# Individual preferences and democratic processes: two theorems with implications for electoral politics

## Abstract

The paper provides a complete characterization of Nash equilibria for games in which *n* candidates choose a strategy in the form of a platform, each from a circle of feasible platforms, with the aim of maximizing the stretch of the circle from where the candidate’s platform will receive support from the voters. Using this characterization, it is shown that if the sum of all players’ payoffs is 1, the Nash equilibrium payoff of each player in an arbitrary Nash equilibrium must be restricted to the interval \( [1/2(n-1),2/(n+1)].\) This implies that in an election with four candidates, a candidate who is attracting less than one-sixth of the voters can do better by changing his or her strategy.

## Notes

## References

- Acemoglu D, Robinson JA (2005) Economic origins of dictatorship and democracy. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Arrow KJ (1950) A difficulty in the concept of social welfare. J Polit Econ 58(4):328–346CrossRefGoogle Scholar
- Arrow KJ (1951) Social choice and individual values. Wiley, New York (second edition, 1963)Google Scholar
- Basu K (1993) Lectures in industrial organization theory. Basil Blackwell, OxfordGoogle Scholar
- Bergson A (1938) A reformulation of certain aspects of welfare economics. Q J Econ 52:310–334CrossRefGoogle Scholar
- Black D (1948) On the rationale of group decision-making. J Polit Econ 56(1):23–34CrossRefGoogle Scholar
- Brander JA, Spencer BJ (2015) Endogenous horizontal product differentiation under Bertrand and Cournot competition: revisiting the Bertrand paradox. NBER Working Paper No. 20966Google Scholar
- Congleton R (2002) The median voter model. In: Rowley CK, Schneider F (eds) The encyclopedia of public choice. Kluwer Academic Press, DordrechtGoogle Scholar
- d’Aspremont C, Gevers L (1977) Equity and the informational basis of collective choice. Rev Econ Stud 46:199–210CrossRefGoogle Scholar
- d’Aspremont C, Gabszewicz JJ, Thisse JF (1979) On Hotelling’s “Stability in competition”. Econometrica 47(5):1145–1150CrossRefGoogle Scholar
- Downs A (1957) An economic theory of political action in a democracy. J Polit Econ 65(2):135–150CrossRefGoogle Scholar
- Fujita M, Thisse J-F (1996) Economics of agglomeration. J Jpn Int Econ 10(4):339–378CrossRefGoogle Scholar
- Gabszewicz JJ, Thisse JF, Fujita M, Schweizer U (1986) Location theory. Harwood, ChichesterGoogle Scholar
- Hotelling H (1929) Stability in competition. Econ J 39(153):41–57CrossRefGoogle Scholar
- Maskin E (1978) A theorem on utilitarianism. Rev Econ Stud 45:93–96CrossRefGoogle Scholar
- Matsushima N (2001) Cournot competition and spatial agglomeration revisited. Econ Lett 73(2):175–177CrossRefGoogle Scholar
- Osborne MJ (1995) Spatial models of political competition under plurality rule: a survey of some explanations of the number of candidates and the positions they take. Can J Econ 28:261–301CrossRefGoogle Scholar
- Pal D (1998) Does Cournot competition yield spatial agglomeration? Econ Lett 60(1):49–53CrossRefGoogle Scholar
- Pattanaik PK (1971) Voting and collective choice. Cambridge University Press, CambridgeGoogle Scholar
- Salop SC (1979) Monopolistic competition with outside goods. Bell J Econ 10(1):141–156CrossRefGoogle Scholar
- Samuelson PA (1947) Foundations of economic analysis. Harvard University Press, CambridgeGoogle Scholar
- Schmalensee R (1978) Entry deterrence in the ready-to-eat breakfast cereal industry. Bell J Econ 9(2):305–327CrossRefGoogle Scholar
- Sen AK (1970) Collective choice and social welfare. Holden-Day/Oliver & Boyd, San Francisco/EdinburghGoogle Scholar
- Stokes DE (1963) Spatial models of party competition. Am Polit Sci Rev 57(2):368–377CrossRefGoogle Scholar
- Suzumura K (1983) Rational choice, collective decisions and social welfare. Cambridge University Press, CambridgeCrossRefGoogle Scholar

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