Greed,Fear, andRushes · 2018-02-16 · Greed,Fear, andRushes∗ Axel Anderson Georgetown Andreas...

Greed, Fear, andRushes∗

Axel Anderson

Georgetown

Andreas Park

Toronto

Lones Smith

Wisconsin

January 6, 2015

Abstract

We develop a tractable continuum player timing game that subsumes wars of attritionand pre-emption games, in whichgreedand fear relax the last and first mover advantages.Payoffs are continuous and single-peaked functions of the stoppingtimeandquantile. Timecaptures the payoff-relevantfundamental, as payoffs “ripen”, peak at a “harvest time”, andthen “rot”. The nonmonotone quantile response rationalizes sudden mass movements ineconomics, and explains when it is inefficiently early or late. With greed, the harvest timeprecedes an accelerating war of attrition ending in a rush; with fear, a rush precedes a slowingpre-emption game ending at the harvest time.

The theory simultaneously predicts the length, duration, and intensity of gradual play, andthe size and timing of rushes, and offers insights for an array of timing games. For instance,matching rushes and bank runs happen before fundamentals indicate, and asset sales rushesoccur after. Moreover,(a) “unraveling” in matching markets depends on early matchingstigma 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 supersedes a primitive early version of this paper by Andreas and Lones, growing out of joint work inAndreas’ 2004 PhD thesis, that assumed multiplicative payoffs. It was presented at the 2008 Econometric SocietySummer Meetings at Northwestern and the 2009 ASSA meetings.While including some results from that paper, themodeling and exposition now solely reflect joint work of Axeland Lones since 2012. Axel and Lones alone bearresponsibility for any errors in this paper. We have profitedfrom comments in seminar presentations at Wisconsin,Western Ontario, Melbourne, and Columbia.

Contents

1 Introduction 1

2 Model 4

3 The Tradeoff Between Time and Quantile 6

4 Monotone Payoffs in Quantile 7

5 Interior Single-Peaked Payoffs in Quantile 8

6 Predictions about Changes in Gradual Play and Rushes 11

7 Economic Applications Distilled from the Literature 157.1 Land Runs, Sales Rushes, and Tipping Models . . . . . . . . . . .. . . . . . . . . . . . . 157.2 The Rush to Match . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 167.3 The Rush to Sell in a Bubble . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 187.4 Bank Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 19

8 The Set of Nash Equilibria with Non-Monotone Payoffs 21

9 Conclusion 24

A Geometric Payoff Transformations 25

B Characterization of the Gradual Play and Peak Rush Loci 26

C A Safe Equilibrium with Alarm 28

D Monotone Payoffs in Quantile: Proof of Proposition 1 28

E Single-Peaked Payoffs in Quantile: Omitted Proofs for §5 29

F Safe is Equivalent to Secure: Proof of Theorem 1 33

G Comparative Statics: Propositions 5 and 6 34

H Bank Run Example: Omitted Proofs (§7) 35

I All Nash Equilibria: Omitted Proofs (§8) 36

1 Introduction

“Natura non facit saltus.” — Leibniz, Linnaeus, Darwin, andMarshall

Mass rushes periodically grip many economic landscapes — such as fraternity rush week;

the “unraveling” rushes of young doctors seeking hospital internships; the bubble-bursting sales

rushes ending asset price run-ups; land rushes for newly-opening territory; bank runs by fearful

depositors; and white flight from a racially tipping neighborhood. These settings are so far re-

moved from one another that they have been studied in wholly disparate fields of economics. Yet

by stepping back from their specific details, this paper offers a unified theory of all timing games

with a large number of players. This theory explains not onlythe size and timing of aggregate

jumps, but also the more commonly studied speed of entry intothe timing game. And finally, we

parallel a smaller number of existing results for finite player timing games, whenever we overlap.

Timing games have usually been applied in settings with identified players, like industrial

organization. But anonymity is a more apt description of themotivational examples, and many

environments. This paper introduces an tractable class of timing games flexible enough for all

of these economic contexts. We therefore assume an anonymous continuum of homogeneous

players, ensuring that no one enjoys any market power; this also dispenses with any strategic

uncertainty. We characterize the Nash equilibrium of a simultaneous-move (“silent”) timing game

(Karlin, 1959) — thereby also ignoring dynamics, or any learning about exogenous uncertainty.

We venture that payoffs reflect a dichotomy — they depend solely on the stopping time and

quantile. Time proxies for a payoff-relevantfundamental, while the quantile embodies strategic

concerns. A mixed strategy is then a stopping time distribution function on the positive reals.

When it is continuous, there isgradual play, as players slowly stop; arushoccurs when a mass

of players suddenly stop, and the cdf jumps. We explore a model with no intrinsic discontinuities

— payoffs are smooth and hump-shaped in time, and smooth and single-peaked in the quantile.

This ensures a unique optimalharvest timefor any quantile, when the fundamental peaks, and a

unique optimalpeak quantilefor any time, when stopping is most strategically advantageous.

Two opposing flavors of timing games have been studied. Awar of attrition entails gradual

play in which the passage of time is fundamentally harmful and strategically beneficial. The

reverse holds in apre-emption game— the strategic and exogenous delay incentives oppose,

balancing the marginal costs and benefits of the passage of time. In other words, standard timing

games assume a monotone increasing or decreasing quantile response, so that the first or last

mover is advantaged overall other quantiles. But in our class of games, the peak quantilemay

be interior. The game exhibitsgreed if the very last mover eclipses theaveragequantile, and

1

fear if the very first mover does. So the war of attrition is the extreme case of greed, with later

quantiles always more attractive than earlier ones. Likewise, pre-emption games capture extreme

fear. Greed and fear are mutually exclusive, so that a game either exhibits greed or fear or neither.

Mixed strategies require constant payoffs in equilibrium,balancing quantile and fundamen-

tal payoff forces. With an interior peak quantile, to sustain indifference at all times, a mass of

consumers must stop at the same moment (Proposition 2). For purely gradual play is impossible

— otherwise, later quantiles would enter before the harvesttime and early quantiles after, which

is impossible. Apropos our lead quotation, despite a continuously evolving world, both from

fundamental and strategic perspectives,aggregate behavior must jump.Only in the well-studied

case with a monotone quantile response can equilibrium involve gradual play for all quantiles

(Proposition 1). But even in this case, an initial rush must happen whenever the gains of immedi-

ately stopping as an early quantile dominates the peak fundamentals payoff growth. We call this

extreme case with early rushes eitheralarmor —- if the rush includes all players —panic.

By the above logic, equilibria are eitherearly or late — namely, transpiring entirely before

or entirely after the harvest time. Absent fear, slow wars ofattrition start precisely at the harvest

time and are followed by rushes. Absent greed, initial rushes are followed by slow pre-emption

games ending precisely at the harvest time. Sorushes occur inefficiently early with fear, and

inefficiently late with greed.With neither greed nor fear, both types of equilibria arise.This

yields a useful big picture insight for our examples: the rush occurs before fundamentals peaks

in a pre-emption equilibrium, and after fundamentals peaksin a war of attrition equilibrium.

Proposition 7 characterizes all Nash equilibria: By realistically assuming slight uncertainty

about the accuracy of the time clock,secure equilibriumrefines Nash equilibria. Theorem 1

proves that secure equilibria aresafe, namely, without a time interval with no entry — except im-

mediately after time zero. When the stopping payoff is monotone in the quantile, Nash equilibria

are secure. But otherwise, the refinement is strict. We introduce a graphical apparatus that allows

us to depict all safe equilibria by crossing two curves: one locus equates the rush payoff and the

adjacent quantile payoff, and another locus imposes constant payoffs in gradual play. Apart from

panic or alarm, Proposition 3 implies at most two safe equilibria: a rush and then gradual play, or

vice versa, as described above. A third locus extends our graphical apparatus to describe non-safe

Nash equilibria; these involve larger rushes, separated from gradual play by an “inaction phase”.

Proposition 4 sheds light on gradual play. Under a common log-concavity assumption on fun-

damental payoffs, any pre-emption game gradually slows to zero after the early rush, whereas any

war of attrition accelerates from zero towards its rush crescendo. So inclusive of the rush, stop-

ping rates wax and wane, respectively, after and before the harvest time. Our payoff dichotomy

2

therefore allows for the identification of timing games fromdata on stopping rates.

We derive and graphically depict general comparative statics of all our equilibria. Changes

in fundamentals or strategic structure simultaneously affect the timing, duration, and stopping

rates in gradual play, and rush size and timing. Proposition5 considers a monotone ratio shift

in the fundamental payoffs that postpones the harvest time —like a faster growing stock market

bubble. With payoff stakes magnified in this fashion, stopping rates during any war of attrition

phase attenuate before the swelling terminal rush; less intuitively, stopping rates during any pre-

emption game intensify,but the initial rush shrinks. All told, an inverse relation between stopping

rates in gradual play and the rush size emerges — stopping intensifies as rushes shrink.

We next explore how monotone changes in the strategic structure influence play. Notably, a

log-supermodular payoff shift favoring later quantiles spans our entire class of games, allowing

extremefear to slowly transition into extremegreed. Proposition 6 reports how as greed rises in

the war of attrition equilibrium, or oppositely, as fear rises in the pre-emption equilibrium, gradual

play lengthens; in each case stopping rates fall and the rushshrinks. So perhaps surprisingly, the

rush is smaller and farther from the harvest time the greateris the greed or fear (Figure 7).

While our model is analytically simple, one might worry thatcomparative statics analysis

requires modeling dynamics or information. We now show thatwe both offer many new testable

insights, and agree with existing comparative statics for explored economic timing models.

Consider the “tipping point” phenomenon. In Schelling’s 1969 model, whites respond my-

opically to thresholds on the number of black neighbors. Butin our timing game, whites choose

when to exit a neighborhood, and a tipping rush occurs even though whites have continuous

preferences. Also, this rush occurs early, before fundamentals dictate, due to the fear.1

We next turn to a famous and well-documented timing game thatarises in matching contexts.

We create a reduced form model incorporating economic considerations found in Roth and Xing

(1994). All told, fear rises when hiring firms face a thinner market, while greed increases in the

stigma of early matching. Firms also value learning about caliber of the applicants. We find that

matching rushes occur inefficiently early provided the early matching stigma is not too great.

By assuming that stigma reflects recent matching outcomes, our model delivers the matching

unravelling without appeal to a slow tatonnement process (Niederle and Roth, 2004).

Next, consider two common market forces behind the sales rushes ending asset bubbles: a

desire for liquidity fosters fear, whereas a concern for relative performance engenders greed.

Abreu and Brunnermeier (2003) (also a large timing game) ignores relative performance, and

so finds a pre-emption game with no rush before the harvest time. Their bubble bursts when

1Meanwhile, tipping models owing to the “threshold” preferences of Granovette (1978) penalize early quantiles,and so exhibit greed. Their rushes are late, as our theory predicts.

3

rational sales exceed a threshold. Like them, we too deduce alarger and later bubble burst with

lower interest rates. Yet by conventional wisdom, the NASDAQ bubble burst in March 2000 after

fundamentals peaked. Our model speaks to this puzzle, for with enough relative compensation,

the game no longer exhibits fear, and thus a sales rush after the harvest time is an equilibrium.

We conclude by exploring timing insights about bank runs. Inspired by the two period model

of Diamond and Dybvig (1983), in our simple continuous time model, a run occurs when too

many depositors inefficiently withdraw before the harvest time. With the threat of a bank run,

we find ourself in our alarm or panic cases, and payoffs monotonically fall in the quantile. By

Proposition 1, either a slow pre-emption game arises or a rush occurs immediately. We predict

that a reserve ratio increase shifts the distribution of withdrawals later, shrinks the bank run, but

surprisingly increases the withdrawal rate during any pre-emption phase.

L ITERATURE REVIEW. The applications aside, there is a long literature on timing games.

Maynard Smith (1974) formulated the war of attrition as a model of animal conflicts. Its biggest

impact in economics owes to the all-pay auction literature (eg. Krishna and Morgan (1997)). We

think the economic study of pre-emption games dates to Fudenberg, Gilbert, Stiglitz, and Tirole

(1983) and Fudenberg and Tirole (1985). More recently, Brunnermeier and Morgan (2010) and

Anderson, Friedman, and Oprea (2010) have experimentally tested it. Park and Smith (2008) ex-

plored a finite player game with rank-dependent payoffs in which rushes and wars of attrition

alternated; however, slow pre-emption games were impossible. Ours may be the first timing

game with all three timing game equilibria.

2 Model

A continuum of identical risk neutral players[0, 1] each chooses a mixture over stopping timesτ

on [0,∞), whereτ ≤ t with chanceQ(t) onR+. Either choices are made irrevocably at time-0,

or players do not observe each other over time, perhaps sincethe game transpires quickly.

A player’s stopping quantileq summarizes the anonymous form of strategic interaction in a

large population, and the timet captures fundamentals. By Lebesgue’s decomposition theorem,

thequantile functionQ(t) is the sum of continuous portions, calledgradual play, and atoms. If

Q is continuous att, then thestopping payoffis u(t, Q(t)). Stopping at anatomt of Q, with

p = Q(t) > Q(t−) = q, earns the average quantile payoff∫ p

qu(t, x)dx/(p− q) for quantiles in

[q, p]. The aggregate outcome corresponds to arush, where a massp− q of agents stops att.

Payoffsu(q, t) are continuous, and for fixedq, are quasi-concave int, strictly rising from

t = 0 (“ripening”) until an idealharvest timet∗(q), and then strictly falling (‘rotting”). For fixedt,

4

payoffsu are either monotone or log-concave inq, with unique interiorpeak quantileq∗(t). We

embed strategic interactions by assuming the payoff function u(t, q) is log-submodular — such

as multiplicative.2 Since higherq corresponds to stochastically earlier stopping by the population

of agents, this yields proportional complementarity — the proportional gains to postponing until

a later quantile are larger, or the proportional losses smaller, the earlier is the stopping time.

Indeed,the harvest timet∗(q) is a decreasing function ofq, while the peak quantile functionq∗(t)

is falling in t. To ensure that players eventually stop, we assume that waiting forever is dominated

by stopping att∗(0):

limt→∞

u(t, q∗(t)) < u(t∗(0), 0). (1)

Formally, letT (q) ≡ inf{t ∈ R+|Q(t) ≥ q} be generalized inverse distribution function.

Then the payoff toτ = t equals the Radon Nikodym derivativew(t) = dW/dQ, given the

following running integral:

W (t) =

∫ Q(t)

0

u(T (x), x)dx

A Nash equilibriumis a cdfQ whose support contains only maximizers of the functionw(t).3

We assume agents can perfectly time their actions. But one might venture that even the best

timing technology is imperfect. If so, agents may be wary of equilibria in which tiny timing

mistakes incur payoff losses. In Theorem 1, we will see that concern for such timing mistakes

leads tosafe equilibria, namely, Nash equilibria in which the quantile functionQ is an atom at

timet = 0, or a cdf whose support is a non-empty time interval, or a mixture of both possibilities.

While we characterize all Nash equilibria, we introduce a trembling refinement for timing

games that prunes the equilibrium set, and pursue sharper comparative statics predictions. Let

w(t;Q) ≡ u(t, Q(t)) be the payoff to stopping at timet ≥ 0 given cdfQ. Theε-secure payoffat

t is:

max〈 infmax(t−ε,0)≤s<t

w(s;Q), infs∈[t,t+ε)

w(s;Q)〉

This is the minmax payoff in the richer model when individuals have access to two differentε-

accurate timing technologies: One clock never runs late, and one never runs early. An equilibrium

Q is secureif wε(t;Q) = w(t;Q) for all t in the support ofQ, and all small enoughε > 0. These

are the only Nash equilibria that are robust to small timing mistakes.

Theorem 1 A Nash equilibrium is safe if and only if it is secure.

2Almost all of our results only require the weaker complementary condition thatu(t, q) bequasi-submodular, sothatu(tL, qL) ≥ (>)u(tH , qL) impliesu(tL, qH) ≥ (>)u(tH , qH), for all tH ≥ tL andqH ≥ qL.

3For if q = Q′(t) > 0 exists, thenw(t) = u(t, q). If Q(t) = q > p = Q(t−), thenT (x) = t on [p, q], and soW (t)−W (t−) =

∫ p

qu(t, x)dx. Finally, if Q(t) = q on [t1, t2), thent is not in the support ofQ.

5

Fear:u(t, 0) ≥∫ 10 u(t, x)dx

u

0 1

u

quantile rankq

Neither Fear nor Greed

u

u

0 1quantile rankq

Greed:u(t, 1) ≥∫ 10 u(t, x)dx

u

u

0 1quantile rankq

Figure 1: Fear and Greed. Payoffs at any timet cannot exhibit both greed and fear, with firstand last quantile factors better than average, but might exhibit neither (middle panel).

3 The Tradeoff Between Time and Quantile

Since homogeneous players earn the same Nash payoff, indifference must prevail whenever play

is gradual on an interval:W ′(t) = u(t, Q(t)) = w, say. Sinceu is C2, if the stopping rateQ′

exists and isC1 during any gradual play phase, then it obeys the differential equation:

uq(t, Q(t))Q′(t) + ut(t, Q(t)) = 0 (2)

Conversely, we can use (2) in reverse, deducing that whenever Q is absolutely continuous, it

must be differentiable in time. The stopping rate is given bythe marginal rate of substitution, i.e.

Q′(t) = −ut/uq. So the slope signsuq andut must be mismatched throughout any gradual play

phase, i.e. on any time interval with gradual play, two possible timing game phases are possible:

• Pre-emption phase:a connected interval of gradual play on whichut > 0 > uq, so that the

passage of time is fundamentally beneficial but strategically costly.

• War of attrition phase:a connected interval of gradual play on whichut < 0 < uq, so that

the passage of time is fundamentally harmful but strategically beneficial.

To analyze rushes, we must compare average and peak quantilepayoffs. LetV0(t, q) ≡

q−1∫ q

0u(t, x)dx be therunning average payoff function. As key assumption is that fundamental

growth is strong enough that:

maxq

V0(0, q) ≤ u(t∗(1), 1) (3)

When (3) fails, stopping as an early quantile dominates waiting until the harvest time, if one is

last. There are then two mutually exclusive possibilities:alarm whenV0(0, 1) < u(t∗(1), 1) <

maxq V0(0, q), andpanicwhen the harvest time payoff is even lower:u(t∗(1), 1) ≤ V0(0, 1).

In a pure pre-emption game, each quantile has an absolute advantage over all later quantiles:

u(t, q′) > u(t, q) for all q > q′ andt ≥ 0. Fear generalizes this, asking only an advantage of

6

Pure War of Attrition

✻1

q

0 ✲tt∗(0)

ΓW

Pure Pre-Emption Gameu(0, 0) ≤ u(t∗(1), 1)✻

1

q

0 ✲t∗(1)0

ΓP

Pure Pre-Emption Gameu(t, q) > u(t∗(1), 1)✻

1

qq0

0 ✲t∗(1)0

ΓP

Figure 2: Monotone Cases. In the left paneluq > 0, and the equilibrium is a pure war ofattrition following the gradual play locusΓW (t). Whenuq < 0 gradual play follows the pre-emption gradual play locusΓP (t). If u(0, 0) ≤ u(t∗(1), 1), as in the middle panel,ΓP defines apure pre-emption game. In the right panel,u(0, 0) > u(t∗(1), 1), as with alarm and panic. In thiscase, the indifference curveΓP intersects theq-axis atq > 0, implying there cannot be gradualplay for all quantiles. Given alarm, the equilibrium involves a rush of sizeq0 at t = 0, followedby a period of inaction along the blue line, and then gradual play alongΓP (t).

the least quantile over the average quantile — there isfear at timet if u(t, 0) ≥∫ 1

0u(t, x)dx.

In a pure war of attrition, each quantile has an absolute advantage over all earlier quantiles:

u(t, q′) > u(t, q) for all q < q′ andt ≥ 0. Greed is more general; there isgreed at timet if the

last quantile payoff exceeds the average; namely, ifu(t, 1) ≥∫ 1

0u(t, x)dx. Both inequalities are

tight if the payoffu(t, q) is constant inq, and strict fear and greed correspond to strict inequalities

(see Figure 1). Sinceu is single-peaked inq, greed and fear att are mutually exclusive.

4 Monotone Payoffs in Quantile

When the stopping payoff is increasing in quantile, rushes cannot occur, since stopping right after

a rush yields a higher payoff than stopping in the rush. So theonly possibility is a pure war of

attrition. In fact, gradual play must begin at timet∗(0) > 0. For if gradual plays start att > t∗(0),

then quantile 0 would profitably deviate tot∗(0), whereas if it starts att < t∗(0), then the required

differential equation (2) is impossible — forut(t, 0) > 0 anduq(t, 0) > 0. Since the Nash payoff

isu(t∗(0), 0), thewar of attrition gradual play locusΓW (Figure 2) begins att∗(0), and is defined

by the implicit equation:

u(t,ΓW (t)) = u(t∗(0), 0) (4)

Similarly, whenuq < 0, any gradual play interval ends att∗(1), implying an equilibrium value

u(t∗(1), 1). For if gradual play ends att < t∗(1) then quantileq = 1 benefits from deviating to

t∗(1). And if it ends att1 > t∗(1), thenut(t, 1) < 0 for all t∗(1) < t < t1. But sinceuq < 0 this

7

violates the differential equation (2). Thus, during gradual play,Q must satisfy thepre-emption

gradual play locusΓP :

u(t,ΓP (t)) = u(t∗(1), 1) (5)

Whenuq < 0, a rush at timet > 0 is impossible, since pre-empting it dominates stopping in

the rush. But notice that a time zero rush is special, since pre-empting such a rush is impossible,

by assumption. Sinceu(t∗(1), 1) is the Nash equilibrium payoff, inequality (3) precisely rules

out a time zero rush. But the left side of (3) reduces tou(0, 0) whenuq < 0. Absent a rush, there

must be a pure pre-emption game obeying equality (5) (depicted in the middle panel of Figure 2).

Panic rules out all but a unit mass rush at time zero, since it impliesV0(0, q) > u(t∗(1), 1)

for all q; any equilibrium with gradual play has Nash payoffu(t∗(1), 1). Given alarm, stopping

immediately for payoffu(0, 0) dominates stopping during gradual play for payoffu(t∗(1), 1),

and thus a rush must occur. But givenV0(0, 1) < u(t∗(1), 1), any such rush must be of size

q0 < 1, and the indifference conditionV0(0, q0) = u(t∗(1), 1) then pins down the rush size. But

V0(0, q0) > u(0, q0) givenuq < 0, so that stopping just after the time zero rush affords a strictly

lower payoff than stopping in the rush. This forces aninaction phase— a time interval[t1, t2]

with no entry, where0 < Q(t1) = Q(t2) < 1 — as seen in the right panel of Figure 2.

Appendix D completes the proof of the following result whenu is strictly monotone inq.

Proposition 1 Assume the stopping payoff is strictly monotone in quantile. There is a unique

Nash equilibrium, and it is safe. Ifuq>0, a pure war of attrition starts att∗(0). If uq<0, and:

(a) with neither alarm nor panic, there is a pre-emption game forall quantiles ending att∗(1);

(b) with alarm there is a time-0 rush of sizeq0 obeyingV0(0, q0) = u(t∗(1), 1), followed by an

inaction phase, and then a pre-emption game ending att∗(1);

(c) with panic there is a unit mass rush at timet = 0.

Rushes here reflect inadequate fundamentals growth to compensate for the strategic cost of delay.

5 Interior Single-Peaked Payoffs in Quantile

Alarm and panic lead to an initial rush. With an interior peakquantile, two other types of rushes

are possible, likewise engulfing the quantile peak: Aterminal rushhappens when0 < Q(t−) =

q < 1 = Q(t), so that all quantiles in[q, 1] rush at the same timet, collecting terminal rush

payoffV1(q, t) ≡ (1− q)−1∫ 1

qu(t, x)dx. A unit mass rushoccurs when0 = Q(t−) < 1 = Q(t),

so that all quantiles[0, 1] rush at the same timet. The latter is a pure strategy Nash equilibrium.

Since the harvest timet∗(q) is falling in q, the harvest time interval[t∗(1), t∗(0)] is nontrivial.

8

Proposition 2 If payoffs are non-monotone in quantile, only three types ofNash equilibria occur:

(a) An initial rush followed by a pre-emption phase time interval ending at harvest timet∗(1) iff

there is not greed at timet∗(1) and no panic. The equilibrium payoff isu(t∗(1), 1).

(b) A terminal rush preceded by a war of attrition phase time interval starting at harvest time

t∗(0) iff there is not fear at timet∗(0). The equilibrium payoff isu(t∗(0), 0).

(c) A unit rush at anyt in an open interval around[t∗(1), t∗(0)] with no greed att∗(1) and no

fear att∗(0). Unit mass rushes cannot occur at any positive time with strict greed or strict fear.

We define apre-emption equilibriumand awar of attrition equilibriumas a Nash equilibrium

of the type described in parts(a) and(b) of Proposition 2, respectively. Let us distill the logic

underlying Proposition 2. First, with an interior peak quantile, play cannot be entirely gradual.

For if so, then early quantiles withuq > 0 would stop whenut < 0, i.e., later in time; meanwhile,

later quantiles withuq < 0 would stop whenut > 0, i.e., earlier in time. Contradiction. More

strongly, Lemma E.1 proves that equilibrium involves either an initial or terminal rush. Since

players only stop in such a rush if gradual play is not more profitable, we haveVi(t, q) ≥ u(t, q).

Given how marginals and averages interact (see Figure 3), Proposition 2 implies:

Corollary 1 The rush includes the unique quantile maximizer ofVi and the peak quantile ofu.

Let us next understand the roles of greed and fear. Assume a pre-emption phase, where

ut > 0 > uq. This can only happen after the peak quantile, and therefore, after the rush. If there

is greed, then the initial rush payoffV0 is maximized atq = 1, and so must include all quantiles,

ruling out a pre-emption phase. Similarly, fear is incompatible with a war of attrition.

There is a multiplicity of Nash equilibria. For example, by varying the timing and size of

the terminal rush, along with the length of the inaction phase separating gradual play from the

terminal rush, one can construct a continuum of war of attrition equilibria whenever one exists.

Section 8 characterizes the full set of Nash equilibria involving gradual play.

We next argue that there are one or two safe equilibria. To this end, define the early and

latepeak rush locusΠi(t) ∈ argmaxq Vi(t, q), namely,i = 0, 1. Since an average peaks when

the margin equals the average, each locus equates payoffs inrushes and immediately adjacent

gradual play (see Figure 3):

u(t,Πi(t)) = Vi(t,Πi(t)) (6)

Lemma B.2 shows that each locusΠi(t) is decreasing whenu(t, q) is strictly log-submodular.

But in the log-modular (or multiplicative) case that we assume in the example in§7.2, the locus

Πi(t) is constant in timet. We now depict safe equilibria in Figure 4.

9

Peak Initial Rush

Π0(t)q∗(t) q

u(t, q)

V0(t, q)

Peak Terminal Rush

q∗(t) qΠ1(t)

V1(t, q)

u(t, q)

Figure 3:Rushes Include the Quantile Peak. The timet peak rush maximizes the average rushpayoffVi(t, q), and so equates the average and adjacent marginal payoffsVi(t, q) andu(t, q).

Proposition 3 Absent fear at the harvest timet∗(0), there exists a unique safe war of attrition

equilibrium. Absent greed att∗(1), there exists a unique safe equilibrium with an initial rush:

(a) with neither alarm nor panic, an initial rush followed immediately by a pre-emption game;

(b) with alarm, a rush att = 0 followed by a period of inaction and then a pre-emption game;or

(c) with panic, a unit mass rush att = 0.

With this result, we see that panic and alarm have the same implications as in the monotone

decreasing case,uq < 0, analyzed by Proposition 1. In contrast, when neither panicnor alarm

obtains, secure equilibria must have both a rush and a gradual play phase and no inaction: either

an initial rush at0 < t0 < t∗(1), followed by a pre-emption phase on[t0, t∗(1)], or a war of

attrition phase on[t∗(0), t1] ending in a terminal rush att1. In each case, the safe equilibrium is

fully determined by the gradual play locus and peak rush locus (Figure 4).

We now characterize gradual play in all Nash equilibria in Propositions 1 and 2.

Proposition 4 (Stopping during Gradual Play) Assume the payoff function is log-concave int.

In any gradual play phase, the stopping rateQ′(t) is strictly increasing in time from zero during

a war of attrition phase, and decreasing down to zero during apre-emption game phase.

Proof: Wars of attrition begin att∗(0) and pre-emption games end att∗(1), by Proposition 2.

Sinceut(t∗(q), q) = 0 at the harvest time, the first term of the gradual play equation (2) vanishes

at the start of a war of attrition and end of a pre-emption game. Consequently,Q′(t∗(0)) = 0 and

Q′(t∗(1)) = 0 in these two cases, sinceuq 6= 0 at the two quantile extremesq = 0, 1.

Next, assume thatQ′′ exists. Then differentiating the differential equation (2) in t yields:4

Q′′ = −1

uq

[

utt + 2uqtQ′ + uqq(Q

′)2]

=1

u3q

[

2uqtuqut − uttu2q − uqqu

2t

]

(7)

4Conversely,Q′′ exists wheneverQ′ locally does precisely because the right side of (7) must be the slope ofQ′.

10

Safe War of Attrition

✻1

q1

0 ✲t1 tt∗(0)

ΓW

Π1

Safe Pre-Emption GameNo Alarm

✻1

q0

✲t0 t∗(1)0

ΓP

Π0

Safe Pre-Emption GameAlarm

✻1

q0

✲t∗(1)0

ΓP

Π0

Figure 4:Safe Equilibria with Non-Monotone Payoffs. In the safe war of attrition equilibrium(left), gradual play begins att∗(0), following the upward sloping gradual play locus (4), andends in a terminal rush of quantiles[q1, 1] at timet1 where the loci cross. In the pre-emptionequilibrium without alarm (middle), an initial rushq0 at timet0 occurs where the upward slopinggradual play locus (5) intersects the downward sloping peakrush locus (6). Gradual play in thepre-emption phase then follows the gradual play locusΓP . With alarm (right), the initial rushoccurs att = 0 followed by an inaction phase, and then a pre-emption game followsΓP .

after replacing the marginal rate of substitutionQ′ = −ut/uq implied by (2). Finally, the above

right bracketed expression is positive whenu is log-concave int.5 Hence,Q′′ ≷ 0 ⇔ uq ≷ 0. �We see that war of attrition equilibria exhibit waxing exits, climaxing in a rush when payoffs

are not monotone in quantile, whereas pre-emption equilibria begin with a rush in this non-

monotone case, and continue into a waning gradual play. So wars of attrition intensify towards

a rush, whereas pre-emption games taper off from a rush. Figure 4 reflects these facts, since the

stopping indifference curve is(i) concave after the initial rush during any pre-emption equilib-

rium, and(ii) convex prior to the terminal rush during any war of attritionequilibrium.

6 Predictions about Changes in Gradual Play and Rushes

We now explore how the equilibria evolve as the two key aspects of our economic environment

monotonically change:(a) fundamentals adjust to advance or postpone the harvest time, or (b)

the strategic interaction alters to change quantile rewards, increasing fear or greed. To this end,

smoothly index the stopping payoff byϕ ∈ R. Whenu(t, q|ϕ) is strictly log-supermodular in

(t, ϕ) and log-modular in(q, ϕ), greaterϕ raises the rate of change of the stopping payoff, but

leaves unaffected the rate in quantile. We call an increase in ϕ a harvest delay, since the harvest

time t∗(q|ϕ) rises inϕ, by log-supermodularity in(t, ϕ).

5 Sinceuqut < 0, the log-submodularity inequalityuqtu ≤ uqut can be reformulated as2uqtuqut ≥ 2u2

qu2

t/u,or (⋆). But log-concavity int andq implies u2

t ≥ uttu andu2

q ≥ uqqu, and so2u2

qu2

t/u ≥ uttu2

q + uqqu2

t .Combining this with(⋆), we find2uqtuqut ≥ uttu

2

q + uqqu2

t .

11

Pre-Emption Equilibrium✻

1

q

❄

✲✲t0

ΓP

❄

Π0

War of Attrition Equilibrium✻

1

q

❄

0 ✲✲t

ΓW

❄

Π1

Figure 5:Harvest Time Delay. The gradual play locus shifts down inϕ. In a pre-emption game(left): A smaller initial rush occurs later and stopping rates rise during gradual play. In a war ofattrition: A larger terminal rush occurs later, while stopping rates fall during gradual play.

We next argue that a harvest time delay postpones all activity, but nevertheless intensifies

stopping rates during a pre-emption game. We also identify an inverse relation between stopping

rates and rush size, with higher stopping rates during gradual play associated to smaller rushes.

Proposition 5 (Fundamentals) Assume safe equilibria and a harvest delay. Stopping is stochas-

tically later. In a war of attrition, stopping rates fall, and a weakly larger terminal rush occurs

later. In a pre-emption game, stopping rates rise, and a weakly smaller initial rush happens later.

Proof: We focus on the safe pre-emption equilibrium with an interior peak quantile. Figure 5

depicts the graphical logic for that case, and the similar omitted proof for the safe war of attrition.

The proof (for gradual play) with payoffs monotone in quantile and alarm is in Lemma G.2.

Since the marginal payoffu is log-modular in(t, ϕ), so too is the average. The peak rush

locusΠ0(t) ∈ argmaxq V0(t, q|ϕ) is then constant inϕ. Now, rewrite the pre-emption gradual

play locus (5) as:u(t,ΓP (t)|ϕ)

u(t, 1|ϕ)=

u(t∗(1|ϕ), 1|ϕ)

u(t, 1|ϕ)(8)

The LHS of (8) falls inΓP , sinceuq < 0 during a pre-emption game, and is constant inϕ, by

log-modularity ofu in (q, ϕ). Log-differentiating the RHS inϕ, and using the Envelope Theorem:

uϕ(t∗(1|ϕ), 1|ϕ)

u(t∗(1|ϕ), 1|ϕ)−

uϕ(t, 1|ϕ)

u(t, 1|ϕ)> 0

sinceu is log-supermodular in(t, ϕ) and t < t∗(1|ϕ) during a pre-emption game. Since the

RHS of (8) increases inϕ and the LHS decreases inΓP , the gradual play locusΓP (t) obeys

12


1

q

✻

✲✲t0

ΓP

❄

Π0

✻


1

q

✻

0 ✲✲t

ΓW

❄Π1

✻

Figure 6: Monotone Quantile Payoff Changes. An increase in greed (or a decrease in fear)shifts the gradual play locus down and the locus equating thepayoff in the rush to the adjacentgradual play payoff up. In the safe pre-emption equilibrium(left): Larger rushes occur later andstopping ratesriseon shorter pre-emption games. In the safe war of attrition equilibrium: Smallerrushes occur later and stopping ratesfall during longer wars of attrition.

∂ΓP /∂ϕ < 0. Next, differentiate the gradual play locus in (5) int andϕ, to get:

∂Γ′P (t)

∂ϕ= −

[(

∂[ut/u]

∂ϕ+

∂[ut/u]

∂ΓP

∂ΓP

∂ϕ

)

u

uq+

ut

u

(

∂[u/uq]

∂ΓP

∂ΓP

∂ϕ+

∂[u/uq]

∂ϕ

)]

> 0 (9)

The first parenthesized term is negative. Indeed,∂[ut/u]/∂ϕ > 0 sinceu is log-supermodular

in (t, ϕ), and∂[ut/u]/∂ΓP < 0 sinceu is log-submodular in(t, q), and∂ΓP /∂ϕ < 0 (as shown

above), and finallyuq < 0 during a pre-emption game. The second term is also negative because

ut > 0 during a pre-emption game, and∂[u/uq]/∂ΓP ≥ 0 by log-concavity ofu(t, q) in q. �

Next consider pure changes in quantile preferences, by assuming the stopping payoffu(t, q|ϕ)

is log-supermodular in(q, ϕ) and log-modular in(t, ϕ). So greaterϕ inflates the relative return

to a quantile delay, but leaves unchanged the relative return to a time delay. Hence, the peak

quantileq∗(t|ϕ) rises inϕ. We say thatgreed increaseswhenϕ rises, since payoffs shift towards

later ranks asϕ rises; this relatively diminishes the potential losses of pre-emption, and relatively

inflates the potential gains from later ranks. Also, if thereis greed at timet, then this remains

true if greed increases. Likewise, we sayfear increaseswhenϕ falls. Figure 7 partially depicts:

Proposition 6 (Quantile Changes) In the safe war of attrition equilibrium, as greed increases,

gradual play lengthens, stopping rates fall, the stopping distribution shifts later, and terminal

rush shrinks. In the safe pre-emption equilibrium, as fear rises, stopping rates fall and the stop-

ping distribution shifts earlier; also, without alarm the pre-emption game lengthens and the

initial rush shrinks, and with alarm, the pre-emption game shortens and the initial rush grows.

13

0 5 10 15 20

Gre

edIn

crea

ses

/Fea

rD

ecre

ases

✻

Time

Greed

Fear

Figure 7:Rush Size and Timing with Increased Greed. Circles at rush times are proportionalto rush sizes. As fear falls, the unique safe pre-emption equilibrium has a larger initial rush, closerto the harvest timet∗ = 10, and a shorter pre-emption phase. As greed rises, the uniquesafe warof attrition equilibrium has a longer war of attrition, and asmaller terminal rush (Proposition 6)

Proof: Sinceu is log-modular in(t, ϕ), the harvest timest∗(0) andt∗(1) are both constant inϕ.

As with Proposition 5, our proof covers the pre-emption case, and we postpone the proof for a

monotone quantile function with alarm to Lemma G.2.

DefineI(q, x) ≡ q−1 for x ≤ q and 0 otherwise, and thusV0(t, q|ϕ) =∫ 1

0I(q, x)u(t, x|ϕ)dx.

Easily,I is log-supermodular in(q, x), and so the productI(·)u(·) is log-supermodular in(q, x, ϕ).

Thus,V0 is log-supermodular in(q, ϕ) since log-supermodularity is preserved by integration by

Karlin and Rinott (1980). So the peak rush locusΠ0(t) = argmaxq V0(t, q|ϕ) rises inϕ.

Now consider the gradual play locus (8). Its RHS is constant in ϕ sinceu(t, q|ϕ) is log-

modular in(t, ϕ), andt∗(q) is constant inϕ. The LHS falls inϕ sinceu is log-supermodular in

(q, ϕ), and falls inΓP sinceuq < 0 during a pre-emption game. All told, the gradual play locus

obeys∂ΓP /∂ϕ < 0. To see how the slopeΓ′P changes, consider (9). The first term in brackets is

negative. For∂[ut/u]/∂ϕ = 0 sinceu is log-modular in(t, ϕ), and∂[ut/u]/∂ΓP < 0 sinceu is

log-submodular in(t, q), and∂ΓP/∂ϕ < 0 (as shown above), and finallyuq < 0 in a pre-emption

game. The second term is also negative:∂[u/uq]/∂ϕ < 0 asu is log-supermodular in(q, ϕ), and

∂[u/uq]/∂ΓP > 0 asu is log-concave inq, and∂ΓP /∂ϕ < 0, andut > 0 in a pre-emption game.

All told, an increase inϕ: (i) has no effect on the harvest time;(ii) shifts the gradual play

locus (5) down and makes it steeper; and(iii) shifts the peak rush locus (6) up (see Figure 6).

Appendix G explores how the peak rush locus shift determineswhether rushes or shrink. �

14

Finally, consider a general monotone shift, in which the payoff u(t, q|ϕ) is log-supermodular

in both (t, ϕ) and(q, ϕ). We call an increase inϕ a co-monotone delayin this case, since the

harvest timet∗(q|ϕ) and the peak quantileq∗(t|ϕ) both increase inϕ. Intuitively, greaterϕ

intensifies the game, by proportionally increasing the payoffs in time and quantile space. By

the logic used to prove Propositions 5 and 6, such a co-monotone delay shifts the gradual play

locus (5) down and makes it steeper, and shifts the peak rush locus (6) up (see Figure 6).

Corollary 2 (Covariate Implications) Assume safe equilibria with a co-monotone delay. Then

stopping shifts stochastically later, and stopping rates fall in a war of attrition and rise in a

pre-emption game. Given alarm, the time zero initial rush shrinks.

The effect on the rush size depends on whether the interaction between(t, ϕ) or (q, ϕ) dominates.

7 Economic Applications Distilled from the Literature

To illustrate the importance of our theory, we devise reduced form models for several well-studied

timing games, and explore the equilibrium predictions and comparative statics implications.

7.1 Land Runs, Sales Rushes, and Tipping Models

The Oklahoma Land Rush of 1889 saw the allocation of the Unassigned Lands. High noon on

April 22, 1889 was the clearly defined time zero, with no pre-emption allowed, just as we assume.

Since the earliest settlers naturally claimed the best land, the stopping payoff was monotonically

decreasing in quantile. This early mover advantage was strong enough to overwhelm any tempo-

ral gains from waiting, and so the panic or alarm cases in Proposition 1 applied.

Next consider the sociology notion of a “tipping point” — themoment when a mass of people

dramatically discretely changes behavior, such as White Flight from a neighorhood (Grodzins,

1957). Schelling (1969) shows how, with a small threshold preference for same type neighbors

in a lattice, myopic adjustment quickly tips into complete segregation. Card, Mas, and Rothstein

(2008) estimate the tipping point around a low minority share m = 10%. Granovette (1978)

explored social settings explicitly governed by “threshold behavior”, where individuals differ in

the number or proportion of others who must act before one follows suit. He showed that a

small change in the distribution of thresholds may induce tipping on the aggregate behavior. For

instance, a large enough number of revolutionaries can eventually tip everyone into a revolution.

That discontinuous changes in fundamentals or preferences— Schelling’s spatial logic or

Granovetter’s thresholds — have discontinuous aggregate effects arguably might be no puzzle.

15

http://en.wikipedia.org/wiki/Land_Rush_of_1889

But in our model, a rush is unavoidable when preferences are smooth, provided they are single-

peaked in quantile. If whites prefer the first exit payoff over the average exit payoff, then there

is fear, and Proposition 2 both predicts a tipping rush, and why it occurred so early, before pref-

erence fundamentals might suggest. Threshold tipping predictions might often be understood in

our model with smooth preferences and greed — eg., the last revolutionary does better than the

average. If so, then tipping occurs, but one might expect a revolution later than expected from

fundamentals.

7.2 The Rush to Match

We now consider assignment rushes. As in the entry-level gastroenterology labor market in

Niederle and Roth (2004) [NR2004], early matching costs include the “loss of planning flexibil-

ity”, whereas the penalty for late matching owes to market thinness. For a cost of early matching,

we simply follow Avery, et al. (2001) who allude to the condemnation of early match agreements.

So we posit a negative stigma to early matching relative to peers.

For a model of this, assume an equal mass of two worker varieties, A and B, each with a

continuum of uniformly distributed abilitiesα ∈ [0, 1]. Firms have an equal chance of needing

a type A or B. For simplicity, we assume that the payoff of hiring the wrong type is zero, and

that each firm learns its need at fixed exponential arrival rate δ > 0. Thus, the chance that a firm

chooses the right type if it waits until timet to hire ise−δt/2+∫ t

0δe−δsds = 1−e−δt/2.6 Assume

that an abilityα worker of the right type yields flow payoffα, discounted at rater. Thus, the

present value of hiring the right type of abilityα worker at timet is (α/r)e−rt.

Consider the quantile effect. Assume an initial ratio2θ ∈ (0, 2) of firms to workers (market

tightness). If a firm chooses before knowing its type, it naturally selects each type with equal

chance; thus, the best remaining worker after quantileq of firms has already chosen is1−θq. We

also assume astigmaσ(q), with payoffs from early matching multiplicatively scaledby 1−σ(q),

where1>σ(0)≥σ(1)=0, andσ′ < 0. All told, the payoff is multiplicative in time and quantile

concerns:

u(t, q) ≡ r−1(1− σ(q))(1− θq)(

1− e−δt/2)

e−rt (10)

This payoff is log-concave int, and initially increasing provided the learning effect is strong

enough (δ > r). This stopping payoff is concave in quantileq if σ is convex.

The match payoff (10) is log-modular int andq, and so always exhibits greed, or fear, or

6We assume firms unilaterally choose the start datet. One can model worker preferences over start dates bysimply assuming the actual start dateT is stochastic with a distributionF (T |t).

16

neither. Specifically, there is fear whenever∫ 1

0(1 − σ(x))(1 − θx)dx ≤ 1 − σ(0), i.e. when

the stigmaσ of early matching is low relative to the firm demand (tightness) θ. In this case,

Proposition 2 predicts a pre-emption equilibrium, with an initial rush followed by a gradual play

phase; Proposition 4 asserts a waning matching rate, as payoffs are log-concave in time. Likewise,

there is greed iff∫ 1

0(1−σ(x))(1−θx)dx ≤ 1−θ. This holds when the stigmaσ of early matching

is high relative to the firm demandθ. Here, Proposition 2 predicts a war of attrition equilibrium,

namely, gradual play culminating in a terminal rush, and Proposition 4 asserts that matching rates

start low and rise towards that rush. When neither inequality holds, neither fear nor greed obtains,

and so both types of gradual play as well as unit mass rushes are equilibria, by Proposition 4.

For an application, NR2004 chronicle the gastroenterologymarket. The offer distribution

in their reported years (see their Figure 1) is roughly consistent with the pattern we predict for

a pre-emption equilibrium as in the left panel of our Figure 4— i.e., a rush and then gradual

play. NR2004 highlight how the offer distribution advancesin time (“unravelling”) between

2003 and 2005, and propose that an increase in the relative demand for fellows precipitated this

shift. Proposition 6 replicates this offer distribution shift. Specifically, assume the market exhibits

fear, owing to early matching stigma. Since the match payoff(10) is log-submodular in(q, θ),

fear rises in market tightnessθ. So the rush for workers occurs earlier by Proposition 6, andis

followed by a longer gradual play phase (left panel of Figure6). This predicted shift is consistent

with the observed change in match timing reported in Figure 1of NR2004.7

Next consider comparative statics in the interest rater. Since the match payoff is log-

submodular in(t, r), lower interest rates entail a harvest time delay, and a delayed matching

distribution, by Proposition 5. In the case of a pre-emptionequilibrium, the initial rush occurs

later and matching is more intense, whereas for a war of attrition equilibrium, the terminal rush

occurs later, and stopping rates fall. Since the match payoff is multiplicative in (t, q), the peak

rush lociΠi are constant int; therefore, rush sizes are unaffected by the interest rate.The sorority

rush environment of Mongell and Roth (1991) is one of extremeurgency, and so corresponds to

a high interest rate. Given a low stigma of early matching anda tight market (for the best sorori-

ties), this matching market exhibits fear, as noted above; therefore, we have a pre-emption game,

for which we predict an early initial rush, followed by a casual gradual play as stragglers match.

7One can reconcile a tatonnement process playing out over several years, by assuming that early matching in thecurrent year leads to lower stigma in the next year. Specifically, if the ratio (1−σ(x))/(1−σ(y)) for x < y, falls inresponse to earlier matching in the previous year, then a natural feedback mechanism emerges. The initial increasein θ stochastically advances match timing, further increasingfear; the rush to match occurs earlier in each year.

17

7.3 The Rush to Sell in a Bubble

We wish to parallel Abreu and Brunnermeier (2003) [AB2003],only dispensing with asymmetric

information. A continuum of investors each owns a unit of theasset and chooses the timet to

sell. LetQ(t) be the fraction of investors that have sold by timet. There is common knowledge

among these investors that the asset price is a bubble. As long as the bubble persists, the asset

pricep(t|ξ) rises deterministically and smoothly in timet, but once the bubble bursts, the price

drops to the fundamental value, which we normalize to 0.

The bubble explodes onceQ(t) exceeds a thresholdκ(t + t0), wheret0 is a random variable

with log-concave cdfF common across investors: Investors know the length of the “fuse”κ, but

do not know how long the fuse had been lit before they became aware of the bubble at time 0.

We assume thatκ is log-concave, withκ′(t + t0) < 0 andlimt→∞ κ(t) = 0.8 In other words, the

burst chance is the probability1− F (τ(q, t)) thatκ(t + t0) ≤ q, whereτ(q, t) uniquely satisfies

κ(t+ τ(q, t)) ≡ q. The expected stopping price,F (τ(q, t))p(t|ξ), is decreasing inq.9

Unlike AB2003, we allow for an interior peak quantile by admitting relative performance

concerns. Indeed, institutional investors, acting on behalf of others, are often paid for their per-

formance relative to their peers. This imposes an extra costto leaving a growing bubble early

relative to other investors. For a simple model of this peer effect, scale stopping payoffs by

1+ρq, whereρ ≥ 0 measuresrelative performance concern.10 All told, the payoff from stopping

at timet as quantileq is:

u(t, q) ≡ (1 + ρq)F (τ(q, t))p(t|ξ) (11)

In Appendix H, we argue that this payoff is log-submodular in(t, q), and log-concave int

andq. With ρ = 0, the stopping payoff (11) is then monotonically decreasingin the quantile,

and Proposition 1 predicts either a pre-emption game for allquantiles, or a pre-emption game

preceded by a timet = 0 rush, or a unit mass rush att = 0. But with a relatively strong concern

for performance, the stopping payoff initially rises inq, and thus the peak quantileq∗ is interior.

In this case, there is fear at timet∗(0) for ρ > 0 small enough, and so an initial rush followed by a

pre-emption game with waning stopping rates, by Propositions 2 and 4. For highρ, the stopping

payoff exhibits greed at timet∗(1), and so we predict an intensifying war of attrition, climaxing

8By contrast, AB2003 assume a constant functionκ, but that the bubble eventually bursts exogenously even withno investor sales. Moreover, absent AB2003’s asymmetric information oft0, if the thresholdκ were constant intime, players could perfectly infer the burst timeQ(tκ) = κ, and so strictly gain by stopping beforetκ.

9A rising price is tempered by the bursting chance in financialbubble models (Brunnermeier and Nagel, 2004).10When a fund does well relative to its peers, it often experiences cash inflows (Berk and Green, 2004). In partic-

ular, Brunnermeier and Nagel (2004) document that during the tech bubble of 1998-2000, funds that rode the bubblelonger experienced higher net inflow and earned higher profits than funds that sold significantly earlier.

18

in a late rush, as seen in Griffin, Harris, and Topaloglu (2011). Finally, an intermediate relative

performance concern allow for two safe equilibria: a war of attrition or a pre-emption game.

Turning to our comparative statics in the fundamentals, recall that as long as the bubble sur-

vives, the price isp(t|ξ). Since it is log-supermodular in(t, ξ), if ξ rises, then so does the ratept/p

at which the bubble grows, and thus there is a harvest time delay. This stochastically postpones

sales, by Proposition 5, and so not only does the bubble inflate faster, but it also lasts longer, since

the selling pressure diminishes. Both findings are consistent with the comparative static derived

in AB2003 that lower interest rates lead to stochastically later sales and a higher undiscounted

bubble price. To see this, simply write our present value price asp(t|ξ) = eξtp(t), i.e. letξ = −r

and letp be their undiscounted price. Then the discounted price is log-submodular in(t, r): A

decrease in the interest rate corresponds to a harvest delay, which delays sales, leading to a higher

undiscounted price, while selling rates fall in a war of attrition and rise in a pre-emption game.

For a quantile comparative static, AB2003 assume the bubbledeterministically grows until the

rational trader sales exceed a fixed thresholdκ > 0. They show that ifκ increases, then bubbles

last stochastically longer, and the price crashes are larger. Consider this exercise in our model.

Assume any two quantilesq2 > q1. We found in§7.3 that the bubble survival chanceF (τ(q, t))

is log-submodular in(q, t), so thatF (τ(q2, t))/F (τ(q1, t)) falls in t. Since the thresholdκ(t)

is decreasing in time, lowert is equivalent to an upward shift in theκ function. Altogether, an

upward shift in ourκ function increases the bubble survival odds ratioF (τ(q2, t))/F (τ(q1, t)).

In other words, the stopping payoff (11) is log-supermodular in q andκ — so that greaterκ corre-

sponds to more greed. Proposition 6 then asserts that sales stochastically delay whenκ rises. So

the bubble bursts stochastically later, and the price drop is stochastically larger, as in AB2003.11

Our model also predicts that selling intensifies during gradual play in a pre-emption equilibrium

(low ρ or κ), and otherwise attenuates. Finally, since our payoff (11)is log-supermodular inq

and relative performance concernsρ, greaterρ is qualitatively similar to greaterκ.

7.4 Bank Runs

Bank runs are among the most fabled of rushes in economics. Inthe benchmark model of

Diamond and Dybvig (1983) [DD1983], these arise because banks make illiquid loans or in-

vestments, but simultaneously offer liquid demandable deposits to individual savers. So if they

try to withdraw their funds at once, a bank might be unable to honor all demands. In their elegant

11Schleifer and Vishny (1997) find a related result in a model with noise traders. Their prices diverge from truevalues, and this divergence increases in the level of noise trade. This acts like greaterκ in our model, since pricesgrow less responsive to rational trades, and in both cases, we predict a larger gap between price and fundamentals.

19

model, savers deposit money into a bank in period 0. Some consumers are unexpectedly struck

by liquidity needs in period 1, and withdraw their money plusan endogenous positive return. In

an efficient Nash equilibrium, all other depositors leave their money untouched until period 2,

whereupon the bank finally realizes a fixed positive net return. But an inefficient equilibrium also

exists, in which all depositors withdraw in period 1 in a bankrun that over-exhausts the bank

savings, since the bank is forced to liquidate loans, and forego the positive return.12

We adapt the withdrawal timing game, abstracting from optimal deposit contract design.13

Given our homogeneous agent model, we ignore private liquidity shocks. A unit continuum of

players[0, 1] have deposited their money in a bank. The bank divides deposits between a safe and

a risky asset, subject to the constraint that at least fractionR be held in the safe asset as reserves.

The safe asset has log-concave discounted expected valuep(t), satisfyingp(0) = 1, p′(0) > 0

andlimt→∞ p(t) = 0. The present value of the risky asset isp(t)(1 − ζ), where the shockζ ≤ 1

has twice differentiable cdfH(ζ |t) that is log-concave inζ andt and log-supermodular in(ζ, t).

To balance the risk, we assume this shock has positive expected value:E[−ζ ] > 0.

As long as the bank is solvent, depositors can withdrawαp(t), where thepayout rateα < 1,

i.e. the bank makes profit(1 − α)p(t) on safe reserves. Since the expected return on the risky

asset exceeds the safe return, the profit maximizing bank will hold the minimum fractionR in the

safe asset, while fraction1−R will be invested in the risky project. Altogether, the bank will pay

depositors as long as total withdrawalsαqp(t) fall short of total bank assetsp(t)(1− ζ(1− R)),

i.e. as long asζ ≤ (1 − αq)/(1 − R). The stopping payoff to withdrawal at timet as quantileq

is:

u(t, q) = H ((1− αq)/(1− R)|t)αp(t) (12)

Clearly,u(t, q) is decreasing inq, log-concave int, and log-submodular (sinceH(ζ |t) is).

Since the stopping payoff (12) is weakly falling inq, bank runs occur immediately or not at

all, by Proposition 1, in the spirit of Diamond and Dybvig (1983) [DD1983]. But unlike there,

Proposition 1 predicts a unique equilibrium that may or may not entail a bank run. Specifically, a

bank run is avoidediff fundamentalsp(t∗(1)) are strong enough, since (3) is equivalent to:

u(t∗(1), 1) = H((1− α)/(1−R)|t∗(1))p(t∗(1)) ≥ u(0, 0) = H(1/(1− R)|0) = 1 (13)

Notice how bank runs do not occur with a sufficiently high reserve ratio or low payout rate.

When (13) is violated, the size of the rush depends on the harvest time payoffu(t∗(1), 1). When

12As DD1983 admit, with a first period deposit choice, if depositors rationally anticipate a run, they avoid it.13Thadden (1998) showed that the ex ante efficient contract is impossible in a continuous time version of DD1983.

20

the harvest time payoff is low enough, panic obtains and all depositors run. For intermediate

harvest time payoffs, there is alarm. In this case, Proposition 1(b) fixes the sizeq0 of the initial run

via:

q−10

∫ q0

0

H((1− αx)/(1−R)|0)dx = H((1− α)/(1− R)|t∗(1))p(t∗(1)) (14)

Since the left hand side of (14) falls inq0, the run shrinks in the peak asset valuep(t∗(1)) or in

the hazard rateH ′/H of risky returns.

Appendix H establishes a log-submodular payoff interaction between the payoutα and both

time and quantiles. Hence, Corollary 2 predicts three consequences of a higher payout rate:

withdrawals shift stochastically earlier, the bank run grows (with alarm), and withdrawal rates

fall during any pre-emption phase. Next consider changes inthe reserve ratio. The stopping

payoff is log-supermodular in(t, R), sinceH(ζ |t) log-supermodular, and log-supermodular in

(q, R) provided the elasticityζH ′(ζ |t)/H(ζ |t) is weakly falling inζ (proven in Appendix H).14

Corollary 2 then predicts that a reserve ratio increase shifts the distribution of withdrawals later,

shrinks the bank run, and increases the withdrawal rate during any pre-emption phase.15

8 The Set of Nash Equilibria with Non-Monotone Payoffs

In any Nash equilibrium with gradual play and a rush, playersmust be indifferent between stop-

ping in the rush and during gradual play. Thus, we introduce the associated theinitial rush locus

RP and theterminal rush locusRW , which are the largest, respectively, smallest solutions to:

V0(t,RP (t)) = u(t∗(1), 1) and V1(t,RW (t)) = u(t∗(0), 0) (15)

No player can gain by immediately pre-empting the initial peak rushΠ0(t) in the safe pre-emption

equilibrium, nor from stopping immediately after the peak rushΠ1(t) in the safe war of attrition.

For with an interior peak quantile, the maximum average payoff exceeds the extreme stopping

payoffs (Figure3). But for larger rushes, this constraint may bind. An initial timet rush islocally

optimalif V0(t, q) ≥ u(t, 0), and a terminal timet rush islocally optimalif V1(t, q) ≥ u(t, 1).

If players can strictly gain from pre-empting any initial rush att > 0, there is at most one

pre-emption equilibrium. Sinceu(t∗(1), 1) equals the initial rush payoff by (15) andut(0, t) > 0

14Equivalently, the stochastic return1 − ζ has an increasing generalized failure rate, a property satisfied by mostcommonly used distributions (see Table 1 in Banciu and Mirchandani (2013)).

15An increase in the reserve ratio increases the probability of being paid at the harvest time, but it also increasesthe probability of being paid in any early run. Log-concavity of H is necessary, but not sufficient, for the formereffect to dominate: This requires our stronger monotone elasticity condition.

21

for t < t∗(1), the following inequality is necessary for multiple pre-emption equilibria:

u(0, 0) < u(t∗(1), 1) (16)

We now define the time domain on which each rush locus is defined. Recall that Proposition 3

asserts a unique safe war of attrition equilibrium exactly when there is no fear at timet∗(0). Let

its rush include the terminal quantiles[qW , 1] and occur at timet1. Likewise, letqP

andt0 be the

initial rush size and time in the unique safe pre-emption equilibrium, when it exists.

Lemma 1 (Rush Loci) Given no fear att∗(0), there existt1 ≤ t, both in (t∗(0), t1), such that

RW is a continuously increasing map from[t1, t1] onto [0, qW ], withRW (t) < ΓW (t) on [t1, t1),

andRW locally optimal exactly on[t, t1] ⊆ [t1, t1]. With no greed att∗(1), no panic, and (16),

there existt ≤ t0 both in(t0, t∗(1)), such thatRP is a continuously increasing map from[t0, t0]

onto[qP, 1], withRP (t) > ΓP (t) on(t0, t0], andRP (t) locally optimal exactly on[t0, t] ⊆ [t0, t0].

Figure 8 graphically depicts the message of this result, with rush loci starting att0 andt1.

We now construct two sets ofcandidate quantile functions: QW andQP . The setQW is

empty given fear att∗(0). Without fear att∗(0), QW contains all quantile functionsQ such that

(i) Q(t) = 0 for t < t∗(0), and for anyt1 ∈ [t, t1]: (ii) Q(t) = ΓW (t) ∀t ∈ [t∗(0), tW ] wheretWuniquely solvesΓW (tW ) = RW (t1); (iii) Q(t) = RW (t1) on (tW , t1); and(iv) Q(t) = 1 for all

t ≥ t1. The setQP is empty if there is greed att∗(1) or panic. Given greed att∗(1), no panic, and

not (16),QP contains a single quantile function: the safe pre-emption equilibrium characterized

by Proposition 3. Finally, if we have no greed att∗(1), no panic, and inequality (16), thenQP

contains all quantile functionsQ such that(i) Q(t) = 0 for t < t0, and for somet0 ∈ [t0, t]:

(ii) Q(t) = RP (t0) ∀t ∈ [t0, tP ) wheretP solvesΓP (tP ) = RP (t0); (iii) Q(t) = ΓP (t) on

[tP , t∗(1)]; and(iv)Q(t) = 1 for all t > t∗(1). By Proposition 2 and Lemma 1,QW is non-empty

iff there is not fear att∗(0), whileQP is non-emptyiff there is not greed att∗(1) and no panic.

We highlight two critical facts about this construction:(a) there is a one to one mapping from

locally optimal rush times in the domain ofRW (RP ) to quantile functions in the candidate sets

QW (QP ); and (b) all quantile functions inQW (resp.QP ) have the same gradual play locus

ΓW (resp.ΓP ) on the intersection of their gradual play intervals.Among all pre-emption (war of

attrition) equilibria, the safe equilibrium involves the smallest rush.

Proposition 7 (Nash Equilibria) The set of war of attrition equilibria is the candidate setQW .

As the rush time postpones, the rush shrinks, and the gradualplay phase lengthens. The set of

pre-emption equilibria is the candidate setQP . As the rush time postpones, the rush shrinks, and

the gradual play phase shrinks. Gradual play intensity is unchanged on the common support.

22

War of Attrition Equilibria

✻1

✲tt∗(0)

ΓW

RW

t1t t1t1

q1

Π1

Pre-Emption EquilibriaNo Alarm✻

1

✲t∗(1)0

ΓP

RP

t0

tt0 t0

q0

Π0

Pre-Emption EquilibriaAlarm

✻1

✲t∗(1)0

ΓP

Π0

RP

t t0

Figure 8:All Nash Equilibria with Gradual Play. Left: All wars of attrition start at timet∗(0),and are given byQ(t) = ΓW (t): Any terminal rush timet1 ∈ [t, t1] determines a rush sizeq1 = RW (t1), which occurs after an inaction phase(Γ−1

W (q1), t1) following the war of attrition.Middle: All pre-emption games start with an initial rush of size Q(t) = q0 at timet0 ∈ [t0, t],followed by inaction on(t0,Γ

−1P (q0)), and then a slow pre-emption phase given byQ(t) = ΓP (t),

ending at timet∗(1). Right: The set of pre-emption equilibria with alarm is constructed similarly,but using the interval of allowable rush times[0, t].

Across both pre-emption and war of attrition equilibria: larger rushes are associated with shorter

gradual play phases. The covariate predictions of rush size, timing, and gradual play length

coincide with Proposition 6 for all wars of attrition and pre-emption equilibria without alarm.

The correlation between the length of the phase of inaction and the size of the rush implies that

the safe war of attrition (pre-emption) equilibrium has thesmallest rush and longest gradual play

phase among all war of attrition (pre-emption) equilibria.In the knife-edge case when payoffs

are log-modular in(t, q), the inaction phase is monotone in the time of the rush, as in Figure 8.

But in the general case of log-submodular payoffs, this monotonicity need not obtain.

Comparative statics prediction with sets of equilibria is aproblematic exercise: Resolving

this, Milgrom and Roberts (1994) focus on extremal equilibria. Here, safe equilibria are extremal

— the safe pre-emption equilibrium has the earliest starting time, and the safe war of attrition

equilibrium the latest ending time. Our comparative statics predictions Propositions 5 and 6 for

safe equilibria robustly extend to all Nash equilibria for the payoffsu(t, q|ϕ) introduced in§6.

Proposition 5∗ Consider a harvest delay withϕH > ϕL and no panic atϕL. Then

(a) For all QL ∈ QW (ϕL), there existsQH ∈ QW (ϕH) such that:(i) QL(t) ≥ QH(t); (ii) The

rush forQH is later and no smaller than forQL; (iii) Gradual play forQH starts later and ends

later than that forQL; and (iv)Q′H(t) < Q′

L(t) in the intersection of the gradual play intervals.

(b) For all QH ∈ QP (ϕH), there existsQL ∈ QP (ϕL) such that:(i) QL(t) ≥ QH(t); (ii) The

rush forQH is later and no larger than forQL; (iii) Gradual play forQH starts later and ends

later than that forQL; and (iv) Q′H(t) > Q′


23

War of Attrition Equilibria✻

1

✲t

ΓLW

ΓHW

RLW

RHW

Π1


1

✲0

ΓHP

ΓLPRL

P

RHP

Π0

Figure 9: Harvest Delay Revisited. With a harvest delay, from lowL to highH, the gradualplay lociΓj

i = Γi(·|ϕj) and rush lociRji ≡ Ri(·|ϕj) shift down, wherei = W,P andj = L,H.

As argued in the proof of Proposition 5,ΓP shifts down and steepens inϕ, ΓW shifts down and

flattens inϕ, while the initial peak rush lociΠ0 andΠ1 are unchanged. Meanwhile, as shown in

Lemma G.2, the rush lociRi(·|ϕ) fall in ϕ for any co-monotone delay, and thereby for a harvest

delay. Thus, the three loci of Figure 8 shift as seen in Figure9.

We now extend our comparative statics for quantile changes to Nash equilibria.

Proposition 6∗ Consider an increase in greed withϕH > ϕL and no panic atϕL, then

(a) ∀QL ∈ QW (ϕL), there existsQH ∈ QW (ϕH) such that:(i) QL ≥ QH ; (ii) The rush forQH

is smaller and occurs later than that forQL; (iii) Gradual play forQH starts later than that for

QL; and (iv) Q′H(t) < Q′


(b) No alarm atϕL: ∀QH ∈ QP (ϕH), there existsQL ∈ QP (ϕL) such that:(i) QL ≥ QH ; (ii)

The rush forQH is no smaller and occurs later than that forQL; (iii) Gradual play forQH starts

later than that forQL; and (iv) Q′H(t) > Q′


(c) Alarm atϕL: ∀QH ∈ QP (ϕH), there existsQL ∈ QP (ϕL) such that:(i) QL ≥ QH ; (ii) The

rush forQH is no larger and occurs no later than that forQL; (iii) Gradual play forQH starts

later than that forQL; and (iv)Q′H(t) > Q′


The gradual play and peak rush loci shift as in Figure 6, whilethe rush loci shift down as in

Figure 9. Altogether, Figure 10 illustrates the effect of anincrease in greed (or decrease in fear).

9 Conclusion

We have developed a novel and unifying theory of large timinggames that subsumes pre-emption

games and wars of attrition. Under a common assumption, thatindividuals have hump-shaped

24

War of Attrition Equilibria✻

1

✲t

ΓLW

ΓHW

RHW

ΠH1

ΠL1


1

✲0

ΓLP

ΓHP

RLP

ΠL0

ΠH0

Figure 10:Quantile Changes Revisited. With an increase in greed or decrease in fear, from lowL to highH, the gradual play lociΓj

i = Γi(·|ϕj) and rush lociRji ≡ Ri(·|ϕj) shift down, where

i = W,P andj = L,H, while the peak rush lociΠ0 andΠ1 shift up.

preferences over their stopping quantile, a rush is inevitable. When the game tilts toward reward-

ing early or late ranks compared to the average — fear or greed— this rush happens early or late,

and is adjacent to a pre-emption game or a war of attrition, respectively. Entry in this gradual

play phase monotonically intensifies as one approaches thisrush when payoffs are log-concave in

time. We derive robust monotone comparative statics, and offer a wealth of realistic and testable

implications. While we have ignored information and a formal treatment of dynamics, our theory

is tractable and identifiable, and rationalizes predictions in several classic timing games.

A Geometric Payoff Transformations

We have formulated greed and fear in terms of quantile preference in the strategic environment.

It is tempting to consider their heuristic use as descriptions of individual risk preference — for

example, as a convex or concave transformation of the stopping payoff. For example, if the stop-

ping payoff is an expected payoff, then concave transformations of expected payoffs correspond

to ambiguity aversion (Klibanoff, Marinacci, and Mukerji,2005).

We can show(♣): for the specific case of a geometric transformation of payoffs u(t, q)β, if

β > 0 rises, then rushes shrink, any pre-emption equilibrium advances in time, and any war of

attrition equilibrium postpones, while the quantile function is unchanged during gradual play.

A comparison to Proposition 6 is instructive. One might musethat greater risk (ambiguity)

aversion corresponds to more fear. We see instead that concave geometric transformations mimic

decreases in fear for pre-emption equilibria, and decreases in greed for war of attrition equilibria.

Our notions of greed and fear are therefore observationallydistinct from risk preference.

25


1

q

✻

0 ✲✲t

ΓW

ΠH1

ΠL1

✻


1

q

❄

✲✛t0

ΓP

ΠL0

ΠH0

❄

Figure 11:Geometric Payoff Transformations. Assume a payoff transformationu(t, q)β. The(thick) gradual play locus is constant inβ, while the (thin) peak rush locus shifts up inβ for awar of attrition equilibrium (right) and down inβ for a pre-emption equilibrium (left).

To prove(♣), consider anyC2 transformationv(t, q) ≡ f(u(t, q)) with f ′ > 0. Thenvt =

f ′ut andvq = f ′uq andvtq = f ′′utuq + f ′utq. Sovtvq − vvtq = [(f ′)2− ff ′′]utuq − ff ′utq yields

vtvq − vvtq = [(f ′)2 − ff ′′ − ff ′/u]utuq − ff ′[utq − utuq/u] (17)

Since the termutuq changes sign, givenutuq ≥ uutq, expression (17) is always nonnegative when

(f ′)2 − ff ′′ − ff ′/u = 0, which requires our geometric formf(u) = cuβ, with c, β > 0. So the

proposed transformation preserves log-submodularity. Log-concavity is proven similarly.

Clearly,f ′ > 0 ensures a fixed gradual play locus (5) in a safe pre-emption equilibrium. Now

consider the peak rush locus (6). Given any convex transformationf , Jensen’s inequality implies:

f(u(t,Π0)) = f(V0(Π0, t)) ≡ f

(

Π−10

∫ Π0

0

u(t, x)dx

)

≤ Π−10

∫ Π0

0

f(u(t, x))dx

So to restore equality, the peak rush locusΠ0(t) must decrease. Finally, any two geometric

transformations withβH > βL are also related by a geometric transformationuβH = (uβL)βH/βL.

B Characterization of the Gradual Play and Peak Rush Loci

Lemma B.1 (Gradual Play Loci) If q∗ > 0, there exists finitetW > t∗(0)withΓW : [t∗(0), tW ] 7→

[0, q∗(tW )] well-defined, continuous, and increasing. Ifq∗ < 1, there existstP ∈ [0, t∗(1)) with

ΓP well-defined, continuous and increasing on[tP , t∗(1)], whereΓP ([tP , t

∗(1)]) = [q∗(tP ), 1]

whenu(0, q∗(0)) ≤ u(t∗(1), 1), and otherwiseΓP ([0, t∗(1)]) = [q, 1] for someq ∈ (q∗(0), 1].

26

STEP 1: ΓW CONSTRUCTION. First, there exists finitetW > t∗(0) such thatu(tW , q∗(tW )) =

u(t∗(0), 0). For q∗ > 0 impliesu(t∗(0), q∗(t∗(0))) > u(t∗(0), 0), while (1) asserts the opposite

inequality for t sufficiently large: existence oftW then follows from continuity ofu(t, q∗(t)).

Next, sinceut < 0 for all t > t∗(0), we haveu(t, 0) < u(t∗(0), 0) andu(t, q∗(tW )) > u(t∗(0), 0),

there exists a uniqueΓW (t) ∈ [0, q∗(tW )] satisfying (4) for allt ∈ [t∗(0), tW ]. Sinceuq > 0,

ut < 0 on (t∗(0), tW ]× [0, q∗(tW )], andu is c2, Γ′W (t) > 0 by the Implicit Function Theorem.

STEP 2: ΓP CONSTRUCTION. First assumeu(0, q∗(0)) ≤ u(t∗(1), 1). Then q∗(t) < 1 ⇒

u(t∗(1), q∗(t∗(1))) > u(t∗(1), 1), while the continuous functionu(t, q∗(t)) is strictly increasing

in t ≤ t∗(1). So there exists a uniquetP ∈ [0, t∗(1)) such thatu(tP , q∗(tv)) ≡ u(t∗(1), 1), with

u(t, q∗(t)) > u(t∗(1), 1) for all t ∈ (tP , t∗(1)]. Also,u(t, 1) < u(t∗(1), 1) andut(t, q) > 0 for all

t < t∗(1) ≤ t∗(q). In sum, there is a uniqueΓP (t) ∈ (q∗(t), 1) solving (5), fort ∈ (tP , t∗(1)). If

instead, the reverse inequalityu(0, q∗(0)) > u(t∗(1), 1) holds, thenu(t, q∗(t)) ≥ u(0, q∗(0)) >

u(t∗(1), 1) > u(t, 1), for all t ≤ t∗(1). Again byut > 0, there is a uniqueΓP (t) ∈ (q∗(t), 1]

satisfying (5) for allt ∈ [0, t∗(1)], i.e. tP = 0. All told, ΓP (t) ≥ q∗(t), so thatuq(t,ΓP (t)) <

0 < ut(t,ΓP (t)), whileu is C2, so thatΓ′P (t) > 0 by the Implicit Function Theorem. �

Lemma B.2 (Peak Rush Loci) Absent greed at timet∗(1), the peak rush locusΠ0 : [0, t∗(1)] 7→

(q∗(t), 1) is well-defined, continuous, and non-increasing with∂V0(t, q)/∂q ≷ 0 as q ≶ Π0(t).

Absent fear at timet∗(0), the peak rush locusΠ1 : [t∗(0),∞) 7→ (q∗(t), 1) is well-defined,

continuous, and non-increasing. Moreover,∂V1(t, q)/∂q ≷ 0 asq ≶ Π1(t).

Proof: We focus onΠ0; the peak terminal rushΠ1 admits similar reasoning.

STEP 1: GREED AND FEAR OBEY SINGLE CROSSING. Sinceu(t, q) is log-submodular, the

ratiou(t, y)/u(t, x) is non-increasing int for all y ≥ x. So if there is no greed att∗(1), then there

is no greed at anyt ≤ t∗(1), and if there is no fear att∗(0), then there is no fear at anyt ≥ t∗(0).

STEP 2: Π0 IS WELL-DEFINED AND CONTINUOUS. Sinceu is log-concave with peak quantile

q∗(t) ∈ (0, 1), we haveuq(t, q) ≷ 0 asq ≶ q∗(t). Thus, for sufficiently smallq > 0, the stopping

payoff u exceeds the averageV0. So by continuity, a solutionΠ0(t) ∈ (0, 1) to (6) existsiff

u(t, 1) < V0(t, 1) (i.e. no greed att). But sinceuq(t, q) ≷ 0 as q ≶ q∗(t), there can be at

most oneq ∈ (0, 1) where the payoffu and associated averageV0 coincide, necessarily at the

unique global maximizerV0, i.e.Π0(t) = argmaxq V0(t, q) with ∂V0(t, q)/∂q ≷ 0 asq ≶ Π0(t).

Continuity ofΠ0(t) follows from the Implicit Function Theorem.

STEP 3: Π0 IS NON-INCREASING. DefineI(q, x) ≡ q−1 for x ≤ q and 0 otherwise, andℓ ≡

t∗(1)− t, and thusV0(t∗(1)−ℓ, q) =

∫ 1

0I(q, x)u(t∗(1)−ℓ, x)dx. Easily,I is log-supermodular in

(q, x), and so the productI(·)u(·) is log-supermodular in(q, x, ℓ). Thus,V0 is log-supermodular

27

in (q, ℓ) since log-supermodularity is preserved by integration by Karlin and Rinott (1980). So

the peak rush locusΠ0(t∗(1)− ℓ) = argmaxq V0(t

∗(1)− t, q) rises inℓ, i.e. falls int. �

C A Safe Equilibrium with Alarm

Assuming no greed att∗(1) and alarm, we construct a quantile function with a rush att = 0

and a pre-emption phase. To this end, let quantileqA ∈ (Π(0), 1) be the largest solution to

V0(0, qA) = u(t∗(1), 1). First,Π(0) = argmaxq V0(0, q) < 1 is well-defined, by Lemma B.2.

By the definition of alarm,V0(0, 1)<u(t∗(1), 1)<maxq V0(0, q) ≡ V0(0,Π(0)). So the unique

qA ∈ (Π(0), 1) follows fromV0(0, q) continuously decreasing inq > Π(0) (Lemma B.2). Then,

givenqA, defineΓP (tA) = qA. To see that such a timetA ∈ (0, t∗(1)) uniquely exists, observe

thatu(0, q∗(0)) ≥ maxq V0(0, q) > u(t∗(1), 1) (by alarm); so that the premise of Lemma B.1

part (b) is met. Thus,ΓP is continuously increasing on[0, t∗(1)] with endpointsΓP (t∗(1)) = 1

andΓP (0) < qA, where this latter inequality follows fromqA > Π(0) ⇒ u(0, qA) < V0(0, qA) =

u(t∗(1), 1) = u(0,ΓP (0)). Finally, define the candidate quantile functionQA as:(i) QA(t) = qA

for all t ∈ [0, tA); (ii) QA(t) = ΓP (t) on [tA, t∗(1)]; and(iii) QA(t) = 1 for all t > t∗(1).

Lemma C.1 Assume alarm and no greed att∗(1). ThenQA is a safe Nash equilibrium. It is the

unique equilibrium with at = 0 rush, one gradual play phase ending att∗(1), and no other rush.

PROOF: The quantile functionQA is safe since its support is{0} ∪ [tA, t∗(1)]. By construction,

the stopping payoff isu(t∗(1), 1) on the support ofQA. The payoff on the inaction region(0, tA)

is strictly lower, sinceu(tA, qA) = u(t∗(1), 1) by construction andu(0, qA) < u(t∗(1), 1) by

ut(t, qA) > 0 for t < t∗(1) ≤ t∗(qA). Finally, sinceut(t, 1) < 0 for t > t∗(1), no player can gain

from stopping aftert∗(1). Altogether,QA is a safe Nash equilibrium.

This is the unique Nash equilibrium with the stated characteristics. In any such equilibrium,

the expected payoff isu(t∗(1), 1), and the unique timet = 0 rush size with this stopping value

is qA. GivenqA, the timetA at which the pre-emption game begins follows uniquely fromΓP ,

which in turn is the unique gradual pre-emption locus given payoff u(t∗(1), 1) by Lemma B.1.�

D Monotone Payoffs in Quantile: Proof of Proposition 1

Case 1: uq > 0. In the text, we established that any equilibrium must involve gradual play for all

quantiles beginning att∗(0), satisfying (4), which uniquely exists by Lemma B.1 (andq∗ = 1).

To see that this is an equilibrium, observe that no agent can gain by pre-empting gradual play,

28

sincet∗(0) maximizesu(t, 0). Further, sincet∗(q) is decreasing, we haveut(t, 1) < 0 for all

t ≥ t∗(0), thus no agent can gain by delaying until after the war of attrition ends. �

Case 2: uq < 0. The text established that rushes cannot occur att > 0 and that all equilibria

involving gradual play have expected payoffu(t∗(1), 1).

STEP 1: A t = 0 UNIT MASS RUSH iff PANIC. Without panic,V0(0, 1) < u(t∗(1), 1): Deviating

to t∗(1) offers a strict improvement over stopping in a unit mass rushat t = 0. Now assume

panic, but gradual play, necessarily with expected payoffu(t∗(1), 1). The payoff for stopping at

t = 0 is eitheru(0, 0) (if no rush occurs att = 0) or V0(0, q), given a rush of sizeq < 1. But

sinceuq < 0, we haveu(0, 0) > V0(0, q) > V0(0, 1) ≥ u(t∗(1), 1) (by panic), a contradiction.

STEP 2: EQUILIBRIUM WITH ALARM . First we claim that alarm implies at = 0 rush. Instead,

assume alarm and gradual play for allq. As in Step 1, the expected value of the proposed equilib-

rium must beu(t∗(1), 1). Givenuq < 0 we haveu(0, 0) = maxq V0(0, q), which strictly exceeds

u(t∗(1), 1) by alarm; therefore deviating tot = 0 results in a strictly higher payoff, contradicting

gradual play for all quantiles. But by Step 1, any equilibrium with alarm must also include grad-

ual play, ending att∗(1). Altogether, with alarm we must have at = 0 rush, followed by gradual

play ending att∗(1). By Lemma C.1, there is a unique such equilibrium:QA. �STEP 3: EQUILIBRIUM WITH NO ALARM OR PANIC. We claim that equilibrium cannot involve

a rush. By Step 1, we cannot have a unit mass rush; thus, the equilibrium must involve gradual

play with expected payoffu(t∗(1), 1). But then givenuq < 0 and no alarm or panic (3), we

have: u(t∗(1), 1) ≥ maxx V0(0, x) = u(0, 0) > V0(0, q) for all q > 0. That is, the payoff in

any t = 0 rush is strictly lower than the equilibrium payoffu(t∗(1), 1), a contradiction. Given

gradual play for all quantiles ending att∗(1), there exists a unique equilibrium. Indeed, absent

alarmu(0, q∗(0)) = maxq V0(0, q) ≤ u(t∗(1), 1), and the premise of Lemma B.1 part(a) is

met, implying the gradual play locusΓP is uniquely defined by (5). Thus, the unique candidate

equilibrium isQ(t) = 0 on [0, tP ); Q(t) = ΓP (t) on [tP , t∗(1)]; and,Q(t) = 1 for t > t∗(1).

Sinceut < 0 for t > t∗(1), no player can gain from delaying until after the gradual play phase,

while ut > 0 implies that stopping before gradual play begins is not a profitable deviation. �

E Single-Peaked Payoffs in Quantile: Omitted Proofs for §5

E.1 Necessary Conditions for Nash Equilibria.

Lemma E.1 When payoffs are non-monotone in quantile, any Nash equilibrium must involve:

an initial rush and then an uninterrupted pre-emption phaseending att∗(1), a terminal rush

29

preceded by an uninterrupted war of attrition phase starting at t∗(0), or a unit mass rush.

STEP 1: RUSH NECESSITY. Assume gradual play for allQ, starting att0. Sinceq∗(t0) > 0,

we haveuq(t0, 0) > 0 ⇒ ut(t0, 0) < 0 (by (2)), i.e.t0 > t∗(0). But then byt∗ non increasing,

ut(t, Q(t)) < 0 on the support ofQ, and by (2)uq(t, Q(t)) > 0, i.e.Q(t) < q∗(t) for all t. Since

q∗ non increasing, we haveQ(t) < q∗(t) ≤ q∗(1) < 1 for all t ∈ supp(Q), which is impossible.

STEP 2: AT MOST ONE RUSH. Assume rushes att1 andt2 > t1. Sinceu strictly falls after the

peak quantile, we must haveQ(t2−) < q∗(t2) else players can strictly gain from preempting the

rush. Likewise, to avoid a strict gain from post-empting therush att1, we needQ(t1) > q∗(t1).

Altogether,q∗(t1) < Q(t1) ≤ Q(t2−) < q∗(t2), which violatesq∗ weakly decreasing.

STEP 3: STOPPING ENDS IN A RUSH OR WITH GRADUAL PLAY ATt∗(1). Assume gradual

play ends att 6= t∗(1). If t < t∗(1) then quantile 1 benefits from deviating tot∗(1). If instead,

t > t∗(1), then we haveut(t, 1) < 0 ⇒ uq(t, 1) > 0, which violatesq∗ < 1.

STEP 4: STOPPING BEGINS IN A RUSH OR WITH GRADUAL PLAY ATt∗(0). On the contrary,

assume gradual play begins att 6= t∗(0). If t > t∗(0) then quantile 0 profits by deviating tot∗(0).

If instead,t < t∗(0), thenut(t, 0) > 0 ⇒ uq(t, 0) < 0, violatingq∗ > 0.

STEP 5: NO INTERIOR QUANTILE RUSH. Assume a rush att > 0, i.e.0 < Q(t−) < Q(t) < 1.

By Step 2, all other quantiles must stop in gradual play. Thenby Steps 3 and 4 we must have

Q(t∗(0)) = 0 andQ(t∗(1)) = 1, but this violatesQ weakly increasing andt∗ weakly decreasing.

STEP 6: ONLY ONE GRADUAL PLAY PHASE. Assume gradual play on[t1, t2] < [t3, t4], sat-

isfying Q(t3) = Q(t4) (WLOG by Step 5). By steps 1, 2, and 5, stopping must begin or end

in a rush. If stopping ends in a rush, then by Step 4,t1 ≥ t∗(0), and sincet∗ is non-increasing,

ut(t, Q(t)) < 0 for all t > t1. But thenu(t2, Q(t2)) > u(t3, Q(t3)), contradicting optimal stop-

ping att2 andt3. If instead, stopping begins in a rush, then by step 3,t4 ≤ t∗(1), and sincet∗ is

non-increasing, we must haveut(t, Q(t)) > 0 for all t < t4. But thenu(t2, Q(t2)) < u(t3, Q(t3)),

again contradicting stopping att2 andt3. �

Lemma E.2 Greed att∗(1) rules out pre-emption, while fear att∗(0) rules out wars of attrition.

By Lemma E.1 any pre-emption phase must end att∗(1), implying Nash payoffw = u(t∗(1), 1).

Also, sincet∗ is non-increasing,ut(t, q) > 0 for all (t, q) < (t∗(1), 1); and thus,w is strictly be-

low the average payoff att∗(1) :∫ 1

0u(t∗(1), x)dx. Altogether,w = u(t∗(1), 1) <

∫ 1

0u(t∗(1), x)dx,

which contradicts greed att∗(1). By similar logic, fear att∗(0) is inconsistent with the Nash pay-

off u(t∗(0), 0) in any war of attrition starting att∗(0). �

30

E.2 Existence and Uniqueness of Safe Equilibria: Proof of Proposition 3. We assume no

greed att∗(1). The case for wars of attrition with no fear att∗(0) follows the logic of Step 3.

STEP 1: PANIC. By definition at = 0 unit mass rush is safe, and is an equilibriumiff panic ob-

tains. For without panic,V0(0, 1) < u(t∗(1), 1), and deviating tot∗(1) offers a strict improvement

over stopping in thet = 0 rush, while panic impliesV0(0, 1) ≥ u(t∗(1), 1) ≡ maxt u(t, 1).

We claim that with panic a unit rush att = 0 is the unique safe equilibrium with an initial

rush. By definition, unit rushes att > 0 are not safe, while by Lemma E.1 any equilibrium

with an initial rush of sizeqR < 1 ends with gradual play att∗(1), implying expected payoff

u(t∗(1), 1). Players will only stop in the rush if the rush payoffV0(t, qR) weakly exceeds the

adjacent gradual play payoffu(t, qR), i.e. thatV0(t, q) is non-increasing inq at qR, and thus

qR ≥ Π0(t) = argmaxq V0(t, q) by Lemma B.2. But sinceV0(t, q) is falling in q > Π0(t) (by

Lemma B.2) and increasing int ≤ t∗(1), we haveV0(t, qR) > V0(t, 1) ≥ V0(0, 1) ≥ u(t∗(1), 1).

Altogether, with panicV0(t, qR) > u(t∗(1), 1), i.e. the rush payoff strictly exceeds the payoff in

any equilibrium with a pre-emption phase, ruling out a pre-emption equilibrium.

STEP 2: ALARM . Assume alarm. Step 1 rules out a unit rush att = 0, while a unit rush att > 0

is not safe by definition. Then given Lemma E.1, any equilibrium with an initial rush must end

with gradual play att∗(1). By Lemma C.1, there exists one such safe equilibrium with aninitial

rush att = 0. We claim that any equilibrium with an initial rush att > 0 is not safe. First,

given an initial rush att > 0, safety requires that gradual play must immediately followthe rush.

Thus, any initial rush at timet of sizeq must be on both the gradual play locus (5) and peak rush

locus (6), i.e.q = ΓP (t) = Π0(t), which impliesu(t∗(1), 1) = maxq V0(t, q). But alarm states

u(t∗(1), 1) < maxq V0(0, q), while maxq V0(0, q) < maxq V0(t, q) for t ∈ (0, t∗(1)]; so that,

u(t∗(1), 1) < maxq V0(t, q), a contradiction. Altogether, the unique pre-emption equilibrium

with alarm is that characterized by Lemma C.1. �

STEP 3: NO ALARM OR PANIC. By Lemma E.1 pre-emption equilibria begin with an ini-

tial rush, followed by gradual play ending att∗(1), and thus have expected payoffu(t∗(1), 1).

First, consider the knife-edged case when (3) holds with equality. In this case,u(0, q∗(0)) >

maxq V0(0, q) = u(t∗(1), 1), and Lemma B.1 part(b) yieldsΓP (t) well defined on[0, t∗(1)] with

u(0,ΓP (0)) = u(t∗(1), 1) = V0(0,Π0(0)). That is,ΓP (0) = Π0(0). In fact, t = 0 is the only

candidate for a safe initial rush. For if the rush occurs at any t > 0, safety demands thatt be on

both the gradual play (5) and peak rush locus (6), i.e.ΓP (t) = Π0(t), butΓP (t) is increasing and

Π0(t) non-increasing on[0, t∗(1)]: tP = 0 andqP= Π0(0) is the only possible safe initial rush.

Now assume (3) is strict, which trivially rules out at = 0 rush, since the maximum rush payoff

falls short of the gradual play payoffu(t∗(1), 1). Given a rush att > 0, safety again requires that

31

the rush time must satisfyΓP (t) = Π0(t), which we claim uniquely defines a rush timetP ∈

(0, t∗(1)) and rush sizeqP∈ (q∗(0), 1]. We establish this separately for two cases. First assume

u(0, q∗(0)) ≤ u(t∗(1), 1). In this case, combining Lemma B.1 part(a) and Lemma B.2, we

discoverΓP (tP ) = q∗(tP ) < Π0(tP ), ΓP (t∗(1)) = 1 > Π0(t

∗(1)), whileΓP is increasing andΠ0

is non-increasing on[tP , t∗(1)): There exists a unique solution(tP , qP ) ∈ (tP , t

∗(1))×(q∗(tP ), 1)

satisfyingqP= ΓP (tP ) = Π0(tP ). For the second case, we assume the oppositeu(0, q∗(0)) >

u(t∗(1), 1), then combine Lemma B.1 part(b) and Lemma B.2 to see thatΓP is increasing andΠ0

non-increasing on[0, t∗(1)], again withΓP (t∗(1)) = 1 > Π0(t

∗(1)). To get the reverse inequality

at t = 0, use (3) to get:u(0,Π0(0)) = V0(0,Π0(0)) < u(t∗(1), 1) = u(0,ΓP (0)), and thus

Π0(0) > ΓP (0), since bothΠ0 andΓP exceedq∗(0), whereuq < 0.

In all cases, there is a unique safe pre-emption equilibriumcandidate:(i)Q(t) = 0 for t < tP ;

(ii) Q(t) = ΓP (t) on [tP , t∗(1)]; and(iii) Q(t) = 1 for all t > t∗(1). Sinceut < 0 for t > t∗(1),

no player can gain from delaying until after the gradual playphase. To see that no player can

gain by pre-empting the rush, note thatut > 0 prior to the rush, while the peak rush payoff

V0(tP ,Π0(tP )) > u(tP , 0). Altogether,Q is the unique safe pre-emption equilibrium. �

E.3 Proof of Proposition 2 (c). Lemmas E.1 and E.2 and Proposition 3 yield parts(a) and(b).

STEP 1: NO UNIT RUSHES WITH STRICT FEAR OR GREED Assume a unit mass rush at time

tR > 0. Given strict greed at timetR, i.e.u(1, tR) > V0(1, tR), post-empting the rush is strictly

better than stopping in the rush. Likewise strict fear at time tR > 0, i.e. u(0, tR) > V0(1, tR)

implies pre-empting the rush results is a strict improvement.

STEP 2: THE UNIT RUSH TIME INTERVAL [T , T ]. Since fear satisfies strict single crossing

(Lemma B.2 Step 1) no fear att∗(0) implies∃T1 > t∗(0) such that no fear obtains for allt ≤ T1.

Likewise no greed att∗(1) implies∃T 1 < t∗(1) such that no greed holds for allt ≥ T 1. Since

t∗(1) ≤ t∗(0), we haveT 1 < T ; thus, both no fear and no greed hold on[T 1, T1]. Given t∗

non-increasing,V0(t, 1) is strictly increasing fort < t∗(1). Thus, no greed att∗(1) implies

∃T 2 ∈ [0, t∗(1)), such thatu(t∗(1), 1) ≤ V0(t, 1) for all t ∈ [T 2, t∗(1)]. Likewise,V0(t, 1) is

strictly decreasing for allt ≥ t∗(0) with limt→∞ V0(t, 1) < u(t∗(0), 0) (by inequality (1)). Thus,

if no fear obtains att∗(0), ∃T2 > t∗(0), such thatV0(t, 1) ≥ u(t∗(0), 0) for all t ∈ [t∗(0), T2].

Finally, letT = max{T 1, T 2} andT = min{T1, T2}.

STEP 3: VERIFICATION OF UNIT RUSH EQUILIBRIA . If a unit mass rush occurs attR ∈

[T , t∗(1)), then the stopping payoff rises until the rush, and no fear attR, i.e.u(tR, 0) ≤ V0(tR, 1)

rules out pre-empting the rush. The stopping payoff rises on(tR, t∗(1)] and then falls: Stopping

in the rush is at least as good as stopping after the rushiff u(t∗(1), 1) ≤ V0(tR, 1), which holds

32

Pre-Emption No Inaction

tR t

Pre-Emption with Inaction

tR t

Unit Mass Rush

tR t

Figure 12:Equilibrium Payoffs. We graph equilibrium payoffs as a function of stopping time.On the left is the unique safe pre-emption equilibrium with aflat payoff on an interval. In themiddle, a pre-emption equilibrium with inaction. On the right a unit rush equilibrium.

for all tR ∈ [T , t∗(1)] by Step 2. If, instead,tR ∈ [t∗(1), t∗(0)], then as long as we have no fear

at tR, u(tR, 0) ≤ V0(tR, 1) pre-empting the rush is not optimal, while no greed attR, u(tR, 1) ≤

V0(tR, 1) ensures post-empting the rush is not optimal. Finally, iftR > t∗(0), the stopping payoff

rises untilt∗(0) and falls betweent∗(0) and the rush. Thus,u(t∗(0), 0) ≤ V0(tR, 1) rules out

pre-empting the rush, which holds for allt ∈ (t∗(0), T ] by Step 2. The stopping payoff falls after

the rush; and so, no greed attR, u(tR, 1) ≤ V0(tR, 1) rules out post-empting the rush. �

F Safe is Equivalent to Secure: Proof of Theorem 1

STEP 1: SAFE ⇒ SECURE. Trivially, any rush att = 0 is secure. Now, assume an interval[ta, tb]

of gradual play with constant stopping payoffπ. So for anyε′ < (tb − ta)/2 and anyt ∈ [ta, tb],

one of the two intervals[t, t + ε′) or (t− ε′, t] will be contained in[ta, tb] and thus obtain payoff

π. Security is maintained by adding a rush with payoffπ at ta or tb.

STEP 2: SECURE⇒ SAFE. We show that an unsafe equilibrium is not secure. With a monotone

payoff function there is a unique equilibrium, which is safe(Proposition 1): Thus, we focus on the

non-monotone case characterized by Proposition 2. By LemmaE.1, there are three possibilities:

a pre-emption equilibrium, a unit mass, and a war of attrition equilibrium. We consider the first

two: The war of attrition case follows similar logic. In particular, assume an equilibrium with an

initial rush of q ∈ (0, 1] at time t ∈ (0, t∗(1)], necessarily withQ(t) = 0 for all t < t. If this

equilibrium is not safe, thenQ(t) = q on an interval followingt. Since this is an equilibrium,

V0(t, q) ≥ u(t, 0). Altogether,infs∈(t−ε,t]w(s;Q) = infs∈(t−ε,t] u(s, 0) < V0(t, q) = w(t;Q) for

all ε ∈ (0, t), where the strict inequality follows fromut(t, q) > 0 for all t < t ≤ t∗(1) ≤ t∗(q).

Now consider an interval following the rush[t, t+ ε). If q < 1, gradual play follows the rush

after delay∆ > 0, andV0(t, q) = u(t+∆, q). But, sincet+∆ < t∗(1) we haveut(t, q) > 0 dur-

ing the delay, andw(t;Q) = u(t, q) < V0(t, q) for all t ∈ (t, t+∆). Thus,infs∈[t,t+ε)w(s;Q) <

33

w(t;Q) for all ε ∈ (0,∆). Now assumeq = 1 and consider the two casest < t∗(1) andt = t∗(1).

If t < t∗(1), thenV0(t, 1) > u(t, 1), else stopping att∗(1) is strictly optimal. But then by conti-

nuity, there existsδ > 0 such thatw(t;Q) = V0(t, 1) > u(t, 1) = w(t;Q) for all t ∈ (t, t + δ).

If t = t∗, equilibrium requires the weaker conditionV0(t∗(1), 1) ≥ u(t∗(1), 1), but then we have

ut(t, 1) < 0 for all t > t; and so,w(t;Q) = V0(t, 1) > u(t, 1) = w(t;Q) for all t > t. �

G Comparative Statics: Propositions 5 and 6

Lemma G.1 In the safe war of attrition equilibrium, the terminal rush shrinks in greed. In the

safe pre-emption equilibrium with no alarm, the initial rush shrinks in fear.

We present the proof for the safe pre-emption equilibrium.

STEP 1: PRELIMINARIES. First we claim that:

V0(t, q|ϕ) ≥ u(t, q|ϕ) ⇒ q−1

∫ q

0

ut(t, x|ϕ)dx ≥ ut(t, q|ϕ) (18)

Indeed, usingu(t, q|ϕ) log-submodular in(t, q):

1

q

∫ q

0

ut(t, x|ϕ)

u(t, q|ϕ)dx =

1

q

∫ q

0

ut(t, x|ϕ)

u(t, x|ϕ)

u(t, x|ϕ)

u(t, q|ϕ)dx ≥

ut(t, q|ϕ)

qu(t, q|ϕ)

∫ q

0

u(t, x|ϕ)

u(t, q|ϕ)dx ≥

ut(t, q|ϕ)

u(t, q|ϕ)

Define ν(t, q, ϕ) ≡ u(t, q|ϕ)/u(t, 1|ϕ), ν∗(t, ϕ) ≡ u(t∗(1), 1|ϕ)/u(t, 1|ϕ), andV(t, q, ϕ) =

q−1∫ q

0ν(t, x, ϕ)dx. By u log-modular in(t, ϕ), ν∗

ϕ = 0, while ν∗t ≶ 0 asut ≷ 0. By log-

submodularity in(t, q) and log-supermodularity in(q, ϕ), νt > 0 andνϕ < 0.

STEP 2: Vt ≥ νt AND −Vϕ > −νϕ FOR ALL (t, q) SATISFYING (6), I .E. q = Π0(t).

Vt − νt = q−1

∫ q

0

[

ut(t, x|ϕ)

u(t, 1|ϕ)−

u(t, x|ϕ)ut(t, 1|ϕ)

u(t, 1|ϕ)2

]

dx−

[

ut(t, q|ϕ)

u(t, 1|ϕ)−

u(t, q|ϕ)ut(t, 1|ϕ)

u(t, 1|ϕ)2

]

≥ −q−1

∫ q

0

[

u(t, x|ϕ)ut(t, 1|ϕ)

u(t, 1|ϕ)2

]

dx+u(t, q|ϕ)ut(t, 1|ϕ)

u(t, 1|ϕ)2by (18)

=ut(t, 1|ϕ)

u(t, 1|ϕ)2

[

u(t, q|ϕ)− q−1

∫ q

0

u(t, x|ϕ)dx

]

= 0 by (6)

Sinceu is strictly log-supermodular in(q, ϕ) symmetric steps establish that−Vϕ > −νϕ.

STEP 3: A DIFFERENCE IN DERVIATIVES. Lemma B.1 provedΓ′P (t) > 0, Lemma B.2 es-

tablishedΠ′0(t) ≤ 0, while the in-text proof of Proposition 6 established that∂ΓP /∂ϕ < 0 and

∂Π0/∂ϕ > 0. We now finish the proof that the initial rush rises inϕ by proving that starting from

34

any(t, q, ϕ), satisfyingq = ΓP (t) = Π(t) and holdingq fixed, the changedt/dϕ in the gradual

play locus (5) is smaller than thedt/dϕ in the peak rush locus (6). Evaluating both derivatives,

this entails:νϕ(t, q, ϕ)− Vϕ(t, q, ϕ)

Vt(t, q, ϕ)− νt(t, q, ϕ)>

−νϕ(t, q, ϕ)

νt(t, q, ϕ)− ν∗t (t)

(19)

Sinceut > 0 during a pre-emption game, we haveν∗t < 0, νt > 0, andνϕ < 0 by Step 1; so that

inequality (19) is satisfied ifνt(νϕ − Vϕ) > −νϕ(Vt − νt) ⇔ −Vϕνt > −νϕVt, which follows

from Vt ≥ νt > 0 and−Vϕ > −νϕ > 0 as established in Steps 1 and 2. �

Lemma G.2 Assume a co-monotone delay. The initial and terminal rush loci R0(t),R1(t) fall.

As a consequence, the initial rush with alarmRP (0) falls.

PROOF: Rewriting (15), we see that any initial rush is defined by theindifference equation:

RP (t)−1

∫ RP (t)

0

u(t, x|ϕ)

u(t, 1|ϕ)dx =

u(t∗(1|ϕ), 1|ϕ)

u(t, 1|ϕ)(20)

Since the initial rush includes the peak quantile, the LHS of(20) falls inRP (t), while the LHS

falls inϕ by log-supermodularity ofu in (q, ϕ). Now, (20) shares the RHS of (8), shown increas-

ing in ϕ in the proof of Proposition 5. So, the initial rush locus obeys∂RP /∂ϕ < 0. �

H Bank Run Example: Omitted Proofs (§7)

Claim 1 The payoffu(t, q) ≡ (1 + ρq)F (τ(q, t))p(t|ξ) in (11) is log-submodular in(t, q), and

log-concave int andq.

Proof: Thatκ(t + τ(q, t))≡q yieldsκ′(t+τ(q, t))(1+τt(q, t))=0 andκ′(t+τ(q, t))τq(q, t)=1.

So,τt ≡ −1 andτq < 0 givenκ′ < 0. Hence,τtq = 0, τtt = 0, andτqq = −(κ′′/κ′)(τq)2. Thus,

∂2 log(F (τ(q, t)))

∂t∂qF (τ(q, t))2 =

[

FF ′′ − (F ′)2]

τtτq + FF ′τtq =[

FF ′′ − (F ′)2]

τtτq ≤ 0

Twice differentiatinglog(F (τ(q, t))) in t likewise yields[FF ′′ − (F ′)2]/F 2 ≤ 0. Similarly,

∂2 log(F (τ(q, t)))/∂q2 = (τq)2[FF ′′ − (F ′)2 − (κ′′/κ′)FF ′]/F 2 ≤ 0

where−κ′′/κ′ ≤ 0 follows sinceκ is decreasing and log-concave. �

35

Claim 2 The bank run payoff (12) is log-submodular in(q, α). This payoff is log-supermodular

in (q, R) provided the elasticityζH ′(ζ |t)/H(ζ |t) is weakly falling inζ .

PROOF: By Lemma 2.6.4 in Topkis (1998),u is log-submodular in(q, α), asH is monotone and

log-concave inζ , and1 − αq is monotone and submodular in(q, α). It is log-supermodular in

(q, R):∂2 log(H(·))

∂q∂RH(·)2(1−R)2 =

(

1− αq

1− R

)

(

(H ′)2 −HH ′′)

−HH ′ ≥ 0 (21)

i.e. x(H ′(x)2 −H(x)H ′′(x))−H(x)H ′(x) ≥ 0, namely, withxH ′(x)/H(x) weakly falling. �

I All Nash Equilibria: Omitted Proofs (§8)

Proof of Lemma 1. We focus onRP , and thus assume no greed att∗(1) and no panic.

STEP 1: RP ([t0, t0]) = [qP , 1] IS CONTINUOUS, INCREASING AND EXCEEDSΓP .

By Propositions 2 and 3, the unique safe initial rush(t0, qP ) satisfies (15). And since any equilib-

rium initial rush includes the peak ofV0 (Corollary 1) withV0 falling after this peak (Lemma B.2),

qP

must be the largest such solution att0, i.e.RP (t0) = qP

. Now, for the upper endpointt0, com-

bine the inequalities for no greed att∗(1) and no panic:V0(0, 1) < u(t∗(1), 1) < V0(t∗(1), 1),

which along withV0(t, 1) continuously increasing fort < t∗(1) yields a uniquet0 < t∗(1) sat-

isfying V0(t0, 1) = u(t∗(1), 1). That is,RP (t0) = 1. CombiningRP (t0) = qP< 1 = R(t0),

with V0(t, q) smoothly increasing int ≤ t∗(1) and smoothly decreasing inq ≥ qP

, we discover:

(i) t0 < t0; as well as,(ii) the two inequalitiesV0(t, qP ) > u(t∗(1), 1) andV0(t, 1) < u(t∗(1), 1)

for all t ∈ (t0, t0). So, byV0(t, q) smoothly increasing int ≤ t∗(1) and smoothly decreasing in

q ≥ qP

, the largest solutionRP to (15) uniquely exists for allt, and is continuously increasing

from [t0, t0] onto[qP, 1] by the Implicit Function Theorem.

We claim thatRP (t) > ΓP (t) on(t0, t0]. First,V0(t0, qP ) ≥ u(t0, qP ), else players would not

stop in the safe rush(t0, qP ). Combining this inequality withV0(t, q) ≥ u(t, q) for q ≥ Π0(t) by

Lemma B.2, we findRP (t0) ≥ Π0(t0). But then sinceRP is increasing andΠ0 non-increasing

(Lemma B.2), we haveRP (t) > Π0(t) on (t0, t0]; and thusu(t,RP (t)) < V0(t,RP (t)) =

u(t∗(1), 1) by Lemma B.2 and (15). Altogether, given (5), we haveu(t,RP (t)) < u(t,ΓP (t));

and thus,RP (t) > ΓP (t) by uq(t, q) < 0 for q > q∗(t).

STEP 2: LOCAL OPTIMALITY . Formally, the candidate initial rushRP (t) is locally optimaliff :

V0(t,RP (t)) ≥ max{u(t, 0), u(t,RP (t))} (22)

36

Step 1 establishedV0(t,RP (t)) ≥ u(t,RP (t)) on[t0, t0]. When inequality (16) holds, we trivially

haveV0(0,RP (0)) ≥ u(0, 0): a t = 0 rush of sizeRP (0) is locally optimal. Thus, we henceforth

assume that the lower bound on the domain ofRP is strictly positive:t0 > 0. But, since this

lower bound is defined as the unique safe initial rush time, Proposition 3 asserts that we cannot

have alarm or panic. In this case, Step 3 in the proof of Proposition 3 establishes that the safe

rush size obeysRP (t0) = argmaxq V0(t0, q), but thenV0(t0,RP (t0)) > u(t0, 0). Now, since

V0(t,RP (t)) is constant int on [t0, t0] by assumption (15) andu(t, 0) is increasing int on this

domain, we either haveV0(t,RP (t)) ≥ u(t, 0) for all t ∈ [t0, t0], in which case we sett ≡ t0 or

there existst < t0 such thatV0(t,RP (t)) T u(t, 0) ast S t for t ∈ [t0, t0]. In either caseRP (t)

satisfies (22) for allt ∈ [t0, t], but for anyt ∈ (t, t0] inequality (22) is violated. �

Proof of Proposition 7. Let QNP be the set of pre-emption equilibria. With greed att∗(1) or

panic,QNP ,QP = ∅ by Proposition 2 and assumption. So assume no greed att∗(1) and no panic.

STEP 1: QNP ⊆ QP . By Proposition 2, pre-emption equilibria share valueu(t∗(1), 1) and

involve an initial rush. Thus, any equilibrium rush must satisfy u(t∗(1), 1) = V0(t, q) ≥ u(t, q),

i.e. be of sizeq = RP (t) at a timet ∈ [t0, t0]. Further, by Proposition 2, there can only be a

single inaction phase separating this rush from an uninterrupted gradual play phase obeying (5),

which Lemma B.1 establishes uniquely definesΓP . Finally, by Lemma 1, the interval[t0, t] are

the only times for whichRP (t) is locally optimal.

STEP 2: QP ⊆ QNP . Recall thatw(t;Q) is the payoff to stopping at timet ≥ 0 given quantile

functionQ. Let QS ∈ QP be the unique safe equilibrium and consider an arbitraryQ ∈ QP

with an initial rush att0, inaction on(t0, tP ) and gradual play followingΓP (t) on [tP , 1]. By

construction the stopping payoff isu(t∗(1), 1) for all t ∈ supp(Q). Further, sinceQS is an

equilibrium andQ(t) = QS(t) on [0, t0) and[tP ,∞), we havew(t, Q) = w(t, QS) ≤ u(t∗(1), 1)

on these intervals: No player can gain by deviating to either[0, t0) or [tP ,∞). By Lemma 1,

RP (t) ≥ ΓP (t) on the inaction interval(t0, tP ) and thusV0(t,RP (t)) ≥ u(t, q): No player

can gain from deviating to the inaction interval. Finally, consider the interval[t0, t0) on which

Q(t) = 0. Sinceu(t, 0) is increasing on this interval, no player can gain from pre-empting the

rush att0 providedV0(t0,RP (t0)) > u(t0, 0), which is ensured by local optimality (22). �STEP 3: COVARIATE PREDICTIONS. ConsiderQ1, Q2 ∈ QP , with rush timest1 < t2. The rush

sizes obeyRP (t1) < RP (t2) by RP increasing (Lemma 1). Gradual play start times are ordered

Γ−1P (RP (t1)) < Γ−1

P (RP (t2)) byΓ−1P increasing (Lemma B.1). Thus, gradual play durations obey

t∗(1)− Γ−1P (RP (t1)) > t∗(1) − Γ−1

P (RP (t2)). Finally, by constructionQ1(t) = Q2(t) = ΓP (t)

on the intersection of the gradual play intervals[Γ−1P (RP (t2)), t

∗(1)]. �

37

Lemma I.1 Assume a harvest delay or increase in greedϕH > ϕL, with qH = RP (tH |ϕH)

locally optimal. IftL = R−1P (qH |ϕL) ≥ t0(ϕL), thenRP (tL|ϕL) is locally optimal.

STEP 1: HARVEST DELAY. If qH = RP (tH |ϕH) satisfies local optimality (22), then:

1 ≤

∫ qH

0

u(tH , x|ϕH)

qHu(tH , 0|ϕH)dx =

∫ qH

0

u(tH, x|ϕL)

qHu(tH , 0|ϕL)dx ≤

∫ qH

0

u(tL, x|ϕL)

qHu(tL, 0|ϕL)dx =

V0(tL, qH |ϕL)

u(tL, 0|ϕL)

where the first equality follows from log-modularity in(q, ϕ) and the inequality owes tou

log-submodular in(t, q) and tL < tH by RP falling in ϕ (Lemma G.2). We have shown

V0(tL, qH |ϕL) ≥ u(tL, 0|ϕL), while tL ≥ t0(ϕL) by assumption. Together these two conditions

are sufficient for local optimality ofRP (tL|ϕL), as shown in Lemma 1 Step 2.

STEP 2: INCREASE IN GREED. By Lemma 1 Step 2,RP (t|ϕ) is locally optimal for t ∈

[t0(ϕ), t0(ϕ)] iff u(t, 0|ϕ) ≤ V0(t,RP (t|ϕ)|ϕ). Givenu(t, 0|ϕ) increasing int ≤ t0(ϕ) and

V0(t,RP (t|ϕ)|ϕ) constant int by (15), if the largest locally optimal timet(ϕ) < t0(ϕ)), it solves:

V (t(ϕ),RP (t(ϕ)|ϕ), ϕ) = 1, where V (t, q, ϕ) ≡V0(t, q|ϕ)

u(t, 0|ϕ)(23)

By assumption,tL ≥ t0(ϕL). For a contradiction, assumetL is not locally optimal:tL > t(ϕL).

We claim that starting from any(t, q, ϕ) satisfying both (15), i.e.q = RP (t|ϕ) and (23), i.e.

V (t, q, ϕ) = 1, the change in the rush locusdR−1P (q|ϕ)/dϕ, holdingq fixed, exceeds the change

along (23)dt/dϕ, holding q fixed. Indeed, definingh(t, q, ϕ) ≡ u(t∗(1), 1|ϕ)/u(t, 0|ϕ) and

differentiating, we discoverdR−1P (q|ϕ)/dϕ−dt/dϕ = Vϕ/(ht− Vt)− Vϕ/(−Vt) > 0, where the

inequality follows fromht < 0 (by ut > 0 for t < t∗(1|ϕ)), hϕ = 0 (by u log-modular in(t, ϕ)),

Vt ≤ 0 (by u log-submodular in(t, q)), Vϕ > 0 (by log-supermodular in(q, ϕ)), andht − Vt ≥ 0

(elseR(t|ϕ) falls in t contradicting Lemma 1). Altogether, givenqL ≡ RP (t(ϕL)|ϕ), we have

shownt(ϕL) = R−1P (qL|ϕH) ≥ t(ϕH); and thus,tL > t(ϕH), but this contradictstH > tL (by

RP (·|ϕ) falling in ϕ) andtH ≤ t(ϕH) (by qH = RP (tH |ϕ) locally optimal). �

Proof of Propositions 5∗ and 6∗. Consider the setsQP (ϕH) andQP (ϕL) for a co-monotone

delayϕH > ϕL. The results vacuously hold ifQP (ϕH) is empty. Henceforth assume not. By

Proposition 2,QP (ϕH) non-empty implies no greed att∗(1|ϕH), which in turn implies no greed

at t∗(1|ϕL) by∫ 1

0[u(t, x|ϕ)/u[t, 1|ϕ]dx falling in ϕ (by log-supermodularity in(q, ϕ)), rising in

t (by log-submodularity in(t, q)), andt∗(1|ϕH) ≥ t∗(1|ϕL) (Propositions 5 and 6). Then, since

we have assumed no panic atϕL, QP (ϕL) is non-empty, containing at least the safe pre-emption

equilibrium by Proposition 2. Two results follow. First, byProposition 2, pre-emption games end

38

at t∗(1), while Proposition 5 assertst∗(1|ϕH) > t∗(1|ϕL) for a harvest delay and Proposition 6

claimst∗(1|ϕH) = t∗(1|ϕL) for an increase in greed. Thus, gradual play end times are ordered

as in part(iii) for anyQH ∈ QP (ϕH) andQL ∈ QP (ϕL). Likewise, the exit rates obey part

(iv) for all QH ∈ QP (ϕH) andQL ∈ QP (ϕL), sinceΓ′P rises inϕ by Propositions 5 and 6 and

Q ∈ QP (ϕ) shareQ′(t) = Γ′P (t) on any common gradual play interval by Proposition 7.

By construction, choosingQH ∈ Q(ϕH) is equivalent to choosing sometH in the locally opti-

mal interval[t0(ϕH), t(ϕH)] characterized by Lemma 1. LetqH ≡ RP (tH |ϕH) be the associated

rush. Lett0(ϕL) andqL= RP (t0(ϕL)|ϕL) be the safe rush time and size forϕL.

FINAL STEPS FOR5∗. Assume a harvest delayϕH > ϕL.

CASE 1: qL

> qH . Let QL be the safe pre-emption equilibrium. By Proposition 5 the safe

rush times obeyt0(ϕL) ≤ t0(ϕH), while qL> qH by assumption:QL has a larger, earlier rush

thanQH as in part(ii). SinceΓP (t|ϕ) is increasing int by Lemma B.1 and decreasing inϕ by

Proposition 5, the inverse functionΓ−1P (q|ϕ) is increasing inq and decreasing inϕ. Thus, gradual

play start times obeyΓ−1P (qH |ϕH) > Γ−1

P (qL|ϕL), as in part(iii). Altogether,QL ≥ QH as in

part(i), sinceQL has a larger and earlier rush, an earlier start and end time togradual play, and

the gradual play cdfs are orderedΓP (t|ϕL) > ΓP (t|ϕH) on the common gradual play support.

CASE 2: qL≤ qH . SinceQH is an equilibrium,qH = RP (tH |ϕH) is locally optimal. And

by Lemma 1,RP (·|ϕL) is continuously increasing with domain[qL, 1], which impliestL ≡

R−1P (qH |ϕL) ≥ t0(ϕL) exists. Thus,qH = RP (tL|ϕ) is locally optimal by Lemma I.1, andtL

defines an equilibriumQL ∈ QP (ϕL). To see thatQL satisfies part(ii) note thatQL andQH have

the same size rush by construction, while rush times are orderedtL < tH ≡ R−1P (qH |ϕH) byRP

falling in ϕ (Lemma G.2). Now, sinceΓP (t|ϕ) is increasing int (Lemma B.1) and falling inϕ

(Proposition 5), gradual play start times obeyΓ−1P (RP (tL|ϕL)|ϕL) < Γ−1

P (qH |ϕH) as in part(iii).

Altogether,QL ≥ QH as in part(i), sinceQL has the same size rush, occurring earlier, an earlier

start and end time to gradual play, and the gradual play cdfs are orderedΓP (t|ϕL) > ΓP (t|ϕH)

on any common gradual play interval.

FINAL STEPS FOR6∗. SinceQP (ϕH) is empty with panic (Proposition 2), we WLOG assume no

panic atϕH . The steps for part(c) (alarm atϕL) exactly parallel those for Proposition5∗ above.

Thus, we henceforth assume no alarm atϕL, and since the premise of Proposition6∗ assumes

no panic atϕL, inequality (3) obtains atϕL, i.e. [maxq V0(0, q|ϕL)]/u(t∗(1|ϕL), 1|ϕL) ≤ 1. But

then, inequality (3) also obtains atϕH , sinceV0(0, q|ϕ))/u(t∗(1|ϕ), 1|ϕ) falls in ϕ by u log-

modular in(t, ϕ) and log-supermodular in(q, ϕ). Altogether, neither alarm nor panic obtains at

ϕL andϕH . Proposition 6 then states that safe rush times obeyt0(ϕL) < t0(ϕH) ≤ tH with sizes

qL< RP (t0(ϕH)|ϕH) ≡ q

H. By Lemma 1,RP (·|ϕL) is continuously increasing onto domain

39

[qL, 1] ⊃ [q

H, 1]; and thus,tL ≡ R−1

P (qH |ϕL) > t0(ϕL) uniquely exists, is locally optimal by

Lemma I.1, and satisfiestL < tH by RP (·|ϕ) falling in ϕ (Proposition 6). Altogether,tL defines

QL ∈ QP (ϕL) with an earlier rush of the same size asQH as in part(ii). The functionΓP (t|ϕ) is

increasing int (Lemma B.1) and decreasing inϕ (Proposition 6): Gradual play starts times obey

Γ−1P (RP (tL|ϕL)|ϕL) < Γ−1

P (qH |ϕH) as in part(iii). Altogether,QL ≥ QH as in part(i), since

QL has the same size rush, occurring earlier, an earlier start and same end time to gradual play,

and gradual play cdfs obeyΓP (t|ϕL) > ΓP (t|ϕH) on any common gradual play interval. �

References

ABREU, D., AND M. BRUNNERMEIER (2003): “Bubbles and Crashes,”Econometrica, 71(1),173–204.

ANDERSON, S. T., D. FRIEDMAN , AND R. OPREA (2010): “Preemption Games: Theory andExperiment,”American Economic Review, 100, 1778–1803.

AVERY, C., C. JOLLS, R. A. POSNER, AND A. E. ROTH (2001): “The Market for FederalJudicial Law Clerks,” Discussion paper series, paper 317, Harvard Law School.

BANCIU , M., AND Y. M IRCHANDANI (2013): “New Results Concerning Probability Distribu-tions With Increasing Generalized Failure Rates,”Operations Research, 61, 925–931.

BERK, J. B., AND R. C. GREEN (2004): “Mutual Fund Flows and Performance in RationalMarkets,”Journal of Political Economy, 112, 1269–1295.

BRUNNERMEIER, M. K., AND J. MORGAN (2010): “Clock Games: Theory and Experiments,”Games and Economic Behavior, 68, 532–550.

BRUNNERMEIER, M. K., AND S. NAGEL (2004): “Hedge Funds and the Technology Bubble,”The Journal of Finance, 59(5), 2013–2040.

CARD, D., A. MAS, AND J. ROTHSTEIN (2008): “Tipping and the Dynamics of Segregation,”Quarterly Journal of Economics, 123, 177–218.

DIAMOND , D. W., AND P. H. DYBVIG (1983): “Bank Runs, Deposit Insurance, and Liquidity,”Journal of Political Economy, 91(3), 401–419.

FUDENBERG, D., R. GILBERT, J. STIGLITZ , AND J. TIROLE (1983): “Preemption, Leapfrog-ging, and Competition in Patent Races,”European Economic Review, 22, 3–31.

FUDENBERG, D., AND J. TIROLE (1985): “Preemption and Rent Equalization in the Adoptionof New Technology,”Review of Economic Studies, 52, 383–402.

40

GRANOVETTE, M. (1978): “Threshold models of collective behavior,”American Journal ofSociology, 83(6), 1420–1443.

GRIFFIN, J., J. HARRIS, AND S. TOPALOGLU (2011): “Who Drove and Burst the Tech Bubble?,”The Journal of Finance, 66(4), 1251–1290.

GRODZINS, M. M. (1957): “Metropolitan Segregation,”Scientific American, 197, 33–47.

KARLIN , S. (1959): “Games of Timing of One Action for Each Player,” in Mathematical Meth-ods and Theory in Games, Programming, and Economics, volumeII: The Theory of InfiniteGames, ed. by S. Karlin, chap. 5. Addison-Wesley, Reading, Massachusetts.

KARLIN , S.,AND Y. RINOTT (1980): “Classes of Orderings of Measures and Related CorrelationInequalities, I: Multivariate Totally Positive Distributions,” Journal of Multivariate Analysis,10, 467–498.

KLIBANOFF, P., M. MARINACCI , AND S. MUKERJI (2005): “A Smooth Model of DecisionMaking under Ambiguity,”Econometrica, 73, 1849–1892.

KRISHNA, V., AND J. MORGAN (1997): “An analysis of the war of attrition and the all-payauction,”Journal of Economic Theory, 72(2), 343–362.

MAYNARD SMITH , J. (1974): “The Theory of Games and Evolution in Animal Conflicts,” Jour-nal of Theoretical Biology, 47, 209–221.

M ILGROM, P., AND J. ROBERTS (1994): “Comparing Equilibria,”American Economic Review,84(3), 441–459.

MONGELL, S., AND A. ROTH (1991): “Sorority Rush as a Two-Sided Matching Mechanism,”American Economic Review, 81(3), 441–464.

NIEDERLE, M., AND A. ROTH (2004): “The Gastroenterology Fellowship Match: How It Failedand Why It Could Succeed Once Again,”Gastroenterology, 127(2), 658–666.

PARK , A., AND L. SMITH (2008): “Caller Number Five and Related Timing Games,”TheoreticalEconomics, 3, 231–256.

ROTH, A. E., AND X. X ING (1994): “Jumping the Gun: Imperfections and Institutions Relatedto the Timing of Market Transactions,”American Economic Review, 84(4), 992–1044.

SCHELLING, T. C. (1969): “Models of segregation,”American Economic Review, 59(2), 488–493.

THADDEN, E.-L. V. (1998): “Intermediated versus Direct Investment: Optimal Liquidity Provi-sion and Dynamic Incentive Compatibility,”Journal of Financial Intermediation, 7, 177–197.

TOPKIS, D. (1998):Supermodularity and Complementarity. Princeton University Press, Prince-ton, NJ.

41

Greed,Fear, andRushes · 2018-02-16 · Greed,Fear, andRushes∗ Axel Anderson Georgetown Andreas...

Documents

Transcript of Greed,Fear, andRushes · 2018-02-16 · Greed,Fear, andRushes∗ Axel Anderson Georgetown Andreas...