On 2×2 Games and Braithwaite’s Arbitration Scheme

Abstract

The problem of arbitration of two-person non-zero-sum games has been given much attention by game theorists. Even when meaningful utility comparisons between the two players are possible, a difficulty has arisen. For, if utilities are measured on an interval scale, an arbitrated result should not be affected by the application of different positive linear transformations to the payoffs of the two players. A natural solution is to distinguish certain transformations which would ‘normalize’ the payoff matrices, and thus permit direct numerical comparison of utilities of different players. Then arbitration is reduced to a problem of cooperative bargaining, which can be solved by the use of lines of constant relative advantage. Braithwaite [1] has proposed such a method. As described by Luce and Raiffa ([2], pp. 148–150), Braithwaite’s normalizing transformation operates by equating the gains of the two players when each performs a certain change of strategy. In essence, Braithwaite’s arbitration scheme rests on a particular ‘symmetrization’ of the game.