The Intrinsic Quantum Nature of Nash Equilibrium Mixtures

The concept of Nash equilibrium has become central in game theory, economics, and other social sciences. A Nash equilibrium is defined as an n-tuple of strategies or strategy profile (one strategy for each player) if each player’s strategy is optimal against the others’ strategies. As is well-known by now, the interactive epistemology under which rational individuals play such a “social equilibrium point” is quite demanding. Aumann and Brandenburger (1995) have notably demonstrated that it presumes that each player knows, or correctly guess his opponents’ beliefs about the other players’ strategy choices. In addition, as acknowledged in most of the literature, the notion of a mixed-strategy Nash equilibrium leads to some seemingly insuperable conceptual difficulties (see e.g., Aumann (1987)). The main trouble lies in the fact that, in a Nash equilibrium, each player who selects a mixed-strategy is always indifferent between two pure strategies of the support. The purpose of this paper is to firm up the foundation of Nash equilibrium and provide a compelling (quantum) interpretation of this concept in the original rationalistic framework of Nash. The bulk of the paper is devoted to show how the Nash equilibrium notion can be constructively derived. The gist of our approach builds on the following two observations: (i)The classical game model is complete in the sense that its complete description is given by the strategy sets, the outcome map, and the payoff functions and; (ii)Rationality is a relativistic or relational concept in the sense that it consists of making an optimal choice that has to be justifiable by some beliefs. Taken together, (i) and (ii) imply that absolute statements like A :=“strategy a is optimal in the game G for player i” are generally neither absolutely “true”, nor absolutely “ false” but indeterminate. Hence, a non-classical logic—the threevalued logic of Lukasiewicz (1930)—enters the picture of the game model in its own right because this model does not (generally) contain the answers to questions like “what constitutes a rational behavior?”. So the question naturally arises: How will a player ascribe a relative truth-value, true, to a particular rational strategy? The answer is simple; in the game model, each player possesses only pieces of a puzzle made of contingent statements about the optimality of a strategy. Playing