Multi-player race

Abstract

We present a model of race with multiple players and study players' effort choices and expected prizes in equilibrium. We show that, in equilibrium, once any two players win one battle each, the remaining players do not exert any effort anymore. This turns the continuation game into a two-player race. This is different than the results in previous two-player models of race, which report that all states of the game are reached with positive probabilities. We also provide a set of comparative static results on the effects of the number of players and the victory threshold.

“The ball is round, the game lasts 90 minutes, everything else is pure theory”

“…In their model, two firms simultaneously spend resources in research and development in each stage of the patent contest and the first firm which reaches a certain number of stage victories wins the patent. Klumpp and Polborn (2006) in the context of sequential elections with two candidates, Konrad and Kovenock (2009) with intermediate prizes in each battle, Gelder (2014) with losing penalties, or Dogan et al (2018) with multiple players are other papers on races. In a tug-of-war game, two players compete for reaching first their respective terminal state and the winner of each battle can move one step further toward this player's terminal state.…”

Race meets bargaining in product development

“…and a ∈ [0, ∞). 4 The total compensation for workers' efforts in the development of the prototype is V p ≥ 0, and it will be allocated by the firm using the Nash bargaining solution (see Nash, 1950). Noting that a higher disagreement payoff is a source of bargaining power in the Nash bargaining solution, in our model the winner of the first stage contest is given a more advantageous disagreement point, d r ≥ 0, whereas the loser's disagreement point is normalized to 0.…”

“…Gelder (2014) introduced a losing penalty to race, showed that it can prevent momentum from building up in favor of the front runner, leading to a last‐stand behavior. Doğan, Karagözoğlu, Keskin, and Sağlam (2018) studied a multiplayer version, characterized the unique subgame perfect Nash equilibrium, and investigated how equilibrium changes compared with a two‐player race. The main differences of our model from earlier works are as follows: (i) there is an endogenously and strategically determined winning threshold, as agents can choose to leave the game at the end of the first stage; (ii) winning prizes are state‐dependent, as they are determined in a cooperative bargaining model that takes the contest outcome as an input; and (iii) jumps from some decision nodes to the last decision node are possible, as the firm management enforces a knowledge transfer from the winner in the first stage to the losing side, which indicates a strong cooperative behavior between the agents.…”

Converse Tournamnets

“…In recent years, this branch of game theory has become quite popular, which multiplied the number of contributions on contests and tournaments (see Yildirim, 2005;Münster, 2007;Konrad and Kovenock, 2009;Sela, 2012;Fu et al . , 2015;Brown and Chowdhury, 2017;Keskin and Sağlam, 2017;Chowdhury et al ., 2018;Doğan et al ., 2018;Mago and Sheremeta, 2018 among others).…”