Interval measures of power

Abstract

Standard power indices, introduced by Shapley and Shubik, Banzhaf, Johnston, and others, assign real numbers to the players in a simple game as a quantitative measure of their influence in the voting situation represented by the game. We consider instead two ‘interval notions of power’ that assign intervals of real numbers to the players. The first of these notions originates in the observation that although a particular choice of weights in a weighted game may not provide an accurate measure of power, the interval of all possible weights for a player is, in fact, a reasonable reflection of the player's influence. The resulting weight interval is extended to non-weighted games via the observation that every simple game is an intersection of weighted games. For example, we have shown (Taylor and Zwicker, Games and Economic behavior, 1993, 5, 170–181) that the president's weight interval in the (non-weighted) federal system is ( 15 115 , 17 117 ), suggesting that the president has between 13% and 15% of the power. The development of a combinatorial equivalent, the dispersal interval, yields a uniform method for calculating weight intervals. Our second interval index, the market interval, is based on a modification of Peyton Young's idea that the influence of players can be measured by the sizes of the bribes they can demand when they sell their votes. We show, for example, that the president's market interval in the federal system is (0, 0.2537,…); he can reasonably insist on between 0% and 25% of the total bribe. An iterated version of the market interval, based on rationality assumptions about other players' actions and linked with the notion of a bribery equilibrium, produces potentially smaller intervals that, in some cases, shrink to single points in the limit.