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.
A Nonsuperadditive Game
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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