Rushes in Large Timing Games∗

We develop a tractable continuum player timing game that subsumes wars of attrition and pre-emption games, in which greed and fear relax the last and first mover advantages. Payoffs are continuous and single-peaked functions of the stopping time and quantile. Time captures the payoff-relevant fundamental, as payoffs “ripen”, peak at a “harvest time”, and then “rot”. Whenever payoffs are nonmonotone in the quantile, a sudden mass movement must occur. With greed, play is inefficiently late — the harvest time precedes an accelerating war of attrition ending in a rush; with fear, play is inefficiently late — a rush precedes a slowing pre-emption game, ending at the harvest time. The theory simultaneously predicts the length, duration, and intensity of gradual play, and the size and timing of rushes, and offers insights for many common timing games. For instance, matching rushes and bank runs happen before fundamentals indicate, and asset sales rushes occur after. Also, (a) “unraveling” in matching markets depends on early matching stigma and market thinness; (b) asset sales rushes reflect liquidity and relative compensation; (c) a higher reserve ratio shrinks the bank run, but otherwise increases the withdrawal rate. ∗This follows on a rough manuscript “Greed, Fear, and Rushes” 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. While including some results from that paper, the modeling and exposition now solely reflect joint work of Axel and Lones since 2012. Axel and Lones alone bear responsibility for any errors in this paper. We have profited from comments in seminar presentations at Wisconsin, Western Ontario, Melbourne, Columbia, UCLA, and Penn State. We thank Faruk Gul for an extremely helpful suggestion. This version has reflects suggestions of the Editor and three referees.