Coalition-proof Nash equilibrium

The concept of coalition-proof Nash equilibrium applies to certain "noncooperative" environments in which players can freely discuss their strategies but cannot make binding commitments. It emphasizes the immunization to deviations that are self-enforcing. While the best-response property in Nash equilibrium is necessary for self-enforceability, it is not generally sufficient when players can jointly deviate in a way that is mutually beneficial.

The Strong Nash equilibrium is criticized as too "strong" in that the environment allows for unlimited private communication. In the coalition-proof Nash equilibrium the private communication is limited.

Formal definition. (i) In a single player, single stage game ${\displaystyle \Gamma }$, ${\displaystyle s^{\ast }\in S}$ is a Perfectly Coalition-Proof Nash equilibrium if and only if ${\displaystyle s^{\ast }}$ maximizes ${\displaystyle g^{1}(s)}$. (ii) Let (n,t) ≠ (1,1). Assume that Perfectly Coalition-Proof Nash equilibrium has been defined for all games with m players and s stages, where (m, s) ≤ (n, t), and (m, s) ≠ (n, t). (a) For any game ${\displaystyle \Gamma }$ with ${\displaystyle n}$ players and ${\displaystyle t}$ stages, s^*€S is perfectly self-enforcing if, for all J€J, s_J^* is a Perfectly Coalition-Proof Nash equilibrium in the game Γ/s^*_−J, and if the restriction of s^* to any proper subgame forms a Perfectly Coalition-Proof Nash equilibrium in that subgame. (b) For any game Γ with n players and t stages, s^*€S is a Perfectly Coalition-Proof Nash equilibrium if it is perfectly self-enforcing, and if there does not exist another perfectly self-enforcing strategy vector s€S such that g¹(s)> g¹(s*) for all i= 1,...,n.

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