Scalar Equilibria For N-person Games

Abstract

In this dissertation we develop a scalarization approach for one-shot, n-person games by defining the notion of Scalar Equilibria. We first show that existing solution concepts can be represented as Scalar Equlibria. For example, Regret, Disappointment, and Joint Equilibrium can be determined by defining Regret, Disappointment, and Joint Scalar Equilibria. These scalar equilibria are useful for finding pure strategies when pure Regret, Disappointment, and Joint Equilibria do not exist. Next, we present the Maximin Scalarization Equilibrium to yield maximin solution concept. In addition, we propose other Scalar Equilibria for various notions of rationality. The Aspiration Scalar Equilibrium is developed for an aspiration criterion when players have specified payoff aspiration levels. Then Risk, Greedy and Cooperative Scalar Equilibrium are developed for risk, greed, and cooperative criteria, respectively. Moreover, Sequential, Simultaneous, and Priority Scalar Equilibria are developed as well as Coalition Scalar Equilibria. In a Sequential Scalar Equilibrium we sequentially, in some chosen order, apply other scalarizations to Scalar Equilibrium of the game until we find a unique one if possible. In a Simultaneous Scalar Equilibrium we combine the criteria for various scalarizations into one. Effectively the multiple criteria are applied simultaneously. In a Priority Scalar Equilibrium players are prioritized as their ability to get their highest payoff. A Coalition Scalar Equilibrium consider fixed teams of players seek team payoffs that are then divided among the players. Finally, we presented examples to illustrate the usage and theoretical aspects of these equilibria.