Preference- Versus Nonpreference-Based Concepts

In recent years it has become customary to say that classical power measures, such as the Shapley-Shubik and Banzhaf indices, suffer a major drawback in that they do not take into account player preferences. The corollary of this lack of preferences criticism is that classical power indices—based on simple games rooted in cooperative rather than noncooperative game theory—are insensitive to the strategic aspects of power and therefore are inappropriate for a positive analysis of the distribution of power in institutional structures. The upshot of the criticism has been the development of so-called strategic power indices that are based on noncooperative games in order to fill this apparent lacuna. Such indices also renew attempts to introduce preferences or strategic considerations into the classical power measures and represent an attempt to find a unified framework that brings together cooperative and noncooperative approaches.

To obtain an intuitive grasp of the problem, consider the three-player simple game with winning coalitions {a, b, c}, {a, b} and {a, c}. Assume that a can make the following ultimatum offer to either player b or c: accept a tiny share of the spoils or be prevented from taking part in a winning coalition. If the players are rational and have utility functions that are monotonic in the spoils, and that there is no way to credibly enforce a blocking coalition {b, c} that could extract concessions from a, the noncooperative game-theoretic equilibrium will be that whichever of players b or c that a approaches first will accept A's pittance of an offer. After all, something, however small, is better than nothing for rational self-interested agents. The conclusion that theorists such as Stefan Napel and Mika Widgrén draw from this example is that, given the pittance or nothing at all that b or c will receive under these two solution concepts, it is only reasonable to deduce that they must be more [Page 530]or less powerless because both of these players are robbed of the power commonly associated with their swing potentials. Yet, the absolute Banzhaf index and the Shapley—Shubik Index, for instance, yield respective power vectors of and , values that are radically at odds with a competitive analysis.

Despite the intuitive appeal of the example, it is fundamentally mistaken. The reason hinges on a conceptual issue: what we mean by a power ascription. Ordinarily speaking, a power ascription refers to a person's ability: what a person is able to do. In the game-theoretic context that we are discussing, the ability in question is to effect (or force or determine) outcomes of the game. That is, a player has a strategy that, if chosen, will make a decisive difference to the outcome. This basic definition is the same for a power index based upon a simple game and one that is ostensibly based upon a noncooperative game. The difference lies in the specification of the ability. In a simple game the ability is turning a winning coalition into a losing coalition or vice versa and thereby being decisive for the acceptance or rejection of a bill, while in a noncooperative game the ability is specified in terms of shifting the equilibrium in one's own favor.

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