Possibilistic Games with Incomplete Information

Bayesian games offer a suitable framework for games where the utility degrees are additive in essence. This approach does nevertheless not apply to ordinal games, where the utility degrees do not capture more than a ranking, nor to situations of decision under qualitative uncertainty. This paper proposes a representation framework for ordinal games under possibilistic incomplete information (π-games) and extends the fundamental notion of Nash equilibrium (NE) to this framework. We show that deciding whether a NE exists is a difficult problem (NP-hard) and propose a Mixed Integer Linear Programming (MILP) encoding. Experiments on variants of the GAMUT problems confirm the feasibility of this approach.

Equilibria in Ordinal Games: A Framework based on Possibility Theory.

The present paper proposes the first definition of mixed equilibrium for ordinal games. This definition naturally extends possibilistic (single agent) decision theory. This allows us to provide a unifying view of single and multi-agent qualitative decision theory. Our first contribution is to show that ordinal games always admit a possibilistic mixed equilibrium, which can be seen as a qualitative counterpart to mixed (probabilistic) equilibrium.Then, we show that a possibilistic mixed equilibrium can be computed in polynomial time (wrt the size of the game), which contrasts with pure Nash or mixed probabilistic equilibrium computation in cardinal game theory.The definition we propose is thus operational in two ways: (i) it tackles the case when no pure Nash equilibrium exists in an ordinal game; and (ii) it allows an efficient computation of a mixed equilibrium.

ORDINAL GAMES

We study strategic games where players' preferences are weak orders which need not admit utility representations. First of all, we extend Voorneveld's concept of best-response potential from cardinal to ordinal games and derive the analogue of his characterization result: An ordinal game is a best-response potential game if and only if it does not have a best-response cycle. Further, Milgrom and Shannon's concept of quasi-supermodularity is extended from cardinal games to ordinal games. We find that under certain topological assumptions, the ordinal Nash equilibria of a quasi-supermodular game form a nonempty complete lattice. Finally, we extend several set-valued solution concepts from cardinal to ordinal games in our sense.