What is Game Theory? by 26MqjbG


									    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,
            Available at

     Computational Constructive
           Modeling 1
• As Herbert Simon observes, a computer program can be a
• 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
• 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
• 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
       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
An Example
The Solution
• 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

 Only the
 partition 1,
 is Pareto
          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
                    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
• 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
• For a game of entrepreneurship, however,
  suggestive differences are observed.
The JBC Game
No Coalition is ever formed!
• 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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