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Ordinal dominance and risk aversion

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.

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Notes

  1. We index \(M_{u}\)-dominance by \(u\) to highlight the fact that it depends on the cardinal information embedded in vNM utility functions. In contrast, \(P\)-dominance only depends on the agent’s ordinal state-contingent preferences over actions.

  2. In general, when indifference is allowed, for an action to be strictly dominated by a pure action implies that it is \(P\)-dominated, which implies in turn that it is weakly dominated by a pure action. In the generic case in which all state-contingent preferences are strict, these three notions of pure dominance coincide.

  3. Our research is in a Bayesian framework, so we use “risk” and “uncertainty” as synonyms. Nevertheless, our intuition is closely related to the work of Klibanoff (2001). He asks under which conditions would an uncertainty-averse agent be willing to choose mixed actions. As it turns out, the trade-off between averaging outcomes (uncertainty) and increasing variance (risk) plays a prominent role.

  4. These results are similar in spirit to Lemma 1 in Chen and Luo (2012), which implies that, in “concave-like” games, an action is \(M_{u}\)-dominated if and only if it is strictly dominated by a pure action. However, their lemma is interesting only for uncountable environments (including mixed extensions of finite environments). In finite or countable environments—like the ones we consider—if an agent has concave-like preferences, then there exists a pure action which \(P\)-dominates every other action.

  5. Lo (2000) extends this result to all models of preferences satisfying Savage’s P3 axiom.

  6. This result can be traced back to a result from statistical decision theory in Wald (1947). In the context of rationalizability, it is established for finite games in Pearce (1984), and for compact action spaces and continuous preferences in Zimper (2005) and Daniëls (2008).

  7. Details are available upon request.

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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.

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Correspondence to Bruno Salcedo.

A Proofs

A Proofs

Proof

(Proposition 1) Fix an action \(a\in A\) and a state \(x\in X\), and let \(u\) and \(v\) be compatible vNM utility functions such that

$$\begin{aligned} \{ \alpha \in \Delta (A) \,|\, u(\alpha ,x) \ge u(a,x)\} \subseteq \{ \alpha \in \Delta (A) \,|\, v(\alpha ,x) \ge v(a,x)\}. \end{aligned}$$
(11)

We want to show that \(\tau _u(a,x)\ge \tau _v(a,x)\). If \(a\) is either \({\mathbin \succcurlyeq }_x\)-maximum or \({\mathbin \succcurlyeq }_x\)-minimum, then \(\tau _u(a,x)=+\infty \) and \(\tau _v(a,x)=+\infty \) by definition, and the result is trivial. Hence, we assume for the rest of the proof that \(\underline{u}(x)<u(a,x)<\bar{u}(x)\).

By Assumption 1, there exists an action \(b\in A\) such that \(u(b,x) = u^-(a,x)\), and, consequently, \(v(b,x) = v^-(a,x)\). Let \((a_m)\) be a sequence of actions such that \(a_m{\mathbin \succ }_x a\) for all \(m\), \(\lim _{m\rightarrow \infty } u(a_m,x) = \bar{u}(x)\), and \(\lim _{m\rightarrow \infty } v(a_m,x) = \bar{v}(x)\). Also, for each \(\theta \in [0,1]\) and each \(m\in {\mathbb {N}}\), let \(\alpha _{m,\theta }\) be the mixed action that plays \(a_m\) with probability \(\theta \), and \(b\) with probability \(1-\theta \). For all such \(m\) we have that \(u(\alpha _{m,1},x) > u(a,x) > u(\alpha _{m,0},x)\). Hence, since expected utility is continuous in the mixing probabilities, there exists \(\theta (m)\in (0,1)\) such that \(u(\alpha _{m,\theta (m)},x)= u(a,x)\). After some simple algebra this implies that:

$$\begin{aligned} \dfrac{u(a,x)-u^-(a,x)}{u(a_m,x) - u(a,x)} = \dfrac{\theta (m)}{1-\theta (m)}. \end{aligned}$$
(12)

By (11), we have that \(v(\alpha _{m,\theta (m)},x) \ge v(a,x)\), which implies that:

$$\begin{aligned} \dfrac{v(a,x)-v^-(a,x)}{v(a_m,x) - v(a,x)} \le \dfrac{\theta (m)}{1-\theta (m)}. \end{aligned}$$
(13)

Using (12) and (13) and taking limits as \(m\) goes to infinity thus yield the desired result

$$\begin{aligned} \tau _v(a,x) \!=\! \lim _{m\rightarrow \infty } \dfrac{v(a,x)-v^-(a,x)}{v(a_m,x) - v(a,x)} \!\le \! \lim _{m\rightarrow \infty } \dfrac{u(a,x)-u^-(a,x)}{u(a_m,x) - u(a,x)} \!=\! \tau _u(a,x).\qquad \end{aligned}$$
(14)

\(\square \)

Proof

(Lemma 1) Let \(\beta = \alpha (W_x(a))\). Since \(u(b,x)\le u^-(a,x)\) for \(b\in W_x(a)\), and \(u(b,x)\le \bar{u}(x)\) for \(b\in B\), it follows that:

$$\begin{aligned} u(\alpha ,x) - u(a,x)&\le \beta \big (u^-(a,x) - u(a,x)\big ) + \big (1-\beta \big ) \big ( \bar{u}(x) - u(a,x)\big ) \nonumber \\&= -\beta \left( \dfrac{u^-(a,x) - u(a,x)}{\bar{u}(x) - u(a,x) } \right) \big (\bar{u}(x) - u(a,x)\big )\nonumber \\&+ \big (1-\beta \big ) \big (\bar{u}(x) - u(a,x)\big ) \nonumber \\&= \big (1-\beta \cdot (\tau _u(a,x)+1)\big ) \big ( \bar{u}(x) - u(a,x)\big ) \le 0. \end{aligned}$$
(15)

\(\square \)

Proof

(Proposition 2) Fix a set \(B\subseteq A\), an action \(a \in A\backslash P(B)\), and a mixture \(\alpha \) with \(\alpha (B\backslash \{a\})=1\). There exists some \(Y\subseteq X\) conditional on which \(a\) is not weakly dominated in \(B\). Assume without loss of generality that for all \(b \in B\setminus \{a\}\) there exists some \(x\in Y\) such that \(b\not \sim _x a\). This implies that for all \(b \in B\backslash \{a\}\) there also exists some \(x\in Y\) such that \(a{\mathbin \succ }_x b\), i.e., \(B\backslash \{a\} \subseteq \cup _{x\in Y} W_x(a)\). Since \(K = \min \{\Vert A\Vert ,\Vert X\Vert \}<+\infty \), there exists a finite subset \(Z = \{x_1,\ldots ,x_k\}\subseteq Y\) with cardinality \(k\le K\), and such that \(B\backslash \{a\} \subseteq \cup _{x\in Z} W_x(a)\). Therefore:

$$\begin{aligned} \sum _{x\in Z} \alpha (W_x(a)) \ge \alpha (B\backslash \{a\}) = 1 \ge \dfrac{k}{K} = \sum _{x\in Z} \dfrac{1}{K} \ge \sum _{x\in Z} \dfrac{1}{\tau _u(a,x) + 1}. \end{aligned}$$
(16)

This implies that there exists a state \(x\) such that \((\tau _u(a,x) + 1)\alpha (W_x(a))\ge 1\), and the result thus follows from Lemma 1. \(\square \)

Proof

(Proposition 3) Let \(a\), \(B\), \(\alpha ,\) and \(Y\) be as in the proof of Proposition 2. As before, we know that \(B\setminus \{a\} \subseteq \cup _{x\in Y} W_x(a,B)\), and thus:

$$\begin{aligned} \sum _{x\in Y} \alpha (W_x(a,B)) \ge 1 \ge \sum _{x\in X} \dfrac{1}{1+\tau _u(a,x)} \ge \sum _{x\in Y} \dfrac{1}{1+\tau _u(a,x)}. \end{aligned}$$
(17)

Hence, there exists \(x\in Y\) such that \((\tau _u(a,x) + 1)\alpha (W_x(a))\ge 1\), and the result follows from Lemma 1. \(\square \)

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Gafarov, B., Salcedo, B. Ordinal dominance and risk aversion. Econ Theory Bull 3, 287–298 (2015). https://doi.org/10.1007/s40505-014-0059-z

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Keywords

  • Rationalizability
  • Dominance
  • Risk aversion
  • Ordinal preferences
  • Revealed preferences

JEL Classification

  • D81
  • C72