Title:

Leastsquares regret and partially strategic players

Noncooperative game theory enjoys a vast canon of solution concepts. The predominant solution concept is Nash equilibrium (Nash, 1950a; Nash, 1951). Other solution concepts include generalizations and refinements of Nash equilibrium as well as alternatives to it. Despite their successes, the established solution concepts are in some ways unsatisfactory. In particular, for many games, such as the Centipede Game (Rosenthal, 1981), the pBeauty Contest (Moulin, 1986; Simonsen, 1988), and the notorious Traveler’s Dilemma (Basu, 1994; Basu, 2007), many of the solution concepts yield solutions that are both unreasonable in theory and refuted by the experimental evidence. And when a solution concept manages to yield the expected or reasonable solutions for such games, it often suffers from other difficulties such as unwieldy complexity or reliance on ad hoc or gamespecific constructions that may fail to be generalizable. We propose a new solution concept, which we call leastsquares regret, that yields the expected or reasonable solutions for games that have thus far proved to be problematic, such as the Traveler’s Dilemma; that is simple; that involves no ad hoc or gamespecific constructions and can thus be applied immediately and consistently to any arbitrary game; that exhibits nice properties; and that is grounded in human psychology. Intuitively, we suppose that a player chooses a strategy so as to minimize the divergence from perfect play overall. In particular, we suppose that a player is partially strategic and chooses a strategy so as to minimize the sum, across all partial profiles of strategies of the other players, of the squares of the regrets, where the regret of a strategy with respect to a partial profile is the difference of the bestresponse payoff with respect to the partial profile and the payoff from choosing the strategy with respect to the partial profile. The aim of this work is to develop the solution concept of leastsquares regret; explore its properties; assess its performance with respect to various games of interest; determine its merits and demerits, especially in relation to other solution concepts; review its weaknesses; introduce a refinement, which we call mutual weighted leastsquares regret, that addresses some of the weaknesses; and propose some questions for further research.
