Uncertainty Aversion in Game Theory: Experimental Evidence

Journal of Economic Behavior and Organization 176 (2020) 720–734

Contents lists available at ScienceDirect

Journal of Economic Behavior and Organization

journal homepage: www.elsevier.com/locate/jebo

R Uncertainty aversion in game theory: Experimental evidence

Evan M. Calford a,b a Research School of Economics, Australian National University, Australia b Department of Economics, Krannert School of Management, Purdue University, USA

a r t i c l e i n f o a b s t r a c t

Article history: This paper studies, using a laboratory experiment, the effects of uncertainty aversion (the Received 14 October 2019 union of risk aversion and ambiguity aversion) on behavior in a normal form game. We

Revised 9 June 2020 isolate and identify two components of uncertainty aversion in games: the effect of an Accepted 10 June 2020 agent’s own uncertainty preferences and effect of the agent’s beliefs regarding their opponent’s uncertainty preferences. Uncertainty preferences are correlated with behavior in JEL classiﬁcation: games. Induced beliefs about the risk preferences of an opponent have a larger effect on C92 strategic behavior than induced beliefs about an opponent’s ambiguity preferences, al- C72 though both components have a signiﬁcant effect on behavior. The results support the hy- D81 pothesis that strategic uncertainty is an important determinant of strategic behavior, and

D83 that the response to strategic uncertainty is modulated by subject uncertainty attitudes. Keywords: ©2020 The Author(s). Published by Elsevier B.V. Ambiguity aversion This is an open access article under the CC BY-NC-ND license.

Experimental economics ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) Game theory Uncertainty preferences

1. Introduction

Strategic uncertainty is a ubiquitous feature of human interaction. Whether playing rock-paper-scissors, taking a penalty shot in soccer, playing poker, negotiating a pay raise or purchasing a house, certainty about an opponent’s strategy is a rarity. A wealth of experiments on individual decision making have found that human subjects have non-neutral attitudes towards uncertainty, with a majority of subjects typically displaying either risk or ambiguity aversion. 1 Therefore, understanding the relationship between preferences over uncertainty and strategic behavior is necessary for understanding human behavior in games. There is, however, a relative paucity of experimental studies on the role of risk and ambiguity aversion in game theoretic environments, perhaps reﬂecting the minor role of uncertainty aversion in the game theory literature (see Section 1.2 for a literature review). This paper asks two fundamental questions. Are there differences in behavior in strategic settings between people who are uncertainty averse and those who are not? And, do people change their behavior in strategic settings when

R I am particularly indebted to Yoram Halevy, who introduced me to the concept of ambiguity aversion and guided me throughout this project. Ryan Oprea provided terriﬁc support and advice throughout. Special thanks to Mike Peters and Wei Li are also warranted for their advice and guidance. I also thank Li Hao, Terri Kneeland, Chad Kendall, Tom Wilkening, Guillaume Frechette, Simon Grant and Peter Wakker for helpful comments and discussion. Funding from the University of British Columbia Faculty of Arts is gratefully acknowledged. E-mail address: [email protected]

1 See Harrison and Rutström (2008) for a survey on risk aversion, and section 13.4.2 of Machina and Siniscalchi (2013) for a survey on Ellsberg urn experiments used to measure ambiguity preferences. https://doi.org/10.1016/j.jebo.2020.06.011 0167-2681/© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734 721 they know that their opponent is uncertainty averse? To answer these questions, this paper studies the relationship between uncertainty preferences, beliefs over the uncertainty preferences of others, and behavior in normal form games. This is the first paper to provide experimental evidence on the causal effect of beliefs over uncertainty preferences in games. To avoid potential confusion, given a multiplicity of terminology in the literature, it is important to clarify some terminology. This paper uses the term uncertainty to refer to the union of risk (uncertainty associated with explicit probabilities) and ambiguity (uncertainty without explicit probabilities). Strategic uncertainty, analogously, consists of both strategic risk (uncertainty regarding the realization of an opponent’s mixed strategy) and strategic ambiguity (uncertainty regarding which mixed strategy an opponent may play). We conduct two treatments using a between-subject design. In the first, we elicit the uncertainty preferences of subjects before having the same subjects play a normal form game. To control for the possibility of domain-specific uncertainty preferences we embed the preference measurement tasks into games, and we control for subject understanding via an extensive set of incentivized comprehension questions. In the second treatment we induce beliefs about uncertainty preferences, by showing each subject his opponent’s decisions in the preference measurement tasks, before asking him to play against the same opponent in a normal form game. The payoff structure of our normal form game is critical to the identification strategy used in this paper. The set of rationalizable strategies for the row player varies with the row player’s uncertainty preferences but not row player beliefs about the preferences of the column player. Conversely, the set of rationalizable strategies for the column player varies with the column player’s beliefs about row player uncertainty preferences but not the column player’s uncertainty preferences. Using our measurement of preferences in the first treatment, and induced beliefs in the second treatment, we are able to observe the effects of risk preferences, ambiguity preferences, and beliefs about risk preferences and ambiguity preferences on behavior in games. The results can be summarized as follows:

• Uncertainty preferences, incorporating both risk and ambiguity preferences, are correlated with behavior as the row player in our normal form game. • Beliefs about opponents’ uncertainty preferences causally affect behavior as the column player in our normal form game. • Beliefs about opponents’ risk preferences have a much larger effect on behavior than beliefs about opponents’ ambiguity preferences.

A corollary of our main results is that previous work estimating risk preferences from behavior in games ( Goeree et al. (2003) , for example) may overestimate risk aversion because of an omitted variable bias stemming from unmeasured strategic ambiguity. This is consistent with the observation that risk aversion measured in games is typically greater than risk aversion measured in lottery tasks. The results presented here are broadly consistent with several solution concepts for uncertainty averse agents in the existing literature, including a generalized notion of rationalizability proposed by Epstein (1997) . The experiment was not, however, designed to test these solution concepts directly and we do not attempt to reject specific solution concepts: theory generates a large multiplicity of equilibrium predictions, making direct tests of the theory difficult. Instead, we hope that the results presented here will be helpful in the development and refinement of game theory with non-expected utility agents.

1.1. An illustrative game

We illustrate the role of uncertainty preferences in determining behavior using the game presented in Fig. 1 as an example, where we assume that all relevant payoff information, including risk and ambiguity preferences, is common knowledge. Intuitively the game models a situation where the column player wishes to predict whether the row player will play their safe strategy, C , or not. The row player may either attempt to predict the column player’s behavior or to take their safe option. Note that the strategic uncertainty in this game is not generated by uncertainty regarding which of multiple Nash equilibrium should be played; there is a unique pure strategy Nash equilibrium. The uncertainty is not generated by uncertainty regarding the realization of an opponents mixed strategy, nor is it generated by uncertainty regarding the type or preferences of the opponent. Instead, the uncertainty arises endogenously as a consequence of two agents with common knowledge of each others uncertainty preferences reasoning about behavior in the game. We now establish the intuition behind the strategic implications of this game in two steps, while a more formal discussion is provided in Section 2.2 .

Fig. 1. Testing game, with the proportion of subjects choosing each strategy in the own preference treatment in parentheses. N = 116 , see Section 3 for subject inclusion criteria. 722 E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734

We begin with an observation: strategy C is justifiable for the row player if and only if the row player is sufficiently uncertainty averse. To see this, consider that for a risk and ambiguity neutral row player, A is a best response for any belief such that the column player plays X at least half of the time, and B is the best response for any belief such that the column player plays Y at least half of the time. C is not a best response for any probabilistic beliefs held by the row player. If the row player is risk averse and ambiguity neutral with a Bernoulli utility function such that 2 u (18) > u (25) + u (14) then the row player can justify playing C with the belief that their opponent will play X and Y with equal probabilities. If the row player is sufficiently ambiguity averse and risk neutral then C may also be justified. To illustrate, suppose that the row player has Maxmin Expected Utility preferences ( Gilboa and Schmeidler, 1989 ) and has complete uncertainty about their opponent’s choice of strategy. When evaluating A the row player will consider the worst case scenario, of the column player playing Y with probability 1, which leads to a payoff of 14. Similarly, strategy B also earns 14. C is the best response to this set of beliefs, earning a payoff of 18. Therefore, a row player who is either ambiguity averse or sufficiently risk averse can justify playing strategy C . For the column player, we observe that Y is a best response to a justifiable strategy if and only if the row player is sufficiently uncertainty averse. Clearly, Y is a best response to a justifiable strategy if C is justifiable. If C is not justifiable, then X is the unique best response to any justifiable strategy. Importantly, this argument does not depend on the uncertainty preferences of the column player –all that is required is the standard assumption that the column player prefers more money to less money. We have set identification of the relationship between uncertainty preferences and behavior: A is always justifiable for the row player, and for the column player X is always a best response to a justifiable strategy. This is not a design flaw, but rather an unavoidable feature of uncertainty preferences in games: uncertainty aversion increases the set of strategies that an agent may reasonably be expected to play. 2 It is precisely this feature of uncertainty preferences in games that makes the question of behavior in games under strategic uncertainty an empirical, and experimental, one. Even if all players are strongly uncertainty averse, it is plausible that strategic uncertainty might have no effect on behavior given that the Nash equilibrium of ( A, X ) appears intuitively reasonable. The results in Fig. 1 show that while ( A, X ) is the most common outcome, it is expected to be observed less than half the time among randomly paired row and column subjects.

1.2. Related literature

There are three strands of literature that are directly relevant to this paper. First, there is a theoretical literature on the role of ambiguity aversion and strategic ambiguity in determining behavior in games. These papers can be classified by their treatment of mixed strategies. Dow and Werlang (1994) , Eichberger and Kelsey (20 0 0) , Eichberger and Kelsey (2014) , Groes et al. (1998) , Grant et al. (2016) , Lo (2009) and Marinacci (2000) all restrict agents to playing only pure strategies while using a belief interpretation of mixing. 3 On the other hand Azrieli and Teper (2011) , Bade (2011) , Lo (1996) , Klibanoff (1996) , and Riedel and Sass (2014) explicitly allow agents to play either mixed or ambiguous strategies. 4 Second, there is a smaller but growing experimental literature on the role of ambiguity aversion and strategic uncertainty in determining behavior in games. Camerer and Karjalainen (1994) is the first paper in this literature and provides evidence that subjects, on average, prefer to avoid strategic uncertainty by betting on known probability de- vices rather than on other subjects’ choices. Relatedly, Bohnet and Zeckhauser (2004) find that subjects are more willing to play trust in the trust game when their opponent’s decision is replaced by an objective randomization device. Eichberger et al. (2008) establish that subjects find “grannies” to be a greater source of strategic ambiguity than game theorists. Kelsey and le Roux (2015) find that subjects exhibit higher levels of ambiguity aversion in games than in a 3-color Ellsberg urn task. Kelsey and le Roux (2017) also find a majority of subjects to be ambiguity averse in public goods and weakest link games, although do not find a significant difference in ambiguity attitudes for subjects playing against local or foreign opponents. In related work, Kelsey and le Roux (2018) find that strategic uncertainty has contrasting effects in games with strategic complements and strategic substitutes. Li et al. (2019) apply an insight from Baillon et al. (2018) to separate ambiguity attitudes from (non-additive) beliefs, and find that both ambiguity preferences and beliefs are correlated with behavior in a trust game. Ivanov (2011) estimates ambiguity preferences from behavior in games (rather than eliciting preferences and then studying behavior in games as is the case here). Both Li et al. (2019) and Ivanov (2011) elicit beliefs over opponent’s strategies, in contrast to the current paper which induces beliefs over opponent’s preferences. Finally, this paper presents evidence on the correlation between risk and ambiguity preferences; a topic where the previous evidence has been mixed. Some papers report a positive correlation between risk and ambiguity preferences.

2 Weinstein (2016) establishes that the set of rationalizable strategies (weakly) increases as risk aversion increases. Battigalli et al. (2016) establishes, using the smooth ambiguity aversion model ( Klibanoff et al., 2005 ), that the set of justiﬁable strategies increases as either ambiguity aversion or risk aversion increases. Calford (2020) establishes that the set of rationalizable strategies (weakly) increases as preference for randomization decreases for agents with Saito (2015) preferences.

3 The equilibrium concepts in these papers allow for any outcome to be observed in equilibrium in the game in Fig. 1 .

4 The equilibrium concepts in these papers support only { A, X } as an equilibrium in the game in Fig. 1 . The row player can use a mixed strategy to build a hedge against ambiguity. In equilibrium, they do not face any ambiguity and do not play a mixed strategy –the existence of the potential to hedge against ambiguity is enough to prevent multiple equilibria arising. E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734 723

Fig. 2. Classiﬁcation game 1. This game is used to measure the row player’s ambiguity aversion.

Abdellaoui et al. (2011) presents evidence of strong correlations between ambiguity preferences and non-expected utility risk preferences, while Bossaerts et al. (2010) find a positive correlation between risk and ambiguity preferences in the context of asset markets. Dean and Ortoleva (2019) also find a positive correlation between risk and ambiguity preferences in a laboratory experiment. Other papers find domain specific correlations between risk and ambiguity preferences. Lauriola and Levin (2001) find a positive correlation, and report that the correlation was particularly strong for risks in the loss domain rather than the gain domain. Conversely, Chakravarty and Roy (2009) find a positive correlation in the gain domain but no correlation in the loss domain. Cohen et al. (1987) do not report correlation coefficients, but find that subjects treat risk and ambiguity in a similar fashion in the loss domain but not the gain domain. Yet other papers find little or no correlation between risk and ambiguity preferences. Curley et al. (1986) find no evidence of a correlation between risk and ambiguity preferences. Cohen et al. (2011) find no correlation between risk and ambiguity attitudes in a student population, and find a correlation only for subjects with extreme risk and ambiguity attitudes in a general population. Di Mauro and Maffioletti (2004) find statistically significant correlation coefficients only for extreme probabilities, but not for intermediate probabilities.

2. Experimental design

The experiment consists of three games: the testing game introduced in Section 1.1 and two measurement games described below. The measurement games are used to measure subject preferences, and the testing game is then used to test the mapping from preferences to strategic behavior.

2.1. The measurement games

There are two measurement games. The first game elicits ambiguity preferences, while the second game elicits risk preferences. In each game the row player selects between a set of prospects, whose payoffs depend only on exogenous random events, while the column player earns a positive payoff if and only if she correctly predicts the row player’s choice. The measurement tasks are presented as games to ensure that all of the subjects’ decisions were presented in the same frame throughout the experiment. However, the source of uncertainty differs between the measurement and testing games: the sources of risk and ambiguity in the measurement games are urns, and not other subjects. Using different sources of uncertainty, but the same frame, across measurement and testing games allows this paper to use measurement tasks that are isomorphic to tasks commonly used in individual decision making experiments and therefore provides a point of comparison to the previous literature. Note that, because of the differing nature of risk and ambiguity preferences, the measurement games provide differing amounts of information about the preferences of subjects. Under standard models, where risk preferences are determined by the curvature of the underlying utility function, the degree of risk aversion should remain fixed across the measurement and testing games. In contrast, the observed degree of ambiguity aversion will not necessarily be constant across games: the expression of ambiguity preference depends on both ambiguity attitudes (whether the subject is averse to or loves ambiguity) and ambiguity perception (how much ambiguity a subject perceives in a given environment). 5 It is reasonable to assume that ambiguity attitude should remain fixed across the measurement and testing games, but ambiguity perception may not. We therefore use only a binary measure of ambiguity preference, and do not attempt to measure the degree of ambiguity aversion. We are primarily interested in the row player behavior in the measurement games; the column player is a necessary consequence of converting the preference measurement task into a game form. 6 The first (ambiguity) measurement game is shown in Fig. 2 . For the row player, this game is isomorphic to a standard Ellsberg task with a slight asymmetry in payoffs as recommended by Epstein and Halevy (2018) . The payoff asymmetry ensures that an ambiguity neutral subject has a strict preference to play S . 7 The game involves two ball draws, one from

5 See Baillon et al. (2018) for a discussion of the distinction between ambiguity attitudes and perception.

6 Column player results are available in earlier versions, and the full data set is available in the supplementary material for this article.

7 An implication of this is that agents with extremely slight ambiguity aversion may be erroneously classiﬁed as ambiguity neutral. This is superior to having the expected large number of ambiguity neutral subjects be indifferent between the two options. See Epstein and Halevy (2018) for further discussion. 724 E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734

Fig. 3. U urn. The U urn consists of 10 balls, each of which may be either red or black. The total number of red balls in the urn lies between 0 and 10.

Fig. 4. K urn. The K urn contains 5 red and 5 black balls.

Fig. 5. Classification game 2. This game is used to measure the row player’s risk aversion. the U urn (which contains red and black balls in unknown proportion, as depicted in Fig. 3 ) and one from the K urn (which contains red and black balls in equal proportion, as depicted in Fig. 4 ). 8 Therefore there are four possible states of nature, but only two payoff tables. The left payoff table represents the state red ball drawn from the U urn and black ball drawn from the K urn: ( R U , B K ). The right payoff table represents the state ( B U , R K ). The payoffs for state ( R U , R K ) are found by adding the two payoff tables together, and the payoffs in state ( B U , B K ) are identically 0 for both players. The relationship between the balls drawn and payoffs was carefully explained to the subjects, and understanding was tested via a series of comprehension questions that are discussed further in Section 2.4 . Given that the row player’s payoffs are independent of the column player’s strategy choice, we can view the row player as facing a choice between a bet that pays $30.10 if a red ball is drawn from the U urn and a bet that pays $30 if a red ball is drawn from the K urn. We assume that subjects hold symmetric beliefs about the distribution of balls in the U urn. 9 If a subject has Subjective Expected Utility (SEU) preferences, then she should strictly prefer strategy S (the bet on the U urn). A subject with strict ambiguity averse preferences should prefer strategy M (the bet on the K urn). 10 We note that because the row player is indifferent to her opponent’s strategy, the existence of the column player should have no effect on the row player’s choices. The second classification game is shown in Fig. 5 and has a very similar structure to the first classification game, with the key difference being that the state is now determined by a single draw from the K urn. The row player chooses which

8 Following convention, we refer to the balls as being either red or black. In the experiment, the colors used were red and yellow. Subjects were informed that the U urn had been composed by an economics graduate student who had no knowledge of the design of the experiment and that the experimenter had no control over, nor knowledge of, the composition of the urn. Subjects were also informed that, at the end of the experiment, they would have the option of observing the contents of the U urn. Few subjects exercised this option.

9 Note that this is an assumption regarding the symmetry of beliefs, and not an assumption on the subject’s utility of drawing either color of ball. Chew et al. (2017) state that “the empirical validity of the symmetry assumption has been supported” in their own data as well as the data of Abdellaoui et al. (2011) and Epstein and Halevy (2018) . Further, violations of this assumption would lead to an attenuation against ﬁnding a relationship between behavior in the measurement games and the testing game, implying that the assumption is conservative relative to potential design alternatives.

10 With the exception, noted in footnote 7, that a subject with extremely slight ambiguity aversion may prefer S. E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734 725 risky prospect they would like to hold, and the column player attempts to predict the row player’s preferences. Strategy L has both the highest expected return and highest variance, while strategy H provides a lower but certain payoff. Strategy I has an intermediate expected return and variance. 11

2.2. Behavioral relationships across games

We require some assumptions about preferences to formalize the relationship between behavior in the measurement games and strategies in the testing game. We parametrize the utility function using the Constant Absolute Risk Aversion (CARA) functional form. 12 For ambiguity preferences, as discussed above, we take a binary approach: we classify each subject as being either ambiguity neutral or ambiguity averse and we focus our attention on the Maxmin Expected Utility (MEU) model of Gilboa and Schmeidler (1989). Suppose that the set of strategies for players i and j are denoted by Ai and Aj , = { , , } = { , } ∈ respectively. For the testing game Ai A B C and A j X Y . In general, preferences for player i playing a strategy ai Ai can be written as: ( ) = ( ( , )) φ ( ) U ai minχ ui mR ai a j i a j (1) φ ∈ i i ∈ a j A j

−ρm where m ( a , a ) denotes the monetary prize for player i when the outcome ( a , a ) occurs, u (m ) = 1 − e for ρ > 0, φ i i j χ i j i ⊆ ( ) denotes a probabilistic belief over Aj and i A j is a closed and convex set of probabilistic beliefs. We also deﬁne the conditional preferences ( | ) = ( ( , )) φ ( ) U ai A j minχ ui mR ai a j i a j φ ∈ i i ∈ a j A j

⊆ , which is the utility of player i playing strategy ai when player j is restricted to only the strategies in A A j where each χ j φ ∈ ∈ φ ∈ ( ) i i is naturally restricted to place positive weight only on a j A j (i.e. i A j ). Our treatment of rationalizability is based on Epstein (1997) who provides a general treatment of rationalizability for non-expected utility agents, including MEU agents. Here, we characterize uncertainty preferences by the pair ( ρ, χ) where χ χ χ ∈ { MEU, SEU }. Let P be the set of feasible for a player of type χ. Note that, in the testing game, any closed and convex set of beliefs for the row player can be written as = { φ(X ) : a ≤ φ(X ) ≤ b} for some 0 ≤ a ≤ b ≤ 1. SEU ∈ P SEU P SEU = { ≤ = ≤ We can therefore define the feasible set of beliefs for each type of player as R R where R : 0 a b } , MEU ∈ P MEU P MEU = { ≤ ≤ ≤ } ρ∗ ρ ( ) + 1 and R R where R : 0 a b 1 . Further, we write to denote the value of such that uR 25 13 u R (14) = 2 u R (18) . χ χ ρ χ ∈ P ∈ Definition 1. A strategy, ai , is justifiable for player i of preference type ( , ) if there exists i i such that ai arg max ∈ U(a ) . a Ai Definition 2. Fix a preference type ( ρ, χ) for each player. The set of rationalizable strategy profiles is the largest set χ χ R × R ⊆A × A with the property: For each a ∈ R there exists ∈ P such that a ∈ arg max ∈ U(a | R ) . i j i j i i i i i a Ai j This definition of justifiability is the standard one. A strategy is justifiable if there exists a belief such that the strategy is a best response to that belief. The definition for rationalizability is also standard. Note, however, that the underlying preferences are not standard. We follow the interpretation of Epstein (1997) that an MEU agent entertains all closed and MEU ∈ P MEU 14 convex belief sets (i.e. the agent considers all i i in the current notation). An implication is that there is no scope for restricting the degree of ambiguity aversion of an MEU agent: an MEU agent is ambiguity averse, and may hold any closed and convex set of beliefs regarding opponent behavior. Restrictions on P MEU would restrict not only the degree of ambiguity aversion exhibited by the agent but also the beliefs of the agent about opponent behavior. That restrictions on beliefs are not appropriate is apparent when considering the analagous subjective expected utility case. Note also that, following the discussion in subsection 2.1 , even if such restrictions were admitted, there is no basis upon which to infer the degree of ambiguity aversion in a game given the degree of ambiguity aversion exhibited in Ellsberg urn tasks because the degree of ambiguity perceived in the testing game may differ substantially from that perceived by the same subject in the measurement game. We now state a series of lemmas that establish the relationship between behavior in the measurement games and the set of justifiable strategies in the testing game. Proofs are relegated to an appendix.

Lemma 1. Consider the normal form game in Fig. 1 . C is justiﬁable for the row player if and only if ρ ≥ ρ∗ or χ = MEU.

11 Both the risk measurement game and the testing game have a completely safe strategy, thereby limiting any bias caused by an excessive attraction towards certainty that might be introduced by including a safe strategy in only one case.

12 This functional form assumption is not critical. We could, for example, assume Constant Relative Risk Aversion (CRRA) without affecting the conclusions of the paper. The CRRA formulation is available in an online appendix, and the results are qualitatively the same as presented here. ∗ 13 This implies that ρ ≈ 0.10, see the proof of Lemma 2 .

14 In Appendix A we consider an alternative speciﬁcation of preferences, the -contamination structure also considered by Epstein (1997) . 726 E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734

Table 1 A summary of the relationship between choices in the classiﬁcation games and the set of rationalizable strategies in the own preference treatment. Calculations are provided in Appendix A .

Row Player Strategies S M

L { A } { A, B, C } H { A, B, C } { A, B, C }

Lemma 1 establishes the justifiable set for our testing game as a function of preferences. To finalize our behavioral hypothesis we require a mapping from the revealed preferences of the risk classification game to the antecedents of Lemma 1 .

Lemma 2. If u (25) + u (10) ≥ max{ u (23) + u (11 ) , 2 u (15) } then ρ < ρ∗. If 2 u (15) ≥ max{ u (23) + u (11 ) , u (25) + u (10) } then ρ > ρ∗.

If a subject chooses L in the risk measurement game then ρ < ρ∗. If a subject chooses H in the risk measurement game then ρ > ρ∗. For subjects who choose I in the risk measurement game, ρ may be either greater or less than ρ∗. Thus, we drop subjects who chose I from the main presentation of the results: we do not have a clear hypothesis on their behavior in the testing game. While we have assumed CARA utility functions—which have the desirable property that the proposition would still hold if we assumed that subjects integrated laboratory earnings into their lifetime wealth—in deriving these results similar results hold if we use, for example, CRRA utility. 15 Finally, we extend the results from justiﬁability to rationalizability. Here, we treat rationalizability as requiring rationality and common knowledge of rationality where rationality is deﬁned with respect to the preferences of the agent. For a SEU agent this is the well known Bernheim (1984) and Pearce (1984) rationalizability. Epstein (1997) provides a framework for generalizing Pearce-Bernheim rationalizability to agents with non-neutral ambiguity preferences. In both cases the set of rationalizable strategies can be found by iterated elimination of never-best responses.

Lemma 3. Consider the normal form game in Fig. 1 . B is rationalizable for the row player if and only if C is justiﬁable for the row player.

We impose one additional behavioral assumption: that row players will not play a non-rationalizable strategy. This assumption generates the testable implication that a subject who plays L and S in the measurement games should not play B in the testing game, and this prediction is borne out in the data. 16 We summarize the expected relationship between behavior in the measurement and testing games in Table 1 , with each entry in the table following directly from the three Lemmas above. For subjects who either exhibit high levels of risk aversion or exhibit ambiguity aversion all strategies are justiﬁable (and also rationalizable) in the testing game. For subjects who exhibit low levels of risk aversion and exhibit ambiguity neutrality, strategy C is not justiﬁable and A is the unique rationalizable strategy in the testing game.

2.3. Treatments

There are two treatments, conducted between-subject. In the own preference treatment subjects played the two measurement games ﬁrst, and then played the testing game. Each subject played each game twice: once as the row player and once as the column player. 17 The games were transposed so that subjects always viewed the game from the row perspective. The own preference treatment allows us to infer risk and ambiguity preferences from behavior in the measurement games, and correlate preferences with behavior in the testing game. Following from the discussion in Section 1.1 and analysis in Section 2.1 , we are interested in the behavior of row players in the own preference treatment. The other preference treatment was conducted after the own preference treatment. Each new subject was matched with a single “opponent”, with the opponent being the pre-recorded behavior of a subject from the own preference treatment. 18 Each subject observed the opponent’s behavior in either one or both of the measurement games, before playing as the column player in the testing game against the same opponent. The subject’s payment is then determined by matching his strategy with the (pre-recorded) strategy of his opponent. Subjects in the other preference treatment play only a single game (the testing game as the column player).

15 I thank Simon Grant for highlighting, in private correspondence, this desirable feature of CARA preferences.

16 Restricting to subjects who passed the comprehension tasks, as done throughout Section 3 , only 3 of the 57 subjects who play both L and S also play B .

17 Subjects also played one additional game that is discussed in earlier versions of this paper, and therefore made a total of seven decisions.

18 In the instructions to subjects, the opponent was referred to as the counterpart. Opponents were chosen at random, conditional on observable information. Subjects were told that they were being matched with someone “who participated in a similar experiment last year, at which time we recorded [their] decisions.” E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734 727

In the other preference treatment we assume that the observation of the opponent’s behavior in the measurement games induces beliefs about the opponent’s uncertainty preferences. Column player behavior in the testing game is then interpreted as identifying the effect of beliefs over an opponent’s uncertainty preferences on strategic behavior. From Table 1 we conclude that a subject, in the other preference treatment, who observes his opponent play L and S in the measurement games should expect his opponent to play A in the testing game. The subject is, therefore, expected to play X as the column player in the testing game. If the subject observes anything other than L and S in the measurement games then he may player either X or Y in the testing game.

2.4. Auxiliary design features

In both treatments subjects were restricted to selecting pure strategies in each of the games. While this is a standard design choice in game theory experiments, it is not an innocuous restriction for ambiguity averse subjects –it constrains the set of theoretical models which are compatible with the experimental design to those which allow only pure strategies to be played (see Section 1.2 ). Restricting subjects to only submit pure strategies prevents subjects from using mixed strategies to construct a hedge against their subjective strategic ambiguity. There remains, however, the possibility that subjects might have implemented a mixed strategy by randomizing privately prior to selecting a pure strategy. If such pre-play randomization were to provide a hedge against strategic uncertainty then the theoretical analysis above does not go through. This does not appear to be the case, for two main reasons. First, if pre- play randomization were used as a hedge then strategy C is dominated in the testing game. However, as Fig. 1 shows, more than one quarter of subjects played C . Second, the prior literature suggests that an objective randomization that resolves prior to the resolution of subjective uncertainty need not provide a hedge against the subjective uncertainty. Eichberger and Kelsey (1996) establishes that in a Savage-style framework (distinct from the Anscombe–Aumann framework where subjective uncertainty is resolved prior to objective uncertainty) that there is not a strict preference for randomization, and they outline the implications for game theory with ambiguity averse agents. More recently, Johnson et al. (2019) and Baillon et al. (2019) demonstrate, in the context of individual decision making, that randomization need not hedge uncertainty when the objective randomization is resolved prior to the subject uncertainty. A potential confound for the identification of the relationship between preferences and strategic behavior is subject confusion. For example, a subject who does not understand the game may simply choose the strategy with the largest minimum payment. To rule this out, we implemented a detailed test of comprehension for each game and restrict our analysis to subjects who pass the comprehension tasks. 19 Note that we cannot allow subjects to learn from experience here: repeated play will cause subjects to learn about the distribution of play in the population, which will alter the degree of strategic uncertainty involved in the game. The comprehension questions were presented to subjects as a set of dynamic drop down menus which subjects were required to fill in before they could submit a strategy for the associated game. 20 The comprehension questions were incentivized, with subjects receiving a $1 bonus for each game if they filled in the drop down menus correctly on their first attempt. Subjects were penalized $0.25 for each incorrect attempt, and after 4 incorrect attempts an alert appeared on screen prompting them to seek help from the experimenter before proceeding. 21 There were very few subjects that had substantial difficulty with the drop down menus, with just under 3% of subjects in the own preference treatment earning half or less of the available bonus payment. We used the level of bonus payment earned by the subject as a measure of comprehension, and removed subjects that performed poorly from our sample according to a pre-determined criterion, with details provided in Section 3 . In the other preference treatment only a single game was played, and subjects were paid for that game. In the own preference treatment subjects were paid for one randomly selected game, and the game selected for payment was determined prior to the start of the experiment. On entering the lab, a randomly selected subject chose a flash card from a row of flashcards stuck to the wall of the lab. After the experiment, the chosen flashcard was turned over to reveal the game that was to be paid for that session. This procedure, an implementation of the ideas found in Baillon et al. (2019) and Johnson et al. (2019) , ensures that the (objective) randomization of selecting which game is to be paid was conducted prior to any (subjective) randomization that occurred during the experiment, and prevents subjects from using the objective randomization device as a hedge against subjective uncertainty. This procedure is incentive compatible under the relatively weak assumption of monotonicity ( Azrieli et al., 2018 ). 22

19 An analysis of the full data set is included in the appendix for completeness. The qualitative results are unchanged by the change in sample.

20 For example, for the game in Fig. 2 , the worded problem for strategy S would read “Your earnings for this choice [are/ are not ] affected by your counterpart’s strategy. Your earnings for this choice will be [ $30.10 /$30/$15/$0] if a red ball is drawn from the [ U urn /K urn] and nothing otherwise.” For each set of terms that are square bracketed, the subjects were required to select the correct term (shown in bold here) from a drop down menu. Subjects were required to ﬁll in drop down menus that described the possible outcomes for each of their strategies.

21 These subjects are all excluded from the data set. The experimenter provided the subjects with minimal help, usually only suggesting that the subject double check the response to a particular drop down menu.

22 Note that the theoretical analysis of Cox et al. (2015) and Cox and Sadiraj (2019) is not relevant for the current experiment. Both Cox et al. (2015) and Cox and Sadiraj (2019) assume reduction of compound lotteries (ROCL), but ambiguity averse subjects frequently violate ROCL. Halevy (2007) , for example, found that of 114 ambiguity averse subjects only one subject satisﬁed ROCL. Chew et al. (2017) similarly found that 147 out of 148 ambiguity averse subjects violated ROCL. Abdellaoui et al. (2015) ﬁnd that 56 out of 60 ambiguity averse engineering students, and 25 out of 25 ambiguity averse non-engineering 728 E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734

Table 2 Number of subjects, classified by preferences, restricted to subjects that passed the comprehension tasks. Low risk aversion subjects selected L in the risk classification game, implying a CARA parameter of ρ < 0.05, while high risk aversion subjects selected H , implying a CARA parameter of ρ > 0.12. Ambiguity neutral subjects selected S in the ambiguity classification game, while ambiguity averse subjects selected M . Indepen-

dence of risk and ambiguity preferences is rejected (Pearson’s χ 2 test, p = 0 . 001 ).

Ambiguity preference

Neutral Averse Aggregated

Risk aversion Low 57 19 76 High 18 22 40 Aggregated 75 41 116

2.5. Experimental conditions

All sessions were held in the ELVSE lab at the Vancouver School of Economics. The own preference treatment consisted of 20 sessions held between March and September 2014, with between 8 and 12 subjects per session and a total of 206 subjects. The instructions for included in an online appendix. Sessions lasted between 60 and 90 minutes, including instructions and payment, and the average payment was C$26.60. The other preference treatment was conducted in January 2015 (92 subjects across 8 sessions) and January 2016 (39 subjects across 2 sessions). The other preference sessions were signiﬁcantly shorter (because subjects faced fewer games), lasting around 45 minutes, and subjects earned an average of C$22.31. With one exception, which was removed from the reported data, no subject participated in more than one session. 23 Subjects were recruited from the ELVSE implementation of the ORSEE subject pool ( Greiner, 2015 ), which is overwhelmingly made up of UBC undergraduate students. The experiments were run using an early version of Redwood, which was developed by LEEPS at the University of California Santa Cruz.

3. Results

This section presents the experimental results. There are three key dimensions to the results presented:

• Risk and ambiguity preferences are positively correlated. • There is an effect of preferences on row player behavior in the testing game. • There is an effect of (beliefs over an) opponent’s preferences on column player behavior in the testing game.

We also emphasize one key implication of these results: because risk and ambiguity preferences are correlated, and the expected effects of risk and ambiguity on behavior in games are typically similar, previous studies that have studied only the effects of risk (or ambiguity) aversion on behavior in games have likely overestimated the parameter of interest because of an omitted variable bias effect. Throughout the results section we restrict attention to subjects who recorded a perfect score for the comprehension task on the questions for which we use their data. For example, subjects are included in the results for the own preference treatment if they answered the comprehension questions correctly the ﬁrst time for each of the risk measurement game, the ambiguity measurement game, and the testing game. For the other preference treatment, where subjects played only one game, any mistake would exclude the subject. 24 The full data set is reported in Appendix B , and there are no qualitative differences in the results.

3.1. Preferences

Table 2 presents the classification of subjects into preference types based on their responses as row players in the preference measuring games. 35% of the subjects were classified as ambiguity averse, a figure that is at the lower end of the level of ambiguity aversion reported in previous individual decision making papers. 25 We find that 25% of subjects with low students, fail to reduce compound lotteries. Aydogan et al. (2019) find a similar relationship for students, but do not find an association between ambiguity aversion and compound lottery reduction for professional actuaries.

23 One subject had signed up to the ORSEE subject pool using two different names and was therefore invited to the experiment twice. Unfortunately, we did not realize this until after the subject had participated in both sessions. Because subjects were not matched with each other until the end of the experiment the experienced subject could not contaminate the decisions of other subjects.

24 In the own preference treatment we drop 69 out of 185 subjects. In the other preference treatment we drop 9 out of 124 subjects.

25 Chew et al. (2018) provide an overview of previous Ellsberg urn experimental results in individual decision making environments. The lower rate of ambiguity aversion found here is likely because of the asymmetrical payoffs used in the ambiguity measurement game –the design here picks up only subjects with a strict preference against ambiguity. In a two-urn Ellsberg task Epstein and Halevy (2018) ﬁnd that 29% of subjects exhibit weak ambiguity preference: preferring the risky urn when payments are symmetric yet prefering the ambiguous urn when payments are asymmetric. E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734 729

Table 3 Proportion of subjects that play A in the game in Fig. 1 by preference type, restricted to subjects that passed the comprehension task. Restricted to subjects that passed comprehension. p -values are shown in brackets and are calculated

using a χ 2 test under the null hypothesis that the proportion of subjects playing A in the relevant cell is the same as the proportion of low risk aversion ambiguity neutral subjects playing A . N = 116 .

Ambiguity preference

Neutral Averse Aggregated

Risk aversion Low 0.74 0.58 0.70 (p = 0 . 194) High 0.61 0.41 0.50 (p = 0 . 307) (p = 0 . 006) Aggregated 0.71 0.49

risk aversion are ambiguity averse, while 55% of the subjects with high risk aversion are also ambiguity averse. A Pearson’s χ 2 test rejects the null hypothesis of independence of risk and ambiguity preference at the 1% level ( p = 0 . 001 ), and the coeﬃcient, a measure of association between two binary variables, is 0.30.

Result 1. Risk and ambiguity preferences are positively correlated.

The previous experimental literature on the correlation between risk and ambiguity preferences has produced mixed results. 26 From a theoretical viewpoint, axiomatic models of preferences ( Schmeidler (1989) and Gilboa and Schmeidler (1989) , for example) are typically agnostic regarding the relationship between risk and ambiguity aversion. There are, however, some non-axiomatic theoretical models of behavior under uncertainty that suggest a positive correlation between risk and ambiguity preferences ( Halevy and Feltkamp (2005) , for example).

3.2. Own preference results

Table 3 displays the results from the own preference treatment. A low risk aversion ambiguity neutral subject is 1.8 times more likely to play A in the testing game than a high risk aversion ambiguity averse subject ( p = 0 . 006 , χ 2 test). This represents clear evidence that preferences toward uncertainty, as measured in the classification games, are associated with behavior in the testing game. Table 3 also highlights the inference problems that can occur in data sets that measure only one of risk or ambiguity aversion. Consider the effects of risk aversion in the testing game ( Fig. 1 ). On average (aggregated across ambiguity preferences) high risk aversion subjects play A 20 percentage points less often than low risk aversion subjects. This difference is statistically significant at standard levels ( p = 0 . 036 ) and we might conclude that risk aversion has a significant effect on behavior in the testing game. However, after conditioning on ambiguity preferences, we observe a smaller effect of risk preferences on behavior both for ambiguity averse and ambiguity neutral subjects. For ambiguity neutral subjects, high risk aversion subjects play A 13 percentage points less often than low risk aversion subjects ( p = 0 . 307 ). For ambiguity averse subjects, high risk aversion subjects play A 17 percentage points less often than low risk aversion subjects ( p = 0 . 278 ). This is an illustration of a milder form of Simpson’s paradox. 27 As Table 1 makes clear, there is only a difference in the rationalizable set of strategies between low and high risk aversion subjects when the subject is also ambiguity neutral. The effect of risk aversion on behavior in the testing game is identified for ambiguity neutral subjects through the change in the set of rationalizable strategies. For ambiguity averse subjects, the effect of risk aversion on behavior is not cleanly identified: it could be, for example, that risk averse subjects hold different beliefs about the behavior of their opponent than risk neutral subjects. This difference in beliefs could lead to a shift in behavior within the rationalizable set that is not attributable to the differing preferences. Therefore, we can only identify the effect of risk aversion on rationalizable behavior by comparing the behavior of low risk aversion ambiguity neutral and high risk aversion ambiguity neutral subjects. The logic in the two previous paragraphs still holds if “ambiguity” and “risk” are swapped at every stage. Correspondingly, we only have clean identification of the effect of ambiguity preference on behavior in the testing game for low risk aversion subjects. The point estimate of the marginal effect of risk aversion is therefore approximately 1.5 times larger than the effect of risk aversion conditional on ambiguity neutrality. Compare this to, say, Goeree et al. (2003) which compares risk preferences

26 See Section 1.2 for a literature review.

27 Simpson’s paradox, as typically formulated, requires the sign of the effect to change when conditioning on a confounding variable. Here we observe a reduction in the effect size for both subgroups, but not a change in sign. 730 E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734

Table 4 Percentage of subjects that play X in the game in Fig. 1 , by opponents’ preference type, restricted to subjects that passed the comprehension task, for subjects that observed both their opponents’ risk and ambiguity preference. p -values calculated using Fisher’s exact test under the Null that the proportion of subjects playing X in the relevant cell is equal to the proportion of subjects playing X in the top left cell. N = 77 .

Opponent ambiguity preference

Neutral Averse Aggregated

Opponent risk aversion Low 1.00 0.92 0.97 (p = 0 . 32) High 0.56 0.30 0.41 ( p < 0.001) ( p < 0.001) Aggregated 0.83 0.51

Table 5 Percentage of subjects that play X in the game in Fig. 1 , by opponents’ preference type, restricted to subjects that passed the comprehension task, for subjects that observed their opponents’ ambiguity preference only. The difference in proportion of subjects playing X when their opponent is ambiguity neutral or averse is signiﬁ- cant under Fisher’s exact test ( p = 0 . 03 ). N = 38 .

Opponent’s ambiguity preference

Neutral Averse Aggregated

Pr ( X ) 0.96 0.67 0.84

as measured in various matching pennies games to risk preferences measured using a Holt and Laury (2002) risk task using a within subject design. Using maximum likelihood estimation Goeree et al. (2003) ﬁnd a constant relative risk aversion parameter of 0.44 (standard error 0.07) in the games and 0.31 (0.03) in the Holt–Laury task, a difference that may be attributable to unmeasured ambiguity aversion in the games.

Result 2. Risk and ambiguity preferences have an affect on row player behavior in the testing game.

Result 3. The marginal effects of both risk and ambiguity aversion in the testing game are overstated relative to the well- identiﬁed conditional effects.

3.3. Other preference results

The other preference treatment is, in some sense, a more challenging environment for subjects. The payoff structure used in the own preference treatment was constructed so that an unobserved variable (the subject’s underlying preferences) was expected to systematically inﬂuence behavior in both the preference measuring games and the testing game. As a consequence, we would expect the results in the own preference treatment to persist independently of whether the subject understood the relationship between the preference measuring games and the testing game; as long as a subject is able to understand each game independently, and express their preferences coherently within each game, then we should expect to see the predicted relationships across games. In the other preference treatment, however, we will only observe the predicted effects when the subject understands the relationship between games: the subject is required to extract information from the behavior of her opponent in the measurement games and use that information to inform her behavior in the testing game. It is, of course, unrealistic to think that subjects will have knowledge of the formal relationship between risk preferences, ambiguity preferences, and solution concepts across games. There are, however, some obvious and intuitive patterns that subjects may recognize. For example, in the risk measurement game ( Fig. 5 ) it is clear that, in layman’s terms, strategies L and I are uncertain, and strategy H is safe. In the testing game ( Fig. 1 ) it is also clear that strategies A and B are uncertain, and strategy C is safe. If an opponent is observed to be willing to take on uncertainty in one situation, then it seems reasonable to suppose that they might take risks in another situation; indeed, the data suggests that subjects agree. Nevertheless, this is a rather sophisticated pattern of behavior that requires a deep understanding of strategic interaction. The data for this treatment is summarized in Tables 4 and 5 , with each table showing the proportion of subjects that played X as a function of their opponent’s preferences (i.e. as a function of the choices the subject observed in the measurement games). Recall that subjects that observe a low risk aversion and ambiguity neutral opponent should only play X , while for other subjects both X and Y are rationalizable. Subjects represented in Table 4 observed both their opponent’s risk and ambiguity preferences, while subjects in Table 5 observed only their opponent’s ambiguity preferences.

Remarkably, subjects that observed a low risk aversion ambiguity neutral opponent played X , their unique best response to a justiﬁable strategy, 100% of the time. There is a large effect of opponent’s risk preferences on behavior, with X being E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734 731 chosen only 56% of the time by subjects with a high risk aversion ambiguity neutral opponent ( p < 0.001). In contrast, there is very little evidence of any effect of observing an opponent’s ambiguity preferences on behavior in the testing game: subjects that observe a low risk aversion ambiguity averse opponent play X 92% of the time ( p = 0 . 32) . As a point of comparison, we can use the column player data from the own preference treatment as a measure of subject behavior in the absence of information about their opponent. In that treatment 114 subjects passed the comprehension task when playing the testing game as the column player, and 78 (68%) of those subjects played X . Therefore subjects who observed a risk neutral opponent played X more often than uninformed subjects, who in turn played X more often than subjects who observed a risk averse opponent. Because subjects in Table 4 observed two signals of their opponent’s preferences, it is not possible to determine whether subjects view the ambiguity preference signal as completely uninformative or merely less informative than the stronger risk preference signal. Consequently, we ran additional sessions where subjects only observed their opponent’s ambiguity preferences. The results are reported in Table 5 . Again, subjects that observe an ambiguity neutral opponent almost always (22 out of 23 subjects) play X while subjects that observe an ambiguity averse opponent play X less than 70% of the time ( p = 0 . 03 , one-sided Fisher exact test). 28 It is obvious that the results for subjects with risk and ambiguity neutral opponents are exactly equal to the pure strategy Nash equilibrium of the game. A natural follow up question is: are the results for subjects with risk averse ambiguity neutral opponents also rationalizable by a Nash equilibrium? 29 The answer is yes: the proportion of subjects playing X when they observe their opponent to be ambiguity neutral but risk averse (9 out of 16 subjects, see Table 4 ) is consistent with behavior in a Nash equilibrium of the game (with row players mixing between A and C ) when the row player has a CARA utility function with risk aversion parameter ρ ≈ 0.15. Given that the choice of H in the risk measurement game implies ρ > 0.12 the results are consistent with potential equilibrium play at reasonable levels of risk aversion.

Result 4. Induced beliefs over the uncertainty preferences of others have an affect on behavior as the column player in the game in Fig. 1 : subjects who face an uncertainty averse opponent are more likely to best respond to their opponent’s safe strategy.

Result 5. Column player behavior in the game in Fig. 1 , among subjects who face ambiguity neutral opponents, is consistent with Nash equilibrium.

4. Conclusion

This paper presents an experiment designed to investigate the role of uncertainty preferences and beliefs over the uncertainty preferences of others on behavior in a normal form game. Subjects who exhibit both risk and ambiguity aversion behave differently than subjects who exhibit risk and ambiguity neutrality. In a second treatment subjects were matched with previous participants and beliefs regarding their opponent’s preferences were induced by displaying their opponent’s choices in preference measurement tasks. Subjects who observed a risk averse opponent, or observed an ambiguity averse opponent when risk preferences were unobservable, were signiﬁcantly more likely to best respond to their opponent’s “safe” strategy. While the results presented here were motivated by an analysis of justiﬁable and rationalizable strategies, and may also seem intuitive, they are not uniquely predicted by theories of strategic uncertainty in games. Even if the row player is highly uncertainty averse she should still play A in the testing game if she believes her opponent will play X . Similarly, even if the column player knows the row player is highly uncertainty averse he should play X if he believes his opponent will play A (perhaps because he believes his opponent believes he will play X ). The results in this paper suggest that this hypothetical structure of beliefs does not persist among uncertainty averse subjects: uncertainty aversion is empirically relevant in strategic environments, and not just a theoretical possibility.

Declaration of Competing Interest

I, Evan Calford, have no conﬂicts of interest to declare with respect to the submission of my paper titled Uncertainty Aversion in Game Theory: Experimental Evidence.

Appendix A. Proofs

Proof of lemma 1. Suppose that the row player has preferences such that ρ < ρ∗ and χ = SEU. Note that ρ < ρ∗ implies u (25) + u (14) > 2 u (18) and we can write the belief set for this row player as (X ) = { φ : φ(X ) = p} . The best response for

> 1 , < 1 = 1 the row player is A when p 2 is B when p 2 and is either A or B when p 2 . C is never a best response.

28 We do not report the χ 2 statistic here because of its poor properties in small and skewed samples. Instead, we report a more conservative Fisher exact test.

29 Because of the large set of equilibrium probabilities for subjects with ambiguity averse opponents we do not repeat the analysis for those subjects – almost any behavior is consistent with equilibrium play under ambiguity aversion in this game. 732 E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734

Now, consider the case where ρ ≥ ρ∗ so that u (25) + u (14) ≤ 2 u (18) . C is a best response to the belief set (X ) = { φ :

φ( ) = 1 } X 2 . Finally, consider the case where χ = MEU. C is a best response to the belief set (X ) = { φ : 0 ≤ φ(X ) ≤ 1 } . Following Deﬁnition 1 , the existence of a belief set such that C is a best response establishes that C is justiﬁable.

Proof of Lemma 2. First, note that u (25) + u (14) = 2 u (18) is equivalent to − ρ∗ − ρ∗ − ρ 1 − e 25 + 1 − e 14 = 2(1 − e 18 ) which implies ρ∗ ≈ 0.10. For the first part, u (25) + u (10) ≥ max{ u (23) + u (11 ) , 2 u (15) } is equivalent to both − ρ − ρ − ρ − ρ 1 − e 25 + 1 − e 10 ≥ 1 − e 23 + 1 − e 11 and − ρ − ρ − ρ 1 − e 25 + 1 − e 10 ≥ 2(1 − e 15 ) and both inequalities are satisfied whenever ρ ≤ 0.05 < ρ∗ (correct to 2 decimal places). For the second part, 2 u (15) ≥ max{ u (23) + u (11 ) , u (25) + u (10) } is equivalent to both − ρ − ρ − ρ 2(1 − e 15 ) ≥ 1 − e 23 + 1 − e 11 and − ρ − ρ − ρ 2(1 − e 15 ) ≥ 1 − e 25 + 1 − e 10 and both inequalities are satisfied whenever ρ ≥ 0.12 > ρ∗.

Proof of Lemma 3. Notice that Y is a best response to a justifiable strategy if and only if C is justifiable. Suppose that C is not justifiable. Therefore Y is not a best response to a justifiable strategy. Eliminate Y. A is the unique best response to X for any row player uncertainty preferences. Therefore B is not rationalizable. Suppose that C is justifiable. A and B are also justifiable. X and Y are justifiable. All strategies are justifiable, therefore all strategies are rationalizable.

A.1. An alternative framework for justiﬁability

In this section we consider justiﬁability with respect to the -contamination model considered in Epstein (1997) . In this model an ambiguity averse agent considers only contaminations of her underlying probabilistic belief, rather than allowing for any closed and convex as in the main text. Recognizing that an arbitrary closed and convex set of beliefs for the row player can be written as R = { a ≤ φ(X ) ≤ b} with 0 ≤ a ≤ b ≤ 1, any feasible -contaminated belief set can be written as = { ≤ φ( ) ≤ } = ( − ) + = ( − ) + ≤ ≤ R a X b with a 1 p a and b 1 p b where 0 p 1 is an underlying probabilistic belief. In this framework, may be interpreted as capturing the degree of ambiguity perceived by the agent. ρ − ρ e −14 −e 18 It is convenient to normalize row player utility such that u (25) = 1 and u (14) = 0 . This implies that u (18) = ρ − ρ e −14 −e 25

4 < ( ) ≤ 1 < ρ ≤ ρ∗ and that 11 u 18 2 for 0 . Lemma 4. C is justiﬁable for a row player with contaminated beliefs if and only if ≥ 1 − 2 u (18) . Proof. Consider a belief set = { a ≤ φ(X ) ≤ b } . C is a best response if and only if a ≤ u (18) and b ≥ 1 − u (18) . R Notice that b − a ≤ . Therefore, if < 1 − 2 u (18) then either b < 1 − u (18) or a > u (18). If ≥ 1 − 2 u (18) then we

≤ ≥ − ( ) = 1 , = = construct a set R with a u(18) and b 1 u 18 by ﬁxing p 2 a 0 and b 1.

ρ = ≥ 3 Notice that if 0 then C is justifiable if 11 . Recall that a choice of L in the risk measurement game implies that ρ ≤ 0.05 and, given our normalization, u (18) ≈ 0.43. Therefore, C is never a best response for a subject who chooses L in the risk measurement game if < 0.14. We provide the contamination formulation here as reference for readers who wish to have a framework to think about the relationship between the degree of ambiguity perceived by the subject and the set of rationalizable strategies. We do not use this framework in the main text, however, because is not observable and the contamination framework does not encode any empirically verifiable information beyond the MEU framework used in the main text. The reason is that may vary within subject across games and environments, and the variation in cannot be identified without implausibly strong assumptions about behavior.

Appendix B. Additional results

This section presents results tables that include subjects who failed the comprehension tasks. Fig. 6 is the analog of Fig. 1 in the main text. Table 6 is the analog of Table 2 , and Table 7 is the analog of Table 3 . Table 8 and Table 9 are the analogs of Table 4 and Table 5 , respectively. In all cases the results are not materially changed by the increase in sample size. E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734 733

Fig. 6. Testing game, with proportion of subjects choosing each strategy in parentheses. N = 206 , full sample.

Table 6 Number of subjects, classified by preferences. Low risk aversion subjects selected L in the risk classification game, implying a CARA parameter of ρ < 0.05, while high risk aversion subjects selected H , implying a CARA parameter of ρ > 0.12. Ambiguity neutral subjects selected S in the ambiguity classification game, while ambiguity averse subjects selected M . The null of in-

dependence is rejected at the 1% level, using Pearson’s χ 2 test ( p = 0 . 002 ).

Ambiguity preference

Neutral Averse Aggregated

Risk aversion Low 85 36 121 High 30 34 64 Aggregated 115 70 185

Table 7 Proportion of subjects that play A in the game in Fig. 1 by preference type. p -values shown are

calculated using a χ 2 test under the null hypothesis that the proportion of subjects playing A in the relevant cell is the same as the proportion of Low Risk Aversion Ambiguity Neutral subjects playing A . N = 185 .

Ambiguity preference

Neutral Averse Aggregated

Risk aversion Low 0.67 0.56 0.64 (p = 0 . 229) High 0.60 0.32 0.45 (p = 0 . 485) (p = 0 . 001) Aggregated 0.65 0.44

Table 8 Percentage of subjects that play X in the game in Fig. 1 , by opponents’ preference type for subjects that observed both their opponents’ risk and ambiguity preference. p -values calculated using a Fisher exact test under the Null that the proportion of subjects playing X in the relevant cell is equal to the proportion of subjects playing X in the top left cell. N = 85 .

Opponent ambiguity preference

Neutral Averse Aggregated

Opponent risk aversion Low 1.00 0.93 0.98 (p = 0 . 33) High 0.63 0.33 0.47 (p = 0 . 001) ( p < 0.001) Aggregated 0.85 0.55

Table 9 Percentage of subjects that play X in the game in Fig. 1 , by opponents’ preference type, restricted to subjects that passed the comprehension task, for subjects that observed their opponents’ ambiguity preference only. The difference in proportion of subjects playing X when their opponent is ambiguity neutral or averse is signiﬁcant under Fisher’s exact test ( p = 0 . 03 ). N = 39 .

Opponent’s ambiguity preference

Neutral Averse Aggregated

Pr ( X ) 0.96 0.69 0.85 734 E.M. Calford / Journal of Economic Behavior and Organization 176 (2020) 720–734

Supplementary material

Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.jebo.2020.06.011

References

Abdellaoui, M. , Baillon, A. , Placido, L. , Wakker, P.P. , 2011. The rich domain of uncertainty: source functions and their experimental implementation. Am. Econ. Rev. 101 (2), 695–723 . Abdellaoui, M. , Klibanoff, P. , Placido, L. , 2015. Experiments on compound risk in relation to simple risk and to ambiguity. Manag. Sci. 61 (6) . Aydogan, I., Berger, L., Bosetti, V., 2019. Economic rationality: investigating the links between uncertainty, complexity, and sophistication. Mimeo. Azrieli, Y. , Chambers, C.P. , Healy, P.J. , 2018. Incentives in experiments: a theoretical analysis. J. Polit. Econ. 126 (4), 1472–1503 . Azrieli, Y. , Teper, R. , 2011. Uncertainty aversion and equilibrium existence in games with incomplete information. Games Econ. Behav. 73, 310–317 . Bade, S. , 2011. Ambiguous act equilibria. Games Econ. Behav. 71, 246–260 . Baillon, A., Halevy, Y., Chen, L., 2019. Experimental Elicitation of Ambiguity Attitude Using the Random Incentive System. Mimeo. Baillon, A. , Huang, Z. , Selim, A. , Wakker, P.P. , 2018. Measuring ambiguity attitudes for all (natural) events. Econometrica 86 (5), 1839–1858 . Battigalli, P. , Cerreia-Vioglio, S. , Maccheroni, F. , Marinacci, M. , 2016. A note on comparative ambiguity aversion and justifiability. Econometrica 84 (5), 1903–1916 . Bernheim, B.D. , 1984. Rationalizable strategic behavior. Econometrica 52 (4), 1007–1028 . Bohnet, I. , Zeckhauser, R. , 2004. Trust, risk and betrayel. J. Econ. Behav. Org. 55, 467–484 . Bossaerts, P.L. , Ghirardato, P. , Guarnaschelli, S. , Zame, W.R. , 2010. Ambiguity in asset markets: theory and experiment. Rev. Financ. Stud. 23, 1325–1359 . Calford, E., 2020. Mixed Strategies and Preference for Randomization in Games with Ambiguity Averse Agents. Mimeo. Camerer, C.F. , Karjalainen, R. , 1994. Ambiguity aversion and non-additive beliefs in non-cooperative games: experimental evidence. In: Munier, B., Machina, M.J. (Eds.), Models and Experiments in Risk and Rationality. Springer Netherlands, pp. 325–358 . Chakravarty, S. , Roy, J. , 2009. Recursive expected utility and the separation of attitudes towards risk and ambiguity: an experimental study. Theory Decis. 66 (199) . Chew, S.H. , Miao, B. , Zhong, S. , 2017. Partial ambiguity. Econometrica 85 (4), 1239–1260 . Chew, S.H. , Ratchford, M. , Sagi, J.S. , 2018. You have to recognize ambiguity to avoid it. Econ. J. 128, 2480–2506 . Cohen, M. , Jaffray, J.-Y. , Said, T. ,1987. Experimental comparison of individual behavior under risk and uncertainty for gains and for losses. Org. Behav. Hum. Decis. Process. 39 (1), 1–22 . Cohen, M. , Tallon, J.-M. , Vergnaud, J.-C. , 2011. An experimental investigation of imprecision attitude and its relation with risk and impatience. Theory Decis. . Cox, J.C. , Sadiraj, V. , 2019. Incentives. In: Schram, A., Aljaz, U. (Eds.), Handbook of Research Methods and Applications in Experimental Economics. Edward Elgar Publishing . Cox, J.C. , Sadiraj, V. , Schmidt, U. , 2015. Paradoxes and mechanisms for choice under risk. Exp. Econ. 18, 215–250 . Curley, S.P. , Yates, J.F. , Abrams, R.A. , 1986. Psychological sources of ambiguity avoidance. Org. Behav. Hum. Decis. Process. 38 (2), 230–256 . Dean, M. , Ortoleva, P. , 2019. The empirical relationship between nonstandard economic behaviors. Proc. Natl. Acad. Sci. 116 (33), 16262–16267 . Di Mauro, C. , Maffioletti, A. , 2004. Attitudes to risk and attitudes to uncertainty: experimental evidence. Appl. Econ. 36 (4), 357–372 . Dow, J. , Werlang, S.R.d.C. , 1994. Nash equilibrium under Knightian uncertainty: breaking down backward induction. J. Econ. Theory 64, 305–324 . Eichberger, J. , Kelsey, D. , 1996. Uncertainty aversion and preference for randomization. J. Econ. Theory 71, 31–43 . Eichberger, J. , Kelsey, D. , 20 0 0. Non-additive beliefs and strategic equilibria. Games Econ. Behav. 30, 183–215 . Eichberger, J. , Kelsey, D. , 2014. Optimism and pessimism in games. Int. Econ. Rev. 55 (2), 483–505 . Eichberger, J. , Kelsey, D. , Schipper, B.C. , 2008. Granny versus game theorist: ambiguity in experimental games. Theory Decis. 64, 333–362 . Epstein, L.G. , 1997. Preference, rationalizability and equilibrium. J. Econ. Theory 73, 1–29 . Epstein, L.G. , Halevy, Y. , 2018. Ambiguous correlation. Rev. Econ. Stud. . Gilboa, I. , Schmeidler, D. , 1989. Maxmin expected utility with non-unique prior. J. Math. Econ. 18, 141–153 . Goeree, J.K. , Holt, C.A. , Palfrey, T.R. , 2003. Risk averse behavior in generalized matching pennies games. Games Econ. Behav. 45 (1), 97–113 . Grant, S. , Meneghel, I. , Tourky, R. , 2016. Savage games. Theoret. Econ. 11, 641–682 . Greiner, B. , 2015. Subject pool recruitment procedures: organizing experiments with orsee. J. Econ. Sci. Assoc. 1 (1), 114–125 . Groes, E. , Jacobson, H.J. , Sloth, B. , Tranaes, T. , 1998. Nash equilibrium with lower probabilities. Theory Decis. 44, 37–66 . Halevy, Y. , 2007. Ellsberg revisited: an experimental study. Econometrica 75 (2), 503–536 . Halevy, Y. , Feltkamp, V. , 2005. A Bayesian approach to uncertainty aversion. Rev. Econ. Stud. 72, 449–466 . Harrison, G.W. , Rutström, E.E. , 2008. Risk aversion in the laboratory. In: Cox, J.C., Harrison, G.W. (Eds.). Research in Experimental Economics, Risk Aversion in Experiments, 12. Bingley . Holt, C.A. , Laury, S.K. , 2002. Risk aversion and incentive effects. Am. Econ. Rev. 92 (5), 1644–1655 . Ivanov, A. , 2011. Attitudes to ambiguity in one-shot normal-form games: an experimental study. Games Econ. Behav. 71, 366–394 . Johnson, C., Baillon, A., Bleichrodt, H., Li, Z., van Dolder, D., Wakker, P. P., 2019. Prince: An Improved Method for Measuring Incentivized Preferences. Mimeo. Kelsey, D. , le Roux, S. , 2015. An experimental study on the effect of ambiguity in a coordination game. Theory Decis. 79 (4), 667–688 . Kelsey, D. , le Roux, S. , 2017. Dragon slaying with ambiguity: theory and experiments. J. Publ. Econ. Theory 19 (1), 178–197 . Kelsey, D. , le Roux, S. , 2018. Strategic ambiguity and decision making: an experimental study. Theory Decis. . Klibanoff, P., 1996. Uncertainty, Decision and Normal form Games. Mimeo. Klibanoff, P. , Marinacci, M. , Mukerji, S. , 2005. A smooth model of decision making under ambiguity. Econometrica 73 (6), 1849–1892 . Lauriola, M. , Levin, I.P. , 2001. Relating individual differences in attitude toward ambiguity to risky choices. J. Behav. Decis. Mak. 14 (2), 107–122 . Li, C. , Turmunkh, U. , Wakker, P.P. , 2019. Trust as a decision under ambiguity. Exp. Econ. 22, 51–75 . Lo, K.C. , 1996. Equilibrium in beliefs under uncertainty. J. Econ. Theory 71, 443–484 . Lo, K.C. , 2009. Correlated Nash equilibrium. J. Econ. Theory 144, 722–743 . Machina, M.J. , Siniscalchi, M. , 2013. Ambiguity and ambiguity aversion. In: Machina, M.J., Viscusi, W.K. (Eds.), Handbook of the Economics of Risk and Uncertainty. Newnes, pp. 730–807 . Marinacci, M. , 20 0 0. Ambiguous games. Games Econ. Behav. 31, 191–219 . Pearce, D.G. , 1984. Rationalizable strategic behavior and the problem of perfection. Econometrica 52 (4), 1029–1050 . Riedel, F. , Sass, L. , 2014. Ellsberg games. Theory Decis. 76 (4), 469–509 . Saito, K. , 2015. Preferences for flexibility and randomization under uncertainty. Am. Econ. Rev. 105 (3), 1246–1271 . Schmeidler, D. , 1989. Subjective probability and expected utility without additivity. Econometrica 57 (3), 571–587 . Weinstein, J. , 2016. The effect of changes in risk attitude on strategic behavior. Econometrica .