Game Forms, Power in

A game form is a game (in the sense of game theory) with all information about the preferences, payoffs, or utilities of the players stripped out. Put otherwise, a game form is a function that maps strategy profiles into outcomes, over which players' preferences are unspecified. Game forms, rather than games, provide the appropriate representation of voting procedures, electoral systems, constitutions, and other collective decision-making institutions or power relationships. The concept of a game form was first explicitly introduced by Allan Gibbard, who proved that every “straightforward” game form with three or more outcomes is “dictatorial.” (A straightforward game form gives every player, regardless of what his or her preferences may be, an undominated strategy.) This general result implies that every “strategy-proof” voting procedure for choosing among three or more alternatives is also dictatorial. (A voting procedure is strategy-proof if it never gives any voter an incentive to cast an “insincere” or “dishonest” vote.) By first proving the general result for all game forms, Gibbard was able neatly to sidestep the questions of what exactly constitutes a voting procedure and what exactly we mean by insincere or dishonest voting.

Subsequently, Nicholas Miller used the concept of a game form to define and analyze power. If we define power as the capacity of an actor, alone or (more likely) in combination with others, to bring about or preclude outcomes, game forms provide a natural framework for analyzing power abstractly. An n-player game form may be represented by an n-dimensional matrix such that each row corresponds to a strategy for player 1, each column to a strategy for player 2, and so forth, and each cell (or strategy profile) belongs to some outcome. (Different cells may belong to the same outcome.) From the point of view of any focal player i, such a matrix can be contracted into two dimensions, such that rows represent i's strategies and columns represent all possible combinations of strategies for the other players. More generally, rows and columns can represent the strategy combinations for any pair of complementary coalitions of players. Given this setup, a number of definitions, observations, and propositions follow in natural ways. (In what follows, a coalition may refer to any subset of players, including a single player.)

A strategy is potent to the extent that there are outcomes that do not appear in that row or column; such a strategy gives a coalition preclusive power. A strategy is decisive for outcome x if it is maximally potent, that is, if it precludes all outcomes other than x; such a strategy gives a coalition affirmative power. A coalition has veto power if it has a strategy that is decisive for some single outcome x (presumably the status quo or some other default outcome). A coalition is all-powerful if it is decisive for every outcome, and power relations are dictatorial if a single player is all-powerful. The affirmative power of every coalition is limited at least by the preclusive power of its complement. If two disjoint coalitions both have [Page 270]affirmative power, they are both decisive for the same single outcome; that is, they both have veto power.

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