Quarreling Paradox

The quarreling paradox is that a player of a simple game may actually increase his power by rendering himself unable to join any coalition containing certain specific opponents. Moreover, the powers of two players may both increase when they “quarrel,” that is, when each refuses to join any coalition containing the other. This phenomenon contradicts the intuitive notion that power depends on freedom to act; by reducing freedom, quarreling should also reduce power.

Simple cooperative games are useful models of group decision processes in which different individuals have different roles. A subset of players, or coalition, is winning if it can make a decision on behalf of the group; it is losing otherwise. To specify a simple game, simply identify all winning coalitions.

When the Shapley value for cooperative games is applied to a simple game, it called the Shapley-Shubik (SS) index, and measures a player's importance (or power) in the group decision process. A player's SS index value can be interpreted as its expected “points” in a model of play in which an ordering (permutation) of players is chosen at random and the first player whose support converts the set of all preceding players to a winning coalition is awarded one point, while all other players receive zero. A player's (absolute) Penrose-Banzhaf-Coleman (PBC) index can be interpreted as its expected “points” in a model of play in which a subset of the players is chosen at random and each player in the subset is awarded one point if it belongs to the subset and its removal changes the subset from winning to losing; otherwise, it receives no points. Other indices are defined by other models of play.

Models of play can be adapted to a context in which some coalitions cannot form, as the quarreling scenario supposes. When thus recalculated, a quarreling player's value on common indices may increase. For example, consider a weighted voting game in which players A, B, and C have two, one, and one votes respectively, the quota is three, and players B and C quarrel. Then the SS indices of B and C rise from 1/6 to 1/4, because orderings in which B and C both join the coalition before it gains winning status are excluded, and their (normalized) BCP indices rise from 1/5 to 1/4, because coalitions including both B and C are excluded. Steven J. Brams called this phenomenon the paradox of quarreling members.

Dan S. Felsenthal and Moshé Machover point out that the quarrelling paradox is different from other paradoxes of voting power in that it requires an ad hoc extension of the calculation (to remove coalitions containing two or more quarrelers but retain all others). They suggest that quarreling is incompatible with many concepts of power; for example, policy-motivated individuals have preferences about outcomes (e.g., to pass or defeat a bill) and not about each other. They argue also that the paradox may not be so surprising in that the strategy of reducing one's own freedom to act (e.g., by “burning bridges”) is widely practiced. Moreover, if one quarreling player never joins a coalition to which the other belongs, this restricts the coalition opportunities of nonquarreling players as well.

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