A Scalar Compromise Equilibrium for N-Person Prescriptive Games

Abstract

A scalar equilibrium (SE) is defined for n-person prescriptive games in normal form. When a decision criterion (notion of rationality) is either agreed upon by the players or prescribed by an external arbiter, the resulting decision process is modeled by a suitable scalar transformation (utility function). Each n-tuple of von Neumann-Morgenstern utilities is transformed into a nonnegative scalar value between 0 and 1. Any n-tuple yielding a largest scalar value determines an SE, which is always a pure strategy profile. SEs can be computed much faster than Nash equilibria, for example; and the decision criterion need not be based on the players' selfishness. To illustrate the SE, we define a compromise equilibrium, establish its Pareto optimality, and present examples comparing it to other solution concepts.

A Mixed Cooperative Dual to the Nash Equilibrium

“…The Nash equilibrium and the dual equilibrium thus model opposite decision criteria for choosing the player's actions, regardless of whether these actions are independently selected by the players, are coordinated by the players, or are even prescribed. For example, the results here are valid if an arbiter assigns actions to the players as in [16].…”

“…., while for > 2 a DE may not exist-even on the average in the long run, even if the players try to be selfless. Mutual cooperation thus differs from the notion of compromise as defined in [16], which exists for any . In particular, for a given payoff matrix, mutual cooperation is not always possible for strictly mathematical reasons as a consequence of sociological information about the players reflected in their joint von Neumann-Morgenstern utilities.…”