Local Political Equilibria

Part of the Studies in Choice and Welfare book series (WELFARE)


This article uses the notion of a “Local Nash Equilibrium” (LNE) to model a vote maximizing political game that incorporates valence (the electorally perceived quality of the political leaders). Formal stochastic voting models without valence typically conclude that all political agents (parties or candidates) will converge towards the electoral mean (the origin of the policy space). The main theorem presented here obtains the necessary and sufficient conditions for the validity of the “mean voter theorem” when valence is involved. Since a pure strategy Nash equilibrium (PNE), if it exists, must be a LNE the failure of the necessary condition for an LNE at the origin also implies that PNE cannot be at the origin. To further account for the non-convergent location of parties, the model is extended to include activist valence (the effect on party popularity due to the efforts of activist groups). These results suggest that it is very unlikely that Local Nash equilibria will be located at the electoral center. The theoretical conclusions appear to be borne out by empirical evidence from a number of countries. Genericity arguments demonstrate that LNE will exist for almost all parameters, when the policy space is compact, convex, without any restriction on the variance of the voter ideal points or on the party valence functions.


Nash Equilibrium Vote Share Liberal Democrat Party Party Leader Policy Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Aldrich, J. (1983) A spatial model with party activists: implications for electoral dynamics. Public Choice, 41: 63–100.CrossRefGoogle Scholar
  2. [2]
    Aldrich, J. (1983) A Downsian spatial model with party activists. American Political Science Review, 77: 974–990.CrossRefGoogle Scholar
  3. [3]
    Aldrich, J. and M. McGinnis (1989) A model of party constraints on optimal candidate positions. Mathematical and Computer Modelling, 42: 437–450.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Alvarez, M. R. and J. Nagler (1998) When politics and models collide: estimating models of multicandidate elections. American Journal of Political Science, 42: 55–96.Google Scholar
  5. [5]
    Alvarez, M. R., J. Nagler and S. Bowler (2000) Issues, economics and the dynamics of multiparty elections: the British 1987 general election. American Political Science Review, 94: 131–150.CrossRefGoogle Scholar
  6. [6]
    Ansolabehere, S. and J. M. Snyder (2000) Valence politics and equilibrium in spatial election models. Public Choice, 103: 327–336.CrossRefGoogle Scholar
  7. [7]
    Aragones, E and T. Palfrey (2004) Spatial competition between two candidates of different quality: the effects of candidate ideology and private information. This volume.Google Scholar
  8. [8]
    Austen-Smith, D. and J. S. Banks (1998) Social choice theory, game theory and positive political theory. In N. Polsby (ed), Annual Review of Political Science vol I: 259–287.Google Scholar
  9. [9]
    Austen-Smith, D. and J. S. Banks (1999) Positive Political Theory I: Collective Preference Ann Arbor: The University of Michigan Press.Google Scholar
  10. [10]
    Banks, J. S. (1995) Singularity theory and core existence in the spatial model. Journal of Mathematical Economics, 24: 523–536.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Banks, J. S. and J. Duggan (2004) The theory of probabilistic voting in the spatial model of elections: the theory of office-motivated candidates. This volume.Google Scholar
  12. [12]
    Banks, J. S. and J. Duggan (2000) A bargaining model of collective choice. American Political Science Review, 94: 73–88.CrossRefGoogle Scholar
  13. [13]
    Banks, J. S., J. Duggan, and M. Le Breton (2002) Bounds for mixed strategy equilibria and the spatial model of elections. Journal of Economic Theory, 103: 88–105.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Banks, J. S., J. Duggan, and M. Le Breton (2004) Social choice and electoral competition in the general spatial model. Journal of Economic Theory, forthcoming.Google Scholar
  15. [15]
    Brown, R. F. (1970) The Lefshetz Fixed Point Theorem Glenview IL: Scott and Foreman.Google Scholar
  16. [16]
    Clarke, H., M. Stewart and P. Whiteley (1998) New models for new labour: the political economy of labour support: January 1992–April 1997. American Political Science Review, 92: 559–575.CrossRefGoogle Scholar
  17. [17]
    Coughlin, P. (1992) Probabilistic Voting Theory Cambridge: Cambridge University Press.Google Scholar
  18. [18]
    Dierker, E. (1974) Topological Methods in Walrasian Economics Lecture Notes on Economics and Mathematical Sciences, 92, Heidelberg: Springer.Google Scholar
  19. [19]
    Downs, A. (1957) An Economic Theory of Democracy New York: Harper and Row.Google Scholar
  20. [20]
    Duggan, J. (2000) Equilibrium equivalence under expected plurality and probability of winning maximization. Mimeo. University of Rochester.Google Scholar
  21. [21]
    Fan, K. (1961) A generalization of Tychonoff’s fixed point theorem. Mathematische Annalen, 42: 305–310.CrossRefGoogle Scholar
  22. [22]
    Groseclose, T. (2001) A model of candidate location when one candidate has a valence advantage. American Journal of Political Science, 45: 862–886.Google Scholar
  23. [23]
    Hinich, M. (1977) Equilibrium in spatial voting: the median voter result is an artifact. Journal of Economic Theory, 16: 208–219.zbMATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    Hirsch, M. W. (1976) Differential Topology Heidelberg: Springer.Google Scholar
  25. [25]
    Konishi, H. (1996) Equilibrium in abstract political economies: with an application to a public good economy with voting. Social Choice and Welfare, 13: 43–50.zbMATHCrossRefGoogle Scholar
  26. [26]
    Lin, T. M., J. Enelow and H. Dorussen (1999) Equilibrium in multicandidate probabilistic spatial voting. Public Choice, 98: 59–82.CrossRefGoogle Scholar
  27. [27]
    McKelvey, R. (1986) Covering, dominance, and institution-free properties of social choice. American Journal of Political Science, 30: 283–314.Google Scholar
  28. [28]
    Michael, E. (1956) Continuous selection I. Annals of Mathematics, 63: 361–382.zbMATHMathSciNetCrossRefGoogle Scholar
  29. [29]
    Miller, G. and N. Schofield (2003) Activists and partisan realignment in the United States. American Political Science Review, 97: 245–260.CrossRefGoogle Scholar
  30. [30]
    Milnor, J. (1963) Morse Theory Annals of Mathematics Studies, Series 51 Princeton: Princeton University Press.Google Scholar
  31. [31]
    Nash, J. (1951) Non-cooperative games. Annals of Mathematics, 54: 289–295.MathSciNetCrossRefGoogle Scholar
  32. [32]
    Poole, K. and H. Rosenthal (1984) Presidential elections 1968–1980: a spatial analysis. American Journal of Political Science, 28: 283–312.Google Scholar
  33. [33]
    Quinn, K., A. Martin and A. Whitford (1999) Voter choice in multiparty democracies. American Journal of Political Science, 43: 1231–1247.Google Scholar
  34. [34]
    Saari, D. (1997) The generic existence of a core for q-rules. Economic Theory, 9: 219–260.zbMATHMathSciNetGoogle Scholar
  35. [35]
    Schofield, N. (1999) The C1 topology on the space of smooth preference profile. Social Choice and Welfare, 16: 445–470.zbMATHMathSciNetCrossRefGoogle Scholar
  36. [36]
    Schofield, N. (2003a) A valence model of political competition in Britain, 1992–1997. Electoral Studies, forthcoming.Google Scholar
  37. [37]
    Schofield, N. (2003b) Valence competition in the spatial stochastic model. Journal of Theoretical Politics, 15: 371–383.CrossRefGoogle Scholar
  38. [38]
    Schofield, N. (2003c) Mathematical Methods in Economics and Social Choice Heidelberg: Springer.Google Scholar
  39. [39]
    Schofield, N. (2004) Equilibrium in the spatial valence model of politics. Journal of Theoretical Politics, 16: 447–481.CrossRefGoogle Scholar
  40. [40]
    Schofield, N., A. Martin, K. Quinn and A. Whitford (1998) Multiparty electoral competition in the Netherlands and Germany: a model based on multinomial probit. Public Choice, 97: 257–293.CrossRefGoogle Scholar
  41. [41]
    Schofield, N., G. Miller and A. Martin (2003) Critical elections and political realignment in the US: 1860–2000. Political Studies, 51: 217–240.CrossRefGoogle Scholar
  42. [42]
    Schofield, N. and R. Parks (2000) Nash equilibrium in a spatial model of coalition bargaining. Mathematical Social Sciences, 39: 133–174.MathSciNetCrossRefGoogle Scholar
  43. [43]
    Schofield, N. and I. Sened (2002) Local Nash equilibrium in multiparty politics. Annals of Operations Research, 109: 193–210.MathSciNetCrossRefGoogle Scholar
  44. [44]
    Schofield, N. and I. Sened (2003) Multiparty competition in Israel. British Journal of Political Science, forthcoming.Google Scholar
  45. [45]
    Schofield, N., I. Sened and D. Nixon (1998) Nash equilibrium in multiparty competition with ’stochastic’ voters. Annals of Operations Research, 84: 3–27.CrossRefGoogle Scholar
  46. [46]
    Seyd, P. and P. Whitely (2002) New Labour’s Grassroots Basingstoke UK: Macmillan.Google Scholar
  47. [47]
    Smale, S. (1960) Morse inequalities for a dynamical system. Bulletin of the American Mathematical Society, 66: 43–49.zbMATHMathSciNetCrossRefGoogle Scholar
  48. [48]
    Smale, S. (1974) Global analysis and economics I: Pareto optimum and a generalization of Morse theory. In M. Peixoto (ed) Dynamical Systems New York: Academic Press.Google Scholar
  49. [49]
    Stokes, D. (1992) Valence politics. In D. Kavanagh (ed) Electoral Politics Oxford: Clarendon Press.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Washington UniversitySt. Louis

Personalised recommendations