Applications of Shapley-Owen Values and the Spatial Copeland Winner

Abstract

The Shapley-Owen value (SOV, Owen and Shapley 1989, Optimal location of candidates in ideological space.International Journal of Game Theory125–42), a generalization of the Shapley-Shubik value applicable to spatial voting games, is an important concept in that it takes us away froma prioriconcepts of power to notions of power that are directly tied to the ideological proximity of actors. SOVs can also be used to locate the spatial analogue to theCopeland winner, thestrong point, the point with smallest win-set, which is a plausible solution concept for games without cores. However, for spatial voting games with many voters, until recently, it was too computationally difficult to calculate SOVs, and thus, it was impossible to find the strong point analytically. After reviewing the properties of the SOV, such as the result proven by Shapley and Owen that size of win sets increases with the square of distance as we move away from the strong point along any ray, we offer a computer algorithm for computing SOVs that can readily find such values even for legislatures the size of the U.S. House of Representatives or the Russian Duma. We use these values to identify the strong point and show its location with respect to the uncovered set, for several of the U.S. congresses analyzed in Bianco, Jeliazkov, and Sened (2004, The limits of legislative actions: Determining the set of enactable outcomes given legislators preferences.Political Analysis12:256–76) and for several sessions of the Russian Duma. We then look at many of the experimental committee voting games previously analyzed by Bianco et al. (2006, A theory waiting to be discovered and used: A reanalysis of canonical experiments on majority-rule decision making.Journal of Politics68:838–51) and show how outcomes in these games tend to be points with small win sets located near to the strong point. We also consider how SOVs can be applied to a lobbying game in a committee of the U.S. Senate.