Measures of conflict and power in strategic settings

In non-cooperative games, if coordination is allowed, then coalitions can take group actions, and thus become players themselves, each opposed to its complement. There is conflict when for every outcome there is at least one player who strictly prefers another outcome. In this paper, an index measuring conflict is provided. It obtains through a general procedure for turning any non-cooperative game into a [0,1]-ranged coalitional one. This latter quantifies the normalized worth of coordination within coalitions, whose group actions maximize the sum of members’ normalized utilities under the assumption that their complement chooses retaliation among best responses. That is, the max-min criterion is sophisticated by minimizing over best responses only. Taking into account retaliation possibilities enables to measure players’ power in any non-cooperative game through the Shapley value of the induced coalitional game; in fact, both transferable and non-transferable utilities are covered. But perhaps the ideal tools for analyzing power are game forms, where utilities are not specified at all, so that the only available information concerns how action profiles are mapped onto outcomes. These situations are treated by means of blockings (i.e. the dual of effectivity functions), specifying what outcomes each coalition may block. The chosen cardinality-based approach yields that power attains its maximum on Maskin blockings, and here again coalitions’ power may be used to define a coalitional game which, in turn, allows to quantify the power share of each player. 1. Some ideas contained in this paper were firstly presented at Logic, Game Theory and Social Choice 4 (22-24 June 2005, Université de Caen, France), where the participants provided interesting comments. I thank Antonio Bove and Giorgio Nicoletti for very helpful conversations. The usual disclaimer applies. 2. Department of Computer Science, University of Bologna, Mura Anteo Zamboni 7, 40126 Bologna, Italy; e-mail: [email protected], url: www.cs.unibo.it/giorossi.