Coalition formation and potential games (Q700088)

The main result of the present paper establishes an interesting connection between the (Au\-mann-Dreze generalization of the) Shapley value of cooperative games and the multiplicity problem in coalition formation non-cooperative games. More specifically, in the book of von Neumann-Morgenstern 1944 a model of coalition formation for a set \(N\) of players consists of a two-stage procedure given by: a cooperative game \(v\) with an allocation rule \(\gamma\) mapping each partition of \(N\) to an \(N\)-vector of payoffs (second stage); and a non-cooperative game (first stage) in which each player in \(N\) chooses a coalition \(T\subseteq N\) to which she wants to belong in \(v\), with a rule which determines a partition for each profile of choices, the payoff being then determined according to rule \(\gamma\). Two rules mapping coalition choices to partitions of \(N\) are considered; the more `aggregative', by which it is more difficult for a player to remain isolated, states that two players end up in the same coalition iff they choose the same coalition in the first stage. Non-cooperative games have typically multiple equilibria. A game in strategic form \((N,(X_i)\), \((\pi_i))\) is called a potential game if it admits a potential function \(P\), which is a real valued function on strategy profiles \(x\) such that \[ P(x_i,x_{-i})-P(x'_i,x_{-i})= \pi_i(x_i,x_{-i})-\pi_i(x'_i,x_{-i})\quad \forall x,i. \] Potential games are important because they have a natural equilibrium refinement, namely the potential maximizer. The remarkable result of the present paper is that the first-stage game of the coalition-formation model is a potential game if and only if the allocation rule \(\gamma\) is the Aumann-Dreze generalization of the Shapley value to games with coalition structures, which simply consists in assigning Shapley values in each subgame -- that is, partition element -- separately.