Designing Multi-Person Tournaments with Asymmetric Contestants: An Experimental Study. Hua Chen, Sung H. Ham and Noah Lim +

Transcription

1 Designing Multi-Person Tournaments with Asymmetric Contestants: An Experimental Study Hua Chen, Sung H. Ham and Noah Lim + + All authors contributed equally. Chen: Bauer College of Business, University of Houston, [email protected]. Ham: College of Business Administration, Kent State University, [email protected]. Lim: Wisconsin School of Business, University of Wisconsin-Madison, [email protected]. We thank seminar participants at Duke University, University of Houston, University of Southern California and the 2009 INFORMS Conference for helpful comments.

2 Designing Multi-Person Tournaments with Asymmetric Contestants: An Experimental Study Abstract Is the right amount of effort exerted in multi-person tournaments where contestants have two different levels of initial endowments (termed "favorites" and "underdogs")? We develop theoretical predictions for the level of effort and the effect of varying the prize structure. We test these predictions for threeperson tournaments using an economic experiment in a social environment where contest outcomes are publicly announced. We find that both favorites and underdogs overexert effort relative to the theoretical point predictions. Moreover, in the treatment with two favorites and one underdog, favorites increase their effort when the number of prizes is increased from one to two, contrary to the theory prediction. We show that a generalized model that allows for psychological losses from losing for favorites and psychological gains from winning for underdogs due to social comparisons tracks the experimental results better than the standard theoretical model. Keywords: Tournaments, Compensation, Sales Management, Experimental Economics, Behavioral Economics

3 1 1. Introduction Tournaments are relative-based incentives that are applied to many corporate settings. Examples include contests for salespeople, customer service representatives, franchisees and retailers, job promotions within organizations and patent races between product development teams. Despite their widespread use, there is very little theoretical or empirical research that guides managers on how to design an optimal tournament, especially when the tournament involves more than two contestants and when the contestants are asymmetric, i.e., some of them may be advantaged or disadvantaged due to different initial market endowments or abilities. Specifically, in a tournament with three or more contestants and a given mix of contestants with asymmetric endowments, how does changing the number of prizes affect the effort of the advantaged contestants versus that of the disadvantaged contestants? What happens when the endowment-mix of the contestants changes? 1 The above questions have neither been theoretically nor empirically examined. The extant literature on tournaments has focused on examining three issues. First, the seminal papers on tournament theory (Lazear and Rosen 1981; Green and Stokey 1983; Nalebuff and Stiglitz 1983) show that tournaments, which are relative-based contracts, can yield effort levels that are identical to or higher than individual-based piece rate contracts. Second, Nalebuff and Stiglitz (1983) and Kalra and Shi (2001) examine how the prize structure affects effort in a multi-person tournament (i.e., when there are three or more contestants) with contestants who have symmetric endowments. These papers show theoretically that the optimal design of the prize structure in a tournament depends on the contestants level of risk aversion and the distribution of the random shocks that affect contestants output. In particular, Kalra and Shi (2001) show that given a fixed budget, the optimal prize structure in a tournament should have more than one prizewinner with unique rank-ordered prizes when contestants are risk averse and the random shocks that affect contestants output are logistically distributed. Third, Lazear and Rosen (1981) and O Keeffe, Viscusi and Zeckhauser (1984) study how asymmetry in endowments and abilities between contestants in a two-person tournament affects the effort levels relative to the case where there is no asymmetry. They find that asymmetric tournaments lead to a reduction in effort when compared to symmetric tournaments. The only paper that studies multi-person tournaments while allowing for contestant asymmetry is Orrison, Schotter and Weigelt (2004). However, they only examine the case where both the number of advantaged contestants and the number of prizes are exactly equal to one-half of the total number of contestants and focus on how effort varies as the total number of contestants 1 CEMEX executives recently faced similar design issues for their sales contests as described in the Harvard Business Case (Martinez-Jerez, Bellin and Winkler 2006).

4 2 increases. 2 Given that managers often need to design tournaments that involve more than two contestants and that there are usually differences in market endowments or ability among the contestants, it is therefore critical to plug the current theoretical gap in understanding how altering the prize structure in a multi-person tournament with asymmetric contestants affects effort. Moreover, because the decision calculus needed to arrive at the equilibrium predictions of tournament models is relatively complex, it is useful to empirically validate these predictions, particularly using laboratory economic experiments. 3 Much of the extant experimental work on tournaments investigates the predictions of the aforementioned theoretical papers. Bull, Schotter and Weigelt (1987) and Orrison, Schotter and Weigelt (2004) find that contestants choose effort consistent with tournament theory in two-person and in multi-person symmetric tournaments. Lim, Ahearne and Ham (2009) test the theoretical model of Kalra and Shi (2001) and confirm that when there is a fixed budget, a tournament with multiple prizes elicits higher effort than a winner-take-all tournament. Schotter and Weigelt (1992) examine two-person asymmetric tournaments experimentally and find that effort is indeed lower when compared to a symmetric tournament. They also find that the contestants who are disadvantaged (in terms of initial endowments or abilities) overexert effort. Orrison, Schotter and Weigelt (2004) obtain experimental results consistent with the prediction of tournament theory that the reduction in effort due to contestant asymmetry will be attenuated as the size of the tournament (i.e., the number of contestants) increases. While much of the experimental literature indicates broad support for the directional predictions of tournament theory, we note that these experiments are conducted in a social environment where the contestants usually do not know who they are competing against and the tournament outcomes of winning and losing are not publicly announced. However, in most tournaments, contestants usually know the identities of their competitors and whether their competitors win or lose. For example, at the conclusion of sales contests, many companies hold award ceremonies or recognize their top sellers in the lobbies of their corporate headquarters. Also, many organizations hold promotion parties for their employees. In a recent paper, Lim (2010) shows that changing the social environment of a tournament can lead to departures from the predictions of tournament theory when contestants make social comparisons. Hence, 2 This also means that the total number of contestants in the tournament must be an even number. Additionally, in this special case, the equilibrium effort levels of favorites and underdogs are identical. We show that this is not always the case when the endowment-mix of the contestants and the prize structure change. 3 See Amaldoss and Jain (2005b), Ding et al. (2005), Wang and Krishna (2006) for successful applications of this methodology to other business contexts.

5 3 we believe that it is also important to construct a social setting in the laboratory that maps closely to the social environments of tournaments in the real world when testing tournament theory s predictions. 4 This paper makes three contributions to the literature on designing optimal tournaments. First, we use tournament theory to examine the effects of varying the prize structure in a multi-person tournament when there is a mix of contestants with two different levels of initial endowments. We term the advantaged contestants favorites and the disadvantaged contestants underdogs. We show for the standard tournament model with up to four contestants that introducing additional prizes will not increase the effort of favorites depending on the relative proportion of prizes to the number of favorites, the favorites will either reduce or maintain their effort levels. In contrast, increasing the number of prizes will not decrease the effort of underdogs they will either increase or maintain their effort depending on the ratio of prizes to the number of favorites. These results are valuable to managers who are interested in understanding how their decisions on the number of prizes in a multi-person tournament impact the effort levels of different contestant subgroups for a given mix of advantaged and disadvantaged contestants. Second, we test the above theory predictions for a three-person tournament using an economic laboratory experiment (Experiment 1). We do so by varying the number of winning prizes in the tournament (one or two) and the number of favorites (one or two) in our experimental design. A distinguishing feature of our experiment is that unlike the typical economic laboratory experiment, we allow contestants to know who they compete with and reveal the tournament outcomes of winning or losing for each contestant to all participants in the experimental session in every decision round. The results show that the predictive power of the theoretical model is mixed: Both favorites and underdogs exert effort that is higher than the equilibrium predictions of the theoretical model in all of the treatment conditions. When there is one favorite and two underdogs, having an additional prize decreases the effort of favorites as predicted. However, when there are two favorites and one underdog, the favorites effort rises as the number of prizes is increased from one to two, contrary to the theory prediction. For underdogs, introducing incremental prizes increases their effort regardless of the number of favorites in the tournament, which is directionally consistent with the theoretical prediction. Third, to explain the overexertion of effort by the contestants and the observed pattern of behavior by the favorites, we introduce a behavioral economics model that generalizes the standard tournament model by allowing for social comparison effects in the contestants utility function: For favorites, we assume that there is an additional non-pecuniary component that captures the disutility from 4 Nevertheless, we recognize that testing the predictions of tournament theory in standard laboratory conditions remains valuable in understanding behavior in tournaments.

6 4 losing and that this disutility is exacerbated when they lose to an underdog. Conversely, for underdogs, we assume that there is an additional non-pecuniary component that captures the joy of winning a prize and that this utility gain is enhanced if they defeat a favorite in the process of winning. We then econometrically estimate the psychological parameters of the model using the experimental data and show that the generalized model tracks behavior much better than the standard theoretical model and other nested behavioral models. Interestingly, the parameter estimates show that there is social loss aversion when contestants compare outcomes across contestant subgroups the additional disutility favorites suffer from losing to an underdog is about 2.1 times greater than the extra joy underdogs experience from beating a favorite. We also provide further validation of the proposed behavioral economics model through two follow-up experiments. Experiment 2A in this paper shows that the effort decisions of participants are closer to the predictions of the standard tournament model in a social environment where social comparisons are less intense, which lends support to our approach of generalizing the theoretical model to capture social comparison effects. Experiment 2B shows that the behavioral economics model estimated from the data in Experiment 1 can predict behavior in another tournament well. The paper proceeds as follows: The next section presents the theoretical model and results that characterize how changing the prize structure affects contestants effort in a multi-person tournament where contestants are asymmetric. Section 3 presents the hypotheses to be tested, the experimental design and the results of the main experiment. In Section 4, we introduce the behavioral economics model that captures the additional utility loss (gain) the favorite (underdog) experiences from losing (winning) and demonstrate that the generalized model explains behavior better. Section 5 presents two additional experiments that further validate the behavioral economics model. Finally, we conclude with the managerial implications and directions for future research. 2. Theory: Effort in a Multi-Person Tournament with Asymmetric Contestants Consider a multi-person tournament with contestants who are ranked according to their outputs, e.g., sales in dollars, from highest to lowest. The contestants ranked 1 to m each receives a winning prize of monetary value, while the contestants ranked m+1 to N each receives a smaller loser s award of. The output metric of contestant is, where is the contestant s effort level and is uniformly distributed on the interval. The stochastic term, which is i.i.d. across the contestants, reflects the component of output that is influenced by environmental

7 5 shocks that are out of the contestants control. The parameter is the known asymmetry across contestants in output. We capture contestant asymmetry through an additive constant rather than through different marginal returns to effort, so that can be interpreted as an initial endowment rather than ability. These differences in endowments can arise in a number of ways: In a sales contest, salespeople may have the same selling ability but may sell in different markets, which may differ in the number of existing customers who require minimal effort to sell to for example, these customers could have intended to make purchases even without the salesperson contacting them. It can also describe a situation where contestants have received feedback about interim performances in a tournament that is held across multiple periods (O Keeffe, Viscusi and Zeckhauser 1984). Finally, can capture rules made (deliberately or unwittingly) by the tournament organizer that favor (or are perceived to favor) some contestants over others. Furthermore, we assume that there are only two sub-groups of contestants, one that is advantaged and one that is disadvantaged, which are termed as favorites and underdogs respectively, so that 0< < for favorites and for underdogs. Favorites enjoy an advantage over the underdogs in terms of winning the tournament and the number of favorites in the tournament is. The contestants are assumed to be risk neutral and have identical utility functions that are separable in the utility from the tournament outcomes of winning or losing and the cost of effort exerted,, which is assumed to be strictly increasing and convex. Given a prize structure with m winning prizes, each contestant evaluates the tradeoff between the utility of winning the tournament and the cost of effort and chooses the effort level that maximizes her expected utility. Denoting the probability of winning a prize of as, the expected utility of a contestant is given by. (2.1) Furthermore, assuming, the pure-strategy Nash equilibrium for effort can be obtained from the following first order condition: 5. (2.2) where the term is the marginal probability of winning, the increase in the probability of receiving due to an increase in effort. Although there is no general formula for the marginal probability of winning for favorites and underdogs in a multi-person tournament with any number of 5 We assume that the participation constraints for favorites and underdogs are satisfied and confirm that this was the case in the experiments.

8 6 prizes, the expressions for can be derived given the number of contestants, N, the number of favorites, N F, and the number of prizes, m. Tables A1a and A1b show these expressions for favorites and underdogs in tournaments with N=3 and N=4 for all possible combinations of N F and m. Given the expressions of, we can then solve for the equilibrium effort of favorites and underdogs. When N=3, we are able to derive closed-form solutions for equilibrium effort. However, when N>3, we are required to rely on numerical simulations to solve for equilibrium effort in some cases due to complexities in evaluating the higher-order polynomials in the expressions of, as illustrated in Table A1b for the N=4 tournament (e.g., effort of the favorite when there is one favorite and two prizes). Given the scope of this paper, we consider only the cases where N=3 and N=4. For the cases where we could not derive analytic solutions of effort when N=4, we perform extensive numerical simulations to check the predictions. Our goal is to examine the change in the contestants effort due to the introduction of an additional prize so that we can guide managers as they decide on the prize structure of a tournament. We obtain the following results for the N=3 and N=4 scenarios: Result 1A. Regardless of the proportion of favorites in a tournament with three or four contestants, increasing the number of prizes cannot increase the effort of favorites. Result 1B. Regardless of the proportion of favorites in a tournament with three or four contestants, increasing the number of prizes cannot decrease the effort of underdogs. PROOF. See the Appendix. The intuition behind the above results can be readily explained. Although increasing the number of prizes in the tournament raises the probability of winning a prize for all contestants, the effect of adding a prize on the behavior of favorites and underdogs is different, since it is the marginal probability of winning that ultimately determines the effort of contestants. We begin by describing the equilibrium behavior of the favorites. When the number of favorites N F is less than or equal to the number of prizes m, there are enough available prizes for all favorites to win. Increasing the number of prizes in this scenario always decreases the effect of putting in additional effort on the probability of winning for the favorites (i.e., becomes smaller) who already enjoy an advantage for winning, thereby reducing their equilibrium effort. When the number of favorites N F is greater than the number of prizes m, however, at least one of the favorites will always lose in the tournament. In this case, the favorites will compete as if they are competing only with contestants within

9 7 their own subgroup, so that their equilibrium effort level is identical to that of a tournament with symmetric contestants. When the number of prizes is raised to m+1, if the number of favorites, N F, is equal to m+1, there will now be enough prizes for all favorites to win in the tournament and consequently, favorites will reduce their effort. However, and this can be the case when N=4, if N F remains higher than m+1, the effort of favorites does not drop because they still compete within their subgroup. This latter result can be shown analytically for the N=4 tournament note from Table A1b that the marginal probability of winning when there are three favorites remains at when the number of prizes is increased from one to two. Overall, favorites will either reduce or maintain their effort levels, so that increasing the number of prizes will not increase their effort. Next, we examine the equilibrium behavior of underdogs. When the number of prizes m is less than or equal to the number of favorites N F, all underdogs may lose in the tournament since they need to outperform the favorites in order to win. Introducing more prizes in this case will raise the probability of an underdog receiving a prize through expending effort (i.e., becomes larger), leading to an increase in equilibrium effort of underdogs. When the number of prizes is greater than the number of favorites however, at least one of the underdogs will always win. Once this is assured, adding more prizes so that more underdogs are guaranteed to win does not raise their effort further. 6 In our model, this scenario occurs when N=4, when there is only one favorite in the tournament, and the number of prizes is increased from two to three. Note from the first row of Table A1b that the marginal probability of winning for underdogs remains at, so it can be shown analytically that underdogs will maintain their effort level in this case. Notice also that when there are enough prizes for underdogs to win, underdogs compete as if they are competing only with contestants within their own subgroup, so that their equilibrium effort levels are identical to a tournament with symmetric contestants. Overall, underdogs will either increase or maintain their effort levels, so that increasing the number of prizes will not decrease their effort. Tables 1 and 2 show the point predictions for the contestants effort in the 3- and 4-person tournaments for parameter values of =2.2, =1, =80, =40, and a=5000. Notice that favorites always reduce their effort when the number of prizes is increased, except for when N F remains greater than m+1 in this case, the favorites maintain their effort (this can be seen from the first two columns of 6 Orrison, Schotter, and Weigelt (2004) show theoretically that in a multi-person tournament where contestants are equally advantaged, adding another prize does not change effort. The intuition behind our results for favorites and underdogs once they compete for prizes within their own subgroup is similar.

10 8 the last row in Table 2). 7 Conversely, underdogs always increase their effort with an additional, except for when at least one of them is assured of winning (see the second and third columns of the top row in Table 2). Furthermore, numerical simulations for N=5 and N=6 suggest that Results 1A and 1B continue to hold beyond N=4. 8 [Insert Table 1 and Table 2 here] Overall, our theoretical analysis demonstrates that the issue of what is the optimal prize structure in a multi-person tournament with asymmetric contestants is an important one for managers because changing the number of prizes involves trading off a possible reduction in effort by the favorites against a possible increase in effort by the underdogs. Next, we conduct a laboratory economic experiment to test if the theoretical predictions of how contestants effort decisions change due to an increase in the number of prizes can be supported. 3. Experiment 1 Hypotheses and Experimental Design Our experiment tests the theory of how changing the number of prizes affects effort in a multiperson tournament with asymmetric contestants for the N=3 tournament. This yields a 2 2 design that varies the number of favorites (1 or 2) and the number of prizes (1 or 2). We denote the effort levels of favorites and underdogs as and, where s, =1,2 indicates the number of favorites and number of winners in a tournament, respectively. In a three-person tournament, the number of favorites can never remain greater than the number of prizes as we move from one prize to two prizes, so increasing the number of prizes is predicted to reduce the effort of favorites. This is formally stated in Hypothesis 1: Hypothesis 1A. When there is 1 favorite and 2 underdogs in a tournament, increasing the number of prizes will reduce the effort of the favorite, that is,. Hypothesis 1B. When there are 2 favorites and 1 underdog in a tournament, increasing the number of prizes will reduce the effort of the favorites, that is,. 7 In Tables 1 and 2, 37.5 is the equilibrium effort level that contestants exert if they compete in a tournament where all contestants are symmetric in endowments. 8 The full numerical simulation results for N=4, N=5 and N=6 are available upon request from the authors.

11 9 Similarly, when N=3, there can never be a case where more than one underdog is assured of winning, so increasing the number of prizes from 1 to 2 will always result in underdogs increasing their effort. Formally, we have Hypothesis 2A. When there is 1 favorite and 2 underdogs in a tournament, increasing the number of prizes will increase the effort of the underdogs, that is,. Hypothesis 2B. When there are 2 favorites and 1 underdog in a tournament, increasing the number of prizes will increase the effort of the underdog, that is,. The parameter values in the experimental test are identical to those used to generate the point predictions of equilibrium effort for the favorites and underdogs in Table 1. That is, we use =2.2, =1, =80, =40, and a=5000 in every decision round of the experiment. We use this set of parameter values because they ensure that 1) there is sufficient spread in the point predictions in the contestants effort when we vary the prize structure, 2) the effort predictions are not focal numbers, and 3) the participation constraints given the equilibrium effort levels for both favorites and underdogs are satisfied. Experimental Procedure Each of the four treatments consisted of either two or three experimental sessions, with each session having 12, 15 or 18 subjects (depending on the number of students who signed up for that session). Subjects were business undergraduates at a large public research university in the United States. They received course credit for showing up to the experiment on time and made cash earnings that ranged from $7 to $12. Each subject was given a start-up payment of $2 and participated in five decision rounds (after a practice round with no monetary consequences). The experimental procedure is largely based on those used in the extant experimental studies of tournaments (e.g., Bull, Schotter and Weigelt 1987 and Lim, Ahearne and Ham 2009), but also differs in a few ways because as mentioned earlier, we wanted to create a social environment that is closer to most tournaments in the real world. At the start of each experimental session, instead of handing out the instructions that describe the decision tasks and the associated payoffs, we began by seating the subjects in a circle and asking them to introduce themselves to the other subjects by stating their names, academic major and to relate the most interesting event that had happened in their lives. Subjects then voted on the most interesting story and the subject with the story that received the most votes was awarded $5. Only after that were the instructions for the tournament handed out and read out aloud to all subjects by the experimenter. In each treatment, subjects were told that they would compete with two other subjects in a contest for five decision rounds. Depending on the number of favorites, they were next told that one

12 10 (two) of them has (have) been randomly selected to be the frontrunner(s) and that the only difference between frontrunner(s) and other contestants is that the Final Number of the former starts off 40 points (k) higher than those who are not frontrunners. Once subjects were selected to be frontrunners, their role was fixed throughout the five decision rounds. Subjects were then told that their task was to select a Decision Number ( ) which ranges from 0 to 90. They were next told that every Decision Number carries a Decision Cost ( ) which increases with the Decision Number that they choose, and the full set of Decision Costs corresponding to each Decision Number was provided to every subject in the form of a Decision Cost Table. Subjects were told that they were to enter their Decision Number into the computer program and upon doing so the program would generate a Random Number ( ) that ranged from 80 to 80 (q), with each Random Number in this range having an equal chance of being drawn. The computer would then add the Decision Number and the Random Number to form the Final Number ( ), by which they would be ranked from the highest (Rank 1) to the lowest (Rank 3). In the tournament with 1 winner (2 winners), subjects were told that they would earn $2.20 ( ) minus their Decision Costs if they win the contest [ranked 1 st ] ( win the contest [ranked 1 st or 2 nd ] ), and $1.00 ( ) minus their Decision Cost if they lose the contest [ranked 2 nd or 3 rd ] ( lose the contest [ranked 3 rd ] ). The experiment was implemented using the z-tree software (Fischbacher 2007). We randomly selected groups of three contestants, assigned the frontrunner(s) and revealed the grouping results along with the identities of the frontrunner(s) so that subjects knew who they were competing with and who the frontrunner(s) was (were). Each group then competed sequentially the three contestants were seated at computer terminals away from the other subjects while the latter watched the proceedings silently. The computer terminals used by the frontrunners were also marked by cards with the word Frontrunner on them. After the Decision Numbers were selected, each subject viewed an output screen for each round that revealed his or her Random Number, Final Number, overall ranking and their outcome (Win or Lose). Subjects were then asked to announce their outcome aloud to the experimenter, who stood about 15 feet away from the computer terminals, by saying either Win or Lose after every decision round. In this way, the outcomes of every contestant in each round were made public to the experimenter and all of the subjects in the room. Consequently, the range of Decision Numbers and the Decision Costs faced by contestants, the degree of advantage enjoyed by the frontrunner(s), the distribution of Random Numbers and how payoffs would be determined were common knowledge to all contestants. In addition, all subjects knew that the identity of the contestants in each contest group, who the frontrunners were and who the winners and losers were in each round would be publicly revealed to all subjects in the room.

13 11 Only the actual Decision Number chosen, the Decision Costs incurred, the Random Number drawn, the rank and payoffs in each round were private knowledge to each contestant. At the end of the experimental session, subjects were paid privately and directed to leave the room. 9 Experimental Results To present the results, we label the treatment conditions by the number of favorites (1F or 2F) and the number of winners (1W or 2W), so that the four treatments are 1F1W, 1F2W, 2F1W and 2F2W. There were 159 subjects who participated in the experiment with 30 subjects in the 1F1W treatment, 54 subjects in the 1F2W treatment, 39 subjects in the 2F1W treatment and 36 subjects in the 2F2W treatment. The average effort levels of the contestants are displayed in Table 3 and presented graphically in Figure 1. In the 1-favorite treatments, the average effort of the favorites is 58.2 and 37.8 with one and two prizes, respectively. In the 2-favorite tournaments, the favorites average effort is 44.9 with only one prize compared to 57.8 when the number of prizes is increased. This finding is directionally opposite to the theory prediction. For underdogs, their average effort levels in the 1F1W and 1F2W conditions are 37.9 and 52.8, respectively. In the 2-favorite tournaments, average underdog effort is 35.1 and 59.3 with one and two prizes, respectively. We proceed to conduct formal statistical tests of the hypotheses. 10 [Insert Table 3 here] [Insert Figures 1a, 1b, 1c, and 1d here] We begin by testing the actual average effort levels of the contestants against the point predictions of the theoretical model. Because subjects make multiple decisions, we cluster the standard errors at the subject level in all the statistical tests to account for potential within-subject correlation. The results of these t-tests are also reported in Table 3. It shows that across the four treatments, subjects in the experiment overexert effort regardless of whether they are favorites or underdogs, so that the actual average effort levels of the contestants are significantly higher than their respective theoretical predictions. Next, we proceed to test Hypotheses 1 and 2 formally. Table 4 displays the results of the OLS regressions of effort on the number of prizes in a tournament for favorites and underdogs separately (with the treatments with one prize as the base and those with two prizes as the dummy). We begin with the 9 The full set of instructions is available from the authors upon request. 10 We test for potential learning by comparing the effort decisions for favorites and underdogs in the first three rounds to the last two rounds of the experiment and find no learning trend (p=0.100 and p=0.186 for favorites and underdogs, respectively). These findings also hold when we conduct the same analysis for each of the four treatments separately.

14 12 effort of favorites. Hypotheses 1A and 1B predict that favorites effort would decrease with an increase in the number of winners. In the 1-favorite tournaments, we find that the effort levels of favorites are indeed lower in the 1F2W treatment compared to the 1F1W treatment ( =0.001), supporting H1A. However, when there are two favorites in the tournament, favorites increase their effort levels when an additional prize is introduced ( =0.030), contrary to the theory prediction. Therefore, H1B is not supported. [Insert Table 4 here] Hypotheses 2A and 2B predict that underdogs will increase their effort when more prizes are added to the tournament. Both hypotheses are supported by the data: Effort of underdogs in the 1F2W tournament is significantly higher than effort in the 1F1W tournament ( =0.016); similarly, underdog effort in the 2F2W tournament is higher than that in the 2F1W treatment ( =0.001). In summary, while we find that the theoretical model predicts the directional effort responses of underdogs to a change in the prize structure of a tournament correctly, it does not fully capture the effort changes of the favorites. Specifically, for tournaments with two favorites, theory fails to predict that favorites increase their effort when one more prize is added. Moreover, the current model does not track the actual effort levels of both favorites and underdogs well. In the next section, we extend the current theoretical model to allow for psychological losses from losing for the favorites and psychological gains from winning for the underdogs in the tournament due to social comparisons and show that this generalized model tracks the experimental data better. 4. Behavioral Economics Model We extend the theory of tournaments by positing a model that captures some of the nonpecuniary factors that may affect the contestants utility when competing in a tournament where contestants are not only asymmetric, but they know who they compete against and also know who wins and loses in each tournament. Our model assumes that in addition to the monetary outcomes from winning or losing, contestants also care about social comparisons, i.e., how their outcomes of winning or losing relative to others may be perceived by themselves and other contestants. 11 Because the contestants are asymmetric, we allow for 11 We rule out risk aversion as an alternative explanation because adding risk aversion to the standard tournament model for our experimental specification causes effort to be systematically lower than the risk neutral predictions and produces the same pattern of results stated in Results 1A and 1B. This occurs because risk aversion reduces the

15 13 how they evaluate winning or losing to be different. For favorites, we assume that they expect themselves and other favorites to win. Hence, receiving a prize does not provide favorites any additional psychological utility gains. However, losing a tournament causes favorites to suffer a psychological loss because they (and others) expect (them) to win. We allow this psychological loss to be exacerbated when favorites lose to an underdog. Specifically, the generalized utility function of a favorite from winning and losing is: (4.1a) and, (4.1b) where if the favorite loses to an underdog, otherwise. The parameter captures the degree of the psychological disutility from losing in the tournament, while is a term that captures additional utility loss if the favorite loses to an underdog and is not present when the favorite does not expect to lose to an underdog. Consider the N=3 tournament: In the 1F1W case, if the favorite loses, he will definitely lose to an underdog, so both and are present. In the 2F1W tournament, however, if the favorite loses, he expects to lose to another favorite, so only is present. Note that our model also builds in the sensible assumption that the psychological disutility from losing also depends on the spread between the winning prize and the amount a loser receives. The utility specifications for the favorite in all of the four conditions for the 3-person tournament are shown in Table 5. [Insert Table 5 here] Underdogs, on the other hand, are assumed to expect to lose since they compete with an initial disadvantage. Hence, losing in the tournament does not bring any psychological disutility. However, we assume that winning a prize garners additional psychological gains because they defied the odds, and that this joy is enhanced when underdogs win through defeating a favorite. Specifically, the generalized utility function of an underdog from winning and losing is: (4.2a) where if the underdog defeats a favorite, otherwise, difference between the utility of the winning prize versus the losing prize for each contestant and creates downward pressure on the marginal benefit of winning and consequently, effort levels.

16 14 and. (4.2b) The parameter captures the degree of psychological gain in utility from winning, while captures the additional joy from winning if the underdog wins by defeating a favorite in the process the latter is not present when the underdog wins without defeating a favorite. For example, in the 1F1W tournament, if an underdog wins, he will only do so through defeating a favorite, hence both and are present. In the 1F2W tournament however, underdogs expect the favorite to be one of the winners so underdogs do not need to defeat a favorite in order to win. In this case, so that only is present. The utility specifications for the underdog in all four conditions for the 3-person tournament are also shown in Table 5. The proposed model is based on the concepts of reference-dependence (Kahneman and Tversky 1979) and social utility functions (e.g., Loewenstein, Bazerman and Thompson 1989). The model is also similar to the social comparison model of Lim (2010) in that we also incorporate psychological gains from being perceived to be a winner ( ) and psychological losses from being perceived to be a loser ( ) in a tournament. However, our model differs in two ways due to the fact that we study a tournament where contestants are asymmetric whereas his paper focuses only on symmetric tournaments: First, we assume that the reference points of favorites and underdogs depend on their subgroup or type so that the reference points of favorites and underdogs are winning and losing, respectively. In contrast, the reference point in Lim (2010) is the modal prize in the tournament. Second, we model the effects of intergroup social comparisons by allowing for additional gains and losses with respect to defeating or losing to a member of the other subgroup through the parameters and. This approach of allowing intergroup social comparisons to affect utility extends the models of Amaldoss and Jain (2005a; 2005b) who allow consumers valuation of luxury goods to not only depend on the intrinsic value of the product, but also on the purchase decisions of other consumer groups. Finally, our model nests the standard tournament model as a special case which we can recover by setting in our generalized model. Estimating the Behavioral Parameters Given the utility specifications in Table 5 above, we can solve for equilibrium effort of the contestants as a function of the behavioral parameters. We then estimate the values of the proposed behavioral parameters using the entire experimental dataset via Maximum Likelihood with the standard

17 15 errors clustered at the subject level. Specifically, we assume that e ilrv ~ N(e * lv, σ 2 lv ), where i represents contestant i, l denotes whether a contestant i is an underdog or favorite, r is one of the five rounds contestants compete in, v indicates which of the four tournament treatments the contestant competes in, * e lv is the equilibrium effort of subgroup l in treatment v, and σ 2 lv is the variance of subgroup l in tournament v. We sum over all i, l, r and v to obtain the following joint log-likelihood:. (4.3) [Insert Table 6 here] The results of the estimation are presented in the second column of Table 6. All the behavioral parameters of,, and are positive and statistically significant at the 5% level, showing that the contestants behavior is consistent with a model that incorporates favorites psychological disutility from losing and underdogs gain in utility from winning, with the favorites pain exacerbated from losing to an underdog and the underdogs joy enhanced from defeating a favorite. The estimates of =0.109 and =0.73 for the favorites suggest that although favorites dislike losing in general, most of the psychological pain from losing is experienced when they lose to an underdog. In contrast, the estimates of =0.38 and =0.348 indicate that although underdogs derive additional joy from defeating a favorite, they also gain significant joy from just being a winner in the tournament. Interestingly, the extra disutility the favorites suffer from losing to an underdog is about 2.1 times greater than the additional joy the underdogs anticipate from defeating a favorite (i.e., =2.1), so that there is social loss aversion (Camerer 1998) in contestants with respect to other contestant subgroups. The parameter estimates also shed light on why favorites increase their effort when an additional prize is introduced in the tournament with two favorites. In the 2F1W case, although the favorites dislike losing, they expect to lose to another favorite so that their disutility from losing is driven only by, which is relatively mild. However, when there are two prizes, if favorites lose they not only experience but also anticipate immense psychological pain through because they know that they will definitely lose to the underdog (the other favorite will claim one of the prizes). The favorites intense dislike of losing to an underdog, coupled with the knowledge that the underdogs will raise effort because of the greater chance of winning and the higher potential for extra joy through defeating a favorite (since there is now one more prize), cause the favorites to respond with even greater effort. When there is only one favorite in the tournament, it is not necessary for favorites to raise effort in response to one additional prize even when these psychological factors are operative, because the favorite does not compete with other favorites and underdogs do not fight as hard since they expect the favorite to claim one of the prizes.

18 16 The other columns in Table 6 display the fit of various nested models, including the standard tournament model. The results of the Wald tests indicate that the generalized model with all of the behavioral parameters,, and produces the best fit of the data. Table 7 shows that the in-sample predictions of the generalized model track the major pattern of the experimental results well, especially the overexertion of effort by all contestants and the increase in effort from favorites when one more prize is added to the 2-favorite tournament. We also included the predictions of the model with and but not and (i.e., constraining = =0). Note that this model cannot capture the rise in the favorites effort as the prize structure changes in the 2-favorite tournament. [Insert Table 7 here] 5. Validating the Behavioral Economics Model Although the generalized model introduced above describes the major empirical regularities of the experimental data well and the values of the estimated behavioral parameters are psychologically sensible, we recognize that there are alternative utility specifications that could also fit the data well. Hence, it is important to provide further validation of our proposed model. We do so through two experiments: First, in Experiment 2A, we test whether contestants effort levels are indeed driven by social comparisons. Second, in Experiment 2B, we investigate the predictive validity of the proposed behavioral economics model to alleviate concerns of over-fitting the data obtained in Experiment 1. The detailed design and results of Experiments 2A and 2B are reported below. Experiment 2A: Do Social Comparisons Drive Effort Decisions? The experimental design of Experiment 2A is identical to that of Experiment 1, except that the social environment under which participants make effort decisions in the tournaments is different. In Experiment 2A, the social environment is similar to that of the typical laboratory economic experiment. Specifically, unlike Experiment 1, 1) the subjects were immediately seated at separate computer terminals upon entering the laboratory and did not introduce themselves to one another prior to the decision tasks; 2) the experimental instructions were context-free and did not describe the decision task as a contest nor use the words win or lose ; 3) the three participants in each contest group were randomly and anonymously matched; 4) the identities of favorites (frontrunners) and underdogs in each contest group were not publicly revealed to all participants in the experimental session; and 5) the contest outcome of winning or losing for each participant in each decision round was not publicly revealed. Lim (2010)

19 17 showed that because social comparisons are less intense in this setting, subjects reported less social pressure to win the tournament when compared to the social environment they faced in Experiment 1. [Insert Table 8 Here] The effort predictions of the standard tournament model for Experiment 2A are identical to those in Experiment 1. Table 8 shows these predictions and the actual average effort decisions for favorites and underdogs in each of the four treatments in Experiment 2A. First, we note that effort decisions in all cases are directionally higher relative to the predictions of the standard tournament model. This oversupply in effort is statistically significant for the favorites in the 1F1W and 2F2W conditions and the underdogs in the 1F2W and 2F2W treatments. Second, the average effort decisions in Experiment 2A are directionally lower than the corresponding averages in Experiment 1 for both favorites and underdogs in all four treatments and are significantly lower at the 5% level in five out of the eight treatment conditions. [Insert Table 9 Here] Next, we test whether the effort levels are consistent with Hypotheses 1 and 2, which are derived from the predictions of the standard tournament model. Table 9 shows that as in Experiment 1, the hypotheses are supported for underdogs (H2A and H2B) and also for the favorites in 1-favorite condition (H1A). Moreover, similar to the findings of Experiment 1, we do not find support for H1B, that is, when there are two favorites in the tournament, the favorites do not decrease effort when the number of prizes is increased from one to two ( =0.345). Note, however, that unlike Experiment 1, we do not find that favorites increase effort with an additional prize. Finally, we estimate the behavioral parameters that capture social comparison effects in the behavioral economics model on the data in Experiment 2A. Overall, the unconstrained behavioral model (LL= ) performs significantly better than the standard tournament theoretical model (χ 2 (4)=38.64, p=0.000). Specifically, both =0.113 (t=2.46, p=0.016) and =0.151 (t=2.80, p=0.006) are positive and statistically significant, suggesting that underdogs derive utility from winning and that favorites suffer disutility when they know that they will definitely lose to an underdog, even when the identities of the contestants and the outcomes of the tournament are not publicly revealed. However, neither =0.036 (t=0.71, p=0.479) nor =0.065 (t=0.94, p=0.349) is significantly different from zero. We also estimated the parameters of the behavioral economics model on the datasets of Experiments 1 and 2A jointly and the Wald tests indicate 1) that the estimate of is smaller in Experiment 2A when compared to the estimate obtained in Experiment 1 (χ 2 (1)=6.88, p=0.009), 2) a similar finding for (χ 2 (1)=15.57, p=0.000) and 3) that the parameters values of,, and are collectively smaller in Experiment 2A

20 18 (χ 2 (4)=54.09, p=0.000). In sum, Experiment 2A provides further evidence that social comparison effects affect contestants behavior. Experiment 2B: Can the Behavioral Economics Model Predict Effort Decisions Well? The main purpose of this experiment is to assess whether the estimated behavioral economics model captures the true psychological drivers of contestants behavior in tournaments if this is indeed the case, the behavioral economics model should track behavior in tournaments other than the one in Experiment 1 well. The experimental design (including the parameter values used) and the social environment in this experiment is identical to that of Experiment 1, except that we alter the prize spread of the tournament by increasing the value of the winning prize, to $2.80 from $2.20 (in Experiment 1), so that there is a larger spread between the winning and losing prize. The predictions of both the standard tournament model and the behavioral economics model when =$2.80 are given in Table 10. Note that, in this new design, predicted effort levels are quite different from the predicted and actual effort levels of Experiment Moreover, besides predicting an oversupply of effort by contestants in all conditions, the behavioral economics model also predicts that H1B would be violated. [Insert Tables 10 and 11 Here] The results of Experiment 2B are shown in Table 11. The average effort decisions are not statistically different from the behavioral economics model predictions for contestants in all conditions except for underdogs in the 1F1W and 2F1W treatments in both of these cases, underdogs choose effort levels greater than those predicted by the behavioral economics model. 13 Note also that there is an oversupply of effort relative to the standard tournament model ( in all cases except for the favorites in the 2F1W treatment in this case, while actual average effort is directionally higher than the prediction of the standard tournament model, the difference is not statistically significant at the 5% level (p=0.091). Next, we also estimate the values of behavioral parameters implied by the data in Experiment 2B and find that =0.449, =0.065, =0.429 and =0.769, all which are positive and significant at the 5% level. These estimates are very close to the estimates found in Experiment 1. In fact, the Wald test indicates that there are no differences among the two sets of parameter estimates ( (4)=0.739, p=0.946). Finally, the results of the regressions of effort decisions on the number of prizes 12 Given the predictions in Table 10, we also had to expand the range of effort decisions that participants could select (from 0 to 90 to 0 to 130) to avoid ceiling effects. 13 Note that this finding, by itself, should not be viewed as conclusive evidence for the predictive efficacy of the behavioral economics model due to the large standard deviations in Experiment 2B. These large standard deviations are partially due to the fact that we expanded the range of effort decisions for subjects to choose.

21 19 shown in Table 12 demonstrates that underdogs systematically increase effort, while favorites decrease effort in the 1-favorite tournament but increase effort in the 2-favorite tournament with the introduction of an additional prize. Together, these findings show that the estimated behavioral economics model can predict behavior well in another tournament. [Insert Table 12 Here] 6. Discussion and Conclusion In this paper, we examine how changing the prize structure affects the effort of contestants in a multi-person tournament where contestants have different initial endowments. We began by showing in the standard tournament model with up to four contestants that adding more prizes to the tournament is predicted not to increase the effort of the favorites, while doing so never reduces the effort of the underdogs. We then test the theory predictions for the three-person standard tournament model using an incentive-aligned experiment conducted in a social setting where contestants know who they are competing against and the tournament outcomes are publicly revealed and find that both favorites and underdogs oversupply effort relative to the point predictions of the theoretical model. Moreover, when there are two favorites and one underdog, the former increase their effort with the addition of one more prize, in contrast to the theory prediction. We show that these experimental results are consistent with a social comparison model that assumes that favorites suffer psychological disutility from losing (with the pain exacerbated if they lose to an underdog) and underdogs derive joy when winning (with the joy heightened when they defeat a favorite). Finally, we conduct two follow-up experiments that support the social comparison assumption underlying the behavioral economics model. The results of this paper also suggest that when managers design the prize structure of a multiperson tournament with asymmetric contestants, the endowment-mix of the contestants can be an important consideration. Depending on the specific goals for conducting the tournament (e.g., in addition to maximizing total effort, they may want to raise the effort of the underdogs), managers can anticipate how increasing or reducing the number of prizes affects the effort of different contestant subgroups. When favorites are a minority (i.e., the 1-favorite tournaments in our experiment), managers must tradeoff the reduction in effort by the favorites and the effort increase of the underdogs when introducing an additional prize. If the number of underdogs is large enough, then managers may want to have more prizes than the number of favorites in the tournament. When favorites form a majority, the results suggest that managers may not necessarily want to restrict the number of prizes because increasing the number of

22 20 prizes (at least up to the number of favorites in the tournament) will not only raise underdog effort but the favorites will also respond favorably because they dislike losing to underdogs. We acknowledge that contestants behavior may be different in large tournaments for example, if there are a very high proportion of favorites but a small number of prizes, favorites may not feel much (or any) disutility when they lose. At a broader level, our paper shows that the explanatory power of formal tournament models can be improved through incorporating psychological factors that may also affect contestants behavior. In particular, we add to the growing evidence of a possible joy of winning effect in tournaments (Parco, Rapoport and Amaldoss 2005 and Amaldoss and Rapoport 2009) and extend the concept of social loss aversion in tournaments (Lim 2010) by demonstrating that social loss aversion can stem from intergroup comparisons. Furthermore, because the presence and strength of these psychological factors may be a function of the institutional or cultural factors that shape the social environment of tournaments, managers need to be attentive to these factors and consider how the rules of the tournament they set (and also the size of the tournament) may also influence these factors.

23 21 REFERENCES Amaldoss, W. and S. Jain (2005a), Pricing of Conspicuous Goods: A Competitive Analysis of Social Effects, Journal of Marketing Research, 42 (1), Amaldoss, W. and S. Jain (2005b), Conspicuous Consumption and Sophisticated Thinking, Management Science, 51 (10), Amaldoss, W. and A. Rapoport (2009), Excessive Expenditures in Two-stage Contests: Theory and Experimental Evidence. In I. Hangen and A. Nilsen (Eds.), Game Theory: Strategy, Equilibria and Theorems, , Nova Science Publishers, New York. Bull, C., A. Schotter and K. Weigelt (1987), Tournaments and Piece Rates: An Experimental Study, Journal of Political Economy, 95 (1), Camerer, C. F. (1998), Behavioral Economics and Nonrational Organizational Decision Making, in Halpern, J. and R. Stern, eds. Debating Rationality: Nonrational Aspects of Organizational Decision Making, Cornell University Press, Ithaca, New York. Ding, M., J. Eliashberg, J. Huber and R. Saini (2005), Emotional Bidders: An Analytical and Experimental Examination of Consumers Behavior in a Priceline-Like Reverse Auction, Management Science, 51 (3), Fischbacher, U. (2007), z-tree: Zurich Toolbox for Ready-made Economic Experiments, Experimental Economics, 10 (2), Green, J. R. and N. Stokey (1983), A Comparison of Tournaments and Contracts, Journal of Political Economy, 91 (3), Kahneman, D. and A. Tversky (1979), Prospect Theory: An Analysis of Decision under Risk, Econometrica, 47 (2), Kalra, A. and M. Shi (2001), Designing Optimal Sales Contests: A Theoretical Perspective, Marketing Science, 20 (2), Lazear, E. P. and S. Rosen (1981), Rank-Order Tournaments as Optimum Labor Contracts, Journal of Political Economy, 89 (5), Lim, N. (2010), Social Loss Aversion and Optimal Contest Design, Journal of Marketing Research, 47 (4), Lim, N., M. Ahearne and S. H. Ham (2009), Designing Sales Contests: Does the Prize Structure Matter? Journal of Marketing Research, 46 (3), Loewenstein, G., M. Bazerman and L. Thompson (1989), Social Utility and Decision Making in Interpersonal Contexts, Journal of Personality and Social Psychology, 57 (3), Martinez-Jerez, F.A., J. Bellin and C. Winkler (2006), CEMEX: Rewarding the Egyptian Retailers, HBS No Boston, Harvard Business School Publishing. Nalebuff B. J. and J. E. Stiglitz (1983), Prizes and Incentives: Towards a General Theory of Compensation and Competition, The Bell Journal of Economics, 14 (1), O Keeffe, M. W., K. Viscusi and R. J. Zeckhauser (1984), Economic Contests: Comparative Reward Schemes, Journal of Labor Economics, 2 (1), Orrison, A., A. Schotter and K. Weigelt (2004), Multiperson Tournaments: An Experimental Examination, Management Science, 50 (2),

24 22 Parco, J. E., A. Rapoport and W. Amaldoss (2005), Two-stage Contests with Budget Constraints: An Experimental Study, Journal of Mathematical Psychology, 49 (4), Schotter, A. and K. Weigelt (1992), Asymmetric Tournaments, Equal Opportunity Laws, and Affirmative Action: Some Experimental Results, Quarterly Journal of Economics, 107 (2), Wang, Y. and A. Krishna (2006), Timeshare Exchange Mechanisms, Management Science, 52 (8),

25 23 Table 1: Theoretical Predictions for the N=3 Tournament 1W 2W Favorite Underdog Favorite Underdog 1F F Note: The parameter values used are =2.2, =1, =80, =40, and a=5000. Table 2: Theoretical Predictions for the N=4 Tournament 1W 2W 3W Favorite Underdog Favorite Underdog Favorite Underdog 1F F F Note: The parameter values used are =2.2, =1, =80, =40, and a=5000.

26 24 Table 3: Theory Predictions and Results for Experiment 1 1W 2W Favorite Theory Prediction F 37.8 Average Effort 58.2 (18.9) # (26.7) Test against Theory prediction t=8.33, p=0.000 t=2.70, p=0.015 Underdog Theory Prediction Average Effort 37.9 (30.2) 52.8 (23.3) Test against Theory prediction t=2.24, p=0.037 t=5.01, p=0.000 Favorite Theory Prediction Average Effort 44.9 (24.5) 57.8 (28.1) 2F Test against Theory prediction t=2.19, p=0.038 t=6.20, p=0.000 Underdog Theory Prediction Average Effort 35.1 (24.1) 59.3 (22.9) Test against Theory prediction t=5.82, p=0.000 t=4.71, p=0.001 # Numbers in parentheses are the standard deviations. Note: There were 30 subjects in 1F1W, 54 subjects in 1F2W, 39 subjects in 2F1W and 36 subjects in 2F2W.

27 25 Figure 1a. Effort of Favorites in the 1-Favorite Tournaments Figure 1b. Effort of Underdogs in the 1-Favorite Tournaments Theoretical prediction Actual effort Theoretical prediction Actual effort F1W 1F2W 0 1F1W 1F2W Figure 1c. Effort of Favorites in the 2-Favorite Tournaments Figure 1d. Effort of Underdogs in the 2-Favorite Tournaments Theoretical prediction Actual effort Theoretical prediction Actual effort F1W 2F2W 0 2F1W 2F2W

28 26 Table 4: OLS Regressions of Effort of Favorites and Underdogs on the Number of Prizes for Experiment 1 Coefficient Robust standard errors t-stat p-value 1-favorite tournament (#obs=140, #clusters=28, R 2 =0.143) Favorites Constant (Base=1-winner) winner favorite tournament (#obs=250, #clusters=50, R 2 =0.057) Constant (Base=1-winner) winner favorite tournament (#obs=280, #clusters=56, R 2 =0.071) Underdogs Constant (Base=1-winner) winner favorite tournament (#obs=125, #clusters=25, R 2 =0.211) Constant (Base=1-winner) winner

29 27 Table 5: Behavioral Economics Model: Utility Functions for the Experimental Conditions + Number of Prizes 1 2 Favorite Favorite 1 Number of Favorites Underdog Favorite Underdog Favorite 2 Underdog Underdog +To obtain the full expected utilities, one would multiply the utility of winning times the probability of winning then add the utility of losing times the probability of losing and subtract the cost of effort.

30 28 Table 6: Parameter Estimates of the Behavioral Economics Model and Nested Models Parameters Full Model Nested Model 1 (λ u =λ f ) Nested Model 2 (λ u =λ f =0) Nested Model 3 (α=λ u =0) Nested Model 4 (β=λ f =0) Standard Tournament Model (α=λ u =β=λ f =0) * * * * (0.050) # - (0.073) (0.031) (0.058) * * * * (0.041) (0.042) (0.059) (0.064) * (0.036) (0.069) (0.024) λ u * * λ f * * - - (0.132) (0.074) - - σ u1f1w 30.2 * (0.82) 31.3 * (1.22) 30.2 * (2.42) 35.5 * (3.18) 32.0 * (2.35) 32.2 * (1.42) σ f1f1w 18.8 * (2.63) 18.8 * (2.14) 22.0 * (1.80) 18.7 * (1.33) 28.8 * (1.72) 30.5 * (1.66) σ u1f2w 23.2 * (2.04) 23.4 * (1.29) 23.6 * (1.81) 27.8 * (1.77) 23.2 * (1.56) 27.8 * (1.48) σ f1f2w 27.2 * (1.45) 26.7 * (0.69) 26.5 * (1.00) 26.7 * (1.72) 27.6 * (2.76) 29.2 * (2.68) σ u2f1w 24.6 * (2.53) 24.8 * (2.51) 30.4 * (3.68) 33.6 * (2.63) 24.9 * (3.07) 32.2 * (1.86) σ f2f1w 24.6 * (2.39) 24.4 * (1.37) 25.1 * (1.66) 24.5 * (1.30) 25.5 * (1.41) 25.5 * (2.83) σ u2f2w 22.9 * (1.12) 23.8 * (2.63) 23.0 * (0.89) 34.9 * (2.60) 22.7 * (0.58) 32.6 * (0.95) σ f2f2w 28.2 * (1.80) 28.4 * (1.82) 32.0 * (1.33) 29.1 * (0.98) 36.8 * (1.59) 39.7 * (2.03) LL Wald Stat * * * * * *Significant at the 5% level. #Clustered standard errors are shown in the parentheses. +The Wald Statistic is used to test whether the restrictions in the 5 nested models significantly reduces the fit. The likelihood ratio test is not appropriate in this context because the observations are not independent at the subject level.

31 29 Table 7: Behavioral Economics Model Predictions 1W 2W Observed Mean Effort Behavioral Economics Model Predictions Nested Model 2 (λ f =λ u =0) Predictions 1F 2F 1F 2F 1F 2F Favorite Underdog Favorite Underdog Favorite Underdog Favorite Underdog Favorite 46.6 * 38.3 Underdog Favorite * Underdog 16.3 * 55.5 * Observed mean effort is significantly different from the specified behavioral model prediction at the 5% level.

32 30 Table 8: Predicted and Actual Effort Decisions in Experiment 2A 1W 2W Favorite Theory Prediction F 2F 28.2 Average Effort 39.4 (9.1) # (14.1) Test against Theory prediction t=3.05, p=0.016 t=1.51, p=0.175 Test against Corresponding Average Effort in Experiment 1 t=-5.41, p=0.000 t=-1.93, p=0.065 Underdog Theory Prediction Average Effort (20.6) (14.6) Test against Theory prediction t=0.77, p=0.452 t=2.18, p=0.046 Test against Corresponding Average Effort in Experiment 1 t=-1.44, p=0.158 t=-2.86, p=0.006 Favorite Theory Prediction Average Effort 39.5 (18.9) 36.7 (14.5) Test against Theory prediction t=0.77, p=0.449 t=5.16, p=0.000 Test against Corresponding Average Effort in Experiment 1 t=-1.26, p=0.215 t=-4.39, p=0.000 # Numbers in parentheses are the standard deviations. Note: There were 27 subjects in 1F1W, 24 subjects in 1F2W, 36 subjects in 2F1W and 30 subjects in 2F2W. Underdog Theory Prediction Average Effort 20.0 (21.5) 42.1 (16.7) Test against Theory prediction t=2.09, p=0.061 t=2.78, p=0.021 Test against Corresponding Average Effort in Experiment 1 t=-3.06, p=0.005 t=-3.06, p=0.006

33 31 Table 9: OLS Regressions of Effort of Favorites and Underdogs on the Number of Prizes in Experiment 2A Coefficient Robust standard errors t-stat p-value 1-favorite tournament (#obs=85, #clusters=17, R 2 =0.190) Favorites Constant (Base=1-winner) winner favorite tournament (#obs=220, #clusters=44, R 2 =0.007) Constant (Base=1-winner) winner favorite tournament (#obs=170, #clusters=34, R 2 =0.119) Underdogs Constant (Base=1-winner) winner favorite tournament (#obs=110, #clusters=22, R 2 =0.246) Constant (Base=1-winner) winner

34 32 Table 10: Predictions of Experiment 2B Standard Tournament Model (α=β=λ u =λ f =0) Behavioral Economics Model (α=.380, β=.109, λ u =.348, λ f =.730) 1W 2W Favorite Underdog Favorite Underdog 1F F F F Note: The parameter values used are =2.8, =1, =80, =40, and a=5000.

35 33 Table 11: Predicted and Actual Effort Decisions in Experiment 2B 1W 2W 1F 2F Favorite Behavioral Economics Model Prediction Average Effort 93.6 (34.3) # (28.7) Test Against Behavioral Model Prediction t=1.26, p=0.234 t=0.49, p=0.633 Test Against Standard Model Prediction t=5.86, p=0.000 t=4.65, p=0.001 Underdog Behavioral Economics Model Prediction Average Effort (33.9) (27.0) Test Against Behavioral Model Prediction t=2.19, p=0.039 t=0.86, p=0.398 Test Against Standard Model Prediction t=5.59, p=0.000 t=5.98, p=0.000 Favorite Behavioral Economics Model Prediction Average Effort (34.7) (33.5) Test Against Behavioral Model Prediction t=0.73, p=0.472 t=0.45, p=0.658 Test Against Standard Model Prediction t=1.76, p=0.091 t=6.85, p=0.000 Underdog Behavioral Economics Model Prediction Average Effort (36.8) (32.8) Test Against Behavioral Model Prediction t=3.61, p=0.004 t=-1.68, p=0.127 Test Against Standard Model Prediction t=5.80, p=0.000 t=5.03, p=0.001 #Numbers in parentheses are the standard deviations. Note: There were 36 subjects in 1F1W, 39 subjects in 1F2W, 39 subjects in 2F1W and 30 subjects in 2F2W.

36 34 Table 12: OLS Regressions of Effort of Favorites and Underdogs on the Number of Prizes in Experiment 2B Coefficient Robust standard errors t-stat p-value 1-favorite tournament (#obs=125, #clusters=25, R 2 =0.114) Favorites Constant (Base=1-winner) winner favorite tournament (#obs=230, #clusters=46, R 2 =0.077) Constant (Base=1-winner) winner favorite tournament (#obs=250, #clusters=50, R 2 =0.054) Underdogs Constant (Base=1-winner) winner favorite tournament (#obs=115, #clusters=23, R 2 =0.175) Constant (Base=1-winner) winner

37 35 Appendix: Proofs of Results 1A and 1B The logic of the proofs is based on the approach used in Orrison, Schotter and Weigelt (2004). First, we characterize the marginal probability of winning for favorites and underdogs given the proportion of favorites in the tournament and the number of prizes. After that, we solve for the equilibrium effort levels of favorites and underdogs. We focus only on the symmetric pure-strategy Nash Equilibrium for each subgroup of contestants. Case 1: N=3 Step 1. Deriving the Marginal Probability of Winning Consider a tournament with 3 contestants i=1,2,3. The output for contestant i is given by, where is the non-negative effort level, and is uniformly distributed on. Contestant i can be ranked above contestant j only if, or. Without loss of generality, we assume that. Given i s random realization of, i is ranked above j only if. The probability that satisfies the above condition determines the conditional probability of contestant i being ranked above contestant j given a realization of :, (A1) where. In this way, we can define,,,,, and. Since we already assumed that, we know that for all realizations of, the probability that contestant 1 is ranked above contestant 2 is one. We define the region of random realizations as the Sure Win Interval (SWI) of contestant 1 relative to contestant 2. Similarly, we can define the other SWIs between any two contestants. With the above setup, we solve for the probability of winning and the marginal probability of winning in the following tournaments. One-winner tournaments To win the prize in the one-winner tournament, the contestant must be ranked above the other two contestants. We can find the unconditional probability of winning by integrating over all possible realizations of with the density function. The probability of winning for contestant 1 is

38 36. (A2) The first term in equation A2 shows the probability of winning for contestant 1 when the realization of is in the SWI relative to contestant 2. In this case, contestant 1 is ranked above contestants 2 and 3 for sure. The second term indicates the probability of winning for contestant 1 when the realization of is in the SWI relative to contestant 3 but not with respect to contestant 2. In this case, contestant 1 is ranked above contestant 3 for sure but the probability of being ranked above contestant 2 is less than 1. The last term indicates the probability of winning when the realization of is too small to be in the SWI relative to contestant 3 (so that contestant 1 is not ranked above contestants 2 and 3 for sure). The probability of winning for contestant 2 is. (A3) The first term indicates the probability of winning for contestant 2 when the realization of is in the SWI relative to contestant 3, so that she needs only to defeat contestant 1 to win. The second term indicates the probability of winning for contestant 2 when the realization of is not in the SWI relative to contestant 3 (i.e., she needs to defeat both contestants 1 and 3). Note that in this case, the realization of cannot be too small, i.e., less than, otherwise contestant 2 will be ranked below contestant 1 for sure and the probability of winning for contestant 2 will be zero. The probability of winning for contestant 3 is. (A4) The expression above reflects the fact that as long as the realization of is not too small, i.e., less than, contestant 3 still has the chance to be ranked above both contestants 1 and 2 even though she has the least potential output. Formulas A2-A4 yield the following expressions for the marginal probability of winning for each contestant, which are, (A5), and (A6). (A7) 1) If the number of favorites equals 1, we label it as the 1F1W tournament. We set and, which means that contestant 1 is the favorite and contestants 2 and 3 are the underdogs. Let and we also know that the marginal probability of winning is identical for

39 37 contestants 2 and 3 at. We obtain the marginal probability of winning for each contestant: and. 2) If the number of favorites equals 2, we label it as the 2F1W tournament. We set and, which means that contestants 1 and 2 are the favorites and contestant 3 is the underdog. We know that the marginal probability of winning is identical for contestants 1 and 2 at. Also, we let. Therefore, we obtain the marginal probability of winning for each contestant: and. Two-winner tournaments In the two-winner tournament, each contestant will win the prize as long as she does not rank last. The probability of winning for contestant 1 is. (A8) This means that contestant 1 will always win the prize unless the realization of than, and she is ranked below both contestants 2 and 3. is too small, i.e., less The probability of winning for contestant 2 is. (A9) The first term in equation A9 shows the probability of winning when the realization of is in the SWI relative to contestant 3 and the conditional probability of winning is 1. The second term captures the probability of winning when the realization of is not in the SWI relative to contestant 3 but greater than, so that contestant 2 can win the prize by being ranked above contestants 1 or 3, or both. The last term shows the probability of winning when the realization of is small, i.e. less than, such that she is ranked below contestant 1 for sure, but still has the chance being ranked above contestant 3. The probability of winning for contestant 3 is. (A10)

40 38 The first term in A10 captures the probability that contestant 3 will be ranked above contestant 1 or contestant 2, or both, when the realization of is greater than so that she is not ranked below contestant 1 for sure The second term indicates the case where the realization of is less than but greater than so that contestant 3 is ranked below contestant 1 for sure but not below contestant 2 for sure. Hence, she can still win the prize by being ranked above contestant 2. Of course, when the realization of is less than, the probability of winning for contestant 3 is zero. Formulas A8-A10 yield the expressions for the marginal probability of winning for each contestant, which are, (A11), and (A12). (A13) 1) If the number of favorites equals 1, we label it as the 1F2W tournament. We set and, which means that contestant 1 is the favorite and contestants 2 and 3 are the underdogs. Let and we also know that the marginal probability of winning is identical for contestants 2 and 3 at. Thus, the marginal probability of winning for each contestant is: and. 2) If the number of favorites equals 2, we label it as the 2F2W tournament. We se an, which means that contestants 1 and 2 are the favorites and contestant 3 is the underdog. We know that the marginal probability of winning is identical for contestants 1 and 2 a. Also, we le. Therefore, the marginal probability of winning for each contestant is: and. We have found the marginal probability of winning for each contestant for all possible combinations of the number of favorites and the number of prizes for the 3-person tournament. These expressions are also shown in Table A1a. [Insert Table A1a here] Step 2. Effort Given the marginal probability of winning found in Step 1, we can now solve for the equilibrium effort in each tournament by using the first order condition for the contestants in Equation 2.2 of the main text. Let and the effort of favorites and underdogs to be and, where s,

41 39 indicating the number of favorites and number of winners in a tournament, respectively. Finally, we require the following three parametric restrictions to ensure interior solutions for effort: i) ; ii) ; and iii) is in the interval (0,. 1F1W tournament The first order condition for the underdogs in the 1F1W tournament can be written as, so that we can express as. Substituting for into the first order condition for the favorite, we have. Because of the restriction, we obtain the solution of, which is:. (A14) We also confirm that is greater than 0. 1F2W tournament The first order condition for the underdogs in the 1F2W tournament can be written as, we express as. Substituting for in the first order condition for the favorite, we have. Because of the restriction, we obtain the following solution for :. (A15) Again, we can confirm that is positive. Comparison of equilibrium effort across the 1F1W and 1F2W tournaments We now examine the effect of adding one more prize into the tournament on the effort of the favorite. Let, so that we have (A16)

42 40 Solving the equation in terms of, we find that one of the solutions is when 2 It is obvious that the aforementioned condition must hold since the extant literature (Orrison, Schotter and Weigelt 2004) has shown that all contestants will exert identical effort regardless of the number of prizes when all contestants are symmetric (i.e., k=0). Another solution of is, which is always positive given. Note also that this term is also greater than the upper limit of, which is 2. Let be the second derivative of with respect to. Then, (A17) which is always negative given. That means is a concave function of and always positive on the interval, i.e., the favorite s effort in the 1F1W tournament is always greater than the favorite effort in the 1F2W tournament. For underdogs, the marginal probability of winning in the 1F1W tournament must be less than that in the 1F2W tournament given that. Therefore, must be true. 2F1W tournament The first order condition for the favorites in the 2F1W tournament can be written as, so that we obtain. Substituting for in the first order condition for the underdog, we have. Given the restriction that, we obtain the following solution for. (A18) Notice that the restriction of stated above is required for to be positive. We can also confirm that is always positive.

43 41 2F2W tournament The first order condition for the favorite in the 2F2W tournament can be written as, so we can express as. Substituting for in the first order condition for the underdog,, we obtain, which is. (A19) Again, we can see that is positive. Comparison of effort across the 2F1W and 2F2W tournaments For the favorites, the marginal probability of winning in the 2F1W tournament must be greater than that for favorites in the 2F2W tournament given that. Therefore,. Next, we compare the effort of the underdogs across the one and two-winner tournaments. Let be the difference between underdog efforts in the 2F1W and 2F2W tournaments. Thus, we have. (A20) Solving the equation in terms of, we find that one of the solutions is under the condition that and the other solution is, which is positive and greater than the upper limit of, 2. Let be the second derivative of with respect to. Then, (A21) which is always positive. That means is a convex function of and is always negative in the interval, i.e., the underdog effort in the 2F2W tournament is always greater than the underdog effort in the 2F1W tournament. As a result, regardless of the proportion of favorites in a 3-person tournament, increasing the number of winning prizes decreases the effort of favorites and increases the effort of underdogs.

44 42 Case 2: N=4 Step 1. Deriving the Marginal Probability of Winning For N=4, the marginal probability of winning for each contestant can be found in a similar fashion as in the 3-person tournament. We omit the detailed derivation and provide only the final expressions for the marginal probability of winning in Table A1b. [Insert Table A1b here] Step 2. Effort Due to the fact that some cases of effort could not be solved analytically for N=4, we conducted extensive numerical simulations to check that our results extend to N=4. Additionally, we ran similar numerical simulations for N=5 and N=6 and obtain findings that suggest that Results 1A and 1B extend beyond N=4. The full set of the numerical simulation results is available upon request from the authors. Please refer to Table 2 of the main text to see a representative case of the simulation results for N=4. One-Favorite tournaments There are no closed-form solutions of effort for favorites in the 1-favorite tournaments because of the complexity in evaluating the higher-order polynomials for the expressions of. To demonstrate that Result 1A holds, we resort to extensive numerical simulations. We find that increasing the number of winning prizes from 1 to 2, and then to 3, decreases the effort of the favorite. For underdogs, it is straightforward to show analytically that underdogs do not decrease effort as the number of winning prizes increases. Given the assumption of (which is assumed to ensure an interior solution), we can easily determine that in the 1F1W tournament is lower than that in the 1F2W tournament ( ) and consequently, the effort of underdogs in the 1F1W tournament is lower than that in the 1F2W tournament. When the number of winning prizes increases from 2 to 3, remains at so that underdogs maintain their effort in the 1F2W and 1F3W tournaments. Two-Favorite tournaments When the number of winning prizes increases from 1 to 2, decreases (it is clear that ) so that the effort of favorites decreases. Now we examine the effort change of

45 43 favorites when the number of winning prizes increases from 2 to 3. The standard approach to prove the aforementioned result would be to take the difference of favorites effort in the 2F2W and 2F3W tournaments and then show that the second derivative of the difference with respect to k is indeed negative. However, there is an alternative approach that is less onerous. Note that the expressions of in the 2F3W tournament are identical to those in the 1F2W tournament when N=3, for both the favorites and underdogs, respectively. This also means that the effort levels for both favorites and underdogs in the 2F3W N=4 tournament and the 1F2W N=3 tournament are identical. Therefore, to show that Result 1A holds, we only need to prove that the effort of favorites in the 2F2W N=4 tournament is greater than that in the 1F1W N=3 tournament since we have already shown that the effort of the favorite in the 1F1W tournament is greater than that in the 1F2W tournament for the N=3 case (so our proposed comparison is a more stringent test). From the expressions of for the favorite and underdogs in the 1F1W N=3 tournament, we know that the effort of the favorite is indeed higher than the effort of the underdogs in the 1F1W tournament because of. We can express this as or. Then comparing for favorites in the 2F2W N=4 tournament to the favorite in the 1F1W N=3 tournament, we see that since. Consequently, favorites effort in the 2F2W N=4 tournament is greater than that in the 1F1W N=3 tournament and also greater than the effort of favorites in the 2F3W N=4 tournament. Next, we will examine the effort change of underdogs when an additional winning prize is introduced. From the expressions of in the 2F1W tournament, we can tell that the effort of the favorites is higher than that of the underdogs, and as a consequence. The expression of for underdogs in the 2F2W tournament can be rewritten as, which is greater than. Since, must be greater than such that. Therefore, the effort of the underdogs in the 2F2W tournament is higher than that in the 2F1W tournament. When the number of winning prizes increases from 2 to 3, for underdogs increases ( ) and the effort of the underdogs also increases.

46 44 Three-Favorite tournaments When there are 3 favorites in the tournament, increasing the number of winning prizes from 1 to 2 does not change the effort of the favorites since remains at. However, increasing the number of winning prizes from 2 to 3 indeed decreases the effort of favorites since. Unfortunately, it is not analytically possible to derive the solutions for the underdog s effort in 3- favorite tournaments. Again, we perform extensive numerical simulations to solve for the equilibrium effort of the underdog in 3-favorite tournaments. We find that when there are 3 favorites in the tournament, increasing the number of winning prizes always increases the effort of the underdog.

47 45 Table A1a: Marginal Probability of Winning when N=3 Marginal Probability of Winning Number of Prizes Number of Favorites 2 Marginal probability of winning for favorites. Marginal probability of winning for underdogs. Table A1b: Marginal Probability of Winning when N=4 Marginal Probability of Winning Number of Prizes Number of Favorites 2 3 Marginal probability of winning for favorites Marginal probability of winning for underdogs