Strategically Equivalent Contests

Using a two-player Tullock-type contest, we show that intuitively and structurally different contests can be strategically equivalent. Strategically equivalent contests generate the same best response functions and, as a result, the same equilibrium efforts. However, strategically equivalent contests may yield different equilibrium payoffs. We propose a simple two-step procedure to identify strategically equivalent contests. Using this procedure, we identify contests that are strategically equivalent to the original Tullock contest, and provide new examples of strategically equivalent contests. Finally, we discuss possible contest design applications and avenues for future theoretical and empirical research

Fight Alone or Together? The Need to Belong

Alliances often face both free-riding and hold-up problems, which under- mine the effectiveness of alliances in mobilizing joint fighting effort. Despite of these disadvantages, alliances are still ubiquitous in all types of contests. This paper asks if there are non-monetary incentives to form alliances, e.g., intimidating/discouraging the single player(s) who is/are left alone. For this purpose, I compare symmetric (2 vs. 2) and asymmetric (2 vs. 1) contests to their equivalent 4-player and 3-player individual contests, respectively. We find that alliance players in symmetric (2 vs. 2) contests behave the same as those in equivalent 4-player individual contests. However, in asymmetric (2 vs. 1) contests, stand-alone players were strongly discouraged to exert effort (especially the females), compared to the 3-player individual contests. Alliance players may have anticipated this effect and also reduced their effort, if alliances share the prize according to the merit rule. Behavioural factors such as the need to belong can help reconcile the "paradox of alliance formation"

Parallel Contests

A problem of Parallel Contests is raised and modeled. The equilibria in final situations of parallel contests are analyzed and characterized and the behaviours of contestants with different abilities’ parameters are explained. Given that the values of the prizes in the two contests are different, in equilibrium a group of strong players prefer entering into the contest with a higher prize. However, except the group of stronger ones, in equilibrium others will enter into both contests because they obtain equal expected revenue from the two contests, though these weak ones do not have equal probabilities to enter into the two parallel contests. Under the condition of rationalizability, this paper characterizes the respective distributions of contestants’ abilities in the two parallel contests, proves the existence of the equilibrium in parallel contests and completes the analysis of the parallel contests from the perspective of contestants.Parallel contests, Contest, Strategic behaviours

Optimal Crowdsourcing Contests

We study the design and approximation of optimal crowdsourcing contests. Crowdsourcing contests can be modeled as all-pay auctions because entrants must exert effort up-front to enter. Unlike all-pay auctions where a usual design objective would be to maximize revenue, in crowdsourcing contests, the principal only benefits from the submission with the highest quality. We give a theory for optimal crowdsourcing contests that mirrors the theory of optimal auction design: the optimal crowdsourcing contest is a virtual valuation optimizer (the virtual valuation function depends on the distribution of contestant skills and the number of contestants). We also compare crowdsourcing contests with more conventional means of procurement. In this comparison, crowdsourcing contests are relatively disadvantaged because the effort of losing contestants is wasted. Nonetheless, we show that crowdsourcing contests are 2-approximations to conventional methods for a large family of "regular" distributions, and 4-approximations, otherwise.Comment: The paper has 17 pages and 1 figure. It is to appear in the proceedings of ACM-SIAM Symposium on Discrete Algorithms 201

Gambling in Contests

This paper presents a strategic model of risk-taking behavior in contests. Formally, we analyze an n-player winner-take-all contest in which each player decides when to stop a privately observed Brownian Motion with drift. A player whose process reaches zero has to stop. The player with the highest stopping point wins. Contrary to the explicit cost for a higher stopping time in a war of attrition, here, higher stopping times are riskier, because players can go bankrupt. We derive a closed-form solution of the unique Nash equilibrium outcome of the game. In equilibrium, the trade-off between risk and reward causes a non-monotonicity: highest expected losses occur if the process decreases only slightly in expectation

Dynamic contests

Considering several main types of dynamic contests (the race, the tug-of-war, elimination contests and iterated incumbency fights) we identify a common pattern: the discouragement effect. This effect explains why the sum of rentseeking efforts often falls considerably short of the prize that is at stake. It may cause violent conflict in early rounds, but may also lead to long periods of peaceful interaction. -- Unter Berücksichtigung verschiedener Haupttypen dynamischer Wettbewerbe (das Wettrennen, das Tauziehen, Ausscheidungskämpfe und wiederholte Kämpfe um Amtszeiten) identifizieren die Autoren ein gemeinsames Muster: den Entmutigungseffekt. Dieser Effekt erklärt, wieso die Rent-seeking-Bemühungen in Summe oft deutlich nicht an den auf dem Spiel stehenden Preis heranreichen. Der Effekt kann heftige Kämpfe in den ersten Runden des Wettbewerbs auslösen, aber auch zu langen Perioden friedlichen Zusammenspiels führen.

Multi-battle contests

We study equilibrium in a multistage race in which players compete in a sequence of simultaneous move component contests. Players may win a prize for winning each component contest, as well as a prize for winning the overall race. Each component contest is an all-pay auction with complete information. We characterize the unique equilibrium analytically and demonstrate that it exhibits endogenous uncertainty. Even a large lead by one player does not fully discourage the other player, and each feasible state is reached with positive probability in equilibrium (pervasiveness). Total effort may exceed the value of the prize by a factor that is proportional to the maximum number of stages. Important applications are to war, sports, and R&D contests and the results have empirical counterparts there

Contests with Investment

Perfectly discriminating contests (or all pay auction) are widely used as a model of situations where individuals devote resources to win some prize. In reality such contests are often preceded by investments of the contestants into their ability to fight in the contest. This paper studies a two stage game where in the first stage, players can invest to lower their bid cost in a perfectly discriminating contest, which is played in the second stage. Different assumptions on the timing of investment are studied. With simultaneous investments, equilibria in which players play a pure strategy in the investment stage are asymmetric, exhibit incomplete rent dissipation, and expected effort is reduced relative to the game without investment. There also are symmetric mixed strategy equilibria with complete rent dissipation. With sequential investment, the first mover always invests enough to deter the second mover from investing, and enjoys a first mover advantage. I also look at unobservable investments and endogenous timing of investments

A generalized Tullock contest

We construct a generalized Tullock contest under complete information where contingent upon winning or losing, the payoff of a player is a linear function of prizes, own effort, and the effort of the rival. This structure nests a number of existing contests in the literature and can be used to analyze new types of contests. We characterize the unique symmetric equilibrium and show that small parameter modifications may lead to substantially different types of contests and hence different equilibrium effort levels