Symmetric Contests Research Articles

More than a feeling: Effects of competitive asymmetry on human emotions

Competitive interactions have important effects on human emotions. Both victory and defeat can evoke a wide range of emotional reactions, including joy, pride, anger, fear, sadness and shame. However, little is known about what determines this variety of contestants’ affective responses. Therefore, the purpose of this study was to evaluate the effects of competitive asymmetry, a common and ecologically relevant feature of animal conflicts, on human emotional responses to winning or losing a contest. To test this hypothesis, we conducted two experiments, the first with high school students (n = 331) and the second with young athletes (n = 73), in which we manipulated the outcomes of successive matches in a non-athletic competition. Thus, by inducing the competitors’ scores, ranging from closer to more decisive outcomes, we were able to define the degree of competitive asymmetry in victory and defeat conditions. We then assessed participants’ emotional responses to a set of affective stimuli. In the defeat condition, we found in both studies an increase in the occurrence of anger and fear due to more symmetric contests. There were also more frequent reports of shame following more decisive defeats (Experiment 1) and of pride following closer victories (Experiment 2), which were seen neither for sadness nor joy in any of the studies. Supporting our hypothesis, emotional reactions triggered by asymmetries among contestants were consistent with the behavioral patterns commonly seen in symmetric and asymmetric animal conflict, such as dominance/aggressive and defensive/escape behaviors.

Gaining advantage by winning contests

We consider a principal who faces many identical competitors, and who can distribute a prize fund over two consecutive contests. The winner of contest one gains an advantage in contest two where his effort is more productive than all rivals. We identify a tipping point for the productivity parameter, below which it is optimal for an effort-maximizing principal to place the whole prize in the second contest. Above this level, a single symmetric contest is preferred. The institution chosen depends inextricably upon the number of competitors and their valuation of future gains and costs. We identify the optimal setting of the productivity parameter, showing that introducing asymmetry can increase total efforts by as much as one quarter compared to a single symmetric contest.

Two-person pairwise solvable games

A game is solvable if the set of Nash equilibria is nonempty and interchangeable. A pairwise solvable game is a two-person symmetric game in which any restricted game generated by a pair of strategies is solvable. We show that the set of equilibria in a pairwise solvable game is interchangeable. Under a quasiconcavity condition, we derive a complete order-theoretic characterization and some topological sufficient conditions for the existence of equilibria, and show that if the game is finite, then an iterated elimination of weakly dominated strategies leads precisely to the set of Nash equilibria, which means that such a game is both solvable and dominance solvable. All results are applicable to symmetric contests, such as the rent-seeking game and the rank-order tournament, which are shown to be pairwise solvable. Some applications to evolutionary equilibria are also given.

Dynamic Win Effects in Sequential Contests

We consider a principal who faces many identical competitors, and who can distribute a prize fund over two consecutive contests. The winner of contest one gains an advantage in contest two where his effort is more productive than all rivals. We identify a tipping point for the productivity parameter, below which it is optimal for an effort-maximizing principal to place the whole prize in the second contest. Above this level, a single symmetric contest is preferred. The institution chosen depends inextricably upon the number of competitors and their valuation of future gains and costs. We identify the optimal setting of the productivity parameter, showing that introducing asymmetry can increase total efforts by as much as one quarter compared to a single symmetric contest.

Aggressiveness compensates for low muscle strength and metabolic disadvantages in shell fighting: an outcome of the individual’s past

An individual’s metabolic competence is important when escalating to costly behaviours in agonistic encounters. The use of broken shells in the wild induces hermit crabs to reduce their overall metabolic rate; however, these crabs perform better in size-symmetric encounters than crabs that used intact shells in the wild. Given this surprising finding, we explored whether carrying a broken shell increases the resource holding potential (RHP) of hermit crabs by building up their muscle strength. These shells are very heavy, with an unusable portion due to being broken; however, the crabs have to carry the complete structure. Additionally, we explored whether the use of broken shells induces long-lasting consequences in motivation (aggressiveness). We conducted body size asymmetric contests to decrease the relative effect of aggressiveness by increasing the effect of RHP. We predicted a decrease of the agonistic behaviours of crab that occupied broken shells in the wild (CBS) relative to those that occupied intact shells (CIS). Symmetric and asymmetric contests were conducted in pairs formed of a CBS and a CIS, both forced to abandon them and occupy a Nerita shell. The CBS had lower muscle strength, but fought at higher intensity in symmetric contests than CIS. The differences in body mass decreased the expression of agonistic behaviours of CBS relative to CIS. Prolonged poor conditions following the use of a broken shell seem to increase aggressiveness by affecting the long-term subjective resource value. Our findings highlight the role of the recent past associated with shell used on fighting performance. Hermit crabs commonly fight to exchange their inadequate shell for a new one. Although high metabolic rates favour the fighting ability, crabs that occupied broken shells in the wild have low metabolism and perform better in shell fighting than crabs that used intact shells. Broken shells are heavy and larger due to having an unusable portion, but the crab has to carry the complete structure. We explored whether carrying these shells increases muscle strength and/or the aggressiveness of crabs, which might explain their fighting dynamic. Unexpectedly, crabs with broken shells were found to be weaker than those with intact ones. Instead, the use of a broken shell in the wild enhances individual aggressiveness. The results highlight that a previous history under adverse conditions can increase the likelihood of animals with functional disadvantages acquiring better resources.

Risk Aversion and Prudence in Contests

I examine the impact of risk preferences on efforts and winning probabilities in generalised Tullock contests between two players. The theoretical analysis yields two main results. First, I specify a sufficient condition on the agents' comparative prudence under which a higher common level of risk aversion leads to lower aggregate effort in symmetric contests. Second, I show that for a certain range of parameters in asymmetric contests, higher risk-aversion will be a disadvantage if the agent is comparatively prudent.

Asymmetric discouragement in asymmetric contests

We provide new experimental evidence which suggests an asymmetric discouragement effect in lottery contests with heterogeneous abilities. Compared to a symmetric contest, subjects invest less effort when facing a stronger opponent, but they invest the same when facing a weaker opponent. Our results can be explained by a simple model of disappointment aversion.

Private-information group contests: Best-shot competition

We model competing groups when players' values for winning are private information, each group's performance equals the best effort (“best shot”) of its members, and the group with the best performance wins the contest. At the symmetric equilibrium of symmetric contests, increasing the number of competing teams may increase or decrease each team's performance, but it unambiguously increases the overall expected best shot. Depending on the elasticity of the distribution of players' values, individual, team, and contest performance may increase or decrease with team size. Considering just two competing groups that differ only in size, we show members of the smaller group use the more aggressive strategy, but, depending on the nature of uncertainty, either team may be more likely to win. More generally, when teams' value cdfs differ, increasing one team's size decreases (increases) that team's chance of winning if its value cdf is elastic (inelastic).

Asymmetric Discouragement in Asymmetric Contests

We provide new experimental evidence which suggests an asymmetric discouragement effect in lottery contests with heterogeneous abilities. Compared to a symmetric contest, subjects invest less effort when facing a stronger opponent, but they invest the same when facing a weaker opponent. Our results can be explained by a simple model of disappointment aversion.

The payoff matrix of repeated asymmetric 2×2 games

Many researches in game theory have been devoted to study the symmetric contest between the contestants. This is due to the computational advantage in this type of game. In contrast, asymmetric games are more complicated in mathematical computations than the symmetric one. This article is particularly interested in the model of asymmetric games. In this model, we cannot interchange the identities of the players without interchanging the payoff of the strategies. This model differs from the symmetric model, where each player has two different choices from the two choices of the other one. We consider that the round of the game is infinitely repeated (infinitely repeated and asymmetric 2×2 game). The payoff matrix corresponding to all possible strategies of the game is obtained. Furthermore, we will apply the resulting matrix in two examples. During this, we will draw the attention towards the behavior of some of the strategies and the evolutionary stability.

Private-Information Group Contests

We model competing groups when players' values for winning are private information, each group's performance equals the best effort (best shot) of its members, and the group with the best performance wins the contest. At the symmetric equilibrium of symmetric contests, increasing the number of competing teams may increase or decrease each team's performance, but it unambiguously increases the overall expected best shot. Depending on the elasticity of the distribution of players' values, individual, team, and contest performance may increase or decrease with team size. Considering just two competing groups that differ only in size, we show members of the smaller group use the more aggressive strategy, but, depending on the nature of uncertainty, either team may be more likely to win. More generally, when teams' value cdfs differ, increasing one team's size decreases (increases) that team's chance of winning if its value cdf is elastic (inelastic).

Incomplete information and rent dissipation in deterministic contests

In a deterministic contest or all-pay auction, all rents are dissipated when information is complete and contestants are identical. As one contestant becomes “stronger”, that is, values the prize more, total expenditures are known to decrease monotonically. Thus, asymmetry among contestants reduces competition and rent dissipation. Recently, this result has been shown to hold for other, non-deterministic, contest success functions as well, thereby suggesting a certain robustness. In this paper, however, the complete information assumption is shown to be crucial. I examine a tractable incomplete information model for which the complete information model is a special case. With incomplete information — regardless of how little — total expenditures in a deterministic two-player contest increase when one contestant becomes marginally stronger, starting from a symmetric contest. In fact, both contestants expend resources more aggressively; with complete information, neither of them do so. Thus, there is a “discontinuity” in the information structure.

Endogenous Liquidity Constraints in a Dynamic Contest

In this article, I analyze the effects of future liquidity constraints on the investment behavior of two contestants with asymmetric prize valuations in a dynamic contest model. Contestants compete in two consecutive Tullock contests in order to win a contest prize in each period. The loser of the first-period contest can be liquidity constraint in the second period. The model reveils the following four main results: Future liquidity constraints marginally affect today's intensity of competition but rather influence tomorrow's contest. A higher contest prize in both periods surprisingly decreases aggregate second-period investment in a symmetric contest. Counter intuitively, a higher asymmetry with respect to the contest prize valuations increases the first-period investment of both contestants. The effect of a higher asymmetry on second-period investment depends on which contestant won the first-period contest. Further results are derived with respect to the existence and uniqueness of the equilibrium, competitive balance and expected total profits.

Contests with private costs: Beyond two players

It was shown previously that in the symmetric contest game of two players, equilibrium bidding is lower in the case of private information than in the case of public information about the players' costs. I consider symmetric contests of an arbitrary number of players with continuously distributed private costs and discuss the existence and properties of equilibrium bidding functions. I show that with more than two players the relationship between equilibrium bids in the cases of public and private information is no longer universal. While high-cost players still bid less in the private information case, relatively low-cost players may bid above or below their corresponding public information bids.

On being asset-constrained in litigation contests

This paper establishes that being asset-constrained may be advantageous for defendants in litigation contests. An active asset constraint of defendants (i) may increase (decrease) defendant (plaintiff) contest effort, (ii) may lower the sum of plaintiff and defendant contest effort, (iii) lowers expected defendant payment, and (iv) makes a symmetric contest asymmetric.

Risk aversion in symmetric and asymmetric contests

We analyze existence, uniqueness and properties of equilibria in incompletely discriminating Tullock contests with logistic contest success functions, when contestants are risk averse. We prove that a Nash equilibrium for such a contest exists, but give an example of a symmetric contest with both symmetric and asymmetric equilibria, showing that risk aversion may lead to multiple equilibria. Symmetric contests have unique symmetric equilibria but additional conditions are necessary for general uniqueness. We also study the effects on incumbents of additional competitors entering the contest under these conditions and examine the effects of risk aversion on rent dissipation in symmetric and asymmetric contests.

The spatial model with non-policy factors: a theory of policy-motivated candidates

The theoretical literature on two candidate elections is dominated by symmetric contests and vote-maximizing candidates. These models fail to capture two important features of real elections. First, most elections pit a stronger candidate against a weaker one. Second, candidates care not only about holding office, but also about policy outcomes. Ignoring any one of these features means we will fail to capture an important dynamic—strong candidates must balance their desire to change policy with their need to win the election. We provide conditions for the existence of an equilibrium in the spatial model with non-policy factors, when candidates are policy motivated. We provide a characterization of ‘regular’ equilibria and show that there exists at most one regular equilibrium. We provide conditions that guarantee that all equilibria are regular. We derive comparative statics for the model and show that increasing a candidate’s non-policy advantage causes that candidate to move towards his ideal point.

HETEROGENEOUS CONTESTS AND LESS INFORMATIVE SIGNALS*

We consider rank-order contests with heterogeneous agents in which the principal is restricted to using a fair contest (or a symmetric contest), focusing on the optimal accuracy of output signals. As opposed to the absolute performance evaluation, we show that it is optimal for the principal to deliberately make the signals noisier according to the degree of heterogeneity. Some economic interpretations of controlling noise are discussed.

Best-of-Three Contests between Equally-Skilled Players

Many contests feature a best-of-N structure. We focus on best-of-three contests between equally-skilled players. Using a general symmetric contest success function, we determine the relationship between the players' efforts in the various rounds. These results imply that the contests are more likely to end in two rounds rather than in three. Data from professional tennis matches strongly support this prediction. Our findings generalize and reinforce the mechanism Klumpp and Polborn (forthcoming) use to explain the New Hampshire effect in US presidential primaries.

Effort and Performance in Public Policy Contests

AbstractGovernment intervention often gives rise to contests in which the possible “prizes” are determined by the status quo and some new public policy proposal. In this paper we study a general class of such two‐player public policy contests and examine the effect of a change in the proposed policy, a change that may affect the payoffs of the two contestants, on their effort and performance. Our results extend the existing comparative statics studies that focus, in symmetric contests, on the effect of a change in the value of the prize or, in asymmetric contests, on the effect of one contestant's valuation of the prize. Our results hinge on a fundamental equation that specifies the equilibrium relationship between the strategic own‐stake effect and the strategic rival's‐stake effect. This fundamental equation clarifies the role of the three possible types of ability and stake asymmetry in determining the effect of payoff variations on the efforts and performance of the contestants.