# Ordinal dominance and risk aversion

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## Abstract

We find that, for sufficiently risk-averse agents, strict dominance by *pure or mixed* actions coincides with dominance by *pure* actions in the sense of (Börgers in Econometrica 61(2):423–430, 1993), which, in turn, coincides with the classical notion of strict dominance by pure actions when preferences are asymmetric. Since risk aversion is a *cardinal* feature, all finite single-agent choice problems with *ordinal* preferences admit compatible utility functions which are sufficiently risk averse as to achieve equivalence between pure and mixed dominance. This result extends to some infinite environments.

## Keywords

Rationalizability Dominance Risk aversion Ordinal preferences Revealed preferences## JEL Classification

D81 C72## Notes

### Acknowledgments

This paper originated from a conjecture by Edward Green. We are thankful for his guidance and support, as well as the useful comments from Lisa Posey, Nail Kashaev, Lidia Kosenkova, Jonathan Weinstein, two anonymous referees, and the attendants of the 2014 Spring Midwest Trade and Theory Conference at IUPUI, and the 25\(\mathrm {th}\) International Game Theory Conference at Stony Brook University. We gratefully acknowledge the Human Capital Foundation, (http://www.hcfoundation.ru/en/) and particularly Andrey P. Vavilov, for research support through the Center for the Study of Auctions, Procurements, and Competition Policy (http://capcp.psu.edu/) at the Pennsylvania State University. All remaining errors are our own.

## References

- Bernheim, D.: Rationalizable strategic behavior. Econometrica
**52**(4), 1007–1028 (1984)zbMATHMathSciNetCrossRefGoogle Scholar - Bohner, M., Gelles, G.M.: Risk aversion and risk vulnerability in the continuous and discrete case. Decis. Econ. Financ.
**35**(1), 1–28 (2012)zbMATHMathSciNetCrossRefGoogle Scholar - Bonanno, G.: A syntactic approach to rationality in games with ordinal payoffs. In: Bonanno, G., Van Der Hoek, W., Wooldridge, M. (eds.) Logic and the foundatons of game and decision theory (LOFT 7), Texts in logic and games, vol. 3, pp 59–85. Amsterdam University Press (2008)Google Scholar
- Börgers, T.: Pure strategy dominance. Econometrica
**61**(2), 423–430 (1993)zbMATHMathSciNetCrossRefGoogle Scholar - Chambers, C.P., Echenique, F., Shmaya, E.: General revealed preference theory, Social Science Working Paper 1332, California Institute of Technology (2010)Google Scholar
- Chen, Y., Luo, X.: An indistinguishability result on rationalizability under general preferences. Econ. Theory
**51**(1), 1–12 (2012)MathSciNetCrossRefGoogle Scholar - Daniëls, T.: Pure strategy dominance with quasiconcave utility functions. Econ. Bull.
**2008**(3), 54–61 (2008)Google Scholar - Epstein, L.G.: Preference, rationalizability and equilibrium. J. Econ. Theory
**73**(1), 1–29 (1997)zbMATHCrossRefGoogle Scholar - Klibanoff, P.: Characterizing uncertainty aversion through preference for mixtures. Soc. Choice Welf.
**18**(2), 289–301 (2001)zbMATHMathSciNetCrossRefGoogle Scholar - Ledyard, J.O.: The scope of the hypothesis of Bayesian equilibrium. J. Econ. Theory
**39**(1), 59–82 (1986)zbMATHMathSciNetCrossRefGoogle Scholar - Lo, K.C.: Rationalizability and the Savage axioms. Econ. Theory
**15**(3), 727–733 (2000)zbMATHCrossRefGoogle Scholar - Pearce, D.: Rationalizable strategic behavior and the problem of perfection. Econometrica
**52**(4), 1029–1050 (1984)zbMATHMathSciNetCrossRefGoogle Scholar - Rockafellar, R.T.: Convex analysis, Princeton landmarks in mathematics and physics, vol. 28, Princeton University Press (1996)Google Scholar
- Wald, A.: An essentially complete class of admissible decision functions. Ann. Math. Stat.
**18**(4), 549–555 (1947)zbMATHMathSciNetCrossRefGoogle Scholar - Weinstein, J.: The dependence of solution concepts on risk attitude, mimeo (2014)Google Scholar
- Zimper, A.: Equivalence between best responses and undominated strategies: a generalization from finite to compact strategy sets. Econ. Bull.
**3**(7), 1–6 (2005)Google Scholar