Risk and expectations in a-priori time allocation in multi-agent contracting.
In related research we have proposed a market architecture for multi-agent contracting and we have implemented prototypes of both the market architecture and the agents in a system called MAGNET. A customer agent in MAGNET solicits bids for the execution of multi-step plans, in which tasks have precedence and time constraints, by posting a Request for Quotes to the market. The Request for Quotes needs to include for each task its precedence constraints and a time window. In this paper, we study the problem of optimizing the time windows in the Requests for Quotes. Our approach is to use the Expected Utility Theory to reduce the likelihood of receiving unattractive bids, while maximizing the number of bids that are likely to be included in the winning bundle. We describe the model, illustrate its operation and properties, and discuss what assumptions are required for its successful integration into MAGNET or other multi-agent contracting systems.
INTRODUCTION
The MAGNET (Multi-AGent NEgotiation Testbed) [4]
system is designed to support multiple agents in negotiating
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contracts for tasks with complex temporal and precedence
constraints.
We distinguish between two agent roles, the customer and
the supplier. A customer is an agent who needs resources
outside its direct control in order to carry out his plans. It
does so by soliciting the help of other self-interested agents
through a Requests for Quotes (RFQ). A supplier is an agent
who, in response to an RFQ, may o#er to provide the requested
resources or services for specified prices, over specified
time periods. The objective of the agents is to maximize
their profits while predicting and managing their financial
risk exposure.
In this paper, we focus on the decision process a customer
agent needs to go through in order to generate an RFQ. We
study in particular the problem of how to specify the time
windows for the di#erent tasks in the RFQ. This decision
determines an approximate schedule by setting limits on the
start and finish times for each individual task, since the
RFQ includes early start and late finish times for each task.
Because there is a probability of loss as well as a probability
of gain, we must deal with the risk posture of the person or
organization on whose behalf the agent is acting.
We show how to use the Expected Utility Theory to determine
the time windows for tasks in the task network, so that
bids that are close to these time windows form the most preferred
risk-payo# combinations for the customer agent. We
further examine to what extent the behavior of the model
corresponds to our expectations, explain what market information
needs to be collected in order to integrate the model
in MAGNET system and, finally, discuss how to use the
resulting time allocations to construct RFQs.
2. RELEVANCE OF THE PROBLEM
Before presenting our proposed solution, we need to understand
the importance of selecting appropriate time windows
for tasks in RFQs and how this choice a#ects the customer
agent's ability to accomplish the tasks as economically
and rapidly as possible.
Choosing appropriate time windows a#ects the number
and price of the bids received, the ability to compose the
bids into a feasible schedule, and the financial exposure of
the customer agent.
There are two major decisions the agent has to take here:
the relative allocation of time among the di#erent tasks, and
the extent to which the time windows of tasks connected by
precedence relations are allowed to overlap.
We have shown [1] that the time constraints specified in
the RFQ can a#ect the customer's outcome in two major
ways:
1. by a#ecting the number, price, and time windows of
the bids submitted. We assume that bids will reflect
supplier resource commitments, and therefore larger
time windows will result in more bids and better utilization
of resources, in turn leading to lower prices [3].
However, an RFQ that features overlapping time windows
makes the process of winner determination more
complex [2]. Another less obvious problem is that
every extra bid over the minimum needed to cover
all tasks adds one more rejected bid. Ultimately, a
large percentage of rejections will reduce the customer
agent's credibility, which, in turn, will result in fewer
bids and/or higher costs.
2. by a#ecting the financial exposure of the customer
agent. We assume non-refundable deposits are paid
to secure awarded bids, and payments for each task
are made as the tasks are completed. The payment
to the customer occurs only at the completion of all
the tasks. Once a task starts and, in case it is successfully
completed in the time period specified by the
contract, the customer is liable for its full cost, regardless
of whether in the meantime the plan as a whole
has been abandoned due to a failure on some other
branch of the plan.
We define successful plan execution as "completed by the
deadline," and we define successful completion of a task as
"completed without violating temporal constraints in the
plan." Note that a task can be completed successfully even
if it is not finished within the duration promised by the
bidder, as long as the schedule has su#cient slack to absorb
the overrun. If a plan is completed after its deadline, it has
failed, and we ignore any residual value to the customer of
the work completed.
The uncertainty of whether the tasks will be completed
on time as promised by suppliers further complicates the
decision process. Because of the temporal constraints between
tasks, failure to accomplish a task does not necessarily
mean failure of the goal. Recovery might be possible,
provided that whenever a supplier fails to perform or decommits
there are other suppliers willing to do the task and
there is su#cient time to recover without invalidating the
rest of the schedule.
If a task is not completed by the supplier, the customer
agent is not liable for its cost, but this failure can have a
devastating e#ect on other parts of the plan. Having slack
in the schedule increases the probability that tasks will be
completed successfully or that there will be enough time to
recover if one of the tasks fails. However, slack extends the
completion time and so reduces the reward. In made-to-
order products speed is the essence and taking extra time
might prevent a supplier from getting a contract. This complicates
the selection of which bids to accept. The lowest
cost combination of bids and the tightest schedule achievable
is not necessarily the preferable schedule because it is
more likely to be brittle.
Risk can also be reduced by consolidating tasks with fewer
suppliers. Suppliers can bid on "packages" composed of sub-sets
of tasks from the RFQ. In general, the customer is better
o# from a risk standpoint if it takes these packages, assuming
that the supplier is willing to be paid for the whole package
at the time of its completion. In some cases, the customer
may be willing to pay a premium over the individual task
prices in order to reduce risk. The advantage of doing this is
greater toward the end of the plan than near the beginning,
since at that point the customer has already paid a significant
part of the tasks. Having a greater financial exposure
provides an additional incentive to reduce risk.
3. THE MAGNET FRAMEWORK
3.1 General Terms
The customer is a human or artificial agent who wants
to achieve some goal and needs resources or services beyond
her direct control.
The supplier is a human or artificial agent who has direct
control over some resources or services and may o#er to provide
those in response to external request, i.e., may submit
and commit to bids.
The mediator is a MAGNET-assisted human agent who
meets the needs of a customer by negotiating over multiple
goods or services with one or more suppliers. We often refer
to the artificial part of the duo as to the customer agent.
The Request for Quotes is a signal composed by the customer
agent on the basis of the customer's needs and is sent
to solicit suppliers' bids. MAGNET is a mixed initiative
system, so between composing RFQs and sending them out,
there is a stage where a human user can impose her preferences
on the RFQ choices.
3.2 Task Network
The task network (see Figure 1) represents the structure
of the customer's plan. In essence, it is a connected directed
acyclic graph.
Masonry
Roofing
Plumbing
Electric
Exterior Interior24

Figure

1: A task network example.
Mathematically speaking, a task network is a tuple #N, #
of a set N of individual tasks and strict partial ordering on
them. We conveniently abuse N to also denote the number
of tasks. A task network represents a plan to accomplish
the agent's goal.
We define P (n) := {m # N |m # n} to be a set of predecessors
of n # N , where every predecessor m should be
completed before task n might start. Note that, in general,
P (n) is not completely ordered by #.
Similarly, S (n) is to denote a set of successors of n.
3.3 Time Allocation and Probabilities
A task network is characterized by a start time t s and a
which delimit the interval of time when tasks
can be scheduled.
The placement of an individual task n in the schedule is
characterized by its start time t s
n and finish time t f
n , which
are subject to the following constraints:
where P1 (n) is the a set of immediate predecessors of n,
defined similarly to be the set of immediate successors of
task n.
The probability of task n completion by the time t, conditional
on the successful completion of task n, is distributed
according to the cumulative distribution function (CDF)
1. Observe that #n is defined
to be explicitly dependent on the start time t s
n . To see the
rationale, consider the probability of successful mail delivery
in x days for packages that were mailed on di#erent days of
a week.
There is an associated unconditional probability of success
characterizing the percentage of tasks that are
successfully completed given infinite time (see Figure 2).p n

Figure

2: Unconditional distribution for successful
completion probability.
3.4 Monetary Transfers
bears an associated cost 1 . We assume the total
cost of task n has two parts: a deposit, which is paid when
the task starts, and a cost cn which is due at some time after
successful completion of n. Since we never compare plans
with di#erent deposits we assume without loss of generality
the deposit to be 0.
There is a single final payment V scheduled at the plan
conditional on all tasks in n being
successfully completed by that time.
There is an associated rate of return qn 2 that is used to
calculate the discounted present value (PV) for payo# cn due
at time t as
We associate the return q with the final payment V .
4. EXPECTED UTILITY
4.1 General Terms
We represent the customer agent's preferences over pay-
o#s by the von Neumann-Morgenstern utility function u.
1 Hereafter we use words "cost" and "reward" to denote some
monetary value, while referring the same value as "payo# "
or "payment" whenever it is scheduled at some time t.
2 The reason for having multiple qn 's is that individual tasks
can be financed from di#erent sources, thus a#ecting task
scheduling.
We further assume that the absolute risk-aversion coe#-
cient [14] r := -u # /u # of u is constant for any value of
its argument, hence u can be represented as follows:
A gamble is a set of payo#-probability pairs
1. The expectation of the utility
function over a gamble G is the expected utility (EU):
The certainty equivalent (CE) of a gamble G is defined
as the single monetary value whose utility matches the expected
utility of the entire gamble G, i.e. u (CE [G]) :=
Eu [G]. Hence under our assumptions
r log #
Naturally, the agent will not be willing to accept gambles
with less than positive certainty equivalent and the higher
values of the certainty equivalent will correspond to more
attractive gambles.
To illustrate the concept, Figure 3 shows how the certainty
equivalent depends on the risk-aversity of an agent. In this
figure we consider a gamble that brings the agent either 100
or nothing with equal probabilities. Agents with positive
r's are risk-averse, those with negative r's are risk-loving.
Agents with zero risk-aversity zero, i.e. risk-neutral, have a
CE equal to the gamble's weighted mean 50.

Figure

3: Certainty equivalent of a simple gamble
as a function of the risk-aversity.
4.2 Cumulative Probabilities
To compute the certainty equivalent of a gamble we need
to determine a schedule for the tasks and compute the payo#-
probability pairs.
We assume that a payo# cn for task n is scheduled at t f
so its present value -
cn 3 is
3 Hereafter we "wiggle" variables that depend on the current
task schedule, while omitting all corresponding indices for
the sake of simplicity.
We define the conditional probability of task n success as
We also define the precursors of task n as a set of tasks
that finish before task n starts in a schedule, i.e.
The unconditional probability that the task n will be completed
successfully is
pm .
That is, the probability of successful completion of every
precursor and of the task n itself are considered independent
events. The reason this is calculated in such form is because,
if any task in -
fails to be completed, there is no need
to execute task n.
The probability of receiving the final payment V is therefor

pn .
4.3 Example and Discussion
To illustrate the definitions and assumptions above, let's
return to the task network in Figure 1 and consider a sample
task schedule in Figure 4. In this figure the x-axis is time,
the y-axis shows both the task numbers and for each individual
task it also shows the cumulative distribution of the
unconditional probability of completion (compare to Figure
2). Circle markers show start times t s
n . Crosses indicate
both finish times t f
n and success probabilities -
pn (numbers
next to each point). Square markers denote that the corresponding
task cannot span past this point due to precedence
constraints. Finally, the thick part of each CDF shows the
time allocation for the task.

Figure

4: CE maximizing time allocations for the
plan in Figure 1 for
In practice, the customer agent needs a way of collecting
the market information necessary to build and use the
model. The probability of success is relatively easy to observe
in the market. This is the reason for introducing the
cumulative probability of success #n and the probability of
success pn , instead of the average project life span or probability
of failure or hazard function. Indeed, it is rational
for the supplier to report a successful completion immediately
in order to maximize the present value of a payment.
Also it is rational not to report a failure until the last possible
moment due to a possibility of earning the payment by
rescheduling, outsourcing or somehow else fixing the problem

To be specific, the information that the agent needs to
collect is the empirical distribution of how long does it take
from the point of starting some task to the point its completion
is reported. This data, unlike the data on failures or
actual positions in the supplier's schedule is less likely to be
private or unobservable.
5. MAXIMIZATION
5.1 Gamble Calculation Algorithm
Given a schedule, like the one shown in Figure 4, we need
to compute the payo# probability and then maximize the
CE for the gamble. Writing an explicit description of the
expected utility as a function of gambles is overly complicated
and relies on the order of task completions. Instead we
propose a simple recursive algorithm that creates these gam-
bles. We then maximize the CE over the space of gambles.
The proposed algorithm does not depend on the structure
of the task network, but on the number of tasks scheduled
in parallel.
Algorithm: G # calcGamble(T, D)
Requires: T "tasks to process", D "processed tasks"
Returns: G "subtree gamble"
"it's a branch''
"according to some ordering "
pn )})
endfor
I # calcGamble(T, D # {n}) "follow . # n path"
endfor
return G "subtree is processed"
else "it's a leaf ''
tasks are done"
return {(V, 1)}
else "some task failed"
return {(0, 1)}
endif
endif
In the first call the algorithm receives a "todo" task list
and a "done" task list #, all the subsequent calls
are recursive. To illustrate the idea behind this algorithm,
we refer to the payo#-probability tree in Figure 6. This tree
was built for the time allocations in Figure 5 and reflects
the precursor relations for this case.

Figure

5: CE maximizing time allocations for the
plan in Figure 1 for
Looking at the time allocation, we note that with probability
fails, the customer agent does not pay
or receive anything and stops the execution (path - 1 in the
tree). With probability -
p1 the agent proceeds with task
(path 1 in the tree). In turn, task 3 either fails with probability
p3 ), in which case the agent ends up stopping
the plan and paying a total of c1 (path 1 # - 3), or it is
completed with the corresponding probability -
p3 .
In the case where both 1 and 3 are completed, the agent
starts both 2 and 4 in parallel and becomes liable for paying
c2 and c4 respectively even if the other task fails (paths
fail, the resulting path in the tree is 1
the corresponding payo#-probability pair is framed in the
figure.
5.2 Computational Complexity
The computational complexity of the maximization procedure
is determined by two parameters: first, the procedure
itself is a non-linear maximization over 2N choice variables
with internal precedence constraints. Second, to calculate a
certainty equivalent value for every time schedule, the maximization
procedure should be able to build a corresponding
gamble and compute its expected utility.
For maximization we use the Nelder-Mead simplex (direct
search) method from the Matlab optimization toolbox.
The complexity of the calcGamble algorithm shown before
is O # 2 K-1
is the maximum number of
tasks that are scheduled to be executed in parallel.
The complexity estimate is based on the observation that
the depth of the payo#-probability tree is N and that any
subtree following an unsuccessful task execution has a depth
of no more than K- 1. The last statement follows from the
assumption that there are no more than K - 1 tasks running
in parallel to the one that failed and therefore no other
tasks will start after the failure was reported. Whether it is
possible to create an algorithm with significantly lower computational
costs is one of the questions we plan to address
in future research.
In commercial projects the ratio K/N is usually low, since
Figure

Payo#-probability tree for the time allocations
in Figure 5.
not many of these exhibit a high degree of the parallelism.
Our preliminary experiments, reported in Section 5.3, allow
us to conclude that K/N ratio is likely to be lower for risk-averse
agents (presumably, businessmen) than for risk-lovers
(gamblers).
5.3 Preliminary Experimental Results
We have conducted a series of experiments on the CE
maximization. Some of the results are summarized in Figure
7. In this figure, the y-axis shows 11 di#erent risk-
aversity r settings, the bottom x-axis - time t in the plan,
and the top x-axis - maximum CE value for each r setting.
The rounded horizontal bars in each of 11 sections denote
time allocations for each of six tasks with task 1 being on
top. Sections correspond to Figure
4 and Figure 5 respectively. Finally, the vertical bars
show the maximum CE values.
Let's examine the relative placement of time allocations
as a function of r. For this purpose we highlighted task 3
(black bars) and task 4 (white bars). Here task 3 has higher
variance of CDF and lower probability of success than task
4 (0.032 and 0.95 vs. 0.026 and 0.98), also task 3 is more
expensive (-15 vs. -7). There are four di#erent cases in
the experimental data:
1. Risk-loving agents tend to schedule tasks in parallel
and late in time in order to maximize the present value
of expected di#erence between reward and payo#s to
suppliers. This confirms the intuition from Figure 3 -
risk-lovers lean toward receiving high risky payments
rather than low certain payments.
2. Risk neutral and low risk-averse agents place risky task
3 first to make sure that the failure doesn't happen
too far in the project. Note, that they still keep task
2 running in parallel, so, in case 2 fails, they are liable
for paying the supplier of task 4 on success. One can
consider those agents as somewhat optimistic.
3. Moderately risk-averse agents try to dodge the situation
above by scheduling task 3 after task 2 is finished.
r

Figure

7: CE maximizing schedules and CE values for the plan in Figure 1 and r # [-0.03, 0.07].
These agents are willing to accept the plan, but their
expectations are quite pessimistic.
4. Highly risk-averse agents shrink task 1 interval to zero,
thus "cheating" to avoid covering any costs. One may
interpret this as a way of signaling a refusal of accepting
the plan.
6. ISSUES AND FUTURE RESEARCH
6.1 Multiple Local Maxima
One of the open issues is the existence of multiple local
maxima of CE, even in cases where task networks are fairly
simple. The reason for this is that the relative task placement
has two preferred configurations: independent individual
tasks can be either performed in parallel (thus increasing
the probability of successful completion) or they can be
scheduled in sequence to minimize overall payo#s, in case
one of tasks fails.
To illustrate the issue, we constructed a sample task net-work
with two parallel tasks. Task 1 has a higher variance of
completion time probability and lower probability of success
than task 2, everything else is the same. The resulting graph
of CE is shown in Figure 8. There are 3 local maxima in this
figure: one in the left side that corresponds to task 2 being
scheduled first in sequential order, another on the right side
corresponding to task 1 being first, and yet another one in
the furthermost corner of the graph representing both tasks
being scheduled at time 0 and executed in parallel.5010515task 1 start time

Figure

8: Local maxima for two parallel tasks.
In the course of the research, we were able to get around
the issue of multiple local maxima by starting the maximization
procedure from di#erent points. However, one may note
that the number of possible start points grows considerably
with the complexity of the task network and the algorithm
that checks each and every one of them is not scalable. A solution
we are considering is to use Simulated Annealing [20],
where each node in the search queue represents a local maximum
for some particular ordering of tasks.
6.2 Slack Allocation in RFQ
The last issue we want to address in this paper is how
to use the CE maximization procedure to construct RFQs
in the MAGNET framework. The CE maximizing schedule
contains information on what is the most desirable task
scheduling for the customer agent. However, it is hard to
imagine that there will always be bids that cover exactly the
same time intervals as in the maximizing schedule.
We suggest the following specify what
percentile # of the maximum CE value is considered acceptable
by the agent. then define the start time for the task
n as the set of values of t s
n , such that the CE of a schedule
that di#ers from the maximizing one only in the start time
of task n is no less than # of the maximum.
Graphically, this process is represented by building the
projection of the CE #-percentile graph (see Figure 9) on
the task n time axis. Assuming there is only one continuous
interval of t s
n values for every n # N , denote it as # t s-
Finally, submit the interval # t s-
and t f
are times from the maximizing schedule, as a part of
the RFQ.
taskstart
time 95%
90%
50%

Figure

9: Contours of some #-percentile graphs for
the CE graph in Figure 8.
This leaves several open questions for further study. For
instance, there could be more than one interval # t s-
for some tasks, so we need to distinguish them in the RFQ
composition. Also, it might be appropriate to decrease the
acceptable CE percentile for tasks involving goods and services
that are rare and do not attract many bids, and, at
the same time, increase the percentile for those tasks that
receive overly many bids. In addition, the RFQ might have
to be split in two or more parts, so that the requests for
rare goods and services are submitted first and the rest of
RFQ is composed after the bids for those rare products are
received. Deciding how and when to split the RFQs is still
an open question.
Although we do not specifically address the above mentioned
and related issues in the current paper, the CE maximization
approach promises to be powerful and flexible
enough to help us resolve those in our future research.
7. RELATED WORK
Expected Utility Theory [19] is a mature field of Eco-
nomics, that has attracted many supportive as well as critical
studies, both theoretical [12, 13] and empirical [23, 10].
We believe that expected utility will play an increasing role
in automated auctions, since it provides a practical way of
describing risk estimations and temporal preferences.
In our previous work on Expected Utility [1] we were
mostly concerned with computing the marginal expected
utility of completing successfully all the tasks within the
duration promised.
Our long term objective is to automate the scheduling
and execution cycle of an autonomous agent that needs the
services of other agents to accomplish its tasks. Pollack's
DIPART system [18] and SharedPlans [7] assume multiple
agents that operate independently but all work towards
the achievement of a global goal. Our agents are trying to
achieve their own goals and to maximize their profits; there
is no global goal.
Combinatorial auctions are becoming an important mechanism
not just for agent-mediated electronic commerce [8,
26, 22] but also for allocation of tasks to cooperative agents
(see, for instance, [9, 5]).
In [9] combinatorial auctions are used for the initial commitment
decision problem, which is the problem an agent
has to solve when deciding whether to join a proposed col-
laboration. Their agents have precedence and hard temporal
constraints. However, to reduce search e#ort, they
use domain-specific roles, a shorthand notation for collections
of tasks. In their formulation, each task type can be
associated with only a single role. MAGNET agents are
self-interested, and there are no limits to the types of tasks
they can decide to do. In [6] scheduling decisions are made
not by the agents, but instead by a central authority. The
central authority has insight to the states and schedules of
participating agents, and agents rely on the authority for
supporting their decisions. Nisan's bidding language [16] allows
bidders to express certain types of constraints, but in
MAGNET both the bidder and the bid-taker (the customer)
need to communicate constraints.
Inspite of the abundance of work in auctions [15], limited
attention has been devoted to auctions over tasks with
time constraints and interdependencies. In [17],
a method is proposed to auction a shared track line for
train scheduling. The problem is formulated with mixed
integer programming, with many domain-specific optimiza-
tions. Bids are expressed by specifying a price to enter a
line and a time window. The bidding language, which is
similar to what we use in MAGNET, avoids use of discrete
time slots. Time slots are used in [25], where a protocol for
decentralized scheduling is proposed. The study is limited
to scheduling a single resource. MAGNET agents deal with
multiple resources.
Most work in supply-chain management is limited to hierarchical
modeling of the decision making process, which is
inadequate for distributed supply-chains, where each organization
is self-interested, not cooperative. Walsh et al [24]
propose a protocol for combinatorial auctions for supply
chain formation, using a game-theoretical perspective. They
allow complex task networks, but do not include time con-
straints. MAGNET agents have also to ensure the scheduling
feasibility of the bids they accept, and must evaluate risk
as well. Agents in MASCOT [21] coordinate scheduling with
the user, but there is no explicit notion of money transfers
or contracts, and the criteria for accepting/rejecting a bid
are not explicitly stated. Their major objective is to show
policies that optimize schedules locally [11]. Our objective
is to optimize the customer's utility.
8.

ACKNOWLEDGMENTS

Partial support for this research is gratefully acknowledged
from the National Science Foundation under award
NSF/IIS-0084202.
9.



--R

Decision processes in agent-based automated contracting

Evaluating risk: Flexibility and feasibility in multi-agent contracting
A market architecture for multi-agent contracting
distributed control of a multirobot system.
Socially conscious decision-making
AI Magazine

A combinatorial auction for collaborative planning.
Estimating preferences under risk: The case of racetrack bettors.
Coordinated Supply Chain Scheduling.
Choice under uncertainty: Problems solved and unsolved.
Dynamic consistency and non-expected utility models of choice und er uncertainty
Microeconomic Theory.
Auctions and bidding.
Bidding and allocation in combinatorial auctions.
An auction-based method for decentralized train scheduling
Planning in dynamic environments: The DIPART system.
Risk aversion in the small and in the large.
Modern Heuristic Techniques for Combinatorial Problems.
MASCOT: an agent-based architecture for coordinated mixed-initiative supply chain planning and scheduling
An algorithm for winner determination in combinatorial auctions.
An empirical analysis of the economic value of risk changes.
Combinatorial auctions for supply chain formation.
Auction protocols for decentralized scheduling.
The Michigan Internet AuctionBot: A configurable auction server for human and software agents.
--TR
Modern heuristic techniques for combinatorial problems
A market architecture for multi-agent contracting
The Michigan Internet AuctionBot
Evaluating risk
Socially conscious decision-making
Bidding and allocation in combinatorial auctions
Combinatorial auctions for supply chain formation
An auction-based method for decentralized train scheduling
Decision Processes in Agent-Based Automated Contracting
An Algorithm for Optimal Winner Determination in Combinatorial Auctions
A Combinatorial Auction for Collaborative Planning

--CTR
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multi-agent contracting;automated auctions;risk estimation;expected utility