Abstract
We theoretically and experimentally examine the effect of noise variance and prize value on effort in individual contests and in three types of group contests: perfect-substitutes, best-shot, and weakest-link. For all contest types, we use the rank-order contest model, where effort and random noise determine performance. The theoretical model for individual contests predicts that effort will increase with prize value and decrease with noise variance. As expected, all subjects in our experiment decrease their efforts as noise variance rises, regardless of the value of the prize. Prize value, however, has no effect on effort. In group contests, each group consists of two players with different prize values. The player for whom the prize value is higher is referred to as a strong player; the other is referred to as a weak player. The theoretical model also predicts that exerted positive efforts will decrease with noise variance in all group contests. Our experimental results show that in perfect-substitutes and weakest-link contests, noise variance has no effect on either strong or weak subjects’ efforts. In best-shot contests, however, both strong and weak subjects decrease their efforts when noise variance increases. Finally, we compare the efforts of subjects in individual and group contests. We find differences only in perfect-substitutes and best-shot contests when the noise variance is high. Efforts are higher in perfect-substitutes contests and lower in best-shot contests compared to individual contests.
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Notes
Tullock lottery contests have commonly been used to model rent-seeking and R &D races; Rank-order tournaments have been used in the principal-agent, contract design, and labor literature; and All-pay auctions have been used to model the process of litigation, lobbying and military combat.
A similar situation can occur in individual contests. For instance, the probability of a golf professional’s winning might change depending on the physical features of the golf course, which can be thought of as a random noise.
Group effort is characterized as a function of all group members’ efforts, which changes with group impact functions.
In their paper, a simple deterministic winner-take-all contest is similar to Lazear and Rosen (1981)’s rank-order tournament, where the sensitivity parameter in contest success function is \(r = \infty\).
Subjects’ rankings, which are determined by their performance in the real-effort task, determine the prize value that they compete for in the second and third parts of the experiment. In particular, subjects in a given session are divided into two groups according to their rankings in the first part. While subjects in the higher-ranked group compete for a high prize, the subjects in the lower-ranked group compete for a low prize.
The constant b is a restriction on players’ abilities on the quadratic cost function, as in Cason et al. (2020).
The strictly increasing convex cost function ensures the existence and uniqueness of an equilibrium where all players exert positive effort. In the experimental contest literature, a quadratic cost function has been commonly used (Bull et al., 1987; Harbring & Irlenbusch, 2003; Eriksson et al., 2009; Agranov & Tergiman, 2013; Cason et al., 2020). Each set of theoretical results we present is supported by the necessary second-order conditions. The second-order condition evaluated at the equilibrium is \(\partial ^2 E(\pi ) / \partial e^{2}\) = \(-2 / b < 0\) in individual and all group contests.
We should emphasize that the mean of this multiplicative distribution, \(\varepsilon\), is 1, as opposed to the mean of 0 when the noise is additive. The reason is that when the mean of the multiplicative noise is 1, a player’s effort and performance are the same. This occurs with 0-mean when the noise variable is additive.
Player 1A and player 1B are “strong players”, and the others, player 2A and 2B, are “weak players”.
We restrict \(\alpha \geqslant 0.5\) in the experiment because the expected payoff in Equation 5 is nonnegative for any \(\alpha \in [0.5, 1]\).
Table 2 presents theoretical predictions for Case 1 and Case 2 in Subsect. 2.2.2. Case 3 is excluded at both noise variance levels because at the equilibrium subjects’ expected payoffs are non-positive in low noise variance. All predictions for best-shot contests can be found in Table 14 in Appendix B.
The treatment names are determined by the group impact functions and noise variance.
Throughout the experiment, payoffs were in “francs,” which were converted into Turkish Lira (TL) at the rate of 40 francs to 3 TL.
37.90% of the subjects were from the economics department; 51.61% were male. Their ages ranged from 20 to 25 (87.10%).
In 2021, the hourly minimum wage in Turkey was 15.90 TL.
The instructions were in Turkish. An English translation is provided in Appendix A.
We used neutral language in the instructions. In individual (group) contests, effort corresponded to bid, random noise corresponded to a personal random number (group random number), match corresponded to the opponent, strong subject corresponded to player 1, and weak subject corresponded to player 2.
This information appeared at the bottom of the decision screen. Subjects could use an on-screen calculator to determine what a bid would cost.
The personal random number was randomly and independently drawn in each period for each subject. It corresponded to the noise variance, which changed from treatment to treatment. Subjects knew that random noise was drawn from U[0, 2] in the high noise variance treatment, and from U[0.5, 1.5] in the low noise variance treatment.
Strong subjects were randomly and anonymously paired with weak subjects in a group, but their competing group in the third part was composed of both strong and weak competitors from the second part. By using this procedure, we aimed to have a higher number of independent observations.
The group random number was randomly and independently drawn in each period for each group.
In total, there are 124 subjects: 30 (32) are subjects with a high-prize value and 30 (32) are subjects with a low-prize value in the high (low) noise variance treatment.
For the first period, in low noise variance both subjects’ efforts are significantly lower than the equilibrium efforts (two-tailed Wilcoxon signed rank test, p-value \(< 0.01\) for both subjects). In high noise variance, there is no significant difference between observed efforts and equilibrium efforts (two-tailed Wilcoxon signed rank test, p-value \(> 0.4\) for both subjects).
The difference between the equilibrium efforts of subjects with a high-prize value in both high and low noise variance is 22.69. The difference between their observed efforts under the same condition is 6.69. Similarly, the difference between the equilibrium efforts of subjects with a low-prize value in both high and low noise variance is 18.53 while the difference between their observed efforts under the same condition is 7.84. Hence, the decrease in observed efforts is less than the decrease in the equilibrium efforts in individual contests.
These independent variables are used for every regression analysis throughout this paper.
Only the distribution of strong subjects’ effort significantly differs with noise variance in best-shot contests (ksmirnov test, p-value = 0.06).
These interpretations are according to the mode of efforts in each case.
We also notice a high variation in subjects’ efforts, especially in low noise variance.
During the first period, strong subjects exert significantly higher effort than weak ones at both noise variance levels (one-tailed Wilcoxon ranksum test, p-value = 0.009 in high noise variance and p-value = 0.083 in low noise variance).
We used the partner-matching procedure, so for every group contest regression analysis, we cluster the standard errors at the group level, where the two subjects in a given group are counted as one observation.
In all group contests, each group has one subject with a high-value prize and one subject with a low-value prize. To compare how each subject’s effort changes in individual and group contests, we cluster the standard errors at the subject level.
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Acknowledgements
We thank Serkan Küçükşenel, Emin Karagözoğlu, Luke Boosey, Mehmet Yiğit Gürdal, Kübra Gurallar, Mert Kayaaslan and Gizem Mutluoğlu for their helpful comments and suggestions, and the participants at the 8th Annual ‘Contests: Theory and Evidence’ Conference in University of Reading, and the North American ESA meeting in Santa Barbara, California. Finally, we thank two anonymous referees for their valuable feedback. Any errors are our responsibility.
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Appendices
Appendix
A Experimental instructions
Below are the English translations of the instructions for individual and perfect-substitutes contests with high noise variance treatment. The noise variance changes from session to session. The only difference in the low noise variance treatment is that random numbers in contests can take any value between 0.5 and 1.5. We also present the changes in group contests in the third part of the experiment for three group contests: perfect-substitute, best-shot, and weakest-link. These are specified in square brackets with the related contest name.
General instructions
Welcome to our experiment.
In this experiment, we want to examine the process of strategic decision-making. If you follow the instructions closely, you can earn money. Your earnings may differ from other participants’. The amount you earn depends on your own decisions, the decisions of other participants, and chance. You will be paid all the money you have earned in cash immediately after the session ends.
The experiment will proceed in five parts. Each part will be explained just before that part begins. During the experiment, your earnings will be calculated in francs (an experimental currency). The francs will be converted to Turkish Liras at a rate of 40 francs to 3 Turkish Lira. You will also be paid an additional fixed show-up fee of 10 TL for your participation. In the final part of the experiment, you will be asked to answer a few questions.
Thank you for participating.
PART 1
In this part of the experiment, you will be asked to answer 20 general knowledge questions in multiple-choice format. Each question has 5 options, only 1 of which is correct. You will have 25 s to answer each question. If you fail to answer within 25 s, you will automatically proceed to the next question, and the unanswered question will count as incorrectly answered.
You will not earn francs from this part at the end of the session. However, the performance you show here will affect other parts of the experiment. Therefore, please do your best to answer each question. At the end of this part, you will know neither your results nor those of other players.
PART 2
This part of the experiment consists of 10 decision-making periods. At the beginning of the first period, you will be randomly and anonymously paired with another participant. You will remain paired with the same person throughout this part to win a prize. The value of the prize will depend on your performance in the first part. The value will not change during this part. You will see the amount of the prize you are competing for on the screen. That amount will be either 80 or 120 francs. This amount will be the same for both people in the competing group.
Each period, you will have an initial endowment of 100 francs. You may bid any amount between 0 and 100. There is a calculator button for you to do your calculations at the bottom of the box where you will enter a bid.
After you make your bid, the computer will multiply it by a “personal random number” to calculate your final bid. This random number may have a value of anywhere between 0 and 2. This random number is separately and independently drawn for each period and each person.
Your Final Bid = Your bid x Personal random number
There is an associated cost for each bid.
Cost of Bid = \(Bid^2/100\)
After you and the other participant have made bids, the computer will draw the random numbers and compare your final bids. If your final bid is higher than the other participant’s, you will receive a prize of 80 or 120 francs. Otherwise, you will receive 0 francs. In other words:
If you win and the prize value is 120:
Earnings = Initial Endowment + Prize - Cost of Bid = 100 + 120 - Cost of your bid
If you win and the prize value is 80:
Earnings = Initial Endowment + Prize - Cost of Bid = 100 + 80 - Cost of your bid
If you do not win:
Earnings = Initial Endowment - Cost of Bid = 100 - Cost of your bid
An example
Suppose you bid 34 francs and the other participant bids 40 francs. Your personal random number is 1.20, and the other participant’s random number is 0.8. Therefore, your final bid is 40.8 = 34 \(\times\) 1.20, and the other participant’s final bid is 32 = 40 \(\times\) 0.8.
Since your final bid (40.8) is higher than the other participant’s final bid (32), you receive the prize. The cost of your bid (34) is 11.56. If you compete for the prize of 120 francs, your earning is 208.44 = 100 + 120 - 11.56. If you compete for the prize of 80 francs, then your earning is 167.04 = 100 + 80 - 11.56.
At the end of each period, your bid, your personal random number, the cost of your bid, your final bid, your reward, and your earnings for that period are reported.
One of the 10 periods will be randomly chosen for your actual payment for this part of the experiment. The amount will be converted to Turkish Lira.
PART 3 [Perfect-Substitutes]
The third part of the experiment consists of 10 decision-making periods. At the beginning, you will be randomly and anonymously placed into a group of two people (Group 1 or Group 2). Your group of two will be matched with another group of two, and the two groups will compete for a prize. One group—either Group 1 or Group 2—will receive a prize of 200 francs at the end of each period. After the group assignments are made, you will be either Player A or Player B in your group. The labels will be determined by your performance in the first part. The other member in your group, the other group, and the labels in each group will remain the same during this part.
In each period, you will have an initial endowment of 100 francs. Each group member can bid any amount between 0 and 100. At the beginning of each period, you will see which group and player type you are assigned to.
At the bottom of the box where you will enter your bid, there is a calculator button to perform your calculations.
After you and the other member of your group have made your bids, the computer will total these bids and multiply them by a “group random number” to calculate your group’s final bid. The group random number can take any value between 0 and 2. This number is separately and independently drawn for each period and each group.
Your Group Final Bid = (Your bid + Group member’s bid) x Group random number
[Best-Shot: After you and the other member of your group have made bids, the computer will multiply the highest bid by a “group random number” to determine your group’s final bid.
Your Group Final Bid = max{Your bid, Group member’s bid} x Group random number
Weakest-Link: After you and the other member of your group have made bids, the computer will multiply the lowest bid by a “group random number” to determine your group’s final bid.
Your Group Final Bid = min{Your bid, Group member’s bid} x Group random number]
For each bid, there is a cost.
Cost of Bid = \(Bid^2/100\)
After your group and the other group make bids, the computer will draw the random numbers and compare your group’s final bid to the other group’s final bid. If your group’s final bid is higher than the other group’s final bid, your group will receive a prize of 200 francs. Otherwise, your group will receive 0 francs. Each member of the winning group will earn an amount from the reward based on the participant’s “name”. In other words:
If your group wins and you are Player A:
Earnings = Initial Endowment + Amount Player A earns from the group winning - Cost of Bid = 100 + 120 - Cost of your bid
If your group wins and you are Player B:
Earnings = Initial Endowment + Amount Player B earns from the group winning - Cost of Bid = 100 + 80 - Cost of your bid
If your group does not win:
Earnings = Initial Endowment - Cost of Bid = 100 - Cost of your bid
An example
Let’s say you have been placed into Group 1 and labeled as Player A. You bid 36 francs and the other member in your group (Player B) bids 40 francs. The other group’s players make bids of 40 and 60 francs. Your group’s random number is 1.25 and the other group’s random number is 0.8. Your group’s final bid is therefore 95 = (36 + 40) \(\times\) 1.25 and the other group’s final bid is 80 = (40 + 60) \(\times\) 0.8.
Since your group’s final bid (95) is higher than the other group’s final bid (80), your group receives the prize. Since the cost of your bid (36) is 12.96 and the reward is 120 francs, your earning is 207.04 = 100 + 120 - 12.96. The earning of the other group member in your group is 164 = 100 + 80 - 16.
At the end of each period, your bid, the cost of your bid, the bid of the other member in your group, your group random number, your group’s final bid, your prize, and your earnings for the period are reported.
One of the 10 periods will be randomly chosen for your actual payment for this part, and it will be converted to Turkish Lira.
PART 4
This part consists of 1 decision-making period. It is similar to the second part. The only difference is the value of the prize. You will be anonymously paired and will compete to receive a prize of 0 francs.
After you make your bid, the computer will multiply it by a “personal random number” to calculate your final bid. This random number may have a value of anywhere between 0 and 2. It is separately and independently drawn for each person.
Your Final Bid = Your bid x Personal random number
For each bid, there is a cost.
Cost of Bid = \(Bid^2/100\)
After you and the other participant have made bids, the computer will draw random numbers and compare your final bids. If your final bid is higher than the other participant’s, you will receive a prize of 0 francs. In other words:
If you win:
Earnings = Initial Endowment + Prize - Cost of Bid = 100 + 0 - Cost of your bid
If you do not win:
Earnings = Initial Endowment - Cost of Bid = 100 - Cost of your bid
At the end of the period, your bid, your random number, the cost of your bid, your final bid, your prize, and your earning for the period are reported.
Your one-period earning will be converted to Turkish Lira for your actual payment.
PART 5
In this part, you will make a series of choices in decision-making problems. How much you earn will depend on chance and the choices you make. For each line, please state whether you prefer Option A or Option B. There are 15 lines in the table, but only one line will be randomly selected for payment, and you will not know which line will be drawn. Thus, you should be careful about the choice you make in every line.
After you have completed all your choices, the computer will randomly draw a number from 1 to 15 to determine which line of the lottery will be selected for payment. If you chose Option A in that line, you will receive 14 francs. If you chose Option B, you will receive either 40 francs or 0 francs. The computer will randomly draw a number from 1 to 20 to determine this earning. If the number is in the left column, you receive 40 francs. If the number is in the right column, you receive 0 francs.
B Best-shot contests: equilibrium conditions
C Additional analysis
Efforts over time in perfect-substitutes contests. PS-HL and PS-HH represent strong subjects in low and high noise variances, respectively. PS-LL and PS-LH represent weak subjects in low and high noise variances, respectively. The horizontal dashed lines (color-coded to match the solid lines) show their equilibrium predictions
Efforts over time in best-shot contests. BS-HL and BS-HH represent strong subjects in low and high noise variances, respectively. BS-LL and BS-LH represent weak subjects in low and high noise variances, respectively
Efforts over time in weakest-link contests. WL-HL and WL-HH represent strong subjects in low and high noise variances, respectively. WL-LL and WL-LH represent weak subjects in low and high noise variances, respectively. The horizontal dashed lines (color-coded to match the solid lines) show their equilibrium predictions
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İntişah, M., Büyükboyacı, M. The role of noise variance on effort in group contests. Theory Decis (2024). https://doi.org/10.1007/s11238-023-09974-4
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DOI: https://doi.org/10.1007/s11238-023-09974-4