On Nash-solvability of n-person graphical games under Markov's and a priori realizations

TL;DR

This paper investigates the existence of Nash equilibria in n-person graphical games with perfect information under two different probabilistic assumptions, revealing conditions where equilibria may or may not exist.

Contribution

It introduces and analyzes two probabilistic frameworks—Markov's and a priori realizations—for solving graphical games, highlighting their differences and implications for Nash solvability.

Findings

Markov's realization guarantees a uniformly best response but may lack Nash equilibria.

A priori realization allows Nash's theorem but lacks some properties like uniform best responses.

Examples demonstrate the application and limitations of both realizations.

Abstract

We consider graphical $n$-person games with perfect information that have no Nash equilibria in pure stationary strategies. Solving these games in mixed strategies, we introduce probabilistic distributions in all non-terminal positions. The corresponding plays can be analyzed under two different basic assumptions: Markov's and a priori realizations. The former one guarantees existence of a uniformly best response of each player in every situation. Nevertheless, Nash equilibrium may fail to exist even in mixed strategies. The classical Nash theorem is not applicable, since Markov's realizations may result in the limit distributions and effective payoff functions that are not continuous. The a priori realization does not share many nice properties of the Markov one (for example, existence of the uniformly best response) but in return, Nash's theorem is applicable. We illustrate both realizations in details by two examples with $2$ and $3$ players and also provide some general results.