Potential, voting, and power

We suggest and advocate a new index of absolute power for voting games, the potential index. This index exhibits desirable properties with respect to overall power assigned in a voting game, which are not met by the Penrose-Banzhaf index.

A voting game consists of a finite players set and a characteristic function that assigns to any subset (coalition) either the number 1 (winning) or 0 (losing). The characteristic function is monotonic, i.e., adding players to a winning coalition never turns it into a losing one, and superadditive, i.e., the complement of any winning coalition is losing.

An (absolute) power index assigns to any player in any voting game a real number, the player’s (voting) power, where individual powers need not sum up to 1 as for relative indices. To which number the individual powers should sum up?

We suggest that the overall power in a game should equal the game’s potential due to Hart and Mas-Colell (1989, Econometrica 57, 589-614), who feel that “the potential provides the most natural one-number summary of a game.” This implies an intuitive overall power for unanimity games. (i) A unanimity game is characterized by its index coalition. In particular, a coalition is winning only if all members of the index coalition are contained in this coalition. The overall power for a unanimity game under the potential index equals the reciprocal of the number of players in its index set. This is quite intuitive. The more players needed to win the vote the less power is present in a game. (ii) Overall power is greatest only if the game contains a dictator, whose presence is necessary and sufficient for a coalition to win. Overall power is lowest only if all players are needed to win the vote.

The potential index shows a number of monotonicity properties. (i) Strong monotonicity: A coalition is the swing for some player if it is losing, but wins when this player enters the coalition. Strong monotonicity: Whenever the swing set of some player in one game is contained in her swing set of another game, then this player’s power in the latter game is not lower than in the former. (ii) Desirability: Whenever the set of swings for a particular player (without the second) is contained in the set of swings of another player (without the first), then the latter player’s power is not lower than the former player’s.