Coherence of probabilistic constraints on Nash equilibria

TL;DR

This paper investigates the problem of verifying whether a set of probabilistic constraints assigned to a game are consistent with the game's Nash equilibria, analyzing algorithms and complexity for pure and mixed equilibria.

Contribution

It introduces the PCE-Coherence decision problem, providing complexity results and algorithms for checking probabilistic constraint coherence in games with pure and mixed Nash equilibria.

Findings

Complexity results for PCE-Coherence with pure Nash equilibria

Algorithms for computing maximal and minimal coherent probabilities

Extension of coherence analysis to mixed Nash equilibria

Abstract

Observable games are game situations that reach one of possibly many Nash equilibria. Before an instance of the game starts, an external observer does not know, a priori, what is the exact profile of actions that will occur; thus, he assigns subjective probabilities to players' actions. However, not all probabilistic assignments are coherent with a given game. We study the decision problem of determining if a given set of probabilistic constraints assigned a priori by the observer to a given game is coherent, which we call the Coherence of Probabilistic Constraints on Equilibria, or PCE-Coherence. We show several results concerning algorithms and complexity for PCE-Coherence when only pure Nash equilibria are considered. In this context, we also study the computation of maximal and minimal probabilistic constraints on actions that preserves coherence. Finally, we study these problems when mixed Nash equilibria are allowed.