everyone.



A third meaning of rationality, due to Hume (1738), applies to your
beliefs, appraising them in how well they are calibrated with your
experience. If in your experience the existence of one thing is
invariably followed by an experience of another, then believing that
the latter follows the former is rational. We might even go so far as
to say that your expectation of the latter given your experience of
the former is rational. This view of rationality is an evaluation of a
person's commitments, like coherence standards; but unlike
coherence, Hume's notion of rationality seeks to tie the
rational standing of a belief directly to evidence from the world.
Much of contemporary epistemology endorses this concept of rationality
while attempting to specify the conditions under which we can
correctly attribute knowledge to someone.



A fourth meaning of rationality, called *substantive
rationality* by Max Weber (1905), applies to the evaluation of
your aims of inquiry. Substantive rationality invokes a Kantian
distinction between the worthiness of a goal, on the one hand, and how
well you perform instrumentally in achieving that goal, on the other.
Aiming to count the blades of grass in your lawn is arguably not a
rational end to pursue, even if you were to use the instruments of
rationality flawlessly to arrive at the correct count.



A fifth meaning of rationality, due to Peirce (1955) and taken up by
the American pragmatists, applies to the process of changing a belief
rather than the Humean appraisal of a currently held belief. On
Peirce's view, people are plagued by doubt not by belief; we
don't expend effort testing the sturdiness of our beliefs, but
rather focus on those that come into doubt. Since inquiry is pursued
to remove the doubts we have, not certify the stable beliefs we
already possess, principles of rationality ought to apply to the
methods for removing doubt (Dewey 1960). On this view, questions of
what is or is not substantively rational will be answered by the
inquirer: for an agronomist interested in grass cover sufficient to
crowd out an invasive weed, obtaining the grass-blade count of a lawn
might be a substantively rational aim to pursue.



A sixth meaning of rationality appeals to an organism's
capacities to assimilate and exploit complex information and revise or
modify it when it is no longer suited to task. The object of
rationality according to this notion is *effective behavior*.
Jonathan Bennett discusses this notion of rationality in his case
study of bees:



> 
> 
> All our *prima facie* cases of rationality or intelligence were
> based on the observation that some creature's behaviour was in
> certain dependable ways successful or appropriate or apt, relative to
> its presumed wants or needs. ...There are canons of
> appropriateness whereby we can ask whether an apian act is appropriate
> not to that which is particular and present to the bee but rather to
> that which is particular and past or to that which is not particular
> at all but universal. (Bennett 1964: 85)
> 
> 
> 



Like Hume's conception, Bennett's view ties rationality to
successful interactions with the world. Further, like the pragmatists,
Bennett includes for appraisal the dynamic process rather than simply
the synchronic state of one's commitments or the current merits
of a goal. But unlike the pragmatists, Bennett conceives of
rationality to apply to a wider range of behavior than the logic of
deliberation, inquiry, and belief change.



A seventh meaning of rationality resembles the notion of coherence by
defining rationality as the absence of a defect. For Bayesians,
sure-loss is the epitome of *irrationality* and coherence is
simply its absence. Sorensen has suggested a generalization of this
strategy, one where rationality is conceived as the absence of
irrationality *tout court*, just as cleanliness is the absence
of dirt. Yet, owing to the long and varied ways that irrationality can
arise, a consequence of this view is that there then would be no
unified notion of rationality to capture the idea of thinking as one
ought to think (Sorensen 1991).



These seven accounts of rationality are neither exhaustive nor
complete. But they suffice to illustrate the range of differences
among rationality concepts, from the objects of evaluation and the
standards of assessment, to the roles, if any at all, that rationality
is conceived to play in reasoning, planning, deliberation,
explanation, prediction, signaling, and interpretation. One
consequence of this hodgepodge of rationality concepts is a pliancy in
the attribution of irrationality that resembles Victorian methods for
diagnosing the vapors. The time may have come to retire talk of
rationality altogether, or to demand a specification of the objects of
evaluation, the normative standards to be used for assessment, and
require ample attention to the implications that follow from those
commitments.


### 8.2 Normative Standards in Bounded Rationality



What are the standards against which our judgments and decisions ought
to be evaluated? A property like systematic bias may be viewed as a
fault or an advantage depending on how outcomes are scored
 (section 4).
 A full reckoning of the costs of operating a decision procedure may
tip the balance in favor of a model that is sub-optimal when costs are
not considered, even when there is agreement of how outcomes are to be
scored
 (sections 2.1).
 Desirable behavior, such as prosocial norms, may be impossible within
an idealized model but commonplace in several different types of
non-idealized models
 (section 5.2).



Accounts of bounded rationality typically invoke one of two types of
normative standards, a coherence standard or an accuracy standard.
Among the most important insights from the study of boundedly rational
judgment and decision making is that, not only is it possible to meet
one standard without meeting the other, but meeting one standard may
inhibit meeting the other.



Coherence standards in bounded rationality typically appeal to
probability, statistical decision theory, or propositional logic. The
"standard picture" of rational reasoning, according to
Edward Stein,



> 
> 
> is to reason in accordance with principles of reasoning that are based
> on rules of logic, probability theory, and so forth. If the standard
> picture of reasoning is right, principles of reasoning that are based
> on such rules are *normative principles of reasoning*, namely
> they are principles we *ought* to reason in accordance with.
> (Stein 1996: 1.2)
> 
> 
> 



The coherence standards of logic and probability are usually invoked when
there are experimental results purporting to violate those
standards, particularly in the biases and heuristics literature
 (section 7.1).
 However, little is said about how and when our reasoning ought to be
in accordance with these standards or even what, precisely, the applicable 
normative standards of logic and probability are. Stein
discusses the logical rule of *And-Elimination* and a normative
principle for belief that it supports, one where believing the
conjunction *birds sing and bees waggle* commits you rationally
to believing each conjunct. Yet Stein switches to probability to
discuss what principle ought to govern conjoining two beliefs.
Why?



Propositional logic and probability are very different formalisms 
(Haenni, Romeijn, Wheeler, & Williamson 2011). For one thing, the
truth-functional semantics of logic is compositional whereas
probability is not compositional, except when events are
probabilistically independent. Why then is the elimination rule from
logic and the introduction rule from probability the standard rather
than the elimination rule from probability (i.e., marginalization) and the
introduction rule from logic (i.e., adjunction)? Answering this question
requires a positive account of what "based on",
"anchored in", or other metaphorical relationships amount
to. But, in appeals to principles of reasoning, typically there is no analog to the
representation theorems of expected utility theory 
 (section 1.1)
 specifying the relationship between qualitative judgments and their
logical or numerical representation, and no account of the conditions
under which such relationships hold.



The second type of normative standard assesses the accuracy of a
judgment or decision making process, where the focus is getting the
correct answer. Consider the accuracy of a categorical judgment, such
as predicting whether a credit-card transaction is fraudulent (\(Y =
1\)) or legitimate (\(Y = 0\)). *Classification accuracy* is
the number of correct predictions from all predictions made, which is
often expressed as a ratio. But classification accuracy can yield a
misleading assessment. For example, a method that always reported
transactions as legitimate, \(Y = 0\), would in fact yield a very high
accuracy score (>97%) due to the very low rate (<3%) of
fraudulent credit card transactions. The problem here is that
classification accuracy is a poor metric for problems that involve
imbalanced classes with few positive instances (i.e., few cases where
\(Y=1\)). More generally, a model with no predictive power can have
high accuracy, and a model with comparatively lower accuracy can have
greater predictive power. This observation is referred to as the
*accuracy paradox*.



The accuracy paradox is one motivation for introducing other measures
of predictive performance. For our fraud detection problem there are
two ways your prediction can be correct and two ways it can be wrong.
A prediction can be correct by predicting that \(Y=1\) when in fact a
transaction is fraudulent (a *true positive*) or predicting
\(Y=0\) when in fact a transaction is legitimate (a *true
negative*). Correspondingly, one may err by either predicting
\(Y=1\) when in fact \(Y=0\) (a *false positive*) or predicting
\(Y=0\) when in fact a transaction is fraudulent (a *false
negative*). These four possibilities are presented in the
following two-by-two contingency table, which is sometimes referred to
as a *confusion matrix*:




|  |  |
| --- | --- |
|  | **Actual Class** |
|  | \(Y\) | 1 | 0 |
| **Predicted Class** | 1 | *true positive* | *false positive* |
| 0 | *false negative* | *true negative* |



For a binary classification problem involving *N* examples, each
prediction will fall into one of these four categories. The
performance of your classifier with respect to those *N* examples
can then be assessed. A perfectly inaccurate classifier will have all
zeros in the diagonal; a perfectly accurate classifier will have all
zeros in the counterdiagonal. The *precision* of your
classifier is the ratio of true positives to all positive predictions,
that is *true positives* / (*true positives* + *false
positives*). The *recall* of your classifier is the ratio
of true positives to all true predictions, that is *true
positives* / (*true positives* + *false
negatives*).



There are two points to notice. The first is that in practice there is
typically a trade-off between precision and recall, and the costs to
you of each will vary from one problem to another. A trade-off of
precision and recall that suits detecting credit card fraud may not
suit detecting cancer, even if the frequencies of positive instances
are identical. The point of training a classifier on known data is to
make predictions on out of sample instances. So, tuning your
classifier to yield a suitable trade-off between precision and recall
in your training data is no guarantee that you will see this trade-off
generalize.



The moral is that to evaluate the performance of your classifier it is
necessary to specify the purpose for making the classification and
even then good performance on your training data may not generalize.
None of this is antithetical to coherence reasoning per se, as we are
making comparative judgments and reasoning by dominance. But putting
the argument in terms of coherence changes the *objects* of
evaluation, moving from the point of view from the first person 
decision maker to that of a third person decision modeler.


### 8.3 The Perception-Cognition Gap



Do human beings systematically violate the norms of probability and
statistics? Petersen and Beach (1967) thought not. On their view human
beings are intuitive statisticians
 (section 5.1),
 so probability theory and statistics are a good, first approximation
of human judgment and decision making. Yet, just as their optimistic
review appeared to cement a consensus view about human rationality,
Amos Tversky and Daniel Kahneman began their work to undo it. People
are particularly bad at probability and statistics, the heuristics and
biases program
 (section 7.1)
 found, so probability theory, statistics, and even logic do not offer
a good approximation of human decision making. One controversy over
these negative findings concerns the causes of those
effects--whether the observed responses point to minor flaws in
otherwise adaptive human behavior or something much less charitable
about our habits and constitution.



In contrast to this poor showing on cognitive tasks, people are
generally thought to be optimal or near-optimal in performing
low-level motor control and perception tasks. Planning goal-directed
movement, like pressing an elevator button with your finger or placing
your foot on a slippery river stone, requires your motor control
system to pick one among a dizzying number of possible movement
strategies to achieve your goal while minimizing biomechanical costs
(Trommershauser, Maloney, & Landy 2003). The loss function
that our motor control system appears to use increases approximately
quadratically with error for small errors but significantly less for
large errors, suggesting that our motor control system is also robust
to outliers (Kording & Wolpert 2004). What is more, advances
in machine learning have been guided by treating human performance
errors for a range of perception tasks as proxies for Bayes error,
yielding an observable, near-perfect normative standard. Unlike
cognitive decisions, there is very little controversy concerning the
overall optimality of our motor-perceptual decisions. This difference
between high-level and low-level decisions is called the
*perception-cognition gap*.



Some view the perception-cognition gap as evidence for the claim that
people use fundamentally different strategies for each type of task
 (section 7.2).
 An approximation of an optimal method is not necessarily an optimal
approximation of that method, and the study of cognitive judgments and
deliberative decision-making is led astray by assuming otherwise
(Mongin 2000). Another view of the perception-cognition gap is that it
is largely an artifact of methodological differences across studies
rather than a robust feature of human behavior. We review evidence for
this second argument here.



Classical studies of decision-making present choice problems to
subjects where probabilities are described. For example, you might be
asked to choose the prospect of winning EUR300 with probability
0.25 or the prospect of winning EUR400 with probability 0.2. Here,
subjects are given a numerical description of probabilities, are
typically asked to make one-shot decisions without feedback, and their
responses are found to deviate from the expected utility hypothesis.
However, in motor control tasks, subjects have to use internal,
implicit estimates of probabilities, often learned with feedback, and
these internal estimates are near optimal. Are perceptual-motor
control decisions better because they provide feedback whereas
classical decision tasks do not, or are perceptual-motor control
decisions better because they are non-cognitive?



Jarvstad et al. (2013) explored the robustness of the
perception-cognition gap by designing (a) a finger-pointing task that
involved varying target sizes on a touch-screen computer display; (b)
an arithmetic learning task involving summing four numbers and
accepting or rejecting a proposed answer with a target tolerance,
where the tolerance range varied from problem to problem, analogous to
the width of the target in the motor-control task; and (c) a standard
classical probability judgment task that involved computing the
expected value of two prospects. The probability information across
the tasks was in three formats: low-level, high-level, and classical,
respectively.



Once confounding factors across the three types of tasks are
controlled for, Jarvstad et al.'s results suggest that (i) the
perception-cognition gap is largely explained by differences in how
performance is assessed; (ii) the *decisions by experience* vs
*decisions by description* gap (Hertwig, Barron et al. 2004) is
due to assuming that exogenous objective probabilities and subjective
probabilities match; (iii) people's ability to make high-level
decisions is better than the biases and heuristics literature suggests
 (section 7.1);
 and (iv) differences between subjects are more important for
predicting performance than differences between the choice tasks
(Jarvstad et al. 2013).



The upshot, then, is that once the methodological differences are
controlled for, the perception-cognition gap appears to be an artifact
of two different normative standards applied to tasks. If the
standards applied to assessing perceptual-motor tasks are applied to
classical cognitive decision-making tasks, then both appear to perform
well. If instead the standards used for assessing the classical
cognitive tasks are applied to perceptual-motor tasks, then both will
appear to perform poorly.