Greed, Fear, and Rushes∗

We develop a continuum player unified timing game that also rationalizes sudden mass movements. Payoffs in the game are a logsubmodular function of time and the stopping quantile. Time captures the payoff-relevant fundamental — payoffs “ripen”, peak at a “harvest time”, and then “rot”. Capturing the strategic interaction, payoffs are single-peaked in the quantile. We distinguish between greed and fear — i.e., the hunger for greater rewards from outlasting others, and the fear of missing out on early rewards. Three local timing games can arise in equilibrium: a war of attrition, a slow pre-emption game, and a pre-emptive rush. With greed, the harvest time precedes an accelerating war of attrition ending in a rush, whereas with fear, an inefficiently early rush precedes a slowing pre-emption game ending at the harvest time. The theory yields predictions for the rush size and time, as well as stopping intensity and duration when fundamentals or the strategic structure change. Our theory subsumes a wealth of timing games with large numbers of players, yielding old and new predictions. For instance, matching rushes, bank runs, social tipping points happen before fundamentals indicate, and asset sales rushes occur after. We also explain how (a) “unraveling” in matching markets depends on early matching stigma and market thinness; (b) asset sales rushes reflect liquidity and relative compensation; (c) illiquid bank loans yields complete bank runs before incomplete ones. This supersedes a primitive early version of this paper by Andreas and Lones, growing out of joint work in Andreas’ 2004 PhD thesis, that assumed multiplicative payoffs. It was presented at the 2008 Econometric Society Summer Meetings at Northwestern and the 2009 ASSA meetings. Most modeling and results, and all of the analysis and exposition now reflects joint work of Axel and Lones since 2012. It has profited from comments in seminar presentations at Wisconsin, Western Ontario, and Melbourne.