Equilibria in ordinal games: A framework based on possibility theory

Abstract

The present paper proposes the first definition of a mixed equilibrium in an ordinal game. This definition naturally extends possibilistic (single agent) 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 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.