# What is Game Theory? by 26MqjbG

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```									    Solution and Simulation for
Coalitional Games
Roger A. McCain
For presentation at the
16th International Conference on Computing in
Economics and Finance, London, July 15-17,
2010
Available at

http://faculty.lebow.drexel.edu/m
ccainr/top/eco/wps/Solution.pdf
Computational Constructive
Modeling 1
• As Herbert Simon observes, a computer program can be a
theory.
• Theories stated in computational languages provide an
alternative to the oversimplification of theories founded in
mathematical analysis.
• Agent-based simulations are an instance.
• Similarly, we might understand a computational algorithm
as the solution to a game in game theory.
Computational Constructive
Modeling 2
• In general, I would define a computational constructive
theory as a theory which begins from the construction of
computational objects with the properties and propensities
real-world economic objects are believed to have, and in
which their potential interactions and processes are
described in algorithmic terms.
• In this paper, the “cooperative solution” of a game is
considered in this way.
Coalitional Games
• Game theory has historically been divided into
two largely non-overlapping traditions,
– “cooperative”
– “non-cooperative”
• In “cooperative” or coalitional games agents are able to
make trustworthy commitments to form coalitions.
• But noncooperative game theory has been more successful.
Solution of Coalitional Games
• “We live our lives in coalitions.” Maskin
• Among the solution concepts for coalitional games, the
Shapley (1953) value has the advantage that it can be
calculated for any game in coalition function form and is
unique.
• In addition, it is based on a concept of the marginal
contribution of an agent to a coalition.
Games in Partition Function
Form 1
• For a (transferable utility) game in coalition
function form, the total value of a coalition
depends only on its members.
• For a game in partition function form, the value
depends also on the other coalitions formed – the
partition of the players into coalitions.
Games in Partition Function
Form 2
• For superadditive games in coalition function form
(classical games), the grand coalition will form.
• Thus the “solution” needs only determine the
payouts to the agents.
• However, this neglects externalities and holdout
behavior.
• Accordingly we consider games in partition
function form.
Shapley Value
• For classical games, the Shapley value has
desirable properties.
• It can be computed in two ways:
– By a combinatorial formula
– As the expected value of the marginal value of
each agent, when all orders of adding agents are
equiprobable.
Shapley’s Algorithm 1
• let the players in the game be indexed as
i=1, 2, …, N, and let m=fj(i) be a
permutation of the order of the indices. Let
the value of a set S of players be v(S). Let
the set of all distinct permutations be
indexed as j=1, 2, …, K. Now execute
Shapley’s Algorithm 2
Generalized Solution 1
• It computes a probability distribution over the partitions
that may be formed.
• For each partition with a positive probability, it imputes a
value for each agent. The imputation is conditional on the
formation of that partition.
• The value is the expected value of the agent's marginal
contribution to the coalition of which he is a member.
Generalized Solution 2
• The total of the values of agents in a coalition (imputed for
a particular partition) just exhausts the value of the
coalition in the partition.
• At each step for a given permutation, the agent may choose
to join any coalition already formed, or to enter the game
as a singleton coalition.
• When these decisions are considered as a game in
extensive form, the decisions are subgame perfect best
responses.
An Example
The Solution
Inefficiency
• Only the grand coalition is efficient.
• However, there is some probability that partition
{1,4},{2,3} will form. Why?
• Consider permutations 1,4,2,3 or 4,1,2,3
• Once {1,4} has formed, 2 can form {1,2,4} for a
marginal value of 3 or remain {2} for 4.
• 3 then forms {2,3} for 10.
Solution

Only the
fine
partition,
partition 1,
is Pareto
efficient.
A Public Goods Game
• Game 3 is a five-person
game of public goods
production and has five
solutions, in each of which
a four-person coalition is
formed and the remaining
agent remains as a
singleton. Here, agent 5 is
the free-rider.
Simulation 1
• In McCain, Roger A. (2007), “Agent-Based
Simulation of Endogenous Coalitions:
Some Small-Scale Examples,” I presented
some agent-based simulations of coalition
formation, indluding some of the same
games.
Simulation 2
• Simulations were conducted with two kinds of dynamics, a
“naïve” and a boundedly rational “foresightful” version.
• Each simulation is initialized with the fine partition for the
particular game.
• The naïve dynamics is thus based on a simple coalition
dominance criterion.
• This may lead to dominance cycles and consequently may
be considered as lacking in foresight.
Simulation 3
• In the foresightful simulations, a record is kept for each
agent of the partitions, coalitions, and payoffs the agent has
experienced.
• An aspiration index is computed as the agent’s average
payoff when that agent has been a member of a successful
candidate coalition.
• The agent will veto any new coalition with a payoff less
than his aspiration.
A Troubling Contrast
• In many cases, the results were quite
similar.
• For a game of entrepreneurship, however,
suggestive differences are observed.
The JBC Game
Solution
No Coalition is ever formed!
Contrast
• In the simulations, the grand coalition was always
formed, and quite rapidly.
• The failure to coalesce in the computed solution is
a result of marginality. Since agents are added one
by one, it never pays anyone to be the first.
• By contrast, the simulations expressed dominance,
and the efficient grand coalition is dominant.
Negative Result?
• In some sense this is a negative result.
• The focus on the Shapley value was motivated by the
property of marginality.
• However, the intuitively unsatisfying results in these
computations are traceable precisely to marginality.
• Conclusion: marginality is not an appropriate property for
a solution for games in partition function form.
• (See McCain, 2009, for an extension of the nucleolus.)

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