It is often said that the early bird gets the worm, and while it sometimes helps to edge out others in in the pursuit of a prize, there are generally limits to going earlier and earlier. This paper introduces and explores a new continuum player timing game in which payoffs depend on timing and your timing rank. Our model subsumes standard wars of attrition and pre-emption games as special cases. Payoffs are continuous and single-peaked functions of the stopping time (peaking at a “harvest time”) and stopping quantile. We show that if payoffs are hump-shaped in the quantile, then a sudden “rush” of players stops in any Nash or subgame perfect equilibrium.
In our paper, fear relaxes the first mover advantage in pre-emption games, asking that the least quantile beat the average; greed relaxes the last mover advantage in wars of attrition, asking just that the last quantile payoff exceed the average. With greed, play is inefficiently late: an accelerating war of attrition starting at optimal time, followed by a rush. With fear, play is inefficiently early: a slowing pre-emption game, ending at the optimal time, preceded by a rush. The theory predicts the length, duration, and intensity of stopping, and the size and timing of rushes, and offers insights for many common timing games:
- Schelling’s Tipping game exploited tatonnement logic on a lattice (mostly). Here, we explain tipping using hum-shaped quantile preferences
- Al Roth has found that sorority and employment matching rushes get earlier and earlier each year; one of his explanations was a tatonnement story, like Schelling We show that these rushes occur long before matching is efficient since they reflect fear. Moreover, the fear increases as the early matching stigma decreases.
- Stock market bubbles often end in a rush. These fit into our rational story of greed, provided there is enough of a reward for mutual fund managers to “beat the average”.
- Bank runs fall into our fear based rushes, and therefore transpire before the efficient time.
The Simple Economics of Rushes:
Assume that payoffs have the standard “hump shape” as a function of the stopping quantile q, and also separately, as a function of the stopping time t (as in the graph). Assume, towards a contradiction, that there is no rush. Then if people are indifferent about stopping at any time in some intervals, we need at every moment in time (a) payoffs falling in t whenever they are rising in q (our generalization of a pure war of attrition), and (b) payoffs rising in t whenever they are falling in q (our generalization of a preemption game). This is impossible, for then earlier quantile stoppers would have to stop late in time [regions A] and later quantile stoppers would have to stop early in time [region B]. Contradiction