The timing of contests

Abstract

We develop a simple model to analyze the timing of contests. When the odds of winning a contest are exogenously given – we show that if either the players discount the future or if the total cost of contest is smaller in the future – there exist subgame perfect equilibria where both players settle, anticipating a contest in the future. With endogenous efforts, the aggregate efforts expanded in a contest are smaller if the contest occurs in the future when the relative effort productivities remain constant or diverge over time, thus creating scope for delay in contests. When the effort productivities converge over time, the total efforts may be greater under a future contest. As a consequence, players either settle over the two periods, or else they initiate a contest immediately.

Keywords:

• Delay
• Contest
• Dynamic game
• War

AMS Subject Classifications:

• C72
• D7

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The theory of contest has extensively been applied to model the outcome of trials (see Katz 1988; Farmer and Pecorino 1999; Bernardo, Talley, and Welch 2000; Gong and Mcafee 2000; Hirshleifer and Osborne 2001; Robson and Skaperdas 2008).

2 For an overview of dynamic contests, see Konrad (2009).

3 Contests involve rent-dissipation since otherwise productive resources are devoted to improving one’s winning odds (Tullock 1980). Moreover, the value of the resources can be reduced because of the destructiveness of a contest (Herschel 1995; De Luca and Sekeris 2013), or as a consequence of the players’ risk aversion (Skaperdas 1991; Skaperdas and Gan 1995).

4 Their focus is on armed conflicts. They argue that there are cases where the armies are more efficient in defense.

5 The chosen timing whereby players first decide whether or not to enter a contest and then decide the effort level is tantamount to assuming a strong commitment capacity on behalf of the players since we rule out the possibility of reneging on the first-stage decision. From a conceptual viewpoint, this commitment capacity may equally be interpreted as players facing important reputation costs of violating their first-stage decision, or the players’ effort slowly translating in contest capacity so that the opponent is never subject to a sneak attack. From a practical viewpoint, contests such as legal or sports contests whereby the players ought to publicly declare their participation before the contest actually taking place satisfy the model’s timing assumptions. Our setting equally applies to military confrontations since they necessitate military preparation which is likely to be observed – at least to some degree – by one’s opponent.

6 We assume away budget constraints that would burden the analysis without bringing any additional insight. The cost of effort can be interpreted in various ways including foregone production given some linear production function, or a constant marginal utility of leisure time.

7 One could object that the players’ efficiency parameters and their consequent win probabilities should reflect the players’ initial allocation of property rights. This would amount to restricting the analysis to regions , , and on Figure 1. We purposefully decided not to constrain the problem in that direction, since interesting results arise in situations where the players’ relative power does not reflect their relative property endowments.