Stable outcomes in spatial voting games

Abstract

The traditional representation of outcomes for an n-person game as vectors in n-space was superseded in the 1960s when it became clear that, for political and related applications, a lower-dimensional space was generally sufficient for the purpose. Thus, instead of vectors in utility space, vectors in policy space came to be used. Utility to the various decision-makers (usually voters or parliamentarians) could then be defined in terms of distance from given ideal points. A well-known result states that, when the underlying policy space is one-dimensional, a non-empty core (one or more undominated points) will exist. In two or more dimensions, it is known that, for decisive games, where a bare majority is sufficient to carry motions, the core will be non-empty only under very special circumstances (degenerate cases). Stability is then sought in the form of near-core outcomes. Among the best known of the near-core outcomes are (a) the Copeland winner, defined as that outcome which defeats the greatest proportion of alternatives; (b) the yolk center, defined as that alternative which comes closest to satisfying all possible winning coalitions of voters; and (c) the minimal response (finagle) point, defined as the point which comes closest to defeating any other alternative. We show that each of these near-core outcomes has a well-defined analogue for games with side payments (classical von Neumann-Morgenstern games). In particular, the Copeland winner's analogue is the Shapley value and the yolk center's analogue is the nucleolus. The side-payment analogue of the finagle point has not been previously studied; a straightforward adaptation, known also as the finagle point, is however possible.