Expectation-Based Loss Aversion and Rank-Order Tournaments

Expectation-Based Loss Aversion and Rank-Order Tournaments SIMON DATO, ANDREAS GRUNEWALD, DANIEL MÜLLER July 24, 205 Many insights regarding rank-order tournaments rest upon contestants behavior in symmetric equilibria. As shown by Gill and Stone (200), however, symmetric equilibria may not exist if contestants are expectation-based loss averse and have choice-acclimating expectations. We show that under choice-unacclimating i.e., fixed expectations both symmetric and asymmetric equilibria exist for all degrees of loss aversion. Importantly, a symmetric equilibrium always prevails if players follow their preferred credible plan and concerns for psychological gain-loss utility do not strongly outweigh concerns for material utility. Hence, for fixed expectations a focus on symmetric equilibria seems justifiable even if contestants are expectation-based loss averse. JEL classification: Keywords: Expectation-Based Loss Aversion, Rank-Order Tournament, Reference- Dependent Preferences We thank David Gill, Fabian Herweg, Matthias Kräkel, and Takeshi Murooka for helpful comments and suggestions. All errors are of course our own. University of Bonn, Institute for Applied Microeconomics, Adenauerallee 24-42, D-533 Bonn, Germany, E-mail address: simdato@uni-bonn.de, Tel: +49-228-73924 University of Bonn, Institute for Applied Microeconomics, Adenauerallee 24-42, D-533 Bonn, Germany, E-mail address: gruni@uni-bonn.de, Tel: +49-228-73927. University of Bonn, Institute for Applied Microeconomics, Adenauerallee 24-42, D-533 Bonn, Germany, E-mail address: daniel.mueller@uni-bonn.de, Tel: +49-228-73922, Fax: +49-228-73922.

. INTRODUCTION Relative performance evaluation in the form of rank-order tournaments is commonplace electoral competition in politics, contests in professional sport, or promotion tournaments within a particular corporation or the labor market in general. In the light of the widespread applicability of rank-order tournaments, it is hardly surprising that, beginning with the seminal article by Lazear and Rosen (98), economic scholars have studied the strategic interaction of the contestants participating in this particular form of incentive mechanism for over three decades. Recently, several theoretical contributions have enriched the canonic tournament model by incorporating insights gathered in the psychological or experimental economics literature. Gill and Stone (200) investigate the implications for tournament theory if the preferences of contestants are reference dependent and exhibit loss aversion. 2 Assuming that the time span between each contestant s effort choice and the realization of the tournament outcome is rather long, they apply the concept of choice-acclimating personal equilibrium (CPE) as defined in Kőszegi and Rabin (2007); i.e., they posit that each contestant s reference point adapts to the contestant s rational expectations about outcomes, where these expectations correctly incorporate the effect of his own behavior. 3,4 Interestingly, Gill and Stone (200) find that if the degree of loss aversion among homogeneous contestants is strong, a symmetric equilibrium ceases to exist. For example, Santos-Pinto (200) as well as Ludwig, Wichardt, and Wickhorst (20) investigate overconfidence of contestants. Furthermore, contestants preferences have been modified to capture inequity aversion (Grund and Sliwka, 2005; Demougin and Fluet, 2003) or joy of winning (Kräkel, 2008). 2 Reference-dependent preferences and loss aversion have been introduced into the economic discourse by Kahneman and Tversky (979) as key aspects of their prospect theory. 3 More specifically, following the disappointment aversion concept in Bell (985), Gill and Stone (200) posit that a contestant s reference point in the money dimension corresponds to the average prize that he will receive in the tournament given his own and his opponent s effort choice. Since the tournament outcome is binary i.e., a contestant either wins or loses this formulation is equivalent to the notion of CPE as introduced in Kőszegi and Rabin (2007). 4 Evidence supporting the reference point formation according to Kőszegi and Rabin is provided by Abeler, Falk, Goette, and Huffman (20), Crawford and Meng (20), Ericson and Fuster (20), and Gill and Prowse (202).

Intuitively, uncertainty about the tournament outcome is maximized in a symmetric equilibrium in the sense that each contestant faces a 50% probability of being victorious. As expectation-based loss-averse contestants strongly dislike this uncertainty, in equilibrium they exert rather different levels of effort, which lead to unequal winning probabilities and thereby reduce uncertainty. Overall, with reference dependence and loss aversion being widely recognized as relevant determinants of individual risk preferences, this result is an important caveat to many of our insights from tournament theory, which typically rest upon the existence of symmetric equilibria. In rank-order tournaments, however, it is conceivable that the time span between the contestants effort choice and the realization of the tournament outcome is rather short. For instance, the winner of a tennis match is announced immediately after the last rally. Likewise, the counting of votes often begins even while political candidates make their last trips to swing-voter states or districts and the final numbers of votes are made public mere hours after the polling stations close. 5 For situations with not too much time passing between the actual decision and the realization of the consequences of this decision, Kőszegi and Rabin (2007) argue that CPE is not appropriate. Here, expectations will not adapt fully to a contestant s actual decision if she initially expected to behave differently than she actually did thus, expectations are choice-unacclimating or fixed. In a rank-order tournament, a contestant s reference point then reflects the consequences that she expected to prevail according to her originally formulated campaign strategy or game plan rather than the actual consequences of her effort choice. To guarantee internal consistency, Kőszegi and Rabin (2006) impose that a decision maker can only make those plans that she is actually willing to follow through a requirement referred to as personal equilibrium (PE). In this paper, we apply the concept of PE to strategic interaction in rankorder tournaments and show that there always exist multiple combinations of individual plans that contestants are willing to follow through both symmetric and asymmetric ones. Intuitively, under fixed expectations contestants become attached to their prior plans and deviations lead to the experience of a loss be- 5 Also, a short time span before the realization of the tournament outcome seems particularly compelling in later stages of multistage tournaments. 2

cause of an unexpected low winning probability or unexpected high effort costs. For any effort allocation sufficiently close of the symmetric Nash equilibrium, however, a unilateral deviation provides at most a small increase in expected material payoff and therefore is unattractive in the light of the associated loss. Hence, effort allocations close to the symmetric Nash equilibrium constitute plans that both contestants are willing to follow through no matter if symmetric or asymmetric. The multiplicity of personal equilibria raises the issue of equilibrium selection. In order to investigate the robustness of equilibria for fixed expectations, we employ the equilibrium refinement of preferred personal equilibrium (PPE), as proposed in Kőszegi and Rabin (2006). Importantly, while the symmetric Nash equilibrium may cease to constitute a mutual PPE for very strong degrees of loss aversion, the necessary degree of loss aversion for this to happen is strictly higher for fixed expectations than for choice-acclimating expectations. This finding resonates well with the observation in Kőszegi and Rabin (2007) that CPE embodies a stronger notion of risk aversion than PE: While the strong dislike of uncertainty under choice-acclimating beliefs leads to an asymmetric equilibrium with a fairly certain tournament outcome, the lagged expectation to compete in a balanced tournament with a rather uncertain outcome can be a credible plan. In fact, for the symmetric Nash equilibrium not to be a mutual PPE, the contestants concerns for psychological gain-loss utility have to clearly outweigh their concerns for material utility. Such a strong degree of loss aversion, however, implies violations of stochastic dominance (Kőszegi and Rabin, 2007). Therefore, if the time span until the realization of the tournament outcome is rather short and expectations are choice-unacclimating, the focus of tournament theory on symmetric equilibria seems justifiable even if players are expectation-based loss averse. In order to present these findings as concise as possible, we make use of a streamlined version of the canonic tournament model. More specifically, in our model agents effort choices affect the probability distribution over output levels but not the output level itself. We thus use a parameterized distribution formulation to set up our tournament environment rather than a state-space formulation as done by Lazear and Rosen (98). Next to tractability, this modeling choice which according to Hart and Holmström (987) often yields more 3

economic insights (p.78) is primarily an educational one, as it allows to delineate how expectation-based loss aversion enriches the strategic interaction of contestants. 6 Nevertheless, to guarantee a comparison across equilibrium concepts on a level playing field, we fully replicate the findings by Gill and Stone (200) with regard to tournaments with homogeneous contestants. The rest of the paper is organized as follows. In Section 2, we introduce the tournament environment and contestants reference-dependent preferences. This model is analyzed for choice-acclimating expectations in Section 3 and for lagged fixed expectations in Section 4. After addressing equilibrium selection in Section 5, we conclude in Section 6. 2. THE MODEL Two agents A and B compete in a rank-order tournament with winner prize W and loser prize L < W. Both agents share the Bernoulli utility function u(x) for money, where u (x) > 0. Let = u(w ) u(l). Agent i can exert effort e i [0, ] at cost c(e i ), where c(0) = c (0) = 0, c (e) > 0 for e > 0, c (e) > 0, c (e) > 0, and lim e c (e) =. 7 Given agent i exerts effort e i, she produces high output π i = π with probability e i and low output π i = π with probability e i, where π > π. The agent with the higher output wins the tournament and receives the winner prize W, whereas the agent with the lower output receives the loser prize L. In case that both agents produce the same output, the winner of the tournament is determined by the flip of a fair coin. Hence, given effort choices e i and e j, the probability of agent i receiving the winner prize amounts to P i (e i, e j ) = e i ( e j ) + 2 [e ie j + ( e i )( e j )] = + e i e j. 2 6 This parameterized distribution formulation of rank-order tournaments was used recently also by Kräkel and Nieken (205). 7 Our assumptions on the effort cost function are slightly different than the assumptions in Gill and Stone (200), who posit that c (0) 0 and c (e i ) 0. Furthermore, as becomes apparent below, in our model each contestant s winning probability is linear in efforts, whereas Gill and Stone (200) allow for a more general form of a contestant s probability to win the tournament. To guarantee a fair comparison of results in the light of these differences, we fully replicate the findings by Gill and Stone (200) with regard to tournaments with homogeneous contestants. 4

Both agents have reference-dependent preferences and are expectation-based loss averse à la Kőszegi and Rabin (2006). Specifically, utility is additively separable across the money dimension and the effort dimension. Furthermore, in each dimension an agent not only experiences standard material utility but also psychological gain-loss utility from comparing the actual consumption outcome to a reference point. This reference point is shaped by the agent s recently held rational expectations; i.e., she compares the material utility of the actual outcome to the material utility of each outcome that she expected to possibly occur, where each such comparison is weighted with the probability that the agent assigned to the respective reference outcome given her recent expectations. Formally, given the agents exert efforts e i and e j, respectively, and agent i expected herself to exert effort ê i, then agent i s expected utility amounts to U i (e i, ê i, e j ) = P i (e i, e j ) {u(w ) + η[ P i (ê i, e j )]µ( )} + [ P i (e i, e j )] {u(l) ηλp i (ê i, e j )µ( )} c(e i ) + ηµ (c(ê i ) c(e i )). () Here, η 0 is the weight a decision maker attaches to gain-loss utility relative to intrinsic utility and x if x 0 µ(x) = λx if x < 0 is a universal gain-loss function, where λ > captures loss aversion in the sense that a loss looms larger than an equally sized gain. As a benchmark, consider the case of loss-neutral contestants for whom η = 0. Given his opponent s effort choice e j, according to () agent i chooses effort e i to maximize the difference between his expected material utility and his effort cost, U i (e i, ê i, e j ) = P i (e i, e j )u(w ) + [ P i (e i, e j )]u(l) c(e i ), which is strictly concave in e i. Hence, agent i s best response to agent j exerting effort e j is characterized by the first-order condition U i (e i, ê i, e j )/ e i = 0, or, equivalently, c (e i ) = /2. As this first-order condition does not depend on agent j s effort choice, we conclude the following: Observation. Suppose η = 0. The unique Nash equilibrium is symmetric with (e A, e B ) = (e NE, e NE ), where e NE satisfies c (e NE ) = /2 and is a strictly dominant strategy for each contestant. 5

3. CHOICE-ACCLIMATING PERSONAL EQUILIBRIUM Our goal is to compare equilibrium play under the different notions of reference point formation embodied in CPE and PE to the Nash equilibrium of the tournament. We begin with the case of choice-acclimating expectations, which has also been analyzed by Gill and Stone (200) using a state-space formulation of a tournament. If, after the agents made their effort choices, a rather long span of time passes until the outcome of the tournament is realized, expectations will have sufficient time to adjust and e i and ê i coincide. This assumption seems particularly plausible for architectural competitions or employment processes. In these cases, the jury s decision on the winner of a tournament is often taken quite a while after contestants have filed their applications. Given e j, agent i s expected utility is U i (e i, e i, e j ) = P (e i, e j )[u(w ) + η( P (e i, e j )) ] + [ P (e i, e j )][u(l) ηλp (e i, e j ) ] c(e i ). (2) Building on the notion of choice-acclimating personal equilibrium (CPE) by Kőszegi and Rabin (2007), we define the equilibrium in the tournament under choice-acclimating expectations as follows. Definition. The effort choices (ẽ A, ẽ B ) represent a choice-acclimating Nash equilibrium (CPNE) if and only if for all i {A, B} and j {A, B} with j i, and Differentiating (2) yields U i (ẽ i, ẽ i, ẽ j ) U i (e i, e i, ẽ j ) e i [0, ]. (3) U i (e i, e i, e j ) e i = 2 [ + η(λ )(e i e j )] c (e i ) (4) 2 U i (e i, e i, e j ) e 2 i = 2 η(λ ) c (e i ). (5) Since c (e i ) > 0, U i (e i, e i, e j ) has at most one inflection point. Furthermore, lim e U i (e i, e i, e j )/ e i < 0 and lim e 2 U i (e i, e i, e j )/ e 2 i < 0, such that U i (e i, e i, e j ) is either concave for all e i [0, ] or, if an inflection point exists, convex (concave) for values of e i below (above) the inflection point. 6

These observations imply the following: First, e i = never is a best response. Second, U i (e i, e i, e j ) has at most one local maximizer in (0, ), which (if it exists) we denote by e i (e j ), where U i (e i (e j ), e i (e j ), e j )/ e i = 0, de i (e j )/de j < 0, and d 2 e i (e j )/de 2 j < 0. Third, beside e i (e j ), e i = 0 represents the only alternative candidate for a global maximizer of U i (e i, e i, e j ). With d [U i (0, 0, e j ) U i (e i (e j ), e i (e j ), e j )] de j = 2 η(λ )e i (e j ) > 0, (6) we should not be surprised to see e i = 0 being the best response for high levels of e j. For low levels of e j, in contrast, the best response always is e i = e i (e j ) because U i (0, 0, e j )/ e i > 0. Deferring the details to Appendix A, we briefly discuss the qualitative features of agent i s best-response function. If the weight attached to the expected net loss is not overly high, η(λ ) < max{, 2 c (0)}, agent i s best response decreases continuously see the left and middle panel in Figure. If, on the other hand, η(λ ) > max{, 2 c (0)}, the best response may display a discontinuity see the right panel in Figure. Intuitively, agent i s main objective is to reduce the uncertainty regarding the tournament outcome. With regard to only psychological gain-loss utility, if e j is small, agent i reduces the likelihood to experience a loss by e i =, which yields an almost certain win of the tournament. However, as e j increases, there is a threshold above which uncertainty about the tournament outcome is not minimized by e i = but e i = 0. This discontinuity in the level of effort that minimizes uncertainty in the tournament translates into a downward discontinuity in agent i s best response function. With best responses being decreasing, there can exist at most one symmetric CPNE. As psychological concerns are effectively absent from (4) if e i = e j, the symmetric CPNE (if it exists) coincides with the unique Nash equilibrium (e NE, e NE ). The symmetric CPNE does not exist if and only if agent i s bestresponse function discontinuously drops to zero before crossing the 45 -line. This is the case if η(λ ) > 2c [c ( 2 )], which yields that playing e i = e NE no longer constitutes a local interior maximizer but a local minimizer of U i (e i, e i, e NE ). In this case, there exist two asymmetric CPNEs in which one agent exerts strictly positive effort whereas the other agent resigns. 7

BR i (e j ) BR i (e j ) BR i (e j ) e j ē e j ē e j Figure : Player i s best response function BR i (e j ) for different degrees of loss aversion. Proposition. Any symmetric CPNE must be identical to the unique Nash equilibrium (e NE, e NE ). [ ] c (i) For η(λ ) [c ( 2 )], 2c [c ( 2 )], such a symmetric CPNE will be asymptotically unstable. (ii) For η(λ ) > 2c [c ( 2 )], such a symmetric CPNE cannot exist. CP NE (iii) For η(λ ) sufficiently large, there exist two asymmetric CPNEs (e e CP NE B ) = (e (0), 0) and (e CP NE A which are asymptotically stable. A, CP NE, eb ) = (0, e (0)) with e (0) > 0, Proposition shows that the results provided by Gill and Stone (200)also prevail under a parameterized distribution formulation of a tournament. Our slightly more tractable approach, however, allows to complement these important insights by specifying the exact degree of loss aversion that is necessary and sufficient for the symmetric equilibrium to cease to exist. ( ) Proposition 2. There exists χ, 2c (c ( 2 )) such that the symmetric CPNE exists if and only if η(λ ) χ. Proposition 2 demonstrates that a symmetric equilibrium exists for moderate degrees of loss aversion. In fact, the contestants concern for psychological gain-loss utility must outweigh their concern for consumption utility for the 8

symmetric equilibrium not to exist. 8,9 Such a strong degree of loss aversion, however, would also imply violations of stochastic dominance (Kőszegi and Rabin, 2007). 4. PERSONAL EQUILIBRIUM In rank-order tournaments, it is often conceivable to think of the time span between contestants effort decisions and the realization of the tournament outcome to be rather short. For example, tournaments of teams or single players may often be undecided until the very end implying that the tournament outcome realizes shortly after their last piece of effort, political candidates may campaign for election until the moment computer predictions are out, and workers may work hard for a promotion until its announcement. In these cases, agents expectations about their behavior may not have enough time to adapt to their actual behavior leading to a potential divergence of actual decisions and expectations. Hence, agents decisions are taken against the background of fixed expectations about their behavior. The assumption of fixed expectations is particularly compelling for later stages in a multi-stage tournament. Sure enough contestants have to exert effort also on the last stage of a tournament. As a direct consequence, the later the tournament stage the more likely it is that agents expectations about their behavior do not have time to acclimate to their decisions until the realization of the tournament outcome. Formally, for the case of fixed expectations, we assume that agent i makes her actual effort choice e i for expectations ê i regarding her own behavior. To guarantee internal consistency of expectations and actual behavior, we apply the concept of personal equilibrium (PE) as defined in Kőszegi and Rabin (2006); i.e., we require that a person can reasonably expect a particular course of action only if she is willing to follow it through given her expectations. The following definition extends this idea of internal consistency to the outcome of tournament play. 8 Note, however, that the symmetric CPNE may already be asymptotically unstable as it ceases to exist. 9 As suggested by a linear example in Gill and Stone (200), there exist alternative specifications of c(e) c (0) sufficiently large and c ( ) small such that the symmetric CPNE ceases to exist for a critical value of η(λ ). 9

Definition 2. The effort choices (ẽ A, ẽ B ) represent a personal Nash equilibrium (PNE) in the rank-order tournament if and only if for all i {A, B} and j {A, B} with j i, U i (ẽ i, ẽ i, ẽ j ) U i (e i, ẽ i, ẽ j ) e i [0, ]. (7) Essentially, in a PNE each agent s effort choice constitutes a PE given her opponent s effort choice. In order to identify the set of PNEs in the rank-order tournament, we begin by analyzing the set of PEs for agent i for a given effort e j of her opponent. To this end, note that a necessary condition for effort level ẽ i to be a PE is that neither a marginal upward deviation nor a marginal downward deviation is strictly profitable for agent i. Formally, U i (e i, ẽ i, e j ) = e i 2 { + η [ + (λ )P (ẽ i, e j )]} ( + ηλ)c (ẽ i ) 0 (8) ei ẽ i and U i (e i, ẽ i, e j ) e i = 2 { + η [ + (λ )P (ẽ i, e j )]} ( + η)c (ẽ i ) 0, (9) ei ẽ i have to hold simultaneously. Here, 2 U i (e i,ẽ i,e j ) = ( + η)c (e e 2 i ) < 0 for all i e i < ẽ i and 2 U i (e i,ẽ i,e j ) = ( + ηλ)c (e e 2 i ) < 0 for all e i > ẽ i. Thus, given i (8) and (9) are satisfied, the expected utility of player i is strictly increasing in e i for e i < ẽ i and strictly decreasing for e i > ẽ i. Hence, (8) and (9) together constitute not only a necessary but also a sufficient condition for ẽ i to be a PE. For agent i, we denote the resulting set of PEs for a given effort choice e j of her opponents by Θ P E i (e j ) = {ẽ i [0, ] (8) and (9) are satisfied}. (0) In order to characterize this set, define the functions θ(ẽ i) 2c (ẽ i )( + η) η(λ )ẽ i, () 2 and θ(ẽ i ) 2c (ẽ i )( + ηλ) η(λ )ẽ i, (2) 2 [ ψ(e j ) + η + ] η(λ ) ( e j ), (3) 2 0

θ(e i ) θ(e i ) ψ(e j ) ψ() 0 e() e(e j )e() e(e j ) e i Figure 2: Construction of the set Θ P E i (e j ). which are illustrated in Figure 2. Condition (8) can be rewritten as θ(ẽ i ) ψ(e j ). Note that ψ(e j ) is strictly decreasing and that ψ() > 0 such that ψ(e j ) is strictly positive for all e j [0, ]. Next, consider the function θ(ẽ i ). By our assumptions on the effort cost function, we have θ(0) = 0 and lim ei θ(e i ) =. Hence, the intermediate value theorem guarantees that there exists ēi (e j ) (0, ) such that θ(ēi (e j )) = ψ(e j ). Due to the strict convexity of θ(e i ), ēi (e j ) is uniquely determined and effort levels below ēi (e j ) do not constitute a PE for agent i given her opponent exerts effort e j. Similarly, condition (9) can be rewritten as θ(ẽ i ) ψ(e j ). By analogous reasoning, we can establish the existence of ē i (e j ) (0, ) such that θ(ē i (e j )) = ψ(e j ) and effort levels above ē i (e j ) do not constitute a PE for agent i, either. Finally, since θ(e i ) < θ(e i ) for all e i (0, ], we have ēi (e j ) < ē i (e j ). This allows us to establish the following observation. Lemma. Given e j [0, ], Θ P i E (e j ) = [ēi (e j ), ē i (e j )] (0, ). Furthermore, ē i (e j ) and ē i (e j ) are continuous, strictly decreasing, and strictly concave. According to Lemma, agent i can credibly expect only to exert a moderate effort level herself. For any fixed expectation, increasing the effort beyond this

expectation involves a tradeoff for the agent. On the one hand, an increase in effort improves her chances to win the tournament and to experience a gain and at the same time it reduces the probability to obtain the loser prize and to experience a loss. On the other hand, the corresponding increase in effort implies higher effort costs and leads to a certain loss in the effort-cost dimension. Due to this tradeoff, expecting to exert a fairly low effort level, e i < ēi (e j ) is not a credible plan for agent i. In this case, the convexity of the effort cost function implies that the latter drawback is rather small and more than outweighed by the former benefit, such that a deviation to a higher effort level is profitable. Likewise, expecting to exert a fairly high effort level, e i > ē i (e j ), neither is a credible plan for agent i. In this case, the benefit of reducing effort costs by decreasing effort below this expectation more than outweighs the drawbacks associated with the decrease in agent i s winning probability. Agent A s set of personal equilibria in dependence of agent B s effort choice is depicted in the left panel of Figure 3. e A e A (e B ) e A e A (e B ) Θ P E A (e B) Θ P E A (e B) Θ P NE Θ P E B (e A) e B e B Figure 3: The left panel depicts the correspondence of PEs for agent A. The right panel depicts the resulting set of PNEs for the tournament. Finally, we use the preceding characterization of an agent s set of PEs to derive equilibrium behavior. According to Definition 2, the set of PNEs is given by Θ P NE = {(e A, e B ) [0, ] 2 e A Θ P A E (e B ) and e B Θ P B E (e A )}. (4) 2

By the properties of the agents PE correspondences listed in Lemma, it follows immediately that there always exists a PNE. Furthermore, as becomes apparent from the right panel of Figure 3, which depicts the set of PNEs, next to asymmetric PNEs there always i.e., for any degree of loss aversion exist symmetric PNEs in which both agents exert the same level of effort. In particular, the symmetric Nash equilibrium always constitutes a PNE irrespective of the degree of loss aversion, which can be easily verified using (8) and (9). Proposition 3. There exist symmetric and asymmetric PNEs in which the agents exert a moderate level of effort; i.e., there exist ē and ē with 0 < ē < ē <, such that (e, e) Θ P NE for all e [ē, ē]. The existence of symmetric PNEs for all degrees of loss aversion distinguishes the case of fixed expectations from the case of choice-acclimating expectations. As stated in Kőszegi and Rabin (2007), choice-acclimating expectations result in stronger risk aversion than fixed expectations. For the case of a rank-order tournament, this induces contestants with choice-acclimating expectations to dislike the uncertainty in a symmetric equilibrium so intensely that they choose rather different effort levels, with one agent completely resigning and exerting no effort at all. By resigning this agent reduces his chances to win the tournament but at the same time he is able to moderate her expectations and thus to dampen the pain of a potential loss. With fixed expectations, in contrast, exerting an identical, moderate level of effort is a credible plan for both agents irrespective of their degree of loss aversion. Here, when expecting to exert moderate effort, resignation would reduce an agent s chances to win the tournament without moderating his expectations, such that resignation would badly disappoint the agent s hopes of winning the tournament. This inevitable increase in the likelihood to experience a loss makes the potential deviation unattractive and a symmetric equilibrium always exists if expectations are fixed. In the same spirit, contrary to the case of choice acclimating expectations, fixed expectations yield asymmetric PNEs for all levels of loss aversion. Intuitively, fixed expectations cause an attachment effect. That is, agents get attached to the actions they expect to play and are willing to play these even if there exists a deviation that is associated with slightly higher expected material payoff. In particular, for all asymmetric effort allocations in the environment of the symmetric Nash equilibrium a unilateral deviation towards a symmet- 3

ric effort allocation only provides a small increase in expected material payoff. Hence, these effort allocations indeed constitute PNEs. 5. PREFERRED PERSONAL NASH EQUILIBRIUM In the previous sections, we showed that, for the case of fixed expectations there always exist both symmetric and asymmetric equilibria. The multiplicity of equilibria raises the question which of the prevalent equilibria is most suitable to describe the contestants behavior. To answer this question in the context of individual decision making, Kőszegi and Rabin (2007) propose the notion of preferred personal equilibrium (PPE) as an equilibrium refinement. The PPE is the PE that promises the highest expected utility among all PEs. If players are able to select their most preferred personal equilibrium given any strategy of the opponent, this concept can also be adopted for the context of strategic interaction. We define a preferred personal Nash equilibrium (PPNE) such that every player plays a PPE given his opponent s strategy. Definition 3. The effort choices (ẽ A, ẽ B ) represent a preferred personal Nash equilibrium (PPNE) in the rank-order tournament if and only if for all i {A, B} and j {A, B} with j i, ẽ i Θ P i E (ẽ j ) and U i (ẽ i, ẽ i, ẽ j ) U i (e i, e i, ẽ j ) e i Θ P E i (ẽ j ). (5) Recall that the CPE is an agent s most profitable action among all his actions provided that his expectations are consistent with consequences of the action he actually takes. Hence, in the context of individual decision making, if a CPE constitutes a PE for a player it also is a PPE. By the same reasoning, a CPNE that constitutes a PNE is also a PPNE. In Section 4, we showed that any symmetric CPNE indeed constitutes a PNE and thus is a PPNE. Proposition 2 then allows us to conclude that there always exists a symmetric PPNE as long as gain loss utility does not dominate material utility. The persistence of the symmetric PPNE, however, is even stronger than that of the symmetric CPNE: even if loss aversion dominates material utility so intensely that the symmetric CPNE ceases to exist, there may still exist a symmetric PPNE. Proposition 4. There exists χ > χ such that for all η(λ ) < χ the symmetric Nash equilibrium is a PPNE. 4

The stronger persistence of the symmetric PPNE arises because a contestant with fixed expectations is more limited in his choice of effort namely to those effort levels that constitute a PE than a contestant with choice-acclimating expectations. As explained in Section 3, the symmetric CPNE ceases to exist for high degrees of loss aversion, because one player ultimately chooses to resign for the purpose of reducing uncertainty in the tournament. As was established in Section 4, however, for fixed expectations not exerting any effort is never a credible plan. If a player expected not to exert any effort at all, he would always be better off by surprising himself and exerting slightly positive effort, which comes without cost (by c (0) = 0) but strictly increases his chances of winning. Thus, a contestant who is restricted to choose an effort level that constitutes a PE cannot reduce uncertainty to the same extent as a contestant who is not restricted in this regard. In consequence, possible deviations from the symmetric equilibrium are less attractive and the symmetric Nash equilibrium is a PPNE for even stronger degrees of loss aversion than for which it is a CPNE. We conclude that, if the players effort decision and the realization of the tournament outcome take place in quick succession, symmetric equilibria are quite persistent. In particular, they always exist if players are not able to select among their PEs. And even if players can do so, there always exists a symmetric PPNE as long as players concerns for gain loss utility does not clearly outweigh those for material utility. In this case, under PNE and PPNE the behavior of expectation-based loss-averse players resembles the behavior of players with standard utility or players with exogenously given reference points (cf. Gill and Stone (200) and Gill and Prowse (202)). When comparing the results from Proposition 2 and Proposition 4, it becomes apparent that the two solution concepts CPNE and PNE/PPNE may predict substantially different equilibrium outcomes. For a tournament designer it is thus an important issue which equilibrium outcome to expect. As explained before, this depends on the span of time that passes between the agents effort decision and the realization of the tournament outcome. If this time span is rather short, expectations do not acclimate to decisions and it is likely that the PNE/PPNE prevails whereas a longer time span favors the CPNE as the tournament outcome. In many applications of rank-order tournaments, however, it seems plausible that this time span is not exogenously given but can be chosen by the 5

tournament designer herself. While a fully fleshed out analysis of the resulting consequences for tournament design is beyond the scope of this paper, as an example, suppose η(λ ) (χ, χ). In this case, the symmetric Nash equilibrium constitutes a PPNE whereas all existing CPNEs are asymmetric. If the designer then pays prizes right after the effort stage, a symmetric PPNE might arise, whereas postponing the payment of prizes would induce some asymmetric CPNE. As soon as the principal opts for a large price spread which will be the case if output π h is sufficiently high overall effort in the symmetric PPNE strictly exceeds overall effort in the asymmetric CPNE. 0 Hence, when deciding about the timing of events, for π h sufficiently high the tournament designer can achieve higher overall effort if he chooses to reveal the tournament outcome rather sooner than later. 6. CONCLUSION Many of our insights about rank-order tournaments build upon the premise that symmetric equilibria exist. As shown by Gill and Stone (200), the existence of symmetric equilibria may fail if contestants are expectation-based loss averse and have choice-acclimating expectations. However, in rank order tournaments it is conceivable that the time span between the contestants effort choices and the realization of the tournament outcome is rather short, such that expectations have not enough time to adapt to actual decisions and are thus fixed. In this paper, we study the resulting behavior of contestants under such choiceunacclimating expectations. We find that symmetric and asymmetric equilibria exist for all degrees of loss aversion if expectations are choice-unacclimating. The symmetric Nash equilibrium, however, always prevails if each player follows his preferred credible game plan and concerns for psychological gain-loss utility do not clearly outweigh material consumption utility. This makes the persistence of the sym- 0 If the price spread is large enough, we get that e NE 2. The boundary asymmetric CPNEs characterized by only one agent exerting positive effort are clearly worse in terms of overall effort. Moreover, any possible interior asymmetric CPNE (e CP NE CP NE i, ej CP NE ) in which both CP NE agents exert a positive amount of effort is characterized by ei < e NE < ej. Using the contestants first order conditions it is however evident that agent i reduces effort more severely than agent j increases effort. 6

metric Nash equilibrium under fixed expectations unambiguously stronger than under choice acclimating expectation. In particular, for the Nash equilibrium not to be a mutual preferred personal equilibrium, loss aversion has to be so strong that players would also be willing to choose stochastically dominated options. We conclude that under fixed expectations a focus on symmetric equilibria is often justifiable even if players are expectation-based loss averse. This paper also adds to the emerging literature that analyzes strategic interaction of expectation-based loss-averse agents by investigating how the equilibrium concepts of Nash equilibrium, personal Nash equilibrium, and choiceacclimating Nash equilibrium relate to each other. Regarding rank-order tournaments, a desirable next step would be to explore the implications of expectationbased loss aversion in dynamic tournaments à la Rosen (986), where choiceacclimating expectations and lagged fixed expectations do not necessarily represent alternative modeling choices: as tournament play evolves, choice-acclimating expectations might apply in the very first round, whereas decisions in later rounds are taken with a fixed set of expectations. REFERENCES ABELER, J., A. FALK, L. GOETTE, AND D. HUFFMAN (20): Reference Points and Effort Provision, American Economic Review, 0(2), 470 492. BELL, D. (985): Disappointment in Decision Making under Uncertainty, Operations Research, 33, 27. BERGERHOFF, J., AND A. VOSEN (204): How Being Behind Can Get You Ahead, mimeo. CRAWFORD, V. P., AND J. MENG (20): New York City Cab Drivers Labor Supply Revisited: Reference-Dependent Preferences with RationalExpectations Targets for Hours and Income, American Economic Review, 0(5), 92 932. Strategic interaction of expectation-based loss-averse agents has been primarily analyzed in rather specific environments like tournaments (Gill and Stone, 200; Bergerhoff and Vosen, 204), team production (Gill and Stone, 204), team compensation (Daido and Murooka, 204), or auctions (Lange and Ratan, 200). A more general approach is presented in Dato, Grunewald, and Müller (204). 7

DAIDO, K., AND T. MUROOKA (204): Team Incentives and Reference- Dependent Preferences, working paper. DATO, S., A. GRUNEWALD, AND D. MÜLLER (204): Expectation-Based Loss Aversion and Strategic Interaction, mimeo. DEMOUGIN, D., AND C. FLUET (2003): Inequity Aversion in Tournaments, working paper. ERICSON, K. M. M., AND A. FUSTER (20): Expectations as Endowments: Evidence on Reference-Dependent Preferences from Exchange and Valuation Experiments, Quarterly Journal of Economics, 26(4), 879 907. GILL, D., AND V. PROWSE (202): A Structural Analysis of Disappointment Aversion in a Real Effort Competition, American Economic Review, 02(), 469 503. GILL, D., AND R. STONE (200): Fairness and Desert in Tournaments, Games and Economic Behavior, 69, 346 364. (204): Desert and Inequity Aversion in Teams, Journal of Public Economics, forthcoming. GRUND, C., AND D. SLIWKA (2005): Envy and Compassion in Tournaments, Journal of Economics and Management Strategy, 4, 87 207. HART, O., AND B. HOLMSTRÖM (987): The Theory of Contracts, in Advances in Economic Theory Fifth World Congress, ed. by T. F. Bewlwy, pp. 70 55, New York. Cambridge University Press. KAHNEMAN, D., AND A. TVERSKY (979): Prospect Theory: An Analysis of Decision under Risk, Econometrica, 47, 263 292. KŐSZEGI, B., AND M. RABIN (2006): A Model of Reference-Dependent Preferences, Quarterly Journal of Economics, 2, 33 65. (2007): Reference-Dependent Risk Attitudes, American Economic Review, 97, 047 073. KRÄKEL, M. (2008): Emotions in Tournaments, Journal of Economic Behavior and Organization, 67, 204 24. 8

KRÄKEL, M., AND P. NIEKEN (205): Relative Performance Pay in the Shadow of Crisis, European Economic Review, 74, 244 268. LANGE, A., AND A. RATAN (200): Multi-Dimensional Reference- Dependent Preferences in Sealed-Bid Auctions - How (most) Laboratory Experiments Differ from the Field, Games and Economic Behavior, 68, 634 645. LAZEAR, E., AND S. ROSEN (98): Rank-Order Tournaments as Optimum Labor Contracts, Journal of Political Economy, 89, 84 864. LUDWIG, S., P. WICHARDT, AND H. WICKHORST (20): Overconfidence can Improve an Agent s Relative and Absolute Performance, Economics Letters, 0, 93 96. ROSEN, S. (986): Prizes and Incentives in Elimination Tournaments, American Economic Review, 76, 70 75. SANTOS-PINTO, L. (200): Positive Self-Image in Tournaments, International Economic Review, 5, 475 496. A. BEST-RESPONSE FUNCTIONS UNDER CHOICE-ACCLIMATING EXPECTATIONS To prove the statements made in the text, we will formally establish the following result. Lemma 2. Either agent i s best response function under choice-acclimating expectations is given by BR i (e j ) = e i (e j ) for all e j [0, ], or there exists ē (0, ] such that agent i s best response is given by e i (e j ) if e j ē BR i (e j ) =, 0 if e j ē where e i (e j ) (0, ), de i (e j) de j < 0 and d2 e i (e j) de j 2 < 0. 9

Proof of Lemma 2. First, we establish the comparative static results listed in the lemma. Given that e i (e j ) is a local maximizer of U i (e i, e i, e j ), we have 2 U i (e i (e j ), e i (e j ), e j )/ e 2 i < 0. Implicit differentiation of the first-order condition U i (e i (e j ), e i (e j ), e j )/ e i = 0 then yields de i (e j ) de j = 2 U i (e i (e j ), e i (e j ), e j )/ e i e j 2 U i (e i (e j), e i (e j), e j )/ e 2 i = η(λ ) 2 2 U i (e i (e j), e i (e j), e j )/ e 2 i < 0(6) and d 2 e i (e j ) de 2 j = η(λ 2 )c (e i (e j )) de i (e j) de j [ 2 U i (e i (e j), e i (e < 0. (7) j), e j )/ e 2 i ]2 Next, we derive player i s best response function BR i (e j ) for each of the following cases: (i) η(λ ) < ; (ii) η(λ ) 2 c (0); (iii) 2 c (0) < η(λ ); (iv) 2 c (0) < η(λ ) Case (i): η(λ ) < Recall that the derivatives of the expected utility function are given by and U i (e i, e i, e j ) e i = 2 [ + η(λ )(e i e j )] c (e i ) (8) 2 U i (e i, e i, e j ) e 2 i = 2 η(λ ) c (e i ). (9) As η(λ ) <, we have that U i (0, 0, e j )/ e i > 0, i.e., U i (e i, e i, e j ) is strictly increasing for small values of e i irrespective of e j. As explained in the text, this implies that U i (e i, e i, e j ) has an interior local maximum that is also its global maximizer. Hence, BR i (e j ) = e i (e j ) for all e j [0, ]. Case (ii): η(λ ) 2 c (0) By c > 0 and 2 U i (0, 0, e j )/ e 2 i 0, we have 2 U i (e i, e i, e j )/ e 2 i < 0 for all e i (0, ]; i.e., expected utility is strictly concave. First, consider e j < such that U i (0, 0, e η(λ ) j )/ e i > 0. In this case, there exists a unique value e i (e j ) (0, ) such that U i (e i (e j ), e i (e j ), e j ) = 0. By strict concavity of U i (e i, e i, e j ), e i (e j ) is the global maximizer of U i (e i, e i, e j ). For all e j η(λ ), we have U i (0, 0, e j )/ e i 0. By strict concavity of U i (e i, e i, e j ), we then have U i (e i, e i, e j )/ e i < 0 for all e i (0, ] and the global maximizer of U i (e i, e i, e j ) is given by e i = 0. Thus, ē = η(λ ). 20

For cases (iii) and (iv), define the following functions to make the proof more tractable, θ(e i ) = c (e i ) 2 η(λ )e i and ψ(e j ) = 2 [ η(λ )e j], (20) such that U i (e i, e i, e j ) e i = ψ(e j ) θ(e i ) and 2 U i (e i, e i, e j ) e 2 i = θ (e i ). (2) 2 Regarding the function ψ(e j ), note that ψ(0) > 0, ψ (e j ) < 0, and < 0. Furthermore, ψ() 0 if and only if η(λ ). dψ() dη(λ ) = Concerning the function θ(e i ), first note that θ(0) = 0. Furthermore, θ (e i ) > 0, i.e., θ(e i ) is strictly convex, and θ (0) 0 if and only if η(λ ) 2 c (0). Finally, with lim ei c (e i ) = and lim ei c (e i ) =, we have lim ei θ(e i ) = and lim ei θ (e i ) =. Hence, if η(λ ) > 2 c (0), the global minimizer e min (0, ) of θ(e i ) is implicitly defined by θ (e min ) = c (e min ) η(λ ) = 0, in which case 2 θ (e i ) 0 if and only if e i e min. Also, > 0, lim η(λ ) e min =, and dθ(e min) de min dη(λ ) = 2c (e min ) dη(λ ) = e 2 min < 0. Note that e min is an inflection point such that U i (e i, e i, e j ) is strictly convex for e i < e min and strictly concave for e i > e min. See Figure 4 for a graphical representation of θ(e i ) and ψ(e j ). Case (iii): 2 c (0) < η(λ ) In this case, ψ() < 0. With lim η(λ ) 2 c (0) θ(e min) = θ(0) = 0 and dψ() dη(λ ) < 0, we have ψ() < θ(e min ) < 0 < ψ(0). Hence, there exist ēj dθ(e min ) dη(λ ) and ē j, where [ēj, ē j ] (0, ), implicitly defined by ψ(ēj ) = 0 and ψ(ē j ) = θ(e min ). For each e j [ēj, ē j ) there exist two values of e i, one strictly smaller and the other strictly larger than e min, such that ψ(e j ) = θ(e i ) or, equivalently, U i (e i, e i, e j )/ e i = 0. The smaller of these e i values is a local minimizer and the larger one, denoted by e i (e j ), is a local maximizer of U i (e i, e i, e j ). Expected utility from exerting effort e i (e j ) amounts to U i (e i (e j ), e i (e j ), e j ) = U i (0, 0, e j ) + e i (e j ) 0 [ψ(e j ) θ(e i )]de i. (22) With U i (e i (ēj ), e i (ēj ), ēj ) > U i (0, 0, ēj ), U i (e i (ē j ), e i (ē j ), ē j ) < U i (0, 0, ē j ), and d[u i (0, 0, e j ) U i (e i (e j ), e i (e j ), e j )] de j = e i (e j ) 0 ψ (e j )de i = η(λ ) > 0,(23) 2 2

θ(e i ) ψ(0) ψ(ēj ) ψ(ē) ψ(ē j ) ψ() e min e i Figure 4: Graphical representation of the derivation of ē there exists ē (ēj, ē j ) such that the global maximizer of U i (e i, e i, e j ) is e i (e j ) > 0 for e j ē and e i = 0 for e j ē. Case (iv): 2 c (0) < η(λ ) Again, U i (e i, e i, e j ) is convex for e i < e min and concave for e i > e min. First, consider e j [0, η(λ ) ). In this case, with ψ(e j) > 0, U i (e i, e i, e j ) is strictly increasing for small values of e i ; i.e., U i (0, 0, e j )/ e i > 0. There exists a unique value of e i, denoted by e i (e j ) and strictly smaller than, such that ψ(e j ) = θ(e i (e j )). With ψ(e j ) θ(e i ) or, equivalently, U i (e i, e i, e j )/ e i 0 if and only if e i e i (e j ), e i (e j ) is the global maximizer of U i (e i, e i, e j ). Next, consider e j [ η(λ ), ]. Note that for e j > η(λ ), U i (e i, e i, e j ) is strictly decreasing for small values of e i i.e., U i (0, 0, e j )/ e i < 0. For η(λ ) =, ψ() = 0 such that ψ() θ(e min ) > 0. For η(λ ), e min such that lim η(λ ) ψ() θ(e min ) =. From d[ψ() θ(e min )] dη(λ ) = 2 ( e min) < 0 (24) then follows the existence of a threshold χ >, such that ψ() θ(e min ) if and only if η(λ ) χ. For η(λ ) < χ, there are two values of e i, one strictly smaller and the other strictly larger than e min, such that ψ() = θ(e i ). The larger of these e i values, which we denote by e i (), is strictly smaller than 22

and a local maximizer of U i (e i, e i, ). With U i (e i (), e i (), ) = U i (0, 0, ) + and U i (0, 0, ) = u(l), we obtain du i (0, 0, ) U i (e i (), e i (), ) dη(λ ) e i () 0 = 2 e i () 0 [ψ() θ(e i )]de i, (25) ( e i )de i > 0, (26) where we made use of ψ() θ(e i ()) = 0. Furthermore, for η(λ ) = we have ψ() = 0 such that U i (0, 0, ) U i (e i (), e i (), ) = e i () 0 [ θ(e i )]de i < 0. For η(λ ) = χ, on the other hand, we have U i (0, 0, ) U i (e i (), e i (), ) = e i () 0 [θ(e min ) θ(e i )]de i > 0. Hence, by the intermediate value theorem, there exists χ (, χ ) such that U i (e i (), e i (), ) U i (0, 0, ) if and only if η(λ ) χ. By (23), we have d[u i (0,0,e j ) U i (e i (e j),e i (e j),e j )] de j > 0, where e i (e j ) is defined as before. Hence, if η(λ ) < χ and, thus, U i (e i (), e i (), ) > U i (0, 0, ), then U i (e i (e j ), e i (e j ), e j ) > U i (0, 0, e j ) for all e j [0, ]. In this case, BR i (e j ) = e i (e j ) for all e j [0, ]. If, on the other hand, η(λ ) χ, there exists ē (0, ] such that e i (e j ) is the best response to e j [0, ē] and e i = 0 is the best response to e j [ē, ]. B. PROOFS Proof of Proposition. A symmetric equilibrium must be interior. First, (e A, e B ) = (0, 0) cannot constitute a CPNE since U i (e i,e i,0) e i ei =0 > 0. Analogously (e A, e B ) = (, ) is not a CPNE since U i (e i,e i,) e i ei = < 0. Hence, a symmetric CPNE must be characterized by a solution to the first-order condition, which (given e i = e j ) boils down to c (e ) = ( ) 2 e = c = e NE. (27) 2 (i) The symmetric CPNE exists if and only if the best response to e j = e NE is given by e i (e j ). Then the best response curves of player i and j in the symmetric equilibrium are both decreasing with identical slope. The symmetric equilibrium therefore is asymptotically unstable if and only if de i (e j ) < η(λ ) > c (e NE ) de j. (28) ej =e NE 23

(ii) A symmetric equilibrium cannot exist if U i (e i, e i, e NE ) is strictly convex at e i = e NE, i.e. 2 U i (e i, e i, e j ) e 2 i 0 η(λ ) 2c (e NE ). (29) ei =e NE (iii) According to part (ii), a finite value of η(λ ) exists for which the symmetric CPNE ceases to exist. For any value of η(λ ) above this threshold, the CP NE CP NE two asymmetric CPNE (ea, eb ) = (e CP NE CP NE (0), 0) and (ea, eb ) = (0, e (0)) exist. To see this, note that for a symmetric CPNE not to exist, each agent s best response function must display a downward discontinuity at ē (0, ) with e i (ē) = e j(ē) > ē. Since e i (e j ) is decreasing in e j we conclude that BR i (0) = e i (0) > e i (ē) > ē, such that BR j (e i (0)) = 0. Overall, given agent i plays his best response to zero effort, exerting zero effort is indeed a best response for agent j. The considered asymmetric CPNEs are asymptotically stable because BR i (e j ) = 0 for e j [ē, ] and ē < BR j (0) <. Proof of Proposition 2. As outlined in the text, for a given effort level e j of the opponent, agent i s best response is either minimum effort e i = 0 or (in case it exists) the interior local maximizer e i = e i (e j ). Furthermore, given e j = e NE, e i = e NE always satisfies U i (e NE, e NE, e NE )/ e i = 0. Finally, as established in the proof of Proposition, 2 U i (e NE, e NE, e NE ) e 2 i 0 η(λ ) 2c (e NE ). (30) Finally, we have 2 c (e NE ) >, which follows from c (0) = 0, c > 0, and c > 0 together with c (e NE ) = 2 and ene <. For η(λ ), we know that e i = e NE constitutes not only a local but also the global maximum of U i (e i, e i, e NE ): With U i (0, 0, e NE )/ e i > 0, we have U i (e i, e i, e NE )/ e i 0 if and only if e i e NE, such that e i = e NE constitutes the global optimizer of U i (e i, e i, e NE ). In particular, note that U i (e NE, e NE, e NE ) > U i (0, 0, e NE ). For η(λ ) 2c (e NE ), on the other hand, e i = e NE is not a candidate for agent i s best response to e j = e NE : With 2 U i (e NE, e NE, e NE )/ e 2 i 0, e i = e NE is either a local minimum or an interior inflection point of a strictly decreasing function. In either case, since 2 U i (0, 0, e NE )/ e 2 i > 0, it follows that U i (0, 0, e NE )/ e i < 0, such that U i (e NE, e NE, e NE ) < U i (0, 0, e NE ). 24

To conclude the argument, note that [ U i (e NE, e NE, e NE ) U i (0, 0, e NE ) ] η(λ ) = ene2 4 < 0, (3) i.e., as η(λ ) increases, U i (e NE, e NE, e NE ) U i (0, 0, e NE ) monotonically decreases at a constant rate. By the intermediate value theorem, there exists χ (, 2c (e NE )/ ) such that U i (e NE, e NE, e NE ) U i (0, 0, e NE ) if and only if η(λ ) χ. For η(λ ) < 2c (e NE )/, we have 2 U i (e NE, e NE, e NE )/ e 2 i < 0. Hence, for η(λ ) (, 2c (e NE )/ ), e i = e NE is a local maximizer of U i (e i, e i, e NE ) and thus, next to e i = 0, the only candidate for agent i s best response to e j = e NE. Thus, the symmetric CPNE exists if and only if η(λ ) χ. Proof of Lemma. The fact that the set Θ P i E (e j ) is an interval is established in the text. It thus remains to establish the comparative statics with regard to the boundaries of this set. Differentiation of θ(ē i (e j )) = ψ(e j ) with respect to e j yields and d 2 ē i (e j ) de 2 j θ (ē i (e j )) + dē i (e j ) de j θ (ē i (e j )) = ψ (e j ) < 0 (32) ( dēi (e j ) de j ) 2 θ (ē i (e j )) = ψ (e j ) = 0 (33) Due to the convexity of = 0, and ψ(e j ) > 0, we must have θ, θ(0) θ (ē i (e j )) > 0, such that (32) implies dē i(e j ) de j < 0. Since furthermore θ (ē i (e j )) > 0, (33) implies d2 ē i (e j ) < 0. de 2 j By analogous reasoning, differentiation of θ(ēi (e j )) = ψ(e j ) with respect to e j yields d ē i (e j ) de j < 0 and d2 ē i (e j ) < 0 de 2 j Proof of Proposition 4. According to Proposition 2, a symmetric CPNE exists if and only if η(λ ) χ. This symmetric CPNE is identical to the Nash equilibrium and, thus, also a PNE and a PPNE. It remains to show that there exists χ > χ such that the same holds true for η(λ ) ( χ, χ]. For this purpose, note that the smallest effort level that is a PE is strictly positive. For η(λ ) = χ, we have that U i (e NE, e NE, e NE ) = U i (0, 0, e NE ) > U i (e i, e i, e NE ) for all 25

e / {0, e NE }. As playing e i = 0 is not a PE, (e NE, e NE ) is the unique PPNE in this case. For η(λ ) marginally larger than χ, e i = e NE remains a local maximum of U i (e i, e i, e NE ) and only effort levels marginally close to zero provide a higher utility when the player expects to play them than expecting to play and playing e NE. As the smallest effort that constitutes a PE is strictly larger than 0, however, there is no PE that provides higher utility than expecting and playing e i = e NE. As a consequence, there exists χ > χ such that (e i, e j ) = (e NE, e NE ) is a PPNE for all η(λ ) [ χ, χ]. 26