Multiple equilibria in Tullock contests

We find the sufficient conditions for the existence of multiple equilibria in Tullock-type contests, and show that asymmetric equilibria arise even under symmetric prize and cost structures. We then present existing contests where multiple equilibria exist under reasonably weak conditions

Endogenous Timing in General Rent-Seeking and Conflict Models

This paper examines simultaneous versus sequential choice of effort in a two player contest with a general contest success function. The timing of moves, determined in a pre-play stage prior to the contest-subgame, as well as the value of the prize is allowed to be endogenous. Contrary to endogenous timing models with an exogenously fixed prize the present paper finds the following. (1) Players may decide to choose their effort simultaneously in the subgame perfect equilibrium (SPE) of the extended game. (2) The SPE does not need to be unique, (3) in particular, there is no unique SPE with sequential moves if costs of effort are exclusively endogenously determined. (4) If the unique SPE is sequential play, the win probability in the NE is in no way crucial for the determination of an endogenous leadership. (5) Finally, symmetry among players does not rule out incentives for precommitment to effort locally away from the Nash-Cournot level.Contests, Endogenous timing, Endogenous prize

Endogenous Timing in General Rent‐Seeking and Conflict Models

This paper examines simultaneous versus sequential choice of effort in a twoplayer contest with a general contest success function. The timing of moves, determined in a pre‐play stage prior to the contest‐subgame, as well as the value of the prize is allowed to be endogenous. Contrary to endogenous timing models with an exogenously fixed prize the present paper finds the following. (1) Players may decide to choose their effort simultaneously in the subgame perfect equilibrium (SPE) of the extended game, (2) the SPE does not need to be unique, (3) in particular, there is no unique SPE with sequential moves if costs of effort are exclusively endogenously determined, (4) if the unique SPE is sequential play, the win probability in the NE is in no way crucial for the determination of an endogenous leadership, (5) and symmetry among players does not rule out incentives for precommitment to effort locally away from the Nash‐Cournot levelContests, Endogenous timing, Endogenous prize

Equilibrium and Efficiency in the Tug-of-War

We characterize the unique Markov perfect equilibrium of a tug-of-war without exogenous noise, in which players have the opportunity to engage in a sequence of battles in an attempt to win the war. Each battle is an all-pay auction in which the player expending the greater resources wins. In equilibrium, contest effort concentrates on at most two adjacent states of the game, the "tipping states", which are determined by the contestants’ relative strengths, their distances to final victory, and the discount factor. In these states battle outcomes are stochastic due to endogenous randomization. Both relative strength and closeness to victory increase the probability of winning the battle at hand. Patience reduces the role of distance in determining outcomes. Applications range from politics, economics and sports, to biology, where the equilibrium behavior finds empirical support: many species have developed mechanisms such as hierarchies or other organizational structures by which the allocation of prizes are governed by possibly repeated conflict. Our results contribute to an explanation why. Compared to a single stage conflict, such structures can reduce the overall resources that are dissipated among the group of players

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

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

Asymmetric All-Pay Contests with Spillovers

When opposing parties compete for a prize, the sunk effort players exert during the conflict can affect the value of the winner's reward. These spillovers can have substantial influence on the equilibrium behavior of participants in applications such as lobbying, warfare, labor tournaments, marketing, and R&D races. To understand this influence, we study a general class of asymmetric, two-player all-pay contests where we allow for spillovers in each player's reward. The link between participants' efforts and rewards yields novel effects -- in particular, players with higher costs and lower values than their opponent sometimes extract larger payoffs

Equilibrium and Efficiency in the Tug-Of-War.

We characterize the unique Markov perfect equilibrium of a tug-of-war without exogenous noise, in which players have the opportunity to engage in a sequence of battles in an attempt to win the war. Each battle is an all-pay auction in which the player expending the greater resources wins. In equilibrium, contest effort concentrates on at most two adjacent states of the game: the “tipping states”, which are determined by the contestants’ relative strengths, their distances to final victory, and the discount factor. In these states battle outcomes are stochastic due to endogenous randomization. Both relative strength and closeness to victory increase the probability of winning the battle at hand. Patience reduces the role of distance in determining outcomes. Applications range from politics, economics and sports, to biology, where the equilibrium behavior finds empirical support: many species have developed mechanisms such as hierarchies or other organizational structures by which the allocation of prizes are governed by possibly repeated conflict. Our results contribute to an explanation why. Compared to a single stage conflict, such structures can reduce the overall resources that are dissipated among the group of players.winner-take-all ; all-pay auction ; tipping ; multi-stage contest ; dynamic game ; preemption ; conflict ; dominance

Equilibrium and Efficiency in the Tug-of-War

We characterize the unique Markov perfect equilibrium of a tug-of-war without exogenous noise, in which players have the opportunity to engage in a sequence of battles in an attempt to win the war. Each battle is an all-pay auction in which the player expending the greater resources wins. In equilibrium, contest effort concentrates on at most two adjacent states of the game, the "tipping states", which are determined by the contestants' relative strengths, their distances to final victory, and the discount factor. In these states battle outcomes are stochastic due to endogenous randomization. Both relative strength and closeness to victory increase the probability of winning the battle at hand. Patience reduces the role of distance in determining outcomes.Applications range from politics, economics and sports, to biology, where the equilibrium behavior finds empirical support: many species have developed mechanisms such as hierarchies or other organizational structures by which the allocation of prizes are governed by possibly repeated conflict. Our results contribute to an explanation why. Compared to a single-stage conflict, such structures can reduce the overall resources that are dissipated among the group of players.winner-take-all, all-pay auction, tipping, multi-stage contest, dynamic game, preemption, conflict, dominance

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.contests; all pay auctions; investment