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Theory and Decision

17.02.2024

The role of noise variance on effort in group contests

verfasst von: Merve İntişah, Mürüvvet Büyükboyacı

Erschienen in: Theory and Decision

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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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Fußnoten
1
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.
 
2
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.
 
3
Group effort is characterized as a function of all group members’ efforts, which changes with group impact functions.
 
4
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 \(r = \infty\).
 
5
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.
 
6
The constant b is a restriction on players’ abilities on the quadratic cost function, as in Cason et al. (2020).
 
7
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.
 
8
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.
 
9
Player 1A and player 1B are “strong players”, and the others, player 2A and 2B, are “weak players”.
 
10
For incentive compatibility and participation constraint conditions, see Table 13 in Appendix B.
 
11
We restrict \(\alpha \geqslant 0.5\) in the experiment because the expected payoff in Equation 5 is nonnegative for any \(\alpha \in [0.5, 1]\).
 
12
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.
 
13
The treatment names are determined by the group impact functions and noise variance.
 
14
Throughout the experiment, payoffs were in “francs,” which were converted into Turkish Lira (TL) at the rate of 40 francs to 3 TL.
 
15
37.90% of the subjects were from the economics department; 51.61% were male. Their ages ranged from 20 to 25 (87.10%).
 
16
In 2021, the hourly minimum wage in Turkey was 15.90 TL.
 
17
The instructions were in Turkish. An English translation is provided in Appendix A.
 
18
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.
 
19
This information appeared at the bottom of the decision screen. Subjects could use an on-screen calculator to determine what a bid would cost.
 
20
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.
 
21
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.
 
22
The group random number was randomly and independently drawn in each period for each group.
 
23
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.
 
24
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).
 
25
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.
 
26
These independent variables are used for every regression analysis throughout this paper.
 
27
Table 16 in Appendix C shows the distribution of the total number of safe lottery options chosen by all subjects in the experiment.
 
28
Table 15 in Appendix C shows that almost 75% of subjects exert positive efforts in the 0-prize contest. However, there is no significant correlation between efforts for a prize of 0 and efforts for contest prizes in either noise variance treatment.
 
29
For the subjects’ average efforts over 10 periods in perfect-substitutes, best-shot, and weakest-link contests, see Appendix C, Figs. 5, 6, 7, respectively.
 
30
Only the distribution of strong subjects’ effort significantly differs with noise variance in best-shot contests (ksmirnov test, p-value = 0.06).
 
31
These interpretations are according to the mode of efforts in each case.
 
32
We also notice a high variation in subjects’ efforts, especially in low noise variance.
 
33
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).
 
34
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.
 
35
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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Metadaten
Titel
The role of noise variance on effort in group contests
verfasst von
Merve İntişah
Mürüvvet Büyükboyacı
Publikationsdatum
17.02.2024
Verlag
Springer US
Erschienen in
Theory and Decision
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI
https://doi.org/10.1007/s11238-023-09974-4

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