Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-23T22:16:56.712Z Has data issue: false hasContentIssue false
  • English
  • Français

Applications of Shapley-Owen Values and the Spatial Copeland Winner

Published online by Cambridge University Press:  04 January 2017

Joseph Godfrey ,
Bernard Grofman  and
Scott L. Feld
Joseph Godfrey*
Affiliation:
WinSet Group, LLC, 4031 University Drive, Suite 200, Fairfax, VA 22030
Bernard Grofman
Affiliation:
Department of Political Science and Institute for Mathematical Behavioral Sciences, University of California, Irvine, CA 92697-5100
Scott L. Feld
Affiliation:
Department of Sociology, Purdue University, Lafayette, IN 47907
*
e-mail: http://www.winset.com (corresponding author)
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
Political Analysis , Volume 19 , Issue 3 , Summer 2011 , pp. 306 - 324
Copyright
Copyright © The Author 2011. Published by Oxford University Press on behalf of the Society for Political Methodology 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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.

References

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.Google Scholar
Banks, J. S. 1985. Sophisticated voting outcomes and agenda control. Social Choice and Welfare 1: 295–306.CrossRefGoogle Scholar
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: 88–105.CrossRefGoogle Scholar
Black, Duncan. 1958. The theory of committees and elections. New York: Cambridge University Press.Google Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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: 681–703.CrossRefGoogle Scholar
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.Google Scholar
Calvert, Randall. 1985. Robustness of the multidimensional voting model: Candidate motivations, uncertainty, and convergence. American Journal of Political Science 29: 69–95.CrossRefGoogle Scholar
Cox, Gary. 1987. The uncovered set and the core. American Journal of Political Science 31: 408–22.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Feld, Scott L., and Grofman, Bernard. 1988a. The Borda count in n-dimensional issue space. Public Choice 59: 167–76.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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: 71–74.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
Fiorina, Morris P., and Plott, Charles R. 1978. Committee decisions under majority rule: An experimental study. American Political Science Review 72: 575–98.CrossRefGoogle Scholar
Fishburn, Peter C. 1977. Condorcet social choice functions. SIAM Journal of Applied Mathematics 33: 469–89.CrossRefGoogle Scholar
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.Google Scholar
Goldstein, K. M. 1999. Interest groups, lobbying and participation in America. New York: Cambridge University Press.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Grofman, Bernard, and Feld, Scott L. 1992. Group decision making over multidimensional objects of choice. Organizational Behavior and Human Performance 52: 39–63.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Hammond, Thomas H., and Miller, Gary J. 1987. The core of the constitution. American Political Science Review 81: 1155–74.CrossRefGoogle Scholar
Hartley, Richard, and Kilgour, Mark. 1987. The geometry of the uncovered set. Mathematical Social Sciences 1: 175–83.Google Scholar
Koehler, David H. 1990. The size of the yolk: Computations of odd and even-numbered committees. Social Choice and Welfare 7: 231–45.CrossRefGoogle Scholar
Koehler, David H. 1992. Limiting median lines frequently determine the yolk. Social Choice and Welfare 9: 37–41.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
Kramer, Gerald. 1972. Sophisticated voting over multidimensional spaces. Journal of Mathematical Sociology 2: 165–80.CrossRefGoogle Scholar
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.Google Scholar
McKelvey, Richard D. 1986. Covering, dominance, and institution-free properties of social choice. American Journal of Political Science 30: 283–314.CrossRefGoogle Scholar
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: 599–615.CrossRefGoogle Scholar
McKelvey, Richard D., and Ordeshook, Peter C. 1984. An experimental study of the effects of procedural rules on committee behavior. Journal of Politics 46: 182–205.CrossRefGoogle Scholar
Miller, Gary J., Hammond, Thomas H., and Kile, Charles. 1996. Bicameralism and the core: An experimental test. Legislative Studies Quarterly 21: 83–103.CrossRefGoogle Scholar
Miller, Nicholas R. 1980. A new solution set for tournament and majority voting. American Journal of Political Science 24: 68–96.CrossRefGoogle Scholar
Miller, Nicholas R. 2007. In search of the uncovered set. Political Analysis 15: 21–45.CrossRefGoogle Scholar
Miller, Nicholas R., Grofman, Bernard, and Feld, Scott L. 1989. The geometry of majority rule. Journal of Theoretical Politics 1: 379–406.CrossRefGoogle Scholar
Miller, Nicholas R., Grofman, Bernard, and Feld, Scott L. 1990. The structure of the Banks set. Public Choice 66: 243–51.CrossRefGoogle Scholar
Moulin, Herve. 1986. Choosing from a tournament. Social Choice and Welfare 3: 271–91.CrossRefGoogle Scholar
Ordeshook, Peter, and Schwartz, Thomas. 1987. Agendas and the control of political outcomes. American Political Science Review 81: 179–200.CrossRefGoogle Scholar
Owen, G., and Shapley, L. S. 1989. Optimal location of candidates in ideological space. International Journal of Game Theory 18339–56.CrossRefGoogle Scholar
Penn, Elizabeth. 2006a. Alternative definitions of the uncovered set, and their implications. Social Choice and Welfare 27: 83–7.CrossRefGoogle Scholar
Penn, Elizabeth. 2006b. The Banks set in infinite spaces. Social Choice and Welfare 27: 531.CrossRefGoogle Scholar
Saari, Donald. 1994. The geometry of majority rule. Berlin, Germany: Springer-Verlag.Google Scholar
Schofield, Norman. 1995. Democratic stability. In Explaining social institutions, eds. Knight, Jack and Sened, Itai. Ann Arbor: University of Michigan Press.Google Scholar
Shapley, Lloyd S. 1977. A comparison of power indices and a non-symmetric generalization. RAND Corporation Paper P-5872, Santa Monica.Google Scholar
Shepsle, Kenneth A., and Weingast, Barry. 1981. Structure-induced equilibrium and legislative choice. Public Choice 37: 503–519.CrossRefGoogle Scholar
Shepsle, Kenneth A., and Weingast, Barry. 1984. Uncovered sets and sophisticated voting outcomes with implications for agenda institutions. American Journal of Political Science 28: 49–74.CrossRefGoogle Scholar
Stone, R., and Tovey, Craig. 1992. Limiting median lines do not suffice to determine the yolk. Social Choice and Welfare 9: 33–5.CrossRefGoogle Scholar
Straffin, Philip D. Jr. 1980. Topics in the theory of voting. Boston, MA: Birkhauser.Google Scholar
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: 48–75.CrossRefGoogle Scholar
Supplementary material: File

Godfrey et al. supplementary material

Appendix

Download Godfrey et al. supplementary material(File)
File 11.6 MB