Multi-stage Contests Research Articles

Screening in Multistage Contests

Screening in Multistage Contests

Prize formation and sharing in multi-stage contests

Prize formation and sharing in multi-stage contests

Strategic investment decisions in multi-stage contests with heterogeneous players

When heterogeneous players make strategic investment decisions in multi-stage contests, they might conserve resources in a current contest to spend more in a subsequent contest, if the degree of heterogeneity in the current (subsequent) contest is sufficiently large (small). We confirm these predictions using data from German professional soccer, in which players are subject to a one-match ban if they accumulate five yellow cards. Players with four yellow cards facing the risk of being suspended for the next match are (i) less likely to be fielded when the heterogeneity in the current match increases and (ii) more likely to receive a fifth yellow card in the current match when heterogeneity in the next match increases or heterogeneity in the next match but one (when they return from their ban) decreases.

Endogenous alliances in survival contests

Esteban and Sákovics (2003) showed in their three-person game that an alliance never appears in a possibly multi-stage contest game for an indivisible prize when allies’ efforts are perfectly substitutable. In this paper, we introduce allies’ effort complementarity in alliances by using a CES effort aggregator function. We consider an open-membership alliance formation game followed by two contests: one played by alliances, and one within the winning alliance. We show that if allies’ efforts are too substitutable or too complementary, there is no meaningful alliance in equilibrium. However, if allies’ efforts are moderately complementary to each other, then competition between two alliances is a subgame perfect equilibrium, which Pareto-dominates the equilibrium in a no-alliance single-stage contest. We also show that if forming more than two alliances is supported in equilibrium, then it Pareto-dominates two-alliance equilibrium. Nevertheless, the parameter space for such an allocation to be supported as an equilibrium shrinks when the number of alliances increases.

Race meets bargaining in product development

We study product development in a firm, utilizing a multistage contest model (i.e., race) with an endogenous length (with one stage or two stages) between two workers. We model the payments to workers using the normatively appealing Nash bargaining solution. We analytically characterize the equilibrium effort levels of workers and describe the conditions under which a full‐fledged final product (as opposed to, say, a prototype) is developed. We also provide an answer to the firm's problem of optimal incentive provision.

Race Meets Bargaining in Product Development

We introduce a model of product development in a firm. Our model describes the process as a multi-stage contest (i.e., race) with an endogenous length (with one stage or two stages) between two workers. We model the payments to workers from the new product using the normatively appealing Nash bargaining solution (see Nash, 1950). In our model the disagreement payoffs endogenously depend on the contest outcome. More precisely, a bargaining advantage is given to the leading worker in the product development contest. We analytically characterize subgame perfect equilibrium effort levels of workers and describe the conditions under which a full-edged final (as opposed to, say, a prototype) product is developed. Our comparative static analyses reveal economically intuitive insights. Finally, we provide an answer to the firm's problem of optimal incentive provision (considering both collective and individual incentives).

Escalation in conflict games: on beliefs and selection

We study learning and selection and their implications for possible effort escalation in a simple game of dynamic property rights conflict: a multi-stage contest with random resolve. Accounting for the empirically well-documented heterogeneity of behavioral motives of players in such games turns the interaction into a dynamic game of incomplete information. In contrast to the standard benchmark with complete information, the perfect Bayesian equilibrium features social projection and type-dependent escalation of efforts caused by learning. A corresponding experimental setup provides evidence for type heterogeneity, for belief formation and updating, for self-selection and for escalation of efforts in later stages.

Discouragement effect and intermediate prizes in multi-stage contests: Evidence from Davis Cup

The discouragement effect of being the lagging player in multi-stage contests is a well-documented phenomenon. In this study, we utilize data from 2447 Davis Cup matches in team tennis tournaments to test the effect of being behind or ahead on individuals’ performance with and without intermediate prizes. Using several different strategies to disentangle the effect of being ahead in the interim score from the effect of selection, we find the usual discouragement effect. However, the discouragement effect disappears after the introduction of intermediate prizes in the form of ranking points. The lagging favorite had close to a 20-percentage point greater probability of winning compared to matches without such a prize. We show that this result is not driven by the selection of better players into tournaments with intermediate prizes. As predicted by previous theoretical studies, our empirical findings suggest that intermediate prizes may mitigate or even eliminate the ahead–behind effects that arise in multi-stage contests.

Strategic Investment Decisions in Multi-Stage Contests with Heterogeneous Players

We study strategic investment decisions in multi-stage contests with heterogeneous players. Our theoretical model of a round-robin rank-order tournament predicts that players conserve resources in a current contest to spend more in the subsequent contest if the degree of heterogeneity in the current (subsequent) contest is sufficiently large (small). We confirm these predictions using data from German professional soccer, where players are subject to a one-match ban if they accumulate five yellow cards. We find that players with four yellow cards facing the risk of being suspended for the next match are (i) less likely to be fielded when the heterogeneity in the current match increases and (ii) more likely to receive a fifth yellow card in the current match when the heterogeneity in the next match increases or when the heterogeneity in the next match but one (for which they return from their ban) decreases.

First in first win: Evidence on schedule effects in round-robin tournaments in mega-events

The order of actions in contests may have a significant effect on performance. In this study, we examine the role of the schedule in round-robin tournaments with sequential games between three and four contestants. Our empirical analysis, based on soccer FIFA World Cups and UEFA European Championships, and on two Olympic wrestling events, reveals that there is a substantial advantage to the contestant who competes in the first and third matches. Our findings are in line with the hypothesis that winning probabilities in multi-stage contests with sequential games are endogenously depending on the schedule of contests as predicted by game-theoretical models. We also discuss possible ways to reduce the effect of the schedule.

Escalation in Dynamic Conflict: On Beliefs and Selection

We study a dynamic multi-stage contest that resolves in each stage only with a given probability. Assuming that there is unobservable heterogeneity in intrinsic motivations we derive properties of the equilibrium efforts across the different stages. Whereas in the corresponding complete information benchmark equilibrium efforts are stable across the stages, uncertainty about the type distribution of possible opponents generates learning. We identify reasons for dynamic adjustments of efforts caused by belief formation and updating and by selection of certain types into continuing conflict. A corresponding experimental setup provides evidence for escalation of efforts in later stages, for type heterogeneity, for belief formation and belief updating, and for selfselection. Overall, our results suggest the importance of an appropriate benchmark model when testing predictions on behavior in conflict or related strategic interactions.

Coping with complexity – Experimental evidence for narrow bracketing in multi-stage contests

This paper investigates whether decision makers bracket their choices narrowly to facilitate complex decision problems. Evidence from a framing variation of rewards in experimental two-stage pairwise elimination contests indicates that decision makers neglect the option value of participation in future stages of the contest if the reward frame facilitates the separate consideration of stages, but not if the reward frame induces forward-looking behavior. Decision makers account for the option value of participation in future stages of the contest independently of the reward frame, however, when complex strategic interactions in future stages of the contest are replaced by simple lotteries that facilitate the determination of the option value. The results present novel evidence for the prevalence and the determinants of choice bracketing as a means to cope with complexity.

Gender Quotas in a Multistage Tournament

We conduct a laboratory experiment to study the impact of affirmative action on the choice to compete in a multistage elimination contest. In particular, we assess whether gender quotas imposed in the final stage increase the share of female entrants and winners already in the first round of the tournament. In a treatment without quotas we reproduce the standard result from one-shot tournaments that high performing women shy away from competitions they could win. Adding gender quotas in the second stage induces high ability women to compete in both stages of the tournament, while it leads to a small and insignificant reduction in men's tournament entry rates. Response to affirmative action is strongest among women with intermediate guessed ranks. Overall, we find that the affirmative action treatment increases female representation in both stages without harming efficiency.

From Custer to Thermopylae: Last stand behavior in multi-stage contests

In a multi-stage contest known as a two-player race, players display two fundamental behaviors: (1) the laggard will make a last stand in order to avoid the cost of losing; and (2) the player who is ahead will defend his lead if it is threatened. Last stand behavior, in particular, contrasts with previous research where the underdog simply gives up. The distinctive results are achieved by introducing losing penalties and discounting into the racing environment. This framework permits the momentum effect, typically ascribed to the winner of early stages, to be more thoroughly examined. I study the likelihood that the underdog will catch up. I find that neck-and-neck races are common when the losing penalty is large relative to the winning prize, while landslide victories occur when the prize is relatively large. Closed-form solutions are given for the case where players have a common winning prize and losing penalty.

Dynamic Difference‐Form Contests

AbstractThis study presents a multistage contest designed by an organizer who chooses the (indivisible) winner prize together with the ranking scheme. When the organizer's objective is to maximize efforts, the optimal ranking scheme that selects the prize recipient coincides with a version of the piecewise linear difference‐form success function postulated by Che and Gale (2000). Even if the organizer's objective is not necessarily to maximize agents' effort, a noisy ranking of the difference‐form remains optimal.

Myopia in Contest: Accumulating Valuation All-Pay Auction Model and Experiment

Myopia is defined as a cognitive bias to underestimate the potential outcomes of current decision. It is believed to exist widely, and often leads to deviation from optimal decision making. We use a multi-stage contest: accumulating valuation all-pay auction to test if people are myopic. We find clear evidence of underbidding in the first stages of a controlled laboratory experiment, suggesting subjects underestimate the potential benefits of current decisions. Also, our treatment sort foresighted players out from myopic players.

The optimal multi-stage contest

This paper investigates the optimal (effort-maximizing) structure of multi-stage sequential-elimination contests. We allow the contest organizer to design the contest structure using two instruments: contest sequence (the number of stages, and the number of contestants remaining after each stage), and prize allocation. When the contest technology is sufficiently noisy, we find that multi-stage contests elicit more effort than single-stage contests. For concave and moderately convex impact functions, the contest organizer should allocate the entire prize purse to a single final prize, regardless of the contest sequence. Additional stages always increase total effort. Therefore, the optimal contest eliminates one contestant at each stage until the finale when a single winner obtains the entire prize purse. Our results thus rationalize various forms of multi-stage contests that are conducted in the real world.

Contests with Ties

We study two-player all-pay contests in which there is a positive probability of a tied outcome. We show that the players' efforts in equilibrium do not depend on the expected prize in the case of a tie given that this prize is smaller than the prize for winning. The implications of this result are twofold. First, in symmetric one-stage contests, the designer who wishes to maximize the expected total effort should not award a prize in the case of a tie which is larger than one-third of the prize for winning. Second, in multi-stage contests, the designer should not limit the number of stages (tie-breaks) but should allow the contest to continue until a winner is decided.

Orchestrating Rent Seeking Contests

Contests have different, sometimes quite complex, organisational structures. In particular, while most of the existing literature focusses on simultaneous contests, multistage contests are also quite frequently encountered. This paper seeks to provide a rationale for the latter by endogenising the choice of a contest structure, which is made by an organiser of a contest interested in maximising the efforts expended by the contenders.