Rushes in Large Timing Games

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Anderson, A., et al. (2017). "Rushes in Large Timing Games." Econometrica 85(3): 871-913.

We develop a continuum player timing game that subsumes standard wars of attrition and pre‐\emption games, and introduces a new rushes phenomenon. Payoffs are continuous and single‐\peaked functions of the stopping 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.

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.

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