## Abstract

Models of electoral competition between two opportunistic, office-motivated parties typically predict that both parties become indistinguishable in equilibrium. I show that this strong connection between the office motivation of parties and their equilibrium choice of identical platforms depends on two—possibly false—assumptions: (1) Issue spaces are uni-dimensional and (2) Parties are unitary actors whose preferences can be represented by expected utilities. I provide an example of a two-party model in which parties offer substantially different equilibrium platforms even though no exogenous differences between parties are assumed. In this example, some voters’ preferences over the 2-dimensional issue space exhibit non-convexities and parties evaluate their actions with respect to a set of beliefs on the electorate.

This is a preview of subscription content, access via your institution.

## Notes

There is a vast empirical literature on the dimensionality of political issue spaces. This literature was pioneered by Poole and Rosenthal (1985); http://voteworld.berkeley.edu/ links to an extensive array of econometric analysis of political issue spaces and voter preferences in various parts of the world.

In Bade (2010), I described conditions on the set of party beliefs \(\varPsi \), under which games of electoral competition among uncertainty averse parties have equilibria. In that paper I used Gilboa and Schmeidler ’s (1989) maximin expected utilities to model party behavior. The results of that paper transfer to the alternative assumption that party behavior follows Bewley ’s (2002) model.

Barbera et al. (1993) propose the same definition of single-peakedness.

For a proof of this claim see Bade (2010).

In Bade (2005), I provide a characterization of equilibrium sets of games with incomplete preferences.

Suppose \((\mathbf x ,\mathbf y )\) is an equilibrium of \((n,X,\varPsi )\) and there exists no \(\mathbf z \) such that either (\(\mathbf z \ne \mathbf x \) and \(\pi _{\psi }(\mathbf z ,\mathbf y )=\pi _{\psi }(\mathbf x ,\mathbf y ) \) for all \(\psi \in \varPsi \)) or (\(\mathbf z \ne \mathbf y \) and \(\pi _{\psi }(\mathbf x ,\mathbf z )=\pi _{\psi }(\mathbf x ,\mathbf y ) \) for all \(\psi \in \varPsi \)). So for any \(\mathbf z \ne \mathbf x \) there is a prior \(\psi ^*\) such that \(\pi _{\psi ^*}(\mathbf z ,\mathbf y )<\pi _{\psi ^*}(\mathbf x ,\mathbf y )\). Since \(\varPsi \subset \varPsi '\) we have that for all \(\mathbf z \ne \mathbf x \) there exists some \(\psi ^*\in \varPsi \subset \varPsi ^*\) such that \(\pi _{\psi ^*}(\mathbf z ,\mathbf y )<\pi _{\psi ^*}(\mathbf x ,\mathbf y )\). So \(\mathbf x \) is a best reply to \(\mathbf y \) for Party 1 under \((n,X,\varPsi ')\). By the same logic \(\mathbf y \) is a best reply to \(\mathbf x \) under \((n,X,\varPsi ')\) and \((\mathbf x ,\mathbf y )\) is an equilibrium of \((n,X,\varPsi ')\). To see that the conclusion need not hold if for some \(\mathbf z \ne \mathbf x \), \(\pi _{\psi }(\mathbf x ,\mathbf y )= \pi _{\psi }(\mathbf z ,\mathbf y )\) holds for all \(\psi \in \varPsi \) consider the example \((1,[0,1],\{\psi ^*\})\) where \(\psi ^*\) is an electorate with two voters who respectively have their ideal points at 0 and 1. The profile (0, 1) is an equilibrium of \((1,[0,1],\{\psi ^*\})\) and we have \(\pi _{\psi ^*}(0,1)=\pi _{\psi ^*}(1,1)=\frac{1}{2}\). Now consider \((1,[0,1],\{\psi ^*,\psi '\})\) where all voters have their ideal point at .5 according to \(\psi '\). The profile (0, 1) is not an equilibrium in \((1,[0,1],\{\psi ^*,\psi '\})\).

Given that \(\{\psi ^1,\psi ^2\}\) has only two elements it is without loss of generality to assume that \(f(\psi ^1)=\rho ^1\) and \(f(\psi ^2)=\rho ^2\) for some \(\rho ^1,\rho ^2\). I therefore dropped

*f*from the description of the perturbed set of beliefs.Since \(\epsilon \) was chosen such that \(y_i-\epsilon \ge 0\) for \(i=1,2\) we have \((1-y_2+\epsilon ,1-y_1+\epsilon )\in [0,1]^2\).

In this context \(\psi \) has to be interpreted as an electorate. While a party’s problem of maximizing its expected vote share is equivalent to maximizing its vote share according to the expected electorate as discussed in Sect. 2.3 this equivalence does not extend to winning probabilities. A party’s (expected) winning probability does generally not coincide with its winning probability according to the expected electorate.

I would like to thank a referee for this example.

## References

Ahn, D., Choi, S., Gale, D., & Kariv, S. (2014). Estimating ambiguity aversion in a portfolio choice experiment.

*Quantitative Economics*,*5*, 195223.Aragones, E., & Neeman, Z. (2000). Strategic ambiguity in electoral competition.

*Journal of Theoretical Politics*,*12*, 183–204.Bade, S. (2005). Nash equilibrium in games with incomplete preferences.

*Economic Theory*,*26*, 309–332.Bade, S. (2010). Electoral competition with uncertainty averse parties.

*Games and Economic Behavior*,*72*, 12–29.Barbera, S., Gul, F., & Stachetti, E. (1993). Generalized median voter schemes and committees.

*Journal of Economic Theory*,*61*, 262–289.Besley, T., & Coate, S. (1997). An economic model of representative democracy.

*The Quarterly Journal of Economics*,*112*, 85–114.Bewley, T. F. (2002). Knightian decision theory: Part 1.

*Decisions in Economics and Finance*,*25*, 79–110.Brusco, S., Dziubinski, M., & Roy, J. (2012). The Hotelling–Downs model with runoff voting.

*Games and Economic Behavior*,*74*, 447–469.Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity.

*Journal of Risk and Uncertainty*,*5*, 325–370.Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., & Montrucchio, L. (2011). Uncertainty averse preferences.

*Journal of Economic Theory*,*146*, 1275–1330.Davis, O. A., de Groot, M. H., & Hinich, M. J. (1972). Social preferences ordering and majority rule.

*Econometrica*,*40*, 147–157.Downs, A. (1957).

*An economic theory of democracy*. New York: HarperCollins.Gilboa, I., & Schmeidler, D. (1989). Maxmin expected utility with non-unique prior.

*Journal of Mathematical Economics*,*18*, 141–153.Glazer, A. (1990). The strategy of candidate ambiguity.

*American Political Science Review*,*84*, 237–241.Grandmont, J.-M. (1978). Intermediate preferences and the majority rule.

*Econometrica*,*46*, 317–330.Halevy, Y. (2007). Ellsberg revisited: An experimental study.

*Econometrica*,*75*, 503–536.Hotelling, H. (1929). Stability in competition. The Economic Journal,

*39*(153), 41–57. doi:10.2307/2224214.Klibanoff, P., Marinacci, M., & Mukerji, S. (2005). A smooth model of decision making under ambiguity.

*Econometrica*,*73*, 1849–1892.Krasa, S., & Polborn, M. (2012). Political competition between differentiated candidates.

*Games and Economic Behavior*,*76*, 249–271.Levy, G. (2004). A model of political parties.

*Journal of Economic Theory*,*115*, 250–277.Meirowitz, A. (2005). Informational party primaries and strategic ambiguity.

*Journal of Theoretical Politics*,*17*, 107–136.Osborne, M. J., & Slivinsky, A. (1996). A model of political competition with citizen-candidates.

*The Quarterly Journal of Economics*,*111*, 65–96.Palfrey, T. (1984). Spatial equilibrium with entry.

*Review of Economic Studies*,*51*, 139–156.Plott, C. R. (1967). A notion of equilibrium and its possibility under majority rule.

*American Economic Review*,*57*, 787–806.Poole, K., & Rosenthal, H. (1985). A spatial model for legislative roll call analysis.

*American Journal of Political Science*,*29*, 357–384.Roemer, J. E. (1999). The democratic political economy of progressive income taxation.

*Econometrica*,*67*, 1–19.Schmeidler, D. (1989). Subjective probability and expected utility without additivity.

*Econometrica*,*57*, 571–587.Wittman, D. A. (1973). Parties as utility maximizers.

*The American Political Science Review*,*67*, 490–498.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

I would like to thank Jean Pierre Benoit, Herve Cres, Kfir Eliaz, Michael Mandler, Efe Ok, Ronny Razin and Ennio Stacchetti for helpful comments.

## Rights and permissions

## About this article

### Cite this article

Bade, S. Divergent platforms.
*Theory Decis* **80, **561–580 (2016). https://doi.org/10.1007/s11238-015-9515-y

Published:

Issue Date:

DOI: https://doi.org/10.1007/s11238-015-9515-y

### Keywords

- Downs model
- Games with incomplete preferences
- Platform divergence
- Knightian uncertainty
- Uncertainty aversion