## 1. History



John von Neumann and Oskar Morgenstern laid the foundations of
classical game theory in their treatise *Theory of Games and
Economic Behavior* (von Neumann & Morgenstern 1944). Following
a series of refinements published in the 1950s by numerous theorists,
most notably John Nash, game theory then transformed social science
over the course of the next several decades. *Noncooperative game
theory*, the more fundamental branch of game theory, explores
scenarios where the results of an agent's actions depend upon
what other agents do, and where agents lack external mechanisms for
enforcing agreements. More precisely, it provides a model of how
agents satisfying certain criteria of rationality interact in
*games* characterized by the *actions* or
*strategies* available to each of the agents and the
*payoffs* they can achieve. Payoffs in such scenarios are a
product of the *joint strategy profile* that is defined by the
actions independently chosen by the respective agents and the
*state* of the world that obtains which reflects aspects of the
game that are variable and over which no agent has control. Many
scenarios in noncooperative game theory can be fruitfully analyzed
using *ordinal preferences* constructed from agents'
rankings of possible outcomes. However, von Neumann and Morgenstern
also introduced a theory of *cardinal utility* that provides a
more precise account of payoffs by using agents' preferences
over lotteries to reflect how much agents prefer one outcome to
another. Utilizing this framework, a more varied set of solution
concepts was developed to predict how games would be played, and to
determine when individual agents are employing optimal strategies and
what the consequences of this are for the set of all agents.



One of the most widely utilized solution concepts is the *Nash
equilibrium*. Based upon the *Bayesian rationality*
criterion according to which one chooses so as to maximize her
expected utility, a Nash equilibrium is a profile of strategies
adopted by a set of agents in which the strategy adopted by each agent
maximizes her respective payoff given the probabilistically
independent strategies adopted by other agents. In other words, at a
Nash equilibrium each agent's strategy is a best response to the
others' strategies, so that none can do better by acting
differently given what others are doing. Although Nash was not the
first to introduce this equilibrium concept, its broad applicability
was recognized only after Nash refined the concept and proved that if
a noncooperative game has finitely many pure strategy profiles, then
at least one Nash equilibrium always exists. Prior to Nash's
proof it was widely known that many games lack a Nash equilibrium in
*pure strategies* where each agent optimally responds to what
every other agent does by adopting one of the actions available to
her. This initially limited the class of problems to which the concept
could be applied, but Nash showed that in such cases there will always
be a Nash equilibrium in *mixed strategies* were each agent
optimally responds to what other agents do by choosing among a subset
of the strategies available to her with some probability. The
usefulness of this concept would then be further extended by
subsequent refinements, including most notably those made by John
Harsanyi and Reinhard Selten (see, e.g., Harsanyi & Selten
1988).



In addition to his work refining solution concepts in noncooperative
game theory Nash also made important contributions to *cooperative
game theory*, an approach to modeling strategic interactions that
extends things by allowing for the possibility that agents can be
constrained by binding agreements. Among other things, cooperative
game theory provides a set of tools for analyzing questions related to
coalition building, as well as problems that arise in contexts where
agents can rely on external authorities to enforce agreements.
Nash's most important contribution here was to introduce two
approaches to analyzing the *bargaining problem*. This problem
has been important in contractarian approaches to justifying moral and
political orders, a subject we'll return to in
 section 4.



A third approach to using game theoretic tools to model social
interactions explores how *boundedly rational* agents might
achieve certain outcomes as a result of *inductive learning* or
the presence of *focal points*. The inductive learning aspect
of this approach is known as
 *evolutionary game theory*,
 so-called because much of the early work on inductive learning was
applied to the question of why various biological traits or behavioral
patterns emerge in particular populations. John Maynard Smith's
*Evolution and the Theory of Games* (1982) is an early classic
in this tradition, while subsequent work by anthropologists,
economists, and political scientists would apply this approach to
problems of interest to social and political philosophers, including
especially how cooperation can be sustained in large groups. Where
much of the foundational work on inductive learning was done by
biologists, much of the early work on focal points was done by
economists, with Thomas Schelling's book *The Strategy of
Conflict* (1960) being particularly influential. Roughly speaking,
focal points are actions or outcomes that are likely to be
*salient* (Lewis 1969: 35-36) in the sense that they
"stand out" to the agents involved in a game. As Schelling
showed, such points are especially important in games where the
strategy space is large and/or there are many potential
equilibria.



Richard Braithwaite was the first professional philosopher to formally
employ game theory in the analysis of a problem of ethics. In his 1954
lecture *Theory of Games as a Tool for the Moral Philosopher*
(published 1955), Braithwaite used Nash's bargaining problem to
argue for a particular resolution to the *fair division*
problem. He also conjectured that game theory might transform moral
philosophy, much as statistics had transformed the social sciences.
Although this transformation has not taken place in philosophy in the
way that it has for some of the other social sciences, game theory has
made significant inroads into moral and political philosophy in the
years since Braithwaite's lecture. David Gauthier's
*Morals by Agreement* (1986) and Ken Binmore's *Game
Theory and the Social Contract* (2 volumes, 1994, 1998) are
arguably the first two works to present systematic moral theories
built from game-theoretic foundations. Gauthier's theory is
rooted in classical game theory, although he proposes revisions to the
orthodox account of Bayesian rationality that classical theory
typically assumes. Binmore's theory, on the other hand,
integrates elements of both evolutionary and classical game theory.
Other significant uses of game theory in ethics include, but are not
limited to:


1. analyses of social conventions as equilibria of games predicated
upon common knowledge (Lewis 1969; Vanderschraaf 2019);
2. analyses of social norms (Ullmann-Margalit 1977; Basu 2000;
Bicchieri 2006, 2017; Brennan, Eriksson, Goodin & Southwood
2013);
3. explanations of elements of social contracts in terms of
evolutionary game theory (Sugden 1986; Skyrms 1996); and
4. applications of Nash's bargaining theory in alternative
social contract theories (Gaus 2010; Muldoon 2016; Moehler 2018;
Vanderschraaf 2019).



Furthermore, although the formal frameworks associated with game
theory did not emerge until the mid-twentieth century, it's now
widely recognized that the work of many earlier philosophers are
amenable to game theoretic analysis, and that, in many cases, these
works clearly anticipated important developments in game theory. For
instance, Barry (1965), Gauthier (1969), Hampton (1986), Kavka (1986),
and Taylor (1987) provide game-theoretic characterizations of
Hobbes' arguments for the inevitability of war in the State of
Nature and for the benefit of establishing a sovereign. Indeed, this
branch of Hobbes scholarship continues to develop, as evidenced by
recent works by Vanderschraaf (2007 & 2010), Moehler (2009), and
Chung (2015). And, while Hobbes' arguments have attracted the
most attention from game-theoretic minded scholars, Grotius (1625),
Pufendorf (1673), Locke (1689), Hume (1740), and Rousseau (1755) also
offer what can be interpreted as informal game-theoretic arguments in
their discussions of morality, the State of Nature, and the natural
law. Of these, Hume's analysis of convention in the
*Treatise* (Hume 1740: 3.2.2-10) is particularly
noteworthy for its informal account of equilibria and analysis of the
role that mutual knowledge plays in allowing agents to arrive at and
sustain certain equilibria (Vanderschraaf 1998). Moreover,
Hume's discussion of how agents can settle into conventions as
the result of gradual trial and error processes, and his suggestion
that agents follow some conventions as a result of properties that
make certain alternatives conspicuous, anticipates analyses of games
based on both inductive learning and focal points (Skyrms 1996; Sugden
1986).


## 2. The Function of Morality



One of the most widespread uses of game theory in ethics has been to
postulate various *functions* of morality and to identify some
of its important features. An early example of this is Edna
Ullmann-Margalit's *The Emergence of Norms* which argues
that morality can be understood as a system of norms that enable
agents to coordinate their actions in order to achieve mutually
advantageous outcomes in situations where the pursuit of self-interest
would normally prevent this (Ullmann-Margalit 1977). The now classic
example she uses to illustrate this involves two soldiers who must
choose whether to man their mortars in order to stem the advance of
the enemy, or to flee. If both stay, it is certain that the
enemy's advance will be halted, but the soldiers will almost
surely be injured. If both flee, the enemy will not only be able to
take the mountain pass, it will also be able to overtake and capture
the fleeing soldiers. And if just one of them stays while the other
flees, the brave soldier will die trying to hold off the enemy, while
the other will have just enough time to escape to safety.



The scenario Ullmann-Margalit presents is an example of a
 *prisoner's dilemma*
 and is illustrated in
 Figure 1
 below. In this *strategic form game*, each soldier acts
without observing the other's action. Each soldier has the
choice between the strategies of *fleeing* (*F*) or
*staying* (*S*). The cells in the matrix correspond to
the outcome of each possible pair of choices, with the numbers in the
cells representing the utilities of the outcomes for the soldiers.
Specifically, higher numbers correspond to a more preferred outcome,
and the first number represents the payoff of Soldier 1 who chooses
between the actions corresponding to the rows, while the second number
represents the payoff of Soldier 2 who chooses between the actions
corresponding to the columns.





|  |  | Soldier 2 |
| --- | --- | --- |
|  |  | *S* | *F* |
| Soldier 1 | *S* | (2,2) | (0,3) |
| *F* | (3,0) | (1,1) |



*S* = stay, *F* = flee



Figure 1. Ullmann-Margalit's
Mortarmen's Dilemma




In this situation each soldier has a reason to flee. This is because
fleeing provides the only chance of escaping unharmed, which is the
outcome each most prefers. However, the situation is a dilemma,
because if each soldier pursues this course of action, then they will
both end up worse off than if neither had. Furthermore, the dilemma is
magnified by the fact that for each soldier, *F* is
his unique best response to any strategy the other soldier might
choose. That is, *F* is each soldier's
*strictly dominant strategy*, and \((F,F)\) is the unique
*Nash equilibrium* of this game even though both soldiers would
fare better is they could somehow follow \((S,S).\)



Ullmann-Margalit pointed out that if the soldiers understood the
nature of their predicament they might want to do something to prevent
themselves from both fleeing and subsequently being captured. More
importantly, she argued that the prisoner's dilemma that
characterizes their predicament arises in innumerable interactions
that we find ourselves in every day. At first glance, the obvious
solution to these dilemmas might seem to be for the soldiers to simply
pledge to remain at their posts. The problem, however, is that in
scenarios like this promises are unlikely to be credible, and, even
when they are, they are prone to being exploited. Instead, what the
soldiers need is a way of literally or figuratively chaining
themselves to their mortars. Drawing on this observation,
Ullmann-Margalit's suggestion was that it is moral norms that do
this work. More specifically, it's not the promises to remain at
their posts that solves the soldiers' predicament, so much as
their commitment to keeping their promises and to sanctioning those
who don't.



One way of understanding views like this is that the function of
morality is to prevent failures of rationality. In the next section we
will discuss how game theory has been used to analyze the relationship
between morality and prudential rationality. In the more recent
literature, though, there is growing consensus that the role norms of
social morality play is not necessarily to constrain our instrumental
reasoning so that we can overcome failures of prudential rationality,
but rather to transform mixed-motive games like the prisoner's
dilemma into coordination problems where mutually advantageous
outcomes are more stable and easier to arrive at. A classic example of
this approach is Bicchieri (2006). Like Ullmann-Margalit, Bicchieri
characterizes norms in terms of our commitment to certain courses of
action, and also emphasizes the importance of sanctions in this
context. However, Bicchieri draws a distinction between social norms
and other types of norm, and argues that what distinguishes the former
from the latter is the conditional nature of our commitment to them.
In particular, compliance with social norms is grounded in both an
individual's (i) *empirical expectation* that a
sufficiently large percentage of the population conforms to the norm,
and (ii) her *normative expectation* that a sufficiently large
percentage of the population expects others to conform to the norm.
Only when both of these expectations are satisfied will an individual
comply with a social norm. Because some individuals will be more
sensitive to the expectations of others, or more inclined to want to
see a particular norm complied with, though, the relevant expectations
may vary between individuals. Only when expectations are satisfied for
a sufficiently large number of people, then, will the norm succeed in
transforming a social dilemma where mutually advantageous outcomes are
unstable into a more easily solved coordination problem. And while it
might seem strange at first glance to suggest that norms we are only
conditionally committed to are the lynchpin of social morality, given
the significant role social dilemmas play in preventing us from living
well together, this shouldn't be surprising. For a more thorough
discussion of these issues see the entry on
 *social norms*.



So far our discussion has focused on how game theory has been used to
illustrate some ostensible functions of morality and the role social
norms play in it. However, game theory has also been utilized to
identify particular norms that might be important, as well as to
identify important characteristics of norms. And, where the previous
analysis relied primarily on the tools of non-cooperative game theory,
here it is evolutionary game theory that has been especially
illuminating. Particularly influential in this regard was the work of
Robert Axelrod and colleagues in the early 1980s on the evolution of
cooperation, along with the subsequent work of Robert Boyd, Peter
Richerson, and various collaborators (see Axelrod 2006 and Boyd &
Richerson 2005 for summaries of the relevant work). By analyzing the
performance of various fixed strategies in repeated social dilemmas
they were able to illustrate the important role that conditional
strategies and the willingness to sanction bad behavior play in
allowing cooperation and other prosocial behaviors to not only emerge,
but become optimal from the standpoint of the individual. More
specifically, they showed that in environments where individuals have
a long-run interest in bringing about mutually advantageous outcomes
but a short-term interest in exploiting others for maximum gain, it is
essential that (i) individuals not leave themselves vulnerable to
repeated exploitation, and (ii) at least some individuals be willing
to punish exploitative behavior.



For Axelrod, who studied these issues primarily in the context of
repeated prisoner's dilemmas, the two requirements identified
above amount to the same thing. A strategy of unconditional
cooperation allows an individual to be repeatedly taken advantage of.
To prevent this she should only cooperate with those who are similarly
willing to cooperate. More precisely, she should be *nice*
(that is she should initially be willing to cooperate with anyone),
but she should also be *provocable* (that is she should be
prepared to stop cooperating with anyone who takes advantage of her,
at least for some amount of time). And, because sustained cooperation
is in everybody's long-run interest, in this context withdrawing
future cooperation as a consequence for being taken advantage of is a
form of punishment.



As Boyd and Richerson showed, though, in many contexts it's not
essential that everybody be willing to punish bad behavior. What is
essential is that *enough* individuals are willing to do this.
However, this generates a pair of puzzles. First, in many cases
it's costly to police norms. In order to sustain cooperation,
then, groups often need some of their members to be prepared to engage
in *altruistic punishment*. That is, some individuals need to
be willing to incur individual costs associated with punishment for
the benefit of others. Second, the willingness of some individuals to
police norms does not just allow the evolution of cooperation and
other prosocial norms. Instead it potentially allows any norm to
emerge and become stable. So, although we've seen how game
theory can help explain why a disposition to comply with and police
certain norms may be important features of morality, important
questions remain concerning whether this is necessarily rational
*from the perspective of the individual*. We'll turn to
those questions now. And in section 5 we'll return to the
question of how to distinguish good from bad norms.


## 3. Reconciling Morality and Prudential Rationality



Kavka (1984) dubs the body of attempts to establish that compliance
with moral requirements is demanded by, or at least compatible with,
prudential rationality the *Reconciliation Project*. At least
since Plato philosophers have been grappling with this project, and
game theory can help us to distinguish between three approaches they
have taken.



The challenge presented by Hobbes' Foole in *Leviathan*
(Hobbes 1651: chapter 15) nicely characterizes the core issue at stake
in the Reconciliation Project. The Foole maintains that under certain
circumstances one's "most reasonable" course of
action is to commit an injustice. Specifically, the Foole claims one
should sometimes break a promise she has made as part of a covenant if
the other parties to the covenant have already kept (or will be
compelled to keep) their promises (Hobbes 1651: 15:5). This is because
performing as a covenant requires can be costly and rationality
plausibly requires that such costs should be avoided if one
doesn't stand to gain anything further by incurring them. This
challenge is illustrated in the decision tree in
 Figure 2
 below.



![a decision tree: link to extended description below](Figure2.svg)

Figure 2. Foole's Opportunity Game
[An
 extended description of figure 2
 is in the supplement.]




Here Claudia and Laura have created a bilateral covenant by exchanging
promises. But because Claudia has already performed her part of the
covenant, and Laura knows this, Laura stands in the position of the
Foole and can either *perform* (*P*) as the covenant
requires of her or *aggress* (*A*) by breaking her
promise and exploiting Claudia's earlier performance. As the
Foole contends, given this payoff structure, *A* is
Laura's unique Bayesian rational choice.



Importantly, though, the Foole does not just present a challenge to
those who endorse her reasoning. This is because in order to avoid
being taken advantage of by individuals who reason like the Foole,
individuals who might otherwise be inclined to keep their promises can
sometimes find themselves with reason to break their promises. We see
this in
 Figure 3
 below which depicts the *sequential prisoner's dilemma*
that results from extending the decision tree from
 Figure 2
 to include Claudia's initial move.



![a decision tree: link to extended description below](Figure3.svg)

Figure 3. Sequential Prisoner's
Dilemma [An
 extended description of figure 3
 is in the supplement.]




In this *extensive form game*, which makes explicit the
sequential order of the agents' actions, the
*i*th component of each payoff vector is the payoff
of the *i*th agent who moves in the game (so
Claudia's payoff is depicted first, since she acts first). Here
we can suppose that Claudia does not agree with the Foole that one
should break her promise if one's partner in a covenant has kept
her promise, but a dilemma nevertheless arises if Claudia suspects
that Laura might agree with the Foole. This is because Claudia might
reason that if she performs, then Laura, having already benefited and
not standing to gain anything further, will not reciprocate.
Consequently, Claudia, *anticipating* (*A*)
Laura's aggression, decides not to perform, leading Laura, now
understandably, not to perform either, so that each ultimately does
worse than if they had both performed.



To take up the Reconciliation Project is to argue alongside Hobbes
that the Foole is not right, and that we need not worry that for some
among us rationality might counsel against acting as morality
requires. And, as we indicated above, there are three main strategies
for doing so.


### 3.1 The Inseparable Goods Strategy



The *inseparable goods strategy* appeals to certain benefits
one can gain or preserve only by acting morally, that the Foole
apparently fails to consider. Plato's account of the benefits of
justice in the *Gorgias* and *Republic* is one example
of this approach. Plato maintains that if one commits an injustice,
then one's soul will consequently be corrupted, and he further
maintains that to have a corrupt soul is the worst of all possible
misfortunes (*Gorgias* 478c3-478e5, *Republic*
353d3-354a7). Alternatively, one might follow Aquinas in arguing
that an injustice like stealing is a mortal sin which will bring about
terrible consequences in the afterlife (Aquinas 1271: II.II Q. 66 A.
6). Or Sidgwick, who emphasizes the *internal sanctions* of
duty one is likely to suffer on account of their immoral conduct
(Sidgwick 1874 [2011: 164]). Viewed through the lens of game theory,
the inseparable goods approach recommends that those who doubt the
prudential rationality of acting morally should reevaluate their
preferences in light of the totality of goods at stake. For example,
 Figure 4
 below illustrates how the scenario depicted in
 Figure 3
 would change if Laura becomes persuaded that she would suffer a cost
\(\gamma > 1\) as a result of breaking her covenant with Claudia,
perhaps owing to the guilt she might feel for committing an
injustice.




![a decision tree: link to extended description below](Figure4.svg)




Figure 4. Sequential Exchange Game with
Revised Payoffs, [An
 extended description of figure 4
 is in the supplement.]




This extensive form game depicts Laura's true payoffs, including
the cost of guilt that Sidgwick would call an internal sanction. In
this game if Claudia performs as promised, then to follow *P*
as justice requires is now Laura's best response. However, this
isn't because we've simply assumed that one couldn't
ever stand to gain by breaking a promise. To see why, note that if
Claudia breaks her promise first, then *A* continues to be
Laura's best response. As we saw in our discussion of the
evolution of cooperation, morality can't require us to be
suckers. Or, to use Kavka's helpful terminology, justice may
forbid violating a covenant *offensively* by breaking
one's promise unexpectedly, while permitting *defensive*
violations of covenants that others can no longer reasonably expect us
to honor (Kavka 1986: 139). Accordingly, if Claudia offensively
violates her covenant with Laura by following *A*, then if
Laura responds in kind, Laura's violation should not give rise
to the internal sanctions associated with having committed an
injustice. Furthermore, note that the presence of internal sanctions
can change things even when these aren't universally felt. For
instance, even if Claudia herself would feel negligible guilt for
offensively violating their covenant, if Claudia knows that Laura
would suffer the internal sanction cost \(\gamma > 1\) of offensive
violation, then Claudia can expect Laura to perform when she does as
well, and in this case she fares best by following *P* at the
outset.


### 3.2 The Social Sanctions Strategy



Where the previous approach to the Reconciliation Project relied on
sanctions that an agent imposes on herself, the *social sanctions
strategy* relies on costs imposed by others. This strategy is
exemplified by Hobbes' response to the Foole and Hume's
discussion of the harvesting dilemma, the latter of which Skyrms
(1998) and Sobel (2009) identify with the game previously depicted in
 Figure 2.
 Hobbes maintains that to offensively violate a covenant is to make a
choice that tends towards self-destruction, since others will be
unwilling to enter into future covenants with one known to do so
(Hobbes 1651: 15:5). Similarly, Hume argues that we make promises in
order to create expectations of our future conduct, and that we keep
promises in order to preserve the trust of others that is needed if we
hope to enter into and benefit from future covenants (Hume 1740:
3.3.5: 8-12). As our earlier discussion of the evolution of
cooperation anticipated, we can use the theory of *repeated
games* to shed light on this strategy.



When agents engage in a sequence of interactions, each of which can be
characterized as a single noncooperative game, then they can adopt
*history-dependent strategies* where their actions in any
particular iteration of the game depend upon actions taken in prior
iterations. Early on, game theorists recognized that when agents
engage in games that are repeated *indefinitely* they can use
such strategies to arrive at equilibria that include acts that are
never parts of an equilibrium of any of the constituent games that are
being repeated. This insight led to a body of results known as the
*folk theorem* that establishes conditions under which a
profile of history-dependent strategies will be an equilibrium of an
indefinitely repeated game. Sugden (1986) and Skyrms (1998) were among
the first to argue that Hobbes's response to the Foole and
Hume's analysis of promises present proto-folk theorem
arguments.



To illustrate how the folk theorem might support the social sanctions
strategy, suppose that Laura and Claudia are interested in exchanging
with one another and that their situation has the structure of the
prisoner's dilemma depicted in
 Figure 5
 below.





|  |  | Party 2 (Laura) |
|  |  | *P* | *A* |
| Party 1 (Claudia) | *P* | (2,2) | (0,3) |
| *A* | (3,0) | (1,1) |



*P* = perform, *A* = anticipate



Figure 5. Prisoner's Dilemma




This prisoner's dilemma is structurally identical to the
Mortarmen's Dilemma of
 Figure 1,
 but here we interpret the parties' strategies rather
differently. (*A*, *A*) is the unique equilibrium of
this game that characterizes Laura and Claudia's particular
interaction with one another. However, if we suppose that their
interaction is one of an indefinite sequence of similar interactions
that they will enter into with each other and other members of their
community, then they can follow history-dependent strategies. One such
strategy, formulated by Sugden (1986), is the *Humean
strategy*:



\(h^{\*}\): follow *P* if my current partner is innocent, and
follow *A* if she is guilty,



where one is guilty if she has offensively violated a covenant by
anticipating or aggressing against an innocent partner at some point
in the past, otherwise she is innocent. If Claudia expects all of her
partners in the indefinitely repeated prisoner's dilemma to
follow \(h^{\*}\), then \(h^{\*}\) can be Claudia's best response.
For example, if Claudia and Laura are both innocent when they meet in
their prisoner's dilemma, then if Claudia follows \(h^{\*}\), she
will perform and remain innocent, and so she can expect her partners
in future iterations to perform with her since they also follow
\(h^{\*}\). If, on the other hand, Claudia anticipates against Laura,
then Claudia becomes guilty and she should expect future partners to
anticipate against her in future iterations as \(h^{\*}\) requires. In
other words, if Claudia has sufficiently high expectation that the
sequence of iterated prisoner's dilemmas will continue, then the
short-run benefit from offensively violating her covenant with Laura
will be swamped by the long-run payoff from future mutually beneficial
interactions that \(h^{\*}\) enables and anticipating precludes.
Furthermore, the long-run benefit of keeping her reputation intact
means that as long as Claudia's discount rate isn't too
high, then this is likely to be true even if she misread things with
Laura and gets exploited by her. Finally, because their situations are
symmetrical, \(h^{\*}\) can be a best response for Laura and all the
other community members, provided that they all expect most other
community members to follow \(h^{\*}\).



The discussion above shows that obeying the requirement of justice to
refrain from offensively violating covenants can be rational, even
though it may not appear to be rational in any particular instance.
Like the inseparable goods strategy, then, the social sanctions
strategy implies that the mistake the Foole makes is to misconstrue
the games that characterize social interactions where morality makes
demands on us. However, where the inseparable goods strategy suggests
that the Foole ignores some of the goods intrinsic to moral conduct,
the social sanctions strategy warns the Foole that the game they
should be concerned with is not any particular interaction, but rather
the indefinitely repeated series of social interactions in which they
are inevitably engaged. That said, the strategy profile where every
agent adopts \(h^{\*}\) is not the only possible equilibrium of the
indefinitely repeated prisoner's dilemma. In fact, there are an
immense number of possible equilibria of such a game, many of which do
not involve any individuals employing the \(h^{\*}\) strategy, and not
all of these lead to the cooperative outcome where all (or even most)
agents keep their promises.


### 3.3 The Reformed Decision Theory Strategy



In contrast to the inseparable goods and social sanctions strategies
that question the Foole's characterization of the scenarios that
give rise to a tension between morality and rationality, the
*reformed decision theory strategy* recommends that agents like
the Foole revisit their understanding of rational choice. In
particular it proposes an account of rational choice that is more
subtle than the orthodox Bayesian account according to which rational
choice is a fairly straightforward matter of maximizing expected
utility. David Gauthier (1986) and Edward McClennen (1990) are the
most prominent exemplars of this approach, with Gauthier proposing a
principle of *constrained maximization* and McClennen a
principle of *resolute choice*. For ease of exposition we focus
on Gauthier.



In *Morals by Agreement* Gauthier distinguishes between
*straightforward maximizers* and *constrained
maximizers*. A straightforward maximizer is an orthodox Bayesian
who, at any given moment in time, chooses so as to maximize her
expected utility given her beliefs regarding the relevant states of
affairs and the choices her counterparts might make. A constrained
maximizer, on the other hand, departs from orthodox Bayesian reasoning
by doing her part to bring about a fair and optimal outcome if (i) she
expects her counterparts to do their part in bringing about this
outcome, and (ii) she benefits as a result of all following it. A
constrained maximizer thus constrains her pursuit of her interests
some of the time, but reverts to straightforward maximization if she
expects her counterparts to act as straightforward maximizers, or if
she does not stand to benefit from constraining her behavior relative
to the scenario where no one constrains their behavior.



For example, if Claudia is a constrained maximizer, then in the
prisoner's dilemma depicted in
 Figure 5
 Claudia will perform if she believes Laura is also a constrained
maximizer, but will anticipate if she believes Laura is a
straightforward maximizer. One way to understand how constrained
maximization modifies an agent's behavior is to regard adopting
one's principle of rationality as a meta-strategy for engaging
in prisoner's dilemma-like games. If Claudia and Laura each
adopt either the straightforward maximizer (*SM*) or the
constrained maximizer (*CM*) meta-strategy, then the
prisoner's dilemma depicted in
 Figure 5
 is transformed into the game depicted in
 Figure 6
 below.





|  |  | Party 2 (Laura) |
|  |  | *CM* | *SM* |
| Party 1 (Claudia) | *CM* | (2,2) | (1,1) |
| *SM* | (1,1) | (1,1) |



*CM*= constrained maximization, *SM* = straightforward
maximization



Figure 6. Meta-Strategy Game for
Prisoner's Dilemma




Here, whatever strategy the other party follows, one always does at
least as well and sometimes better by following *CM* as opposed
to *SM*. That is, *CM* is a *weakly dominant*
strategy for both Claudia and Laura, and (*CM*, *CM*) is
the unique Pareto-optimal Nash equilibrium. Gauthier argues that there
are a wide variety of social dilemmas where constrained maximizers
fare at least as well as straightforward maximizers, and can often
fare strictly better, so long as agents' dispositions towards
constrained or straightforward maximization are sufficiently clear to
each other. Accordingly, Gauthier concludes that agents should adopt
the constrained maximization principle, and that, when they do,
compliance with the requirements of morality coincides with rational
choice.


### 3.4 Limitations of the Reconciliation Project



As we've seen, game theory has been incorporated into each of
the main approaches to the Reconciliation Project. However,
game-theoretic analysis can also show why each of these strategies may
fall short of fully reconciling morality with prudential rationality.
For instance, for the inseparable goods strategy to be effective,
agents like the Foole must be persuaded that the goods intrinsic to
just conduct are either of such great value, or that the costs
associated with injustice are so substantial, that one would be
foolish to risk acting unjustly. But it's far from obvious that
this is always (or even usually) true.



Similarly, although one can present social sanctions arguments for
acting morally to agents like the Foole, these arguments depend on
members of society being willing to sanction bad behavior and capable
of doing so effectively, and this is by no means guaranteed. In fact
this is a difficulty that the functional analysis of norms clearly
anticipates. For example, in addition to highlighting the importance
of conditional strategies and the willingness to sanction bad behavior
Axelrod (1984) also emphasized the importance of employing strategies
that are easy for others to interpret. While Bicchieri (2006)
emphasizes the problems that are prone to arise in circumstances where
it's not clear which norm applies, and Boyd & Richerson
(2005) have drawn attention to the costs associated with policing
norms. Accordingly, the Foole can object that, even if all the members
of her community were committed to following something like the Humean
strategy to the best of their abilities, it's not obvious that
they would be able to reliably follow it. In fact, the Humean strategy
rests upon some rather demanding epistemic assumptions, including
especially that community members can reliably distinguish between
guilty and innocent partners, something that may become increasingly
unlikely as the size of a community increases.



Furthermore, as the folk theorem implies, although it's
certainly possible that all of the members of a community could follow
a strategy like \(h^{\*}\) in a situation like the Foole finds herself
in, there are likely to be many other equilibria as well. Moreover,
some of these equilibria may be hard to distinguish from one another,
especially if agents like the Foole are sufficiently rare, and in some
of these equilibria it may be in the Foole's interest to
offensively violate covenants. The effectiveness of the social
sanctions strategy, then, is plainly dependent upon the ability of
community members to build and maintain reliable reputations, and to
rely on this information in their interactions with one another.



Finally, the revisionary principles at the heart of the reformed
decision theory strategy proposed by Gauthier and McClennen rest upon
an agent's ability to adopt a disposition to conditionally
follow certain pro-social strategies regardless of whether doing so is
in an agent's best interest at the moment of action. Critics of
this strategy, however, have questioned what it means to bind oneself
to her past commitments in this way, and they have also identified
numerous cases where doing so would be in neither an agent's
short nor long-run interest. Furthermore, while Gauthier's
analysis of the benefits of constrained maximization is analogous in
some ways to Bicchieri's analysis of how social norms can
transform mixed-motive social dilemmas into coordination games, the
latter depend on sanctions (or expectations of sanctions) in a way
that the former do not. As a result, even if agents could credibly
adopt dispositions to constrain their future behavior, for a rule like
constrained maximization to do its practical work, agents must be able
to *recognize* who has and has not adopted the dispositions in
question. However, some critics doubt that we can reliably recognize
who adopts such dispositions. Of course, proponents of the reformed
decision theory strategy are well aware of these criticisms and have
developed careful responses in their works. So far, though, its
critics have not been assuaged and the reformed decision theory
strategy has been the least influential of the approaches described
here.


## 4. Game Theory and Contractarianism



Arguably the most systematic use of game theory in moral and political
philosophy has been in contractarian approaches to justifying moral or
political orders. Philosophers have typically characterized the
*social contract* as a set of rules for regulating the
interactions of members of a community. By providing a formal model of
interactive decision-making, game theory is clearly relevant to the
analysis of such contracts. What has been particularly distinctive of
game theoretic analysis of the social contract, though, is an emphasis
on bargaining problems. As we will discuss below, the Nash bargaining
problem provides a simple, yet flexible and conceptually powerful
framework for identifying which of the rules that might comprise a
contract can be said to be mutually beneficial (or acceptable) to each
member of a community. In the main text of the two subsections that
follow we try to present a relatively non-technical overview of this
framework. Where formal characterizations of concepts are important,
or where discussion of technical issues is likely to be illuminating
for some readers, we've included such discussion in notes that
appear at the end of the respective subsections. For a more thorough
analysis of contractarian approaches to justifying moral and political
orders, including further discussion of some of these technical
issues, see the entry on
 contemporary approaches to the social contract.


### 4.1 The Bargaining Problem



As Gauthier (1986) argues, the bargaining problem provides a framework
for a *contractarian* analysis of the social contract by
modeling how parties with partially conflicting interests might arrive
at a mutually acceptable set of rules they can each follow to their
own advantage. This contrasts with the *contractualist*
approach of Harsanyi (1953, 1955, 1977), Rawls (1958, 1971), and
Scanlon (1982, 1998) that characterizes the social contract as an
object of rational choice made under constraints designed to ensure
that the choice is appropriately unbiased, and which, in principle,
could be made by a single agent. Braithwaite (1955) was the first to
use the Nash bargaining problem to derive a social contract, in his
case a rule for fairly dividing a resource between two claimants.
Later, Hampton (1986) and Kavka (1986) argued that the Hobbesian
device of creating a *commonwealth by design* incorporates
elements of a Nash bargaining problem. Gauthier (1986) and Binmore
(1994, 1998) used the Nash bargaining problem for developing
standalone theories of distributive justice. More recently, Muldoon
(2016) and Moehler (2018) have used the Nash bargaining problem to
model the problem of how members of deeply pluralistic societies can
find rules for living together peacefully.



Nash characterized the bargaining problem as a combination of (i) a
*feasible set* of payoffs that a group of agents might achieve
each of which is associated with a respective set of actions the
agents must jointly agree to follow, and (ii) a *nonagreement
point* that specifies what each agent receives in case they fail
to follow any of these joint action sets. An example of Nash's
bargaining problem is the 2-agent *Wine Problem* depicted in
 Figure 7
 below.





|  |  | Agent 2 (Laura) |
|  |  | *D* | *H* |
| Agent 1 (Claudia) | *D* | (0,0) | (0,1) |
| *H* | (1,0) | ([?]1,[?]1) |



*D*= dove (claim none), *H* = hawk (claim all)



Figure 7. Wine Problem




In this game Claudia and Laura each have two pure strategies:
*hawk* (*H*), which is to claim an available bottle of
wine, or *dove* (*D*), which is to leave the bottle for
the other. The game has two Nash equilibria in pure strategies:
(*H*,*D*), the outcome most favorable to Claudia, and
(*D*,*H*), the outcome most favorable to Laura. If both
try to claim the bottle, at the resulting (*H*,*H*)
outcome they fight and neither gets any wine, the worst outcome for
them both. The Wine Problem is an example of a *conflictual
coordination game* (Vanderschraaf & Richards 1997), so-called
because it has multiple equilibria, any of which the agents achieve
only by coordinating their actions, and where coordination is
complicated by the conflicting preferences of the agents over the
different available
 equilibria.[1]



One might think that a natural solution to the Wine Problem is for
Claudia and Laura to each claim half of the bottle.
 Figure 8
 below thus illustrates an extended version of the Wine Problem in
which each agent now has a third pure strategy, *M*, in which
she claims half the bottle.





|  |  | Agent 2 (Laura) |
|  |  | *D* | *M* | *H* |
| Agent 1 (Claudia) | *D* | \((0,0)\) | \((0,\frac{1}{2})\) | \((0,1)\) |
| *M* | \((\frac{1}{2},0)\) | \((\frac{1}{2},\frac{1}{2})\) | \(({-1},{-1})\) |
| *H* | \((1,0)\) | \(({-1},{-1})\) | \(({-1},{-1})\) |



*D* = dove (claim none), *M* = moderate (claim half),
*H* = hawk (claim all)



Figure 8. Extended Wine Problem




In this game, Laura and Claudia each receives the amount of wine she
claims if their claims are *compatible*, that is, if there is
enough wine to satisfy each claim. However, if their claims are
*incompatible*, they fight. There is now a new Nash equilibrium
in pure strategies, \((M,M)\), where Laura and Claudia each receive
half the bottle. It's not a foregone conclusion that they will
settle into \((M,M)\), though, for \((H,D)\) and \((D,H)\) remain
equilibria, and the former continues to be most preferred by Claudia,
and the latter most preferred by Laura.



Furthermore, if each can claim half the bottle why should they not be
able to claim other shares?
 Figure 9
 below summarizes the bargaining problem that arises when Claudia can
claim any share \(x\_1\) of the wine and Laura can similarly claim any
share \(x\_2\).



![a graph: link to extended description below](Figure9.jpg)

Figure 9. Wine Bargaining Problem [An
 extended description of figure 9
 is in the supplement.]




Here Claudia's payoffs are depicted on the horizontal axis, and
Laura's payoffs are depicted in the vertical axis. The green
shaded region represents the feasible set of payoff vectors
\(\Lambda\) where Laura and Claudia make compatible claims, that is
where \(x\_1 +x\_2 \le 1.\) While the payoff vector \(\bu\_0 =
({-1},{-1})\) corresponding to the (*H*,*H*) outcome
represents their nonagreement point.



In the social contract context we can think of the nonagreement point
\(\bu\_0\) as representing the State of Nature, and the feasible set
\(\Lambda\) as representative of the various contracts parties might
agree to. Each point \(\bu \in \Lambda\) that is weakly Pareto
superior to \(\bu\_0\) thus characterizes a social contract with a
*cooperative surplus* (Gauthier 1986:
 141).[2]
 And, when a set of claims constitutes an equilibrium solution to the
associated bargaining problem, this allows us to say that the parties
to a contract all have good reason to comply with its terms. As we
will see below, though, there are various approaches to identifying
the most attractive equilibrium from a normative point of view. An
advantage of using the Nash bargaining approach to modeling the social
contract, then, is that this formalization helps to clarify the
differences between these
 approaches.[3]



The Basis Game that Braithwaite analyzed in his Cambridge lecture is
ideal for illustrating the axiomatic approach to the bargaining
problem. In this game two agents, Luke and Matthew, vie for shares of
a limited resource. Like the Wine Problem, the Braithwaite Basis Game
is a conflictual coordination game in which each agent has two pure
strategies available, *H* in which the agent claims all of the
good at stake, and *D* in which he claims none. Here
(*H*,*D*) is the Nash equilibrium most favorable to Luke
and (*D*,*H*) is the equilibrium most favorable to
Matthew. As
 Figure 10
 illustrates, though, this game has asymmetries not present in the
Wine
 Problem.[4]





|  |  | Agent 2 (Matthew) |
|  |  | *D* | *H* |
| Agent 1 (Luke) | *D* | \((\frac{1}{6},0)\) | \((\frac{1}{2},1)\) |
| *H* | \((1,\frac{2}{9})\) | \((0,\frac{1}{9})\) |



*D*= dove (claim none), *H* = hawk (claim all)



Figure 10. Braithwaite Basis Game




Again, as before, this game can be extended to accommodate the case
where agents can claim shares of the good. And if the nonagreement
point again results in the payoffs generated by the conflict outcome
(*H*, *H*), as Braithwaite supposed, then Matthew and
Luke are in a Nash bargaining problem with feasible set \(\Lambda\)
and nonagreement point \(\bu\_0 = \left(0,\frac{1}{9}\right)\).
 Figure 11
 depicts the feasible set \(\Lambda\) for such a game (again shaded in
green). This time, though, the figure also identifies the payoff
vectors associated with three solutions concepts that have been
especially influential in in game theoretic analyses of the social
contract (namely the *Nash*, *proportional*, and
*maximin proportionate gain* solutions).



![a graph: link to extended description below](Figure11.jpg)

Figure 11. Braithwaite Bargaining
Problem [An
 extended description of figure 11
 is in the supplement.]




Here Luke's payoffs are depicted on the horizontal axis, while
Matthew's are on the vertical axis. As we can see, although the
solutions differ, each leaves Matthew with a higher payoff than Luke
receives. This is because each solution allocates Matthew a larger
share of the
 good,[5]
 and is due to the asymmetry in the nonagreement point that
characterizes the Braithwaite Problem. Specifically, (*H*,
*H*) is Luke's worst possible outcome, but not
Matthew's, so Matthew has a *threat advantage*. In less
technical terms, because failing to agree is less bad for Matthew he
can use the threat of nonagreement to press for an agreement that is
more favorable to him. And this asymmetry distinguishes the
Braithwaite Bargaining Problem from the Wine Division Problem depicted
in
 Figure 9
 where the three solutions coincide with the point \(\bu^{\*} =
\left(\frac{1}{2}, \frac{1}{2}\right)\) in which the agents involved
each receive equal shares.



In his own analysis Braithwaite favored the *proportional*
solution to the bargaining problem that selects the point on the
Pareto frontier where the payoffs for each agent relative to the
nonagreement point are proportionate to some independently specified
measure of fair
 gains.[6]
 Gauthier (1986, 2013), on the other hand, defends the *maximin
proportionate gain* solution that also selects a point on the
Pareto frontier by comparing the payoffs for agents relative to the
nonagreement point, but which, in addition, takes into account how
agents fare relative to their ideal
 payoffs.[7]
 In the game theory literature, however, the *Nash* solution
has remained the most widely utilized solution concept, and this has
arguably been true among philosophers offering game-theoretic
treatments of the social contract as well (see, e.g., Binmore 1994,
1998; Muldoon 2016). Roughly speaking, this solution picks out the
point that maximizes the cooperative surplus, although some
philosophers have offered refinements on the concept that guarantee
payoffs also meet a minimum threshold (Moehler
 2018).[8]



At the heart of debates utilizing game-theoretic treatments of the
social contract is the fact that each of the solution concepts
mentioned above, and indeed any solution to the bargaining problem,
might satisfy several *but not all* of the intuitively
plausible constraints we might want ]a fair bargaining procedure to
realize. For instance, Nash's solution is the only solution
concept that satisfies all four of:


1. *Pareto optimality*,
2. *symmetry*,
3. *scale (or utility) invariance*, and
4. *reapplication
 stability*.[9]



But it fails to satisfy both:


5. *individual monotonicity*, and
6. *population monotonicity*.



In this context, an outcome (or its associated payoff vector) is
Pareto optimal if there are no other outcomes that are better for some
agents without being worse for
 others.[10]
 A solution is symmetric if the payoffs that agents receive from
identical shares of a good are equivalent when those agents have
equivalent utility functions, nonagreement points, and
 strategies.[11]
 It is scale invariant if the solution does not depend on how the
payoffs are
 represented.[12]
 And it is reapplication stable if the application of a solution
concept to a bargaining problem does not generate downstream
bargaining problems among subsets of agents over the relative shares
of a good they can
 claim.[13]
 While a solution is individually monotone if an agent is guaranteed
to do at least as well when a bargaining problem is modified by
expanding the feasible set to include new outcomes that are all more
favorable to the agent. And it is population monotone if, for a given
set of claimants, none lose while others gain merely because some
members of the set depart and/or new claimants arrive.



The maximin proportionate gain solution, on the other hand, is both
individually and population monotone, but is not reapplication stable.
While the proportional solution is individually monotone, population
monotone, and reapplication stable, but not scale invariant.
Ultimately, then, much like
 Arrow's Impossibility Theorem
 and related results have cast doubt on whether there exist uniquely
best procedures for aggregating preferences or making collective
decisions, there is not a single solution concept for bargaining
problems that is clearly best. That said, the application of
bargaining theory to philosophical analyses of the social contract has
been useful insofar as it has provided a framework for illustrating
what features of a social contract a theorist regards as valuable, and
for identifying what the tradeoffs associated with these features
might be. So, for example, one might conjecture that the reapplication
stability properties associated with the Nash and proportional
solution concepts would allow a social contract based upon one of them
to be better suited to forestalling conflict in the face of social
change. This is because such a contract would be renegotiation-proof
with respect to changes in the benefits of cooperation that might
arise in the future, e.g., as a result of an influx of new members in
society or the discovery of new technologies for resource development.
But this stability would have to come at the price of either using a
privileged representation of agent's payoffs, or accepting that
the parties to the contract will not necessarily be made better off if
they are given more to bargain over. And while this tradeoff may
ultimately be justified, the important point is that any tradeoff the
contract theorist makes commits her to a certain set of normative
judgments, and game theory can help clarify what these are. For
instance, many game theoretically minded philosophers have contended
that any account that privileges certain representations of
agents' utilities cannot avoid incorporating certain moral
judgments into these utilities (Sen 1980; Moehler 2018).


### 4.2 Evolutionary Approaches to the Social Contract



Having focused on the use of bargaining theory in philosophical
analyses of the social contract, we will now briefly turn our
attention to the relevance of evolutionary game theory to this task.
Recall that evolutionary game theory is the formal approach to
modeling how the distribution of traits in a population will evolve
over time when those traits determine how the agents who possess them
will interact. Gaus (2010) and Muldoon (2016) are two recent works
that draw on informal evolutionary models in order to show how and why
the terms of a social contract might evolve over time to accommodate a
diverse polity. While, taking a more formal approach, Alexander (2007)
explores how the equilibria that characterize prosocial norms in
social networks evolve over time. It is the works of Brian Skyrms
(1996 [2014], 2004), though, that have been the most influential in
bringing evolutionary game theory to bear on the social contract.



One of Skyrms' signature uses of evolutionary game theory is to
explain the origins of various "fair" solutions to the
Nash bargaining problem (1996 [2014]: Chapter 1). As we have seen
above, different axiomatic solution concepts can recommend different
solutions to the same bargaining problem. However, even if one
endorses the arguments in favor of a particular solution concept, this
doesn't yet show that the actual claimants engaged in a
bargaining problem would tend to follow this solution. This is because
a typical bargaining problem will often have a large number of
equilibria each of which is likely to be more favorable to some and
less favorable to others. Accordingly, one might doubt that actual
claimants could settle into any equilibrium, let alone an equilibrium
that characterizes an outcome we would regard as fair.



To gain traction on this problem Skyrms considers a simple but
nontrivial 3-piece *Nash Demand Game* that is structurally
similar to the Extended Wine Problem of
 Figure 8.
 In this now familiar game a pair of agents vie for shares of a good.
Each issues a claim and receives a payoff equivalent to her claim when
the pair's claims are compatible, and each receives nothing when
their claims are incompatible. In Skyrms' version a claimant can
issue a minimal claim, *D*, to 1/3 of the good; a maximal
claim, *H*, to 2/3 of the good; or a "middle"
claim, *M*, to 1/2 of it.
 [14]
 As Skyrms notes (1996 [2014: 4]), in games of this sort we tend to
think there is one outcome claimants ought to follow, namely, the fair
outcome where each claims half of the good at stake. This corresponds
to each side following *M*, and, as we saw earlier,
(*M*,*M*) is an equilibrium of the game. But the
question remains, why would claimants opt for the fair outcome when
there are other equilibria that might be preferable?



One answer to this question is that when agents repeatedly have to
come to agreements with one another over the division of a good, but
can only obtain a share of the good when they agree on a feasible
division, agents who restrain themselves from pressing maximal claims
are more likely to encounter other agents with whom feasible bargains
are possible. In other words, as we saw in a slightly different
context in section 2, in the long run the benefits from successfully
coming to terms with like-minded agents outweigh the foregone gains
that can occasionally be gotten by pressing for less equitable
divisions. Skyrms tests this rationale by applying a dynamic model of
evolution, the *replicator dynamic*, to his Nash Demand Game.
 Figure 12
 illustrates a *simplex* that one can use to summarize this
dynamic.



![a triangle: link to extended description below](Figure12.png)

Figure 12. Strategy Simplex for
Skyrms's Nash Demand Game [An
 extended description of figure 12
 is in the supplement.]




Each point in the simplex corresponds to a particular mix of types of
agents in the population that can represented by an ordered triplet
\((z\_1,z\_2,z\_3),\) where \(z\_1\) is the fraction of the population
that follows *D*, \(z\_2\) the fraction that follow *M*,
and \(z\_3\) the fraction that follow *H*. Points closer to a
vertex of the simplex thus represent mixes where more agents follow
the strategy associated with that vertex. The replicator dynamic then
summarizes the evolution of population states where the fractions of
represented strategies change over time in proportion to their
*average reproductive fitness*. Here a strategy's average
reproductive fitness corresponds to its average payoff relative to the
average payoff of other strategies represented in the population at a
given
 time.[15]
 To illustrate, in
 Figure 12,
 the point \(z\_0 = \left(\frac{1}{6}, \frac{1}{3},
\frac{1}{2}\right)\) represents an initial state where \(\frac{1}{6}\)
of the population follow *D*, \(\frac{1}{3}\) follow
*M*, and \(\frac{1}{2}\) follow *H*. Given this starting
point, the replicator dynamic then traces an *orbit* starting
at \(z\_0\) that converges to \(\be\_2 = (0,1,0),\) the vertex point
where the entire population follows the pure strategy *M*.



By applying the replicator dynamic across a wide swath of initial
states in the
 Figure 12
 simplex, Skyrms then showed that the fair outcome, \(\be\_2,\) where
everyone presses equal claims, is the most likely to evolve. We
replicate Skyrms' experiment here.
 Figure 13
 depicts 200 orbits of the replicator dynamic starting from initial
population states chosen at random from the
 Figure 12
 simplex.



![a triangle: link to extended description below](Figure13.png)

Figure 13. Replicator Dynamic Orbits of
Skyrms's Nash Demand Game [An
 extended description of figure 13
 is in the supplement.]




As this figure illustrates most initial distributions of claimant
types eventually converge to an equilibrium where all agents follow
*M* and press claims of 1/2. To use a technical term, we can
say that \(\be\_2\) has the largest *basin of
 attraction*.[16]
 To see why this is the case, we can consider how different pure
strategy types fare when agents of a population meet at random and
engage in the Nash Demand Game. Agents who press claims of *D*
always arrive at feasible bargains, but in particular interactions
they never do as well as partners who press claims of *M* or
*H*. On the other hand, agents who press claims of *H*
only arrive at feasible bargains when matched with agents who press
minimal claims of *D* and can thus be exploited. Meanwhile,
agents who press claims of *M* arrive at feasible bargains when
matched against either *D*-following partners or other
*M*-following partners. On average, then, *M*-followers
do better than both *D*-followers and *H*-followers
unless there are so many *H*-followers and so few
*M*-followers in the population that in an individual encounter
an *M*-follower is more likely to meet an *H*-follower
than either another *M*-follower or a
 *D*-follower.[17]
 Indeed, *M* is an *evolutionarily stable strategy*
(Maynard Smith 1982) of this game. What this means is that, if
\(\be\_2\) is the current population state so that *M* is the
sole *incumbent strategy*, then \(\be\_2\) can repel any limited
invasion of some different *mutant*
 strategy.[18]



In some respects Skyrms' observation shows how Gauthier's
argument for constrained maximization might have been strengthened by
drawing on resources from outside the confines of rational choice
theory. Indeed, Skyrms shows that the basin of attraction for the
equitable equilibrium where all agents press claims of 1/2 expands
when agents become more likely to interact with agents of their own
strategy type, as they might if they are somewhat able to recognize
agents like themselves. For example,
 Figure 14
 depicts a series of orbits of a *correlated replicator
dynamic* Skyrms uses to model such situations. In this dynamic, an
agent meets a counterpart of the same pure strategy type with
probability .2 greater than the random chance probability of the
ordinary replicator
 dynamic.[19]
 And, as we can see, even with this relatively minor increase in the
likelihood of same type interactions, *all* of the modified
replicator dynamic orbits now converge to \(\be\_2\)
 .[20]



![a triangle: link to extended description below](Figure14.png)

Figure 14. Correlated Replicator Dynamic
Orbits of Skyrms's Nash Demand Game [An
 extended description of figure 14
 is in the supplement.]




Alternatively, Skyrms showed how evolved attachments to particular
places or things, and more importantly the ability to *signal*
these attachments can help solve conflictual coordination problems
like the Nash Demand Game. In particular, Skyrms (1996 [2014]: chap.
4) showed that by developing a primitive notion of property rights,
agents involved in such games could correlate their behavior by
playing Hawk at home and Dove away from home, and thereby avoid the
sorts of interactions that lead to conflict or to both agents
abandoning a resource. At the end of the day, though, the evolutionary
game theory utilized by Skyrms can only lend so much support to a view
like Gauthier's. For one thing, the game Skyrms models assumes
that agents have fixed types and cannot revise their claims in light
of the claims made by others. Furthermore, it is one thing to offer an
explanation of why a trait like fair division or constrained
maximization would evolve, or to show why this might be beneficial to
both individual agents and groups, but to show either of these is not
necessarily to show that constraining one's behavior is the
rational course of action.


## 5. Analyzing Morally Significant Social Issues



If contractarian approaches to justifying moral and political orders
have been the area of ethics where the use of game theory has been
most systematic, the area where its deployment is likely to be most
impactful going forward is in the analysis of morally significant
social issues. Examples range from global issues, like how to combat
climate change; to issues that arise within states or communities,
like tax policy or how to manage common pool resources; to issues that
primarily concern individuals and groups, like how to elicit optimal
levels of effort from individuals engaged in cooperative endeavors.
Below we will discuss three examples in more detail.


### 5.1 Nuclear Conflict



Many of the early advances in game theory were made during World War
II and the early stages of the Cold War by individuals working with
organizations interested in advancing our understanding of nuclear
conflict. John von Neumann, for instance, whose early contributions to
game theory we described in section 1, did work for the Manhattan
Project that was instrumental in the development of nuclear weapons,
and he is often credited with having developed the strategy of
mutually assured destruction (MAD) that played a prominent role in
Cold War nuclear policy (Macrae 1992). MAD is a policy governing the
use of nuclear weapons whereby nuclear powers avoid nuclear conflict
by credibly threatening to retaliate against a nuclear first strike
with a nuclear strike that would be destructive enough to negate any
strategic or tactical advantage of a first strike.



Unsurprisingly, ethical issues associated with the use of nuclear
weapons would eventually receive extensive attention in the
philosophical literature. Much of this work focused on how nuclear
ethics raises problems for utilitarian or deontological approaches to
ethics, especially the question of whether it's acceptable to
threaten great evil so that it might be avoided. And, while most of
the philosophers working in this area made little use of game theory,
many of the questions they addressed were framed by the important work
of Gregory Kavka, Russell Hardin, and a few others who made extensive
use of both formal and informal game theoretic tools (see, e.g., Kavka
1987; R. Hardin 1983; Gauthier 1984; MacLean 1984). In particular,
this work helped identify a number of paradoxes associated with MAD
and other theories of deterrence, clarified the case both for and
against unilateral disarmament, and would have implications for more
general debates about rationality (especially debates concerning the
nature of intentions and whether modularity is a demand of
rationality).



The analysis of nuclear conflict was also responsible for many of the
early developments in cooperative game theory. Among the earliest
examples of this kind of work was Lloyd Shapley and Martin
Shubik's effort to predict the behavior of members of the United
Nations Security Council (Shapley & Shubik 1954). The
Shapley-Shubik power index quantifies the likelihood that any given
member of a voting body casts a deciding vote. This provides a measure
of the influence of each member and helps us analyze how likely
various coalitions are to form, how stable those coalitions will be,
and how the gains secured by a winning coalition will be
distributed.



Perhaps the most influential application of game theory to problems
associated with nuclear conflict is found in Schelling's seminal
works *The Strategy of Conflict* (1960)and *Arms and
Influence* (1966) which together reshaped the game-theoretic
analysis of conflict. One of Schelling's most important insights,
which the prospect of nuclear war makes especially clear, was the
observation that only in rare circumstances are the interests of
parties engaged in conflict completely opposed. This led Schelling to
emphasize the ways conflict creates and shapes opportunities for
bargaining and cooperation. These were issues that had increasingly
come to be studied using the tools of cooperative game theory, but
Schelling would revitalize non-cooperative game theory by showing how
such issues could be more productively analyzed through
non-cooperative approaches. Arms races to develop more effective
nuclear weapons, and the extensive efforts that governments make to
guard nuclear secrets, are also emblematic of the importance of
information and timing to the analysis of conflict, and the emphasis
Schelling placed on these issues would lead game theorists to
re-prioritize the extensive form analysis of games that is better able
to capture the influence of incomplete information and the dynamic
nature of conflict. Finally, tying together many of the issues
sketched above is Schelling's early work on commitment and
threats. As Schelling showed, by pre-committing to a costly course of
action (or by limiting one's future options) an agent can
influence the actions of other agents with whom she is engaged in
conflict. In fact, Schelling showed that in some cases irrationality
can even be an advantage, an argument that Kavka (1987) would develop
further in the context of deterrence.


### 5.2 Bad Norms



If nuclear conflict was the first morally significant social issue to
be extensively analyzed using game theoretic tools, in the recent
philosophical literature it has been the analysis of harmful,
destructive, and wasteful norms where game theoretic tools have been
most prevalent. Specifically game theoretic tools have been used to
explain: 1) what bad norms are, 2) when and why they emerge, and 3)
whether they can be changed, and, if so, how.



Of these topics, the question of what distinguishes good norms from
bad norms has received by far the least attention (a notable exception
is Thrasher 2018). One explanation for this is that it is often taken
to be obvious that certain norms are bad. For instance, honor killings
have received extensive attention in the norms literature, and,
perhaps unsurprisingly, such killings are typically assumed to provide
a clear instance of a harm that violates the rights of at least some
of those subjected to the practice. Instead of focusing on what makes
honor norms bad, then, the literature has tended to focus on why such
norms emerge and how they might be changed.



To answer these questions game theoretic tools have been used to
characterize the underlying social dynamics that give rise to the
practice in question, and to then illustrate how the practice is
responsive to these dynamics. A dynamic common to several of these
practices is the role norms play in signaling a group's
commitment to certain values or courses of action. For example, in the
case of honor killing, when these killings target members of an
outgroup, they might deter future slights at the hands of the outgroup
by signaling the ingroup's commitment to seeking revenge for
(real or perceived) slights (Boehm 1986; Nisbett & Cohen 1996;
Skarbek 2014; Thrasher & Handfield 2018). This can be especially
important in the absence of strong institutions that maintain public
order. However, one thing that makes honor norms bad, is that a
group's commitment to seeking revenge for slights can risk
setting off (or perpetuating) a cycle of violence.



Honor killings can also be directed towards members of one's own
group who violate certain norms. For instance, there are many places
where it is not uncommon for members of a family to kill children
(especially girls) who break norms of sexual morality or who otherwise
fail to adhere to traditional gender norms. In these cases the desire
of a group's members to adhere to such norms is typically
explained by the belief that doing so makes members of the group more
attractive marriage partners, and the role honor killings play is to
signal that the family (or group to which the family belongs) is
committed to the norm(s) in question (United Nations Population Fund
2000; Chesler & Bloom 2012; Skarbek 2014; Thrasher & Handfield
2018). In some circumstances, though, a group's commitment to a
norm may be driven by *pluralistic ignorance*. That is, there
are cases where two (or more) groups may each be engaged in a practice
like footbinding or female genital cutting that they themselves would
prefer not to engage in, but where they remain committed to the norm
on the basis of the mistaken belief that other groups are committed to
the norm (Bicchieri & Fukui 1999; Bicchieri 2017). Here
understanding the conditional nature of each group's preferences
is crucial to identifying the role played by pluralistic ignorance in
sustaining the norm in question, while understanding the role of
pluralistic ignorance in sustaining the norm helps explain why it is
bad.



In addition to helping identify why a norm might be bad and what
sustains it, game theoretic analysis can also help us identify when
and how such norms might be changed. For example, in the case of
practices sustained by pluralistic ignorance, the key to changing the
practices lies in changing the beliefs of multiple groups. As game
theoretic analysis illustrates, though, it may be that this is only
possible if each group publicly and simultaneously signals to the
other that they do not in fact have an unconditional preference for
engaging in the practice in question. This is because, without knowing
that their beliefs about the other group's normative commitments
are mistaken, if one group is asked to reveal their preferences
unilaterally, whatever motivates their conditional preference is
likely to give them reason to conceal their true preferences
(Bicchieri 2017).



Alternatively, game theoretic analysis can also help us understand
practices like open defecation which seem to be sustained not by
pluralistic ignorance, but by a strong commitment to norms that
outweigh the concern of individuals or communities for things like
public health. For instance, in rural India where open defecation is
common, the practice tends to be sustained by purity norms that
interact with caste systems in ways that make communities reluctant to
utilize latrines even when they are made available. Combating the
public health crises exacerbated by open defection is thus not only a
matter of building the requisite infrastructure, but of undermining
the prevailing norms that prevent the infrastructure from being
utilized (Coffey & Spears 2017), and of incentivizing individuals
to police new norms that promote better sanitation practices
(Bicchieri 2017). And, in such cases, game theoretic analysis allows
us to see how the relevant behaviors might be incentivized.



The analysis of bad norms isn't just limited to cases where the
norms in question are clearly the culprit in bringing about bad
outcomes, though. Game theory can also tell us about the ways in which
seemingly innocent norms and preferences can give rise to bad
outcomes. Perhaps the most famous example of this is Schelling's
(1971) model of segregation which helps illustrate why segregation can
be so difficult to combat. Schelling showed that even when agents
don't mind living alongside members of other groups, even very
minimal preferences for having some neighbors like oneself can lead to
individuals sorting themselves into relatively segregated communities
over time. For example, Figure 15 below shows how a group of
individual agents randomly distributed around a grid will segregate
themselves over time when individual agents have preferences for being
surrounded by certain thresholds of neighbors like
 them.[21]
 Here Figure 15a depicts a random initial distribution on a \(20\times
20\) grid, where 10% of squares are empty in order to facilitate
movement, and there are two types of agent in equal proportion to one
another. Figure 15b depicts the sorting that occurs if agents prefer
to be surrounded by neighbors of whom at least 20% are like them.
Figure 15c shows how things change when the threshold is increased to
just 30%. And Figure 15d shows what things are like when the threshold
is 50%.




![model 1 of 4: link to extended description below](Figure15.png)

a. Initial Distribution




![model 2 of 4: link to extended description below](Figure16.png)

b. 20% Threshold




![model 3 of 4: link to extended description below](Figure17.png)

c. 30% Threshold




![model 4 of 4: link to extended description below](Figure18.png)

d. 50% Threshold




Figure 15. Schelling Segregation Models
[An
 extended description of figure 15
 is in the supplement.]




If we compare Figures 15a and 15b we see that when agents have a
preference to have at least 1 out of every 5 of their neighbors be
like them, the distribution isn't substantially impacted.
However, if we compare Figures 15a and 15b to Figure 15c we see that
when that threshold is increased from 20% to 30% the agents become
significantly segregated. And by the time the threshold reaches 50%
agents have been sorted into almost completely homogeneous
neighborhoods. In other words, even when agents have no objection to
the vast majority of their neighbors being different from them (in the
case of Figure 15c, as many as 70%), a preference to have just some
neighbors like oneself can have massive distributional consequences,
and in addition to being unintended these may be undesirable. However,
while we might criticize the underlying preferences in light of these
consequences, and there may be thresholds for neighborhood composition
that seem obviously discriminatory, when the preference in question is
to have just 30% of one's neighbors be like oneself (in a world
with only two groups) it's far from obvious that this preference
is objectionable.


### 5.3 Climate Change and the Environment



Finally, at least for our purposes, game theory has been crucial to
the analysis of many important problems in environmental ethics, the
most famous probably being the *tragedy of the commons* (G.
Hardin 1968) where game theory has been used to show how resources can
easily be depleted or degraded when they are held in common and use of
the resource is not regulated. The idea of the tragedy of the commons
is that when access to a scarce resource is not limited in some way
there is a tendency for use of the resource to exceed the
environment's capacity to regenerate the resource. So, for
example, we overgraze land, deplete fish and game stocks, run down the
water level in aquifers, or emit too much pollution into the
environment. Generally speaking, *common pool resource*
problems arise in situations where (i) individuals stand to benefit
from using the resource to the greatest extent possible and (ii) the
actions of any one individual have a negligible impact on the
resource, but (iii) the actions of every individual taken together
have a large impact. In these cases it's often easy for an
individual to see that the actions of the community as a whole are
problematic, but because any individual's decision to refrain
from exploiting a resource is unlikely to make a difference by itself,
individuals reasonably infer that they might as well exploit a
resource for themselves while they can. As such, common pool resource
problems are a species of *collective action problem* and are
analogous in some ways to problems we've discussed in previous
sections (e.g., how to ensure that members of a community are willing
to police norms even when doing so is costly).



Common pool resource problems are generally solved in one of two ways:
(1) by taking the resource in question out of the commons, that is by
creating property rights in it, or (2) by finding ways to collectively
manage the resource, for instance by regulating who can access the
commons at a given time. Which approach is best suited to solving a
particular problem, and what form that approach should take, depends
on many factors, e.g., how excludable the resource is, whether it is
stationary, and how quickly the resource can be regenerated. By
providing a framework for analyzing the strategic behavior of
individuals in different contexts, game theory provides a useful tool
for identifying the solutions most likely to work in a given case.
Here Elinor Ostrom's work on common pool resources, which won
her the 2009 Nobel Prize in Economics, has been especially
influential. Ostrom used game theoretic analysis complemented by
extensive field work to show that, although there is a surprising
amount of diversity in the ways that communities have managed to
sustainably govern common resources, the success of these approaches
tends to be a function of a number of underlying principles (Ostrom
1990). Some of these principles reflect things we've already
discussed in other contexts, e.g., the importance of monitoring and
sanctions. While others are perhaps less obvious. For instance, the
importance of accommodating nested or overlapping jurisdictions of
responsibility. Because the ways in which these principles manifest
themselves is highly dependent on context, though, rather than saying
more about them here, we'll instead turn to another problem that
has received a lot of attention in environmental ethics: climate
change.



Game theory has played an important role in the moral and political
analysis of climate change. In particular, it has been crucial to the
analysis of why addressing climate change is difficult. Here, the
consensus in the literature is that states (and other entities)
interested in addressing climate change confront a massive *free
rider problem* stemming from that fact that many of the things
that can be done to prevent further climate change or mitigate its
effects are either *quasi-public goods* or things which
generate substantial *positive externalities*. For example,
drastic reductions of carbon emissions and increased investment in
carbon sequestration or alternative energy technologies are things
which benefit most countries regardless of who invests in them. And,
because investing in these things is costly (at least in the short
term), we tend to underinvest in them, a problem that is exacerbated
by the fact that the most serious costs of climate change arguably
fall on future generations.



Broadly speaking there are two schools of thought with respect to how
we should confront the problem described above. One school exemplified
by Garvey (2008), Gardiner (2011), and Broome (2012) acknowledges the
collective and political nature of the problem, but maintains that in
addition to state action individuals have responsibilities to do
things like reduce their carbon footprints to mitigate the effects of
climate change. On the other hand, members of the other school of
thought (e.g., Brennan 2009, Nordhaus 2014) tend to be skeptical of
how much unilateral action (at either the individual or state level)
can do, and so argue that our efforts are best spent trying to find
workable collective mechanisms to address climate change.


## 6. Conclusion



Many of the early applications of game theory to ethics were limited
to contractarian theories and work exploring the relationship between
morality and rationality. It has increasingly come to be an important
tool in the study of applied problems in moral and political
philosophy, though, and this promises to become even more true as the
number of scholars working on issues at the intersection of
philosophy, politics, and economics increases. That said, game theory
remains an important tool for analyzing the function of morality and
for identifying some of its necessary features. And new contributions
to the social contract tradition continue to be made using ever more
sophisticated applications of game theoretic tools.



However, even if the use of game theory in ethics has become more
widespread, many moral and political philosophers remain skeptical of
this trend. One worry voiced by skeptics is that game theoretic
analyses often rely on simplifying assumptions that give us reason to
doubt how relevant these analyses are to real world problems. For
instance, individuals are rarely the selfish utility maximizers game
theory so often assumes. This worry is misplaced for two reasons.
First, most philosophers inclined to use formal methods in their work
are acutely aware of the assumptions they rely on. This doesn't
mean that formal work never uses misguided models or assumptions, or
that we shouldn't look critically at the conclusions we draw
from game theoretic analysis and other formal work like it. But when
results are interpreted correctly they can be extremely helpful. For
instance, when simplifying assumptions are made for the sake of
computational tractability, they can provide a useful tool for
narrowing down future lines of inquiry. And sometimes simplifying
assumptions are made so that we might ask speculative questions such
as whether rational utility maximizers could reason their way into
morality, as is the case in the social contract tradition. Second, at
the end of the day, formal models aren't really so different
from the thought experiments that are perhaps more familiar in
philosophy. One advantage of game theory and other formal tools,
though, is that they force the philosopher using them to be explicit
about the assumptions they are making, and they often allow the adept
practitioner to say something precise about where the action is and
how various assumptions affect things.