Computing All Equilibria in Ordinal Graphical Games

Graphical games allow to concisely represent games where the utility received by each player depends on strategies played by a (hopefully small) subset of the players only. Recently, Graphical Ordinal Games have been proposed as a framework for game theory where utility degrees are ordinal. The concept of probabilistic mixed-Nash equilibrium is irrelevant in this framework since ordinal utility degrees cannot be averaged. Instead, possibilistic mixed equilibria have been proposed as a principled solution concept in such games. A generic possibilistic mixed equilibrium computation algorithm has been proposed, which applies to ordinal games, be they in standard normal form or graphical form. However, this algorithm only computes a single least-specific equilibrium. When analyzing ordinal games, one may be interested in finding every least-specific equilibria or in counting them. In this paper, we propose two original algorithms for computing all least-specific possibilistic mixed equilibria in Ordinal Graphical games. We first focus on tree-structured Ordinal Graphical Games and propose the Possibilistic Tree-Nash algorithm (-Tree-Nash), a possibilistic counterpart of the Tree-Nash algorithm proposed by Kearns et al. for (cardinal) graphical games. Then, we propose the Search All Equilibria algorithm (SAE) which computes all least-specific mixed equilibria of an arbitrary Ordinal Graphical Game. We provide algorithmic complexity results as well as an experimental evaluation of both algorithms.