Abstract
An algorithm is presented in this note for determining all Berge equilibria for an n-person game in normal form. This algorithm is based on the notion of disappointment, with the payoff matrix (PM) being transformed into a disappointment matrix (DM). The DM has the property that a pure strategy profile of the PM is a BE if and only if (0,…,0) is the corresponding entry of the DM. Furthermore, any (0,…,0) entry of the DM is also a more restrictive Berge-Vaisman equilibrium if and only if each player’s BE payoff is at least as large as the player’s maximin security level.
Highlights
In a Berge equilibrium (BE) for an n-person game, every n − 1 player has pure strategies that maximized the remaining player’s payoff
The BE was intuitively defined for pure strategies in [1] as a refinement to the Nash equilibrium (NE) [2]
The disappointment incurred by player i for a strategy profile s = is the difference between the best payoff that player i could obtain by choosing si and the actual payoff that player i would obtain for the strategy profile
Summary
In a Berge equilibrium (BE) for an n-person game, every n − 1 player has pure strategies that maximized the remaining player’s payoff. The BE was intuitively defined for pure strategies in [1] as a refinement to the Nash equilibrium (NE) [2]. A simple algorithm is presented here for computing all BEs for an n-person game G = (N, (Si)i∈N, (ui)i∈N) in normal form, with N = {1, . Sn) the von Neumann-Morgenstern utility of player i for a pure strategy profile s =
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