Skip to main content Accessibility help
×
Home

Applications of Shapley-Owen Values and the Spatial Copeland Winner

  • Joseph Godfrey (a1), Bernard Grofman (a2) and Scott L. Feld (a3)

Abstract

The Shapley-Owen value (SOV, Owen and Shapley 1989, Optimal location of candidates in ideological space. International Journal of Game Theory 125–42), a generalization of the Shapley-Shubik value applicable to spatial voting games, is an important concept in that it takes us away from a priori concepts 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 the Copeland winner, the strong 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 Analysis 12:256–76) 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 Politics 68:838–51) 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.

Copyright

Corresponding author

e-mail: http://www.winset.com (corresponding author)

Footnotes

Hide All

Authors' note: The authors wish to thank Nicholas Miller for his encouragement and support; Guillermo Owen and Keith Dougherty for their helpful comments regarding an earlier version of this paper; Itai Sened and Michael Lynch for making available their compilation of data from experimental committee voting games; Fuad Aleskerov, Director, Departement of Higher Mathematics, Higher School of Economics, State University of Russia (Moscow) for making available data on party locations in the Russian Duma; and Sue Ludeman for bibliographic assistance. Supplementary materials for this article are available on the Political Analysis Web site.

Footnotes

References

Hide All
Aleskerov, Fuad, and Otchour, Olga. n.d. Extended Shapley-Owen indices and power distribution in the 3rd State Duma of the Russian Federation Unpublished manuscript. State University of Russia, Higher School of Economics.
Banks, J. S. 1985. Sophisticated voting outcomes and agenda control. Social Choice and Welfare 1: 295306.
Banks, J. S., Duggan, J., and Le Breton, M. 2002. Bounds for mixed strategy equilibria and the spatial model of elections. Journal of Economic Theory 103: 88105.
Black, Duncan. 1958. The theory of committees and elections. New York: Cambridge University Press.
Bianco, William T., Jeliazkov, Ivan, and Sened, Itai. 2004. The limits of legislative actions: Determining the set of enactable outcomes given legislators preferences. Political Analysis 12: 256–76.
Bianco, William T., Lynch, Michael S., Miller, Gary J., and Sened, Itai. 2006. A theory waiting to be discovered and used: A reanalysis of canonical experiments on majority-rule decision making. Journal of Politics 68: 838–51.
Brauninger, Thomas. 2003. When simple voting doesn't work: Multicameral systems for the representation and aggregation of interests in international organizations. British Journal of Political Science 33: 681703.
Brunell, Thomas, and Grofman, Bernard. 2008. Evaluating the impact of redistricting on district homogeneity, political competition and political extremism in the U. S. House of Representatives, 1962-2006. In Designing democratic government, eds. Levi, Margaret, Johnson, James, Knight, Jack, and Stokes, Susan, 117–40. New York: Russell Sage Foundation.
Calvert, Randall. 1985. Robustness of the multidimensional voting model: Candidate motivations, uncertainty, and convergence. American Journal of Political Science 29: 6995.
Cox, Gary. 1987. The uncovered set and the core. American Journal of Political Science 31: 408–22.
Endersby, James W. 1993. Rules of method and rules of conduct: An experimental study on two types of procedure and committee behavior. Journal of Politics 55: 218–36.
Feld, Scott L., and Grofman, Bernard. 1988a. The Borda count in n-dimensional issue space. Public Choice 59: 167–76.
Feld, Scott L., and Grofman, Bernard. 1988b. Majority rule outcomes and the structure of debate in one-issue-at-a-time decision making. Public Choice 59: 239–52.
Feld, Scott L., and Grofman, Bernard. 1990. A theorem connecting Shapley-Owen power scores and the radius of the yolk in two dimensions. Social Choice and Welfare 7: 7174.
Feld, Scott L., Grofman, Bernard, Hartley, Richard, Kilgour, Mark O., and Miller, Nicholas. 1987. The uncovered set in spatial voting games. Theory and Decision 23: 129–56.
Feld, Scott L., Grofman, Bernard, and Godfrey, Joseph. 2007. Putting a Spin on it: Geometric insights into how candidates with seemingly losing positions can still win. Paper presented at the Annual Meeting of the American Sociological Association New York, August.
Fiorina, Morris P., and Plott, Charles R. 1978. Committee decisions under majority rule: An experimental study. American Political Science Review 72: 575–98.
Fishburn, Peter C. 1977. Condorcet social choice functions. SIAM Journal of Applied Mathematics 33: 469–89.
Godfrey, Joseph, and Grofman, Bernard. 2008. Pivotal voting theory: The 1993 Clinton Health Care Reform Proposal in the U. S. Congress. In Power, freedom, and voting, eds. Braham, Matthew and Steffen, Frank, 139–58. Berlin, Germany: Springer Verlag.
Goldstein, K. M. 1999. Interest groups, lobbying and participation in America. New York: Cambridge University Press.
Golosov, Grigorii V. 2006. The structure of party alternatives and voter choice in Russia: Evidence from the 2003-4 regional elections. Party Politics 12: 707–25.
Grofman, Bernard, and Feld, Scott L. 1992. Group decision making over multidimensional objects of choice. Organizational Behavior and Human Performance 52: 3963.
Grofman, Bernard, Owen, Guillermo, Noviello, Nicholas, and Glazer, Amihai. 1987. Stability and centrality of legislative choice in the spatial context. American Political Science Review 81: 539–53.
Hammond, Thomas H., and Miller, Gary J. 1987. The core of the constitution. American Political Science Review 81: 1155–74.
Hartley, Richard, and Kilgour, Mark. 1987. The geometry of the uncovered set. Mathematical Social Sciences 1: 175–83.
Koehler, David H. 1990. The size of the yolk: Computations of odd and even-numbered committees. Social Choice and Welfare 7: 231–45.
Koehler, David H. 1992. Limiting median lines frequently determine the yolk. Social Choice and Welfare 9: 3741.
Koehler, David H. 2001. Instability and convergence under simple-majority rule: Results from simulation of committee choice in two-dimensional space. Theory and Decision 50: 305–32.
Koehler, David H. 2002. Convergence and restricted preference maximizing under simple majority rule: Results from a computer simulation. American Political Science Review 95: 155–67.
Kramer, Gerald. 1972. Sophisticated voting over multidimensional spaces. Journal of Mathematical Sociology 2: 165–80.
Laing, James D., and Olmstead, Scott. 1978. An experimental and game-theoretic study of committees. In Game theory and political science, ed. Ordeshook, Peter C., 215–81. New York: New York University Press.
McKelvey, Richard D. 1986. Covering, dominance, and institution-free properties of social choice. American Journal of Political Science 30: 283314.
McKelvey, Richard D., Winer, Mark D., and Ordeshook, Peter. 1978. The competitive solution for n-person games without transferable utility, with an application to committee games. American Political Science Review 72: 599615.
McKelvey, Richard D., and Ordeshook, Peter C. 1984. An experimental study of the effects of procedural rules on committee behavior. Journal of Politics 46: 182205.
Miller, Gary J., Hammond, Thomas H., and Kile, Charles. 1996. Bicameralism and the core: An experimental test. Legislative Studies Quarterly 21: 83103.
Miller, Nicholas R. 1980. A new solution set for tournament and majority voting. American Journal of Political Science 24: 6896.
Miller, Nicholas R. 2007. In search of the uncovered set. Political Analysis 15: 2145.
Miller, Nicholas R., Grofman, Bernard, and Feld, Scott L. 1989. The geometry of majority rule. Journal of Theoretical Politics 1: 379406.
Miller, Nicholas R., Grofman, Bernard, and Feld, Scott L. 1990. The structure of the Banks set. Public Choice 66: 243–51.
Moulin, Herve. 1986. Choosing from a tournament. Social Choice and Welfare 3: 271–91.
Ordeshook, Peter, and Schwartz, Thomas. 1987. Agendas and the control of political outcomes. American Political Science Review 81: 179200.
Owen, G., and Shapley, L. S. 1989. Optimal location of candidates in ideological space. International Journal of Game Theory 18339–56.
Penn, Elizabeth. 2006a. Alternative definitions of the uncovered set, and their implications. Social Choice and Welfare 27: 83–7.
Penn, Elizabeth. 2006b. The Banks set in infinite spaces. Social Choice and Welfare 27: 531.
Saari, Donald. 1994. The geometry of majority rule. Berlin, Germany: Springer-Verlag.
Schofield, Norman. 1995. Democratic stability. In Explaining social institutions, eds. Knight, Jack and Sened, Itai. Ann Arbor: University of Michigan Press.
Shapley, Lloyd S. 1977. A comparison of power indices and a non-symmetric generalization. RAND Corporation Paper P-5872, Santa Monica.
Shepsle, Kenneth A., and Weingast, Barry. 1981. Structure-induced equilibrium and legislative choice. Public Choice 37: 503519.
Shepsle, Kenneth A., and Weingast, Barry. 1984. Uncovered sets and sophisticated voting outcomes with implications for agenda institutions. American Journal of Political Science 28: 4974.
Stone, R., and Tovey, Craig. 1992. Limiting median lines do not suffice to determine the yolk. Social Choice and Welfare 9: 33–5.
Straffin, Philip D. Jr. 1980. Topics in the theory of voting. Boston, MA: Birkhauser.
Wuffle, A., Feld, Scott L., Owen, Guillermo, and Grofman, Bernard. 1989. Finagle's law and the Finagle point, a new solution concept for two-candidate competition in spatial voting games. American Journal of Political Science 33: 4875.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.
Type Description Title
WORD
Supplementary materials

Godfrey et al. supplementary material
Appendix

 Word (11.6 MB)
11.6 MB

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed