Contest design: an experimental investigation.
Costly competitions between economic agents are often portrayed as contests. Examples range from college admissions and competition for promotions to global relationships in which different countries and political parties expend resources to lobby their own interests (Krueger 1974; Tullock 1980). The variety of economic situations that can be described as contests has attracted enormous attention from economic theorists. The main focus of this literature is the relationship between the setup of rent-seeking contests and the strategic behavior of contestants. It is well recognized that strategic behavior is sensitive to different contest rules. Therefore, depending on the objective, a careful design of each contest is required.
Despite the abundance of theoretical work on contest design, no experimental research has specifically compared alternative contest mechanisms. (1) To begin to bridge this gap, this study investigates and compares the performance of four simultaneous contests: a grand contest (GC), two multi-prize settings (equal and unequal prizes), and a contest which consists of two subcontests (SCs). Consistent with the theory, we find that the GC generates the highest effort levels among all simultaneous contests. In multi-prize settings, equal prizes produce lower efforts than unequal prizes. Our results also provide strong empirical support for the argument that joint contests generate higher efforts than an equivalent number of SCs. However, contrary to the theory, we find significant over-dissipation in all contests. This over-dissipation can be partially explained by strong endowment size effects. Subjects who receive higher endowments tend to over-dissipate, whereas such over-dissipation disappears when the endowments are lower. This behavior is consistent with the predictions of a quantal response equilibrium (QRE). Finally, there is a strong heterogeneity between subjects and individual expenditures over time, which is clearly inconsistent with the symmetric pure strategy equilibrium. The deviations from the symmetric equilibrium can be explained to some extent by differences in risk preference and probabilistic nature of lottery contests.
A number of theoretical papers have been devoted to the design of an optimal contest that generates the highest revenue--the total amount of effort expended by the contestants. A common motivation for such research is the objective of various agencies (political parties, lottery administrators, and economic groups) to maximize earnings by extracting the highest effort from the contestants. Gradstein and Konrad (1999), for example, provide a rationale for a multistage contest design by endogenizing the choice of contest structure. They show that, depending on a return to scale parameter of the contest success function, a multistage contest may induce higher effort by the participants than a one-stage contest. In the same line of research, Baik and Lee (2000) study a two-stage contest with effort carryovers. They show that, in the case of player-specific effort carry-overs, the rent-dissipation rate (defined as the ratio of the expended total effort to the value of the prize) increases in the carryover rate and the rent is fully dissipated with carryover rate equal to one. Finally, Fu and Lu (2007) investigate the optimal structure of a multistage sequential-elimination contest with pooling competition in each stage. They demonstrate that the optimal contest eliminates one contestant at each stage until the finale in which a single winner takes the entire prize.
Overall, it is generally observed in the contest literature that pooling competition generates higher dissipation rates (Amegashie 2000; Clark and Riis 1998; Fu and Lu 2009; Moldovanu and Sela 2006). (2) Clark and Riis (1998) show that the income maximizing contest administrator obtains the highest rent-seeking effort when, instead of many small prizes, a large prize is provided. Fu and Lu (2009) demonstrate that the rent-dissipation rate increases when the number of contestants and prizes are scaled up. Therefore, the authors conclude that a GC generates higher revenue than any set of SCs. Moldovanu and Sela (2006) investigate a similar problem under the structure of all-pay auctions where all players know their own abilities and the distribution of abilities in the population. The major finding of Moldovanu and Sela (2006) is that independently of the number of contestants and the distribution of abilities, a GC generates the highest revenue when the cost function is either linear or concave. However, it is not always the case that pooling competition generates the highest efforts. For example, if the contestants have convex costs, several prizes may be optimal (Krakel 2006; Moldovanu and Sela 2001). The nonoptimality of a single large prize can also occur in a contest where players have commonly known but different abilities (Szymanski and Valletti 2005).
The empirical evidence for contest design theory is mixed (Szymanski 2003). Maloney and McCormick (2000), for example, analyze responses of individual runners to prizes in foot races. They find a significant relation between the performance and the prize value. Consistent with Lazear and Rosen (1981), higher prize values cause higher effort levels. Similar to Maloney and McCormick (2000), Lynch and Zax (2000) examine data on road races in the United States. They find that the performance increases in response to larger prize spreads. However, when controlled for ability factor, the impact of the prize spread disappears. The authors thus conclude that the larger prize spreads produce better performance not because they encourage all runners to run faster but because they attract faster runners.
To complement the existing empirical studies and to further investigate contest design problem, we conduct a controlled experiment. The experiment is based on the theoretical model presented in Section II. Section III provides experimental design and testable hypotheses. Section IV reports the results of the experiment. Section V offers alternative explanations for over-dissipation and heterogeneity observed in the experiment and Section VI concludes.
II. THEORETICAL MODEL
Denote by C [equivalent to] C(N, [<[V.sub.s]>.sup.K.sub.s=1]) a contest with N identical risk-neutral players who are competing for K prizes of a common value [V.sub.s], s = 1, ..., K. (3) No player may win more than one prize, and there are more players than available prizes. Each player i chooses irreversible effort level of [e.sub.i] to influence the probability of winning. Let [[OMEGA].sub.s] be the set of remaining (N - s + 1) players who have not won one of the (s - 1) prizes. Then, the conditional probability that a contestant i wins the s-th prize is given by a lottery contest success function:
(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The efforts are often raised to an exponent term to indicate the sensitivity of a contest. Our reasons for choosing this specific contest success function is that it is simple enough for subjects to understand, and it is also commonly used in most of the rent-seeking contest literature, including virtually all of the experimental contest literature. It is important to emphasize, however, that the simplicity of (l) does not affect the comparative statics predictions of the theory (Clark and Riis 1998; Fu and Lu 2009).
We concentrate our analysis on the symmetric pure strategy Nash equilibrium of the game. The expected payoff of player i, E([[pi].sub.i]), is derived by multiplying player i's probability of winning each prize, [p.sub.i]([e.sub.i], [e.sub.-i]; [[OMEGA].sub.s]), by its value, [V.sub.s]. Because we are considering symmetric equilibrium, the efforts made by other players i [not equal to] j can be denoted as e. Therefore, the probability that i wins the first prize is [e.sub.i]/([e.sub.i] + (N - 1)e). If i does not win the first prize, his/her conditional probability of winning the second prize is the product of the probability that i does not win the first prize and the probability that he/she does win the second prize. Applying this reasoning, we can write player i's expected payoff as:
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The expected payoff (2) is based on the assumptions that players are risk-neutral and have linear costs. However, by relaxing the linearity of costs assumption the comparative statics predictions of the theory are not affected. In fact, in the derivation of the equilibrium, Clark and Riis (1998) use a nonlinear cost function [e.sup.l/r.sub.i] instead of [e.sub.i], where r > (<) 1. Differentiating (2) with respect to [e.sub.i] leads to the equilibrium effort level in the contest C(N, [<[V.sub.s]>.sup.K.sub.s=1]): (4)
(3) [e.sup.*] [[summation].sup.K.sub.s=1] [V.sub.S] (1 - [[summation].sup.s-1.sub.h=0] 1/(N - h))/N.
Formula (3) is the building block of the experimental design used in this study. It shows that the effort level of each contestant depends on the number of contestants, the number of prizes, the value of prizes, and the ordering of prizes. Especially interesting is the "placement effect": the contest administrator can increase the effort level (3) by reducing the value of an early prize [V.sub.s] and increasing the value of a later prize [V.sub.s-1] by the same amount. Taking into account that the revenue collected by the administrator is simply the summation of all individual efforts, the placement effect justifies the use of a large single prize to maximize the revenue collected in the contest.
III. EXPERIMENTAL DESIGN AND PROCEDURES
A. Treatments and Hypothesis
Consider a situation in which there are N players who are willing to participate in a contest. The administrator has a budget V and he/she wants to maximize total revenue extracted from contestants. The administrator must choose how to organize this contest. The simplest way to do this is a simultaneous move GC, in which all players are pooled into one large group with only one large prize. This type of contest is the baseline treatment of this study.
Treatment GC. The first contest is a GC [C.sub.1] (N, V) in which all N contestants are in the same group and they compete for a single prize of value V. Applying (3) and summing over all contestants' efforts, the total revenue collected in [C.sub.1] is
(4) [TR.sub.GC] = V(1 - 1/N).
If the prize V is divisible, the administrator must choose how to divide it. He can divide the prize into several unequal prizes or he can make all prizes equal. The next two treatments investigate these alternatives.
Treatment UC. In contest [C.sub.2](N, ([V.sub.1], [V.sub.2]>) all contestants are competing for two unequal prizes [V.sub.1] = 3V/4 and [V.sub.2] = V/4. A 3:1 ratio of splitting the prize has been proposed by Galton (1902). Note that the sum of [V.sub.1] and [V.sub.2] yields the combined prize of value V. The total revenue generated by this contest is
(5) [TR.sub.UC] = V(1 - 1/N - 1/4(N - 1)).
Treatment EC. In the third contest [C.sub.3](N, ([V.sub.1], [V.sub.2]>), all contestants compete for two prizes of the same value [V.sub.1] = [V.sub.2] = V/2. The total revenue collected is derived from formula (3):
(6) [TR.sub.SC] = V(1 - 1/N - 1/2(N - 1)).
Frequently, instead of putting the contestants into one large group, they are split into several subgroups. In these cases, the competition goes on within each group. As a result, the contest organizer collects the revenue from each SC separately.
Treatment SC. This last simultaneous contest treatment consists of two separate and identical contests [C.sub.41] = [C.sub.42] = C(N/2, V/2). The SC treatment resembles the EC treatment, but instead of competition within the same group, contestants are split into two equal size groups N/2 and the winner of each group receives a prize value V/2. The total revenue collected in both [C.sub.41] and [C.sub.42] is
(7) [TR.sub.SC] = V(1 - 2/N).
Based on the four treatments, we can formalize the following three hypotheses:
Hypothesis 1. GC generates the highest revenue among all simultaneous contests.
This hypothesis follows directly from the four treatments listed earlier. It can also be derived from Clark and Riis (1998), who showed that an administrator who wishes to maximize the revenue should combine all of the prizes into one grand prize.
Hypothesis 2. In multi-prize settings, equal prizes (EC) produce lower efforts than unequal prizes (UC).
This hypothesis comes from the observation that increasing the value of the first prize, whereas decreasing the value of the second prize by the same amount, increases effort expenditures. Therefore, the UC treatment should generate higher revenue than the EC treatment, because in the UC treatment the first prize is [V.sub.1] = 3V/4, whereas in the EC treatment the first prize is [V.sub.2] = V/2. Our final hypothesis is based on a recent study by Fu and Lu (2009), who showed that the joint contest generates higher revenue than any set of SCs.
Hypothesis 3. Joint contest (EC) generates higher efforts than equivalent number of SCs.
In summary, the four contests can be ranked by the total revenue collected: [TR.sub.GC] >[TR.sub.UC] > [TR.sub.EC] > [TR.sub.SC]. If revenue maximization is the objective of the administrator, then the GC should be preferred over all other contests, unequal-prize splitting should be preferred over equal-prize splitting, and a joint contest should be preferred over two equivalent SCs.
B. Experimental Procedures
The experiment consists of four different contests. Table 1 shows the equilibrium effort levels, revenue generated by each contest, and dissipation rates, defined as the total expenditures divided by the total value of the prize, for N = 4 and V = 120.
The experiment used 132 subjects drawn from the population of undergraduate students at Purdue University. Computerized experimental sessions were run using z-Tree (Fischbacher 2007) at the Vernon Smith Experimental Economics Laboratory. We ran 11 experimental sessions with two treatments in each session as in Table 2. There were 12 subjects in the lab during each session. Each experimental session proceeded in three parts. Subjects were given instructions, available in the Appendix, at the beginning of each part and the experimenter read the instructions aloud. In the first part subjects made a series of choices in simple lotteries, similar to Holt and Laury (2002). The second and the third parts of the experiment corresponded to two out of four treatments. For example, in GC-UC, each subject played in a GC for 30 periods, then played for 30 periods in an unequal-prize contest. In each period, subjects were randomly and anonymously placed into a group of four players in GC, UC, and EC treatments or into a group of two players in SC treatment.
At the beginning of each period, each subject received an endowment of 60 experimental francs. Subjects could use their endowments to expend efforts (place bids) in order to win a prize. Subjects were informed that by increasing their efforts, they would increase their chance of winning the prize and that, regardless of who wins the prize, all subjects would have to pay for their efforts. After all subjects submitted their efforts, the computer assigned the winner through a simple lottery. At the end of each period, the sum of all efforts in the group, the result of the random draw, and personal period earnings were reported to all subjects. After completing all 60 decision periods, ten periods were randomly selected for payment (five periods for each treatment). The earnings were converted into U.S. dollars at the rate of 50 francs to $1. On average, subjects earned $18 each and this was paid in cash. The experimental sessions lasted for about 70 min.
Table 3 summarizes average efforts and pay-offs over all treatments, and shows that subjects over-expend effort relative to the risk-neutral Nash prediction. As a result, payoffs are lower than expected. Note that on average players competing in the GC do not earn any positive payoffs. The dissipation rate is defined as the ratio of the expended total effort (revenue) to the value of the prize. In the GC, 100% of the rent is dissipated by four players, whereas only 66% of the rent is dissipated by four players in the two SCs. Actual dissipation rates are significantly higher than what is predicted by the theory. (5)
Result 1. Significant over-dissipation is observed in all treatments.
Table 3 also reports the total revenue collected in each contest. This revenue can be calculated by summing up all efforts within a given contest or by multiplying dissipation rate by the prize value. The data indicate that all four revenues are ranked consistently with the theory. (6) The revenue collected in the EC treatment is higher than the revenue collected in the SC treatment. A random-effect (RE) regression of effort on the treatment dummy-variable, session dummy-variables, and a period trend indicates that the revenue difference is significant (p value < .05). (7) This finding is consistent with Hypothesis 3. The actual difference between the revenue collected in the EC and SC treatments is about 8 (=86-78), which is very close to the theoretical prediction of 10 (=70-60).
Result 2. The equal-prize joint contest generates significantly greater effort and revenue than the two equivalent SCs.
The next result, which supports Hypothesis 2, is that the revenue collected in the UC treatment exceeds the revenue collected in the EC treatment. Based on the estimation of a RE model with standard errors clustered at the session level, the difference in revenues is significant (p value < .05). Although this finding supports Hypothesis 2, the difference in revenues of 31 (= 117-86) is much higher than the theoretical difference of 10 (= 80-70).
Result 3. The unequal-prize contest generates significantly greater effort and revenue than the equal-prize contest.
The GC is designed to produce the highest competition from the contestants and, therefore, generates the highest revenue for the administrator. Table 3 shows that the GC indeed generates the highest effort level, the highest revenue, and the highest dissipation rate. This provides support for Hypothesis 1. Based on the estimation of a RE model with standard errors clustered at the session level, the effort expended in the GC treatment is significantly higher than the effort expended in the EC treatment (p value < .05) and the SC treatment (p value < .05). The difference in effort between the GC and UC treatments is significant only for the last 15 periods of the experiment (p value < .05). (8)
Result 4. The GC generates somewhat higher efforts and revenue than unequal-prize contest and considerably higher efforts and revenue than either equal-prize contest or two equivalent SCs.
Overall, Results 2, 3, and 4 provide strong empirical support for the theoretical findings of contest design: the most rent-seeking efforts are obtained when a large prize is provided instead of several small prizes and the joint contest generates higher revenue than a set of SCs. The support for comparative statics comes from aggregate rather than individual analysis of the data. Figures 1A and 1B display the full distribution of efforts made in the first 15 periods of the experiment. It is striking to see that, instead of following a unique pure strategy Nash equilibrium, subjects' efforts are distributed on the entire strategy space. In the SC treatment, for example, all efforts should be concentrated at 15, but instead they range from 0 to 60. Similar behavior is observed in GC, UC, and EC treatments.
[FIGURE 1 OMITTED]
Result 5. The actual efforts are distributed on the entire strategy space.
It is often argued that subjects need to get some experience in order to learn how to play the equilibrium (Camerer 2003). For that reason, Figures 1C and 1D display the distribution of efforts in the final 15 periods of the experiment. The fraction of the equilibrium efforts in the SC and EC treatments is around 13%-16%, and the fraction of equilibrium efforts in the GC and UC treatments is approximately 4%-11%. There is a minor difference between the distribution of efforts in periods 1-15 and periods 16-30; nevertheless, some learning takes place. The fraction of efforts which are higher than the equilibrium decreases and the fraction of efforts which are lower than the equilibrium increases with the periods played. This can be seen by the leftward shift of distributions in Figures 1A and 1C versus Figures 1B and 1D (note that there is no leftward shift in the GC treatment). In Section V, we provide more formal analysis of the learning trends that occur in our experiment.
Another argument that is commonly made in the experimental and theoretical literature is that players may play an asymmetric equilibrium instead of a symmetric equilibrium (Dechenaux et al. 2006). Although Clark and Riis (1998) do not prove the uniqueness of the pure strategy equilibrium (3), in our specific case the equilibrium is indeed unique (Comes and Hartley 2005; Szidarovszky and Okuguchi 1997). (9)
[FIGURE 2 OMITTED]
Figure 2 displays the average efforts by all subjects who participated in the experiment. On the left side, each subject is ranked by the average effort he/she expended in the EC treatment and on the right side each subject is ranked by the average effort he expended in the GC treatment. Some subjects never enter the competition and expend zero effort in all periods, whereas others expend substantial effort, averaging about 50. (10)
Result 6. There is a strong heterogeneity in efforts between the subjects.
Uniqueness of the pure strategy equilibrium and findings in Results 1, 5, and 6 produce a challenge for contest theory. Nevertheless, Results 2, 3, and 4 support the major comparative static predictions. Why individual behavior is different across subjects is a separate question. There are many behavioral and demographic factors that may cause these differences. The next section explores in more detail the possible behavioral and demographic factors that cause subjects to deviate from the theoretical predictions.
V. EXPLORING OVER-DISSIPATION
A. Quantal Response Equilibrium
Although the comparative statics predictions hold in the experiment, there is a significant over-dissipation in all treatments (Result 1) which is not captured by the theory. Potters et al. (1998) conjectured that most subjects are likely to make mistakes. These mistakes add noise to the Nash equilibrium solution and thus may cause over-dissipation in contest games. We check this hypothesis by applying a QRE developed by McKelvey and Palfrey (1995). The crucial parameter of this model is the error parameter, [mu], which determines the sensitivity of the choice probabilities with respect to payoffs. The maximum likelihood estimates of [mu] for each treatment are shown in Table 4. (11) The table also reports the corresponding value of the likelihood function. The level of mistakes made in the GC and UC treatments is very high. We cannot reject the random play hypothesis for either of the treatments. This conclusion stands even when we estimate the model based on the data from the last 15 periods of the experiment. However, the behavior in the EC and SC treatments can be captured by the QRE with a reasonable level of mistakes.
Figure 3 illustrates the average effort at the QRE as a function of g for each treatment. On the vertical axis, we find the average effort for each player. When [mu] is zero, the behavior is consistent with the Nash equilibrium. With increasing level of mistakes, all players over-expend average effort relative to the Nash equilibrium. As players move closer to random play, that is, putting equal weights on each strategy, the average effort approaches 30 (one-half of the endowment). Even without additional computation, one can see how the QRE can account for the over-dissipation in all treatments of the experiment. For example, the average effort of 19.6 in the SC treatment falls perfectly on the bottom curve around [mu] approximately 9 (left panel of Figure 3).
It is important to emphasize that computation of QRE is heavily dependent on the initial endowment which subjects receive to play the contest game. In our experiment, each period all subjects receive an endowment of 60. Given this endowment, according to the QRE, at each level of mistakes subjects can only expend effort which is higher than the Nash equilibrium (left panel of Figure 3). Therefore, one may argue that the over-dissipation in contests can always be explained by the QRE. (12) However, this argument is not necessarily true because lower endowments may lead to under-dissipation relative to the Nash equilibrium prediction. For example, when the endowment is 40, the QRE predicts that higher level of mistakes in the GC treatment should result in under-dissipation (right panel of Figure 3). The intuition behind this prediction is straightforward: when subjects have large endowments then their mistakes are more likely to result in over-dissipation, whereas small endowments are more likely to result in under-dissipation.
[FIGURE 3 OMITTED]
To make a definite conclusion, we conducted one more session with GC (40) and UC (40) treatments. This time each subject was given an endowment of 40 instead of 60. We were very surprised to discover that the average effort in GC (40) treatment indeed fell from 30.0 to 21.6 which is below the Nash equilibrium prediction of 22.5. In the UC (40) treatment, average effort fell from 29.3 to 21. This finding is a strong support for QRE. (13) It is also consistent with Sheremeta (2008), who conducted one treatment equivalent to the GC treatment. In that study, subjects were given the endowment of 120 francs instead of 60 and as a result the average effort was 34.1 instead of 30. A strong effect of the endowment on subjects' behavior can explain why some experimental studies (Schmidt et al. 2005; Shupp 2000) find less rent-seeking expenditures than what is predicted by the equilibrium. (14)
B. Risk Aversion
The QRE model can account for the general trend of over-dissipation in the experiment; however, it cannot explain the heterogeneity in efforts between the subjects (Result 6). In the experimental literature, it is believed that this heterogeneity is mainly caused by heterogeneity of risk preferences. Previous experimental studies found a significant effect of risk aversion on the dissipation rate (Millner and Pratt 1991). In our experiment, rather than estimating risk aversion from the observed choices in contest games (Goeree et al. 2002; Schmidt et al. 2005), in the first stage we used a simple lottery to elicit risk aversion from the subjects.
Following Holt and Laury (2002), subjects were asked to state whether they preferred safe option A or risky option B. In the experiment, the majority of subjects chose the safe option A when the probability of the high payoff in option B was small, and then crossed over to option B. (15) Table 5 presents a summary of A choices made by all subjects in the experiment. Risk neutrality corresponds to the switching point of either seven or eight safe choices A. The majority of subjects show a tendency toward risk-averse behavior. Based on the observed switching point for each subject, we can estimate their degree of risk aversion. (16) To be consistent with other studies, we calculate risk aversion parameters, r, based on the assumption that all subjects have constant relative risk aversion. The estimates are shown in Table 5. Higher r corresponds to lower number of safe choices A. Conventionally, subjects are considered to be risk-seeking when r > 1. Risk neutrality corresponds to the case when r z 1. As r decreases, subjects become more risk-averse and prefer more safe options A.
Theoretical work by Hillman and Katz (1984) showed that risk-averse players should exert lower efforts than the prediction for risk-neutral players and risk-seeking players should exert higher efforts. Thus, if risk aversion is a crucial factor for explaining heterogeneity between the subjects, then the efforts expended in the contest should be negatively correlated with the number of safe choices made. The last column of Table 5 displays an average effort corresponding to the number of safe choices A made by all subjects. Consistent with the theory, there is significant negative correlation between these two variables. The Spearman's rank correlation coefficient, 13, is -0.81 and it is significantly different from zero (p value < .05).
C. Lag Dependence and Assessment of the Random Draw
So far, we have discussed several explanations for over-dissipation (Result 1) and heterogeneity between the subjects (Result 6). Another question that needs to be addressed is why actual efforts are distributed on the entire strategy space (Result 5). One explanation may come from the probabilistic nature of lottery contests. The random draw made by the computer in period t-1 may affect the individual behavior in period t. To capture this dynamic, we estimated several RE models as in Table 6. In the estimation, we used the data only from the eight main sessions. The estimation results were very similar when using the data from all 11 sessions.
Specification (1) is a simple RE regression of individual efforts made in all periods of the experiment on experimentally relevant explanatory variables. The coefficient capturing risk aversion is significant and has the expected sign. The variable inconsistency is intended to capture the subjects who demonstrated inconsistency in their risk preferences. Time spent on making a decision has a positive effect on over-dissipation. One explanation for this may be that subjects who take more time to make their decisions are actually confused about what they should do and therefore they make more mistakes (from section "Quantal Response Equilibrium," more mistakes corresponds to higher over-dissipation). We also find that the inverse of a time trend is positive and significant, which suggests that individual learning is taking place and that, with the repetition of the game, subjects expend lower efforts. The quiz variable is measured by the number of correct quiz answers (a measure of how well subjects understand the instructions) and is designed to capture the ability factor. (17) Specifications (2) and (3) indicate that subjects who understand the instructions better expend significantly lower efforts in contests. Therefore, this is another evidence that the over-dissipation is caused by subjects who make mistakes and who do not understand the game.
To capture the dynamics of the game, we include a win-lag variable. This is a dummy-variable which takes on the value of 1 if the player won the prize in period t-1 and is 0 otherwise. In Specification (1), this variable has a significant positive effect on effort. One explanation for this finding is due to the income effect: subjects who won in period t-I have higher income in period t and therefore expend higher efforts. (18) In Specification (2), instead of using win-lag variable, we use fairwin-lag and unfairwin-lag variables. The fairwin-lag (unfairwin-lag) variable takes on the value of 1 if subject wins the prize in period t-1 and the random draw in period t-1 is fair (unfair). The fair draw is defined as a random draw that favors the player whose effort is higher than the average effort in the group. However, the unfair draw favors a player with a low effort. From the estimation, we find that the subjects who expend high efforts and win raise their efforts in the consecutive period, whereas the subjects who expend low efforts and win reduce their efforts in the consecutive period. One may argue that this is simply because subjects who expend higher (lower) efforts in one period are also more likely to expend higher (lower) efforts in the next period. To address this argument, we run Specification (3) in which we include daverage-lag variable. This variable is equal to the difference between player i's effort and the average effort in the group in period t-1. From the estimation, we find that this variable is indeed significant, that is, subjects whose efforts are above the average in the past exert higher efforts in the current period. Even though the magnitudes of fairwin-lag and unfairwin-lag variables dropped, both variables are still significant. The response to fair and unfair draw by the subjects is intuitive, but it is not rational. Because the nature of winning the contest is probabilistic, the perception of fair and unfair draw is important in explaining why subjects vary their efforts across periods and why actual efforts are distributed on the entire strategy space.
In this study, we use experimental methods to test several theoretical predictions of contest design literature. We investigate and compare the performance of four simultaneous contests: a GC, two multi-prize settings (equal and unequal prizes), and a contest which consists of two SCs. Consistent with the theory, we find that the GC generates the highest revenue among all simultaneous contests. We also find that in multi-prize settings, equal prizes produce lower efforts than unequal prizes. Finally, our experiment supports the argument that joint contests generate higher efforts than the equivalent number of SCs.
Although the comparative statics predictions hold in our experiment, consistent with the other experimental studies, there is a strong heterogeneity between the subjects and significant over-dissipation of rent (Davis and Reilly 1998; Millner and Pratt 1989, 1991; Potters et al. 1998). Subjects' heterogeneity can be explained to some extent by differences in risk preferences. Significant over-dissipation can be partially explained by strong endowment size effects.
We argue that because of the probabilistic nature of lottery contests it is important to control for lag of winning and misperception of the random draw. Subjects who expend high efforts and win the prize in period t-1 raise their efforts in the consecutive period, while subjects who expend low efforts and win in period t-1 substantially decrease their efforts in period t. These findings are attributed to the misperception of the random draw, and they can partly explain why actual efforts in contests are distributed on the entire strategy space.
EC: Equal Contest
GC: Grand Contest
QRE: Quantal Response Equilibrium
UC: Unequal Contest
This is an experiment in the economics of strategic decision-making. Various research agencies have provided funds for this research. The instructions are simple. If you follow them closely and make appropriate decisions, you can earn an appreciable amount of money.
The experiment will proceed in three parts. Each part contains decision problems that require you to make a series of economic choices which determine your total earnings. The currency used in Part 1 of the experiment is U.S. dollars. The currency used in Parts 2 and 3 of the experiment is francs. Francs will be converted to U.S. dollars at a rate of 50 francs to 1 dollar. At the end of today's experiment, you will be paid in private and in cash. Twelve participants are in today's experiment.
It is very important that you remain silent and do not look at other people's work. If you have any questions, or need assistance of any kind, please raise your hand and an experimenter will come to you. If you talk, laugh, exclaim out loud, etc., you will be asked to leave and you will not be paid. We expect and appreciate your cooperation. At this time, we proceed to Part 1 of the experiment.
Instructions for Part 1
Your decision. In this part of the experiment, you will be asked to make a series of choices in decision problems. How much you receive will depend partly on chance and partly on the choices you make. The decision problems are not designed to test you. What we want to know is what choices you would make in them. The only right answer is what you really would choose.
For each line in the table in the next page, please state whether you prefer option A or option B. Notice that there are a total of 15 lines in the table but just one line will be randomly selected for payment. You ignore which line will be paid when you make your choices. Hence you should pay attention to the choice you make in every line. After you have completed all your choices, a token will be randomly drawn out of a bingo cage containing tokens numbered from 1 to 15. The token number determines which line is going to be paid.
Your earnings for the selected line depend on which option you chose: If you chose option A in that line, you will receive $1. If you chose option B in that line, you will receive either $3 or $0. To determine your earnings in the case you chose option B, there will be a second random draw. A token will be randomly drawn out of the bingo cage now containing 20 tokens numbered from 1 to 20. The token number is then compared with the numbers in the line selected (see the table). If the token number shows up in the left column you earn $3. If the token number shows up in the right column, you earn $0.
Instructions for Part 2
Your Decision. The second part of the experiment consists of 30 decision-making periods. At the beginning of each period, you will be randomly and anonymously placed into a group of four participants. The composition of your group will be changed randomly every period. Each period, you and all other participants will be given an initial endowment of 60 francs. You will use this endowment to bid for a reward. The reward is worth 120 francs to you and the other three participants in your group. You may bid any integer number of francs between 0 and 60. An example of your decision screen is shown below.
After all participants have made their decisions, your earnings for the period are calculated. These earnings will be converted to cash and paid at the end of the experiment if the current period is one of the five periods that is randomly chosen for payment. If you receive the reward, your period earnings are equal to your endowment plus the reward minus your bid. If you do not receive the reward, your period earnings are equal to your endowment minus your bid.
If you receive the reward: Earnings = Endowment + Reward-Your Bid = 60 + 120-Your Bid
If you do not receive the reward: Earnings = Endowment-Your Bid = 60-Your Bid
The more you bid, the more likely you are to receive the reward. The more the other participants in your group bid, the less likely you are to receive the reward. Specifically, for each franc you bid you will receive one lottery ticket. At the end of each period, the computer draws randomly one ticket among all the tickets purchased by four participants in the group, including you. The owner of the drawn ticket receives the reward of 120 francs. Thus, your chance of receiving the reward is given by the number of francs you bid divided by the total number of francs all four participants in your group bid.
Chance of receiving the reward = Your bid/Sum of all four bids in your group
In case all participants bid zero, the reward is randomly assigned to one of the four participants in the group.
Example of the Random Draw. This is a hypothetical example used to illustrate how the computer is making a random draw. Let us say participant 1 bids 10 francs, participant 2 bids 15 francs, participant 3 bids 0 francs, and participant 4 bids 40 francs. Therefore, the computer assigns 10 lottery tickets to participant 1, 15 lottery tickets to participant 2, 0 lottery tickets to participant 3, and 40 lottery tickets for participant 4. Then the computer randomly draws one lottery ticket out of 65 (10 + 15 + 0 + 40). As you can see, participant 4 has the highest chance of receiving the reward: 0.62 = 40/65. Participant 2 has 0.23 = 15/65 chance, participant 1 has 0.15 = 10/65 chance, and participant 3 has 0 = 0/65 chance of receiving the reward.
After all participants make their bids, the computer will make a random draw which will decide who receives the reward. Then, the computer will calculate your period earnings based on your bid and whether you received the reward or not.
At the end of each period, your bid, the sum of all bids in your group, whether you received the reward or not, and the earnings for the period are reported on the outcome screen as shown below. Once the outcome screen is displayed, you should record your results for the period on your Personal Record Sheet under the appropriate heading.
You will not be told which of the participants in this room are assigned to which group. At the beginning of each period, you will be randomly re-grouped with three other participants to form a four person group. You can never guarantee yourself the reward. However, by increasing your contribution, you can increase your chance of receiving the reward. Regardless of who receives the reward, all participants will have to pay their bids.
At the end of the experiment we will randomly choose five of the 30 periods for actual payment in Part 2 using a bingo cage. You will sum the total earnings for these five periods and convert them to a U.S. dollar payment, as shown on the last page of your record sheet. Are there any questions?
Instructions for Part 3
The third part of the experiment consists of 30 decision-making periods. The rules for part 3 are almost the same as the rules for part 2. At the beginning of each period, you will be randomly and anonymously placed into a group of four participants. The composition of your group will be changed randomly every period. Each period you will be given an initial endowment of 60 francs. The only difference is that in part 3, you will use this endowment to bid for two rewards (instead of one reward). The first reward is worth 90 francs and the second reward is worth 30 francs to you and the other three participants in your group. You may bid any integer number of francs between 0 and 60. After all participants have made their decisions, your earnings for the period are calculated in the similar way as in part 2.
If you receive the first reward: Earnings = Endowment + First Reward-Your Bid = 60+90 - Your Bid
If you receive the second reward: Earnings = Endowment + Second Reward-Your Bid = 60+30 - Your Bid
If you do not receive either reward: Earnings = Endowment - Your Bid = 60 - Your Bid
The more you bid, the more likely you are to receive either first or second reward. The more the other participants in your group bid, the less likely you are to receive any reward. Specifically, for each franc you bid you will receive one lottery ticket. At the end of each period, the computer draws randomly one ticket among all the tickets purchased by four participants in the group, including you. The owner of the drawn ticket receives the first reward of 90 francs. Thus, your chance of receiving the first reward is given by the number of francs you bid divided by the total number of francs all four participants in your group bid.
Chance of receiving the first reward = Your bid/Sum of all four bids in your group
If you do not receive either reward: Earnings = Endowment - Your Bid = 60 - Your Bid
In case you do not receive the first reward, there is a second draw for the second reward. For the second draw, computer draws randomly one ticket among all the tickets purchased by three participants in the group who did not receive the first reward (the participant who received the first reward is excluded from the second draw). The owner of the drawn ticket receives the second reward of 30 francs. Your chance of receiving the second reward is given by the number of francs you bid divided by sum of three bids made by the participants who did not receive the first reward.
Chance of receiving the second reward = Your bid/Sum of three bids (made by participants who did not receive the first reward)
Each participant can win at most one reward. In case all participants bid zero, the first and the second reward is randomly assigned to two of the four participants in the group.
Example of the Random Draw. This is a hypothetical example used to illustrate how the computer is making a random draw. Let us say participant I bids 10 francs, participant 2 bids 15 francs, participant 3 bids 0 francs, and participant 4 bids 40 francs. Therefore, the computer assigns 10 lottery tickets to participant 1, 15 lottery tickets to participant 2, 0 lottery tickets to participant 3, and 40 lottery tickets to participant 4. Then, for the first random draw, the computer randomly draws one lottery ticket out of 65 (10 + 15 + 0 + 40). As you can see, participant 4 has the highest chance of receiving the first reward: 0.62 = 40/65. Participant 2 has 0.23 = 15/65 chance, participant 1 has 0.15 = 10/65 chance, and participant 3 has 0 = 0/65 chance of receiving the first reward.
After all participants make their bids, the computer makes a first random draw which decides who receives the first reward. Let us say that participant 4 has received the first reward. Then, for the second random draw, the computer randomly draws one lottery ticket out of 25 (10 + 15 + 0). Since participant 4 has already received first reward he is excluded from the second draw. Now, as you can see, participant 2 has the highest chance of receiving the second reward: 0.6 = 15/25. Participant 1 has 0.4 = 15/25 chance and participant 3 has 0 = 0/25 chance of receiving the second reward.
To summarize, all participants will make only one bid. After all participants have made their decisions, the computer will make two consecutive draws which will decide who receives the first and the second reward. Regardless of who receives the first and the second reward, all participants will have to pay their bids. Then the computer will calculate your period earnings based on your bid and whether you received either reward.
At the end of each period, your bid, the sum of all bids in your group, whether you received the first reward or not, whether you received the second reward or not, and the earnings for the period are reported on the outcome screen. Once the outcome screen is displayed, you should record your results for the period on your Personal Record Sheet under the appropriate heading.
At the end of the experiment, we will randomly choose five of the 30 periods for actual payment in Part 3 using a bingo cage. You will sum the total earnings for these five periods and convert them to a U.S. dollar payment, as shown on the last page of your record sheet. Are there any questions?
Participant ID-- Please Decision choose no. Option A Option B A or B 1 $1 $3 never $0 if 1,2,3,4,5,6,7, 8,9,10,11,12, 13,14,15,16,17, 18,19,20 2 $ l $3 if I comes out $0 if of the bingo 2,3,4,5,6,7,8, cage 9,10,11,12,13, 14,15,16,17,18, 19,20 3 $1 $3 if 1 or 2 $0 if 3,4,5,6,7, comes out 8,9,10,11,12, 13,14,15,16,17, 18,19,20 4 $1 $3 if 1,2 or 3 $0 if 4,5,6,7,8, 9,10,11,12,13, 14,15,16,17,18, 19,20 5 $1 $3 if 1,2,3,4 $0 if 5,6,7,8,9, 10,11,12,13,14, 15,16,17,18, 19,20 6 $1 $3 if 1,2,3,4,5 $0 if 6,7,8,9,10, 11,12,13,14,15, 16,17,18,19,20 7 $1 $3 if 1,2,3,4,5,6 $0 if 7,8,9,10, 11,12,13,14,15, 16,17,18,19,20 8 $1 $3 if 1,2,3,4,5, $0 if 8,9,10,11, 6,7 12,13,14,15,16, 17,18,19,20 9 $1 $3 if 1,2,3,4,5, $0 if 9,10,11,12, 6,7,8 13,14,15,16,17, 18,19,20 10 $1 $3 if 1,2,3,4,5, $0 if 10,11,12, 6,7,8,9 13,14,15,16,17, 18,19,20 11 $1 $3 if 1,2,3,4,5, $0 if 11,12,13, 6,7,8,9,10 14,15,16,17,18, 19,20 12 $1 $3 if 1,2,3,4,5, $0 if 12,13,14, 6,7,8,9,10,11 15,16,17,18, 19,20 13 $1 $3 if 1,2,3,4,5, $0 if 13,14,15, 6,7,8,9,10, 16,17,18,19,20 11,12 14 $1 $3 if 1,2,3,4,5, $0 if 14,15,16, 6,7,8,9,10,11, 17,18,19.20 12,13 15 $1 $3 if 1,2,3,4,5, $0 if 15,16,17, 6,7,8,9,10,11, 18,19,20 12,13,14
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(1.) Several experimental studies looked at the design of rank order tournaments (Harbring and Irlenbusch 2003, 2005; Orrison et al. 2004) and fund-raising lotteries (Lange et al. 2007; Morgan and Sefton 2000). The most closely related work to ours is performed by Muller and Schotter (2009) who analyze the influence of prizes in all-pay auction with heterogeneous agents following the theoretic work by Moldovanu and Sela (2001). The main focus of Muller and Schotter (2009) is the effect of heterogeneity in abilities on the efforts expended.
(2.) For more multiple prize contests, see Glazer and Hassin (1988), Barut and Kovenock (1998), and Che and Gale (2003).
(3.) For theoretical and experimental analysis of heterogeneous agents in lottery contests, see Harbring et al. (2007), Lange et al. (2007), and Sheremeta (2009).
(4.) General conditions for existence of the equilibrium and detail derivations are shown in Clark and Riis (1998).
(5.) To support this conclusion, we estimated a simple panel regression for each treatment, where the dependent variable is effort and independent variables are a constant and session dummy-variables. The model included a random effects error structure, with the individual subject as the random effect, to account for the multiple decisions made by individual subjects. Based on a standard Wald test, conducted on estimates of a model, we found that for all treatments the constant coefficients are significantly higher than the predicted theoretical values in Table 3 (p value <.05). The same conclusion also stands after clustering standard errors at the session level (p value <.05).
(6.) We also checked for a possible order effect since subjects consecutively played in two of the tour possible contests. No significant difference was found. In fact, the averages presented in Table 3 are almost identical to the averages when we consider only the first treatment in each session. In GC, UC, EC, and SC the average efforts without the order effect are 30.2, 29.9. 21.5, and 18.5.
(7.) When clustering standard errors at the session level, the difference is significant only for the last 15 periods of the experiment (p value < .05).
(8.) It is important to emphasize that although the average efforts are similar in both GC and UC treatments, the strategic behavior of individual subjects is very different. By analyzing individual efforts, we find that in the GC treatment subjects choose 0 and 60 more often than in the UC treatment (Figure 1D). As a result, there are more subjects who compete too much and at the same time more subjects who drop out of the competition in the GC treatment than in the UC treatment.
(9.) Because of experimental design, all players are restricted to choose integer effort levels from 0 to 60. Therefore, one can look at the 4-player contest as four-dimensional normal form game with nearly 1.4E+07 possible outcomes. We ran computer simulation to check for all possible pure strategy equilibria and the only one that was found is unique and symmetric. Because of the restriction on the strategy space, in the equilibrium of the GC (EC) treatment two players expend 23 (18) francs and two players expend 22 (17) francs. It is also important to emphasize that because of the concavity of payoff functions the pure strategy equilibrium is also the unique mixed strategy equilibrium. We performed computer simulation for the SC treatment to confirm this.
(10.) Evidently, the participants who bid more in the EC treatment are also more likely to bid more in the SC treatment. We can track this correlation since each subject participated in two different treatments in each session. The Spearman's rank correlation coefficient, [rho], is 0.58, and it is significantly greater than zero (p value < .05). The correlation between the GC and UC treatment is even higher, [rho] = 0.81 (p value < .05). This important observation shows the consistency of each subject's behavior across different experimental contests. Therefore, it strongly confirms the presence of heterogeneity between the subjects.
(11.) The estimation procedure followed Goeree et al. (2002). A more detail description of the estimation procedure is available from the author upon a request.
(12.) Bullock and Rutstrom (2007) find that observed behavior in the Tullock-type model of political competition is fully captured by QRE predictions. Anderson et al. (1998) develop a theoretical model of the all-pay auction based on the QRE. The model predicts that overbidding in the all-pay auction occurs due to the mistakes and that overbidding should increase with the size of the bidders' group. Nevertheless, Gneezy and Smorodinsky (2006) found that the over-dissipation in the all-pay auction is independent of the group size in later periods.
(13.) With the restriction on the endowment, the estimated level of mistakes, g, also decreased in both treatments (Table 4). However, in the GC (40) treatment, we still cannot reject the random play hypothesis.
(14.) In Schmidt et al. (2005) and Shupp (2000), subjects were given a budget which allowed them to bid up to $20, whereas $13.5 was the Nasb equilibrium prediction. Assuming a substantial level of mistakes made by subjects, the QRE approaches $10 and is below the Nash equilibrium. Therefore, taking into account our findings about the effect of the endowment on the behavior of the rent-seekers, we can explain why these authors find significant under-dissipation in their experiments.
(15.) Option A yielded $1 payoff with certainty, whereas option B yielded a payoff of either $3 or $0. The probability of receiving $3 or $0 varied across all 15 lotteries. The first lottery offered a 5% chance of winning $3 and a 95% chance of winning $0, whereas the last lottery offered a 70% chance of winning $3 and a 30% chance of winning $0.
(16.) Note that switching from A to B only gives us an interval of risk aversion coefficient; however, for statistical computations, we will use a mid-point approximation.
(17.) Before the actual experiment, subjects completed the quiz on the computer to verify their understanding of the instructions. If a subject's answer was incorrect, the computer provided the correct answer. The experiment started only after all participants had answered all quiz questions.
18. It is rather surprising since we tried to avoid this effect by using random payment. It is also possible that subjects derive utility from winning (Goeree et al. 2002; Sheremeta forthcoming). Thus, subjects who win the prize in period t-I expend higher efforts to ensure that they win the prize in period t.
ROMAN M. SHEREMETA *
* I am particularly grateful to Tim Cason for excellent guidance and support. I thank Jason Abrevaya, Marco Casari, Subhasish Modak Chowdhury, Dan Kovenock, Jingjing Zhang, anonymous referees, and the associate editor of this journal for their constructive comments, as well as seminar participants at Purdue University for helpful suggestions. This research has been supported by National Science Foundation Grant (SES-0751081). Any remaining errors are mine.
Sheremeta: Assistant Professor of Economics, Argyros School of Business and Economics, Chapman University, 1 University Drive, Orange, CA 92868, USA. Phone (714) 744-7604, Fax (714) 532-6081, E-mail: firstname.lastname@example.org
TABLE 1 Experimental Design and Nash Equilibrium Predictions Number of Players per Prizes per Value of the Treatment Groups Group, N Group, s Prize, V GC 1 4 1 120 UC 1 4 2 90.30 EC 1 4 2 60.60 SC 2 2 1 60 Equilibrium Revenue, Dissipation Treatment Effort, e TR Rate GC 22.5 90 0.75 UC 20.0 80 0.67 EC 17.5 70 0.58 SC 15.0 60 0.50 TABLE 2 Summary of Treatments and Sessions Number of Total Number of Design Sessions Participants Decisions GC-UC 2 24 1440 UC-GC 2 24 1440 EC-SC 2 24 1440 SC-EC 2 24 1440 GC-UC (40) 1 12 720 Pilot sessions 2 24 960 Number of Design Periods Endowment GC-UC 30 60 UC-GC 30 60 EC-SC 30 60 SC-EC 30 60 GC-UC (40) 30 40 Pilot sessions 20 60 TABLE 3 Average Statistics Effort, e Revenue, Treatment Equilibrium Actual TR GC 22.5 30.0 (0.5) 120.1 UC 20.0 29.3 (0.5) 117.4 EC 17.5 21.6 (0.3) 86.4 SC 15.0 19.7 (0.3) 78.6 Payoff, [pi] Dissipation Rate Treatment Equilibrium Actual Equilibrium Actual GC 7.5 0.0 (1.3) 0.75 1.00 UC 10.0 0.7 (0.9) 0.67 0.98 EC 12.5 8.4 (0.7) 0.58 0.72 SC 15.0 10.3 (0.8) 0.50 0.66 Note. Standard error of the mean in parentheses. The total number of observations in each treatment is 1,440. TABLE 4 QRE Computation Based on All Periods LR Test LR Test Treatment [mu], Error LL (Random) (Equilibrium) GC 1.21E+07 -5919.7 p = .99 p < .01 UC 729.4 -5919.7 p = .77 p < .01 EC 9.3 -5654.6 p < .01 p < .01 SC 9.3 -5557.9 p < .01 p < .01 GC (40) 113.4 -1336.7 p = .56 p < .01 UC (40) 3.2 -1274.5 p < .01 p < .01 TABLE 5 Classification of Subjects by Risk Aversion (All Treatments) Number of Number of Relative Risk Average A Choices Subjects Aversion, r Effort, e 4 1 1.73 30.07 5 2 1.46 33.03 6 4 1.26 34.94 7 18 1.10 25.68 8 12 0.96 24.74 9 10 0.83 27.06 10 19 0.73 21.86 11 14 0.63 26.59 12 8 0.54 22.93 13 5 0.46 25.11 14 1 0.39 9.55 15 2 0.32 10.07 TABLE 6 Random-Effect Models Dependent Variable, Effort (1) RE (2) RE (3) RE Risk aversion -1.34 *** -1.28 *** -0.83 *** [# of safe options A] (0.30) (0.21) (0.09) Inconsistency 0.85 0.7 0.18 [1 if subject is inconsistent in (2.37) (1.62) (0.71 the lottery choices] Bidding time 0.09 ** 0.11 ** 0.14 *** [time spent to make a (0.04) (0.04) (0.04) decision] Time trend 5.06 *** 4.68 *** 4.59 ** [inverse of a time trend 1/t] (1.68) (1.68) (1.78) Quiz -0.80 -0.67 * -0.57 *** [# correct quiz answers] (0.56) (0.38) (0.17) Win-lag 2.17 [I if subject won in t-1] (0.36) Fairwin-lag 5.57 *** 2.00 [I if subject wins in period (0.41) (0.50) t-1 and the draw is fair] Unfairwin-lag -3.11 *** -2.61 *** [I if subject wins in period (0.54) (0.56) t-1 and the draw is unfair] Daverage-lag 0.51 *** [deviation from average effort (0.02) in t-1] Treatment-GC 17.34 *** 5.16 *** 18.56 [I if treatment is GC] (2.48) (1.71) (0.82) Treatment-UC 16.01 *** 3.85 ** 17.71' [I if treatment is UC] (2.48) (1.71) (0.81) Treatment-EC 2.06 *** 2.05 *** 2.07 *** [I if treatment is EC] (0.48) (0.48) (0.51) Constant 35.51 *** 34.08 *** 28.14 (5.28) (3.62) (1.78) Observations 5568 5568 5568 Note. Standard errors in parentheses. In each regression, we also include session dummy-variables to control for session effects. * Significant at 10%; ** Significant at 5%; *** Significant at 1%.
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|Author:||Sheremeta, Roman M.|
|Date:||Apr 1, 2011|
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