Greed, Fear, and Rushes∗
Axel Anderson
Andreas Park
Lones Smith
Georgetown
Toronto
Wisconsin
January 6, 2015
Abstract
We develop a tractable continuum player timing game that subsumes wars of attrition
and pre-emption games, in which greed and fear relax the last and first mover advantages.
Payoffs are continuous and single-peaked functions of the stopping time and quantile. Time
captures the payoff-relevant fundamental, as payoffs “ripen”, peak at a “harvest time”, and
then “rot”. The nonmonotone quantile response rationalizes sudden mass movements in
economics, and explains when it is inefficiently early or late. With greed, the harvest time
precedes an accelerating war of attrition ending in a rush; with fear, a rush precedes a slowing
pre-emption game ending at the harvest time.
The theory simultaneously predicts the length, duration, and intensity of gradual play, and
the size and timing of rushes, and offers insights for an array of timing games. For instance,
matching rushes and bank runs happen before fundamentals indicate, and asset sales rushes
occur after. Moreover, (a) “unraveling” in matching markets depends on early matching
stigma and market thinness; (b) asset sales rushes reflect liquidity and relative compensation;
(c) a higher reserve ratio shrinks the bank run, but otherwise increases the withdrawal rate.
∗
This supersedes a primitive early version of this paper by Andreas and Lones, growing out of joint work in
Andreas’ 2004 PhD thesis, that assumed multiplicative payoffs. It was presented at the 2008 Econometric Society
Summer Meetings at Northwestern and the 2009 ASSA meetings. While including some results from that paper, the
modeling and exposition now solely reflect joint work of Axel and Lones since 2012. Axel and Lones alone bear
responsibility for any errors in this paper. We have profited from comments in seminar presentations at Wisconsin,
Western Ontario, Melbourne, 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
7.1 Land Runs, Sales Rushes, and Tipping Models . .
7.2 The Rush to Match . . . . . . . . . . . . . . . .
7.3 The Rush to Sell in a Bubble . . . . . . . . . . .
7.4 Bank Runs . . . . . . . . . . . . . . . . . . . . .
15
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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
36
All Nash Equilibria: Omitted Proofs (§8)
1 Introduction
“Natura non facit saltus.” — Leibniz, Linnaeus, Darwin, and Marshall
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 removed 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 only the size and timing of aggregate
jumps, but also the more commonly studied speed of entry into the 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 the motivational 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-relevant fundamental, 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 is gradual play, as players slowly stop; a rush occurs 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 optimal harvest time for any quantile, when the fundamental peaks, and a
unique optimal peak quantile for any time, when stopping is most strategically advantageous.
Two opposing flavors of timing games have been studied. A war of attrition entails gradual
play in which the passage of time is fundamentally harmful and strategically beneficial. The
reverse holds in a pre-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 over all other quantiles. But in our class of games, the peak quantile may
be interior. The game exhibits greed if the very last mover eclipses the average quantile, 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 fundamental 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 harvest time 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 immediately stopping as an early quantile dominates the peak fundamentals payoff growth. We call this
extreme case with early rushes either alarm or —- if the rush includes all players — panic.
By the above logic, equilibria are either early or late — namely, transpiring entirely before
or entirely after the harvest time. Absent fear, slow wars of attrition 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. So rushes 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 peaks in a war of attrition equilibrium.
Proposition 7 characterizes all Nash equilibria: By realistically assuming slight uncertainty
about the accuracy of the time clock, secure equilibrium refines Nash equilibria. Theorem 1
proves that secure equilibria are safe, namely, without a time interval with no entry — except immediately 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 fundamental 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, stopping rates wax and wane, respectively, after and before the harvest time. Our payoff dichotomy
2
therefore allows for the identification of timing games from data 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. Proposition 5 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 preemption 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
extreme fear to slowly transition into extreme greed. 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 rush shrinks. So perhaps surprisingly, the
rush is smaller and farther from the harvest time the greater is the greed or fear (Figure 7).
While our model is analytically simple, one might worry that comparative statics analysis
requires modeling dynamics or information. We now show that we 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 myopically to thresholds on the number of black neighbors. But in 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 that arises 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
1
Meanwhile, 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 a larger 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 R EVIEW. 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) explored 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 times τ
on [0, ∞), where τ ≤ t with chance Q(t) on R+ . Either choices are made irrevocably at time-0,
or players do not observe each other over time, perhaps since the game transpires quickly.
A player’s stopping quantile q summarizes the anonymous form of strategic interaction in a
large population, and the time t captures fundamentals. By Lebesgue’s decomposition theorem,
the quantile function Q(t) is the sum of continuous portions, called gradual play, and atoms. If
Q is continuous at t, then the stopping payoff is u(t, Q(t)). Stopping at an atom t of Q, with
Rp
p = Q(t) > Q(t−) = q, earns the average quantile payoff q u(t, x)dx/(p − q) for quantiles in
[q, p]. The aggregate outcome corresponds to a rush, where a mass p − q of agents stops at t.
Payoffs u(q, t) are continuous, and for fixed q, are quasi-concave in t, strictly rising from
t = 0 (“ripening”) until an ideal harvest time t∗ (q), and then strictly falling (‘rotting”). For fixed t,
4
payoffs u are either monotone or log-concave in q, with unique interior peak quantile q ∗ (t). We
embed strategic interactions by assuming the payoff function u(t, q) is log-submodular — such
as multiplicative.2 Since higher q 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 time t∗ (q) is a decreasing function of q, while the peak quantile function q ∗ (t)
is falling in t. To ensure that players eventually stop, we assume that waiting forever is dominated
by stopping at t∗ (0):
lim u(t, q ∗ (t)) < u(t∗ (0), 0).
t→∞
(1)
Formally, let T (q) ≡ inf{t ∈ R+ |Q(t) ≥ q} be generalized inverse distribution function.
Then the payoff to τ = t equals the Radon Nikodym derivative w(t) = dW/dQ, given the
following running integral:
Z Q(t)
W (t) =
u(T (x), x)dx
0
A Nash equilibrium is a cdf Q whose support contains only maximizers of the function w(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 to safe equilibria, namely, Nash equilibria in which the quantile function Q is an atom at
time t = 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 time t ≥ 0 given cdf Q. The ε-secure payoff at
t is:
maxh
inf
max(t−ε,0)≤s<t
w(s; Q),
inf
s∈[t,t+ε)
w(s; Q)i
This is the minmax payoff in the richer model when individuals have access to two different εaccurate timing technologies: One clock never runs late, and one never runs early. An equilibrium
Q is secure if wε (t; Q) = w(t; Q) for all t in the support of Q, and all small enough ε > 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.
2
Almost all of our results only require the weaker complementary condition that u(t, q) be quasi-submodular, so
that u(tL , qL ) ≥ (>)u(tH , qL ) implies u(tL , qH ) ≥ (>)u(tH , qH ), for all tH ≥ tL and qH ≥ qL .
3
For if q = Q′ (t)
R p> 0 exists, then w(t) = u(t, q). If Q(t) = q > p = Q(t−), then T (x) = t on [p, q], and so
W (t) − W (t−) = q u(t, x)dx. Finally, if Q(t) = q on [t1 , t2 ), then t is not in the support of Q.
5
Fear: u(t, 0) ≥
u
R1
0
u(t, x)dx
ū
0
quantile rank q
1
u
Greed: u(t, 1) ≥
u
ū
ū
Neither Fear nor Greed
0
quantile rank q
1
0
R1
0
u(t, x)dx
quantile rank q
1
Figure 1: Fear and Greed. Payoffs at any time t cannot exhibit both greed and fear, with first
and 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)) = w̄, say. Since u is C 2 , if the stopping rate Q′
exists and is C 1 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 by the marginal rate of substitution, i.e.
Q′ (t) = −ut /uq . So the slope signs uq and ut 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 which ut > 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 which ut < 0 < uq , so that
the passage of time is fundamentally harmful but strategically beneficial.
To analyze rushes, we must compare average and peak quantile payoffs. Let V0 (t, q) ≡
Rq
q
u(t, x)dx be the running average payoff function. As key assumption is that fundamental
0
growth is strong enough that:
−1
max V0 (0, q) ≤ u(t∗ (1), 1)
q
(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 when V0 (0, 1) < u(t∗ (1), 1) <
maxq V0 (0, q), and panic when 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 ′ and t ≥ 0. Fear generalizes this, asking only an advantage of
6
Pure War of Attrition
1
✻
1
✻
Pure Pre-Emption Game
u(0, 0) ≤ u(t∗ (1), 1)
1
✻
Pure Pre-Emption Game
u(t, q) > u(t∗ (1), 1)
ΓP
q
q
ΓW
0
t∗ (0)
✲
t
q
q0
ΓP
✲
0
t∗ (1)
0
✲
0
0
t∗ (1)
Figure 2: Monotone Cases. In the left panel uq > 0, and the equilibrium is a pure war of
attrition following the gradual play locus ΓW (t). When uq < 0 gradual play follows the preemption gradual play locus ΓP (t). If u(0, 0) ≤ u(t∗ (1), 1), as in the middle panel, ΓP defines a
pure pre-emption game. In the right panel, u(0, 0) > u(t∗ (1), 1), as with alarm and panic. In this
case, the indifference curve ΓP intersects the q-axis at q > 0, implying there cannot be gradual
play for all quantiles. Given alarm, the equilibrium involves a rush of size q0 at t = 0, followed
by a period of inaction along the blue line, and then gradual play along ΓP (t).
R1
the least quantile over the average quantile — there is fear at time t if u(t, 0) ≥ 0 u(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 ′ and t ≥ 0. Greed is more general; there is greed at time t if the
R1
last quantile payoff exceeds the average; namely, if u(t, 1) ≥ 0 u(t, x)dx. Both inequalities are
tight if the payoff u(t, q) is constant in q, and strict fear and greed correspond to strict inequalities
(see Figure 1). Since u is single-peaked in q, greed and fear at t 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 the only possibility is a pure war of
attrition. In fact, gradual play must begin at time t∗ (0) > 0. For if gradual plays start at t > t∗ (0),
then quantile 0 would profitably deviate to t∗ (0), whereas if it starts at t < t∗ (0), then the required
differential equation (2) is impossible — for ut (t, 0) > 0 and uq (t, 0) > 0. Since the Nash payoff
is u(t∗ (0), 0), the war of attrition gradual play locus ΓW (Figure 2) begins at t∗ (0), and is defined
by the implicit equation:
u(t, ΓW (t)) = u(t∗ (0), 0)
(4)
Similarly, when uq < 0, any gradual play interval ends at t∗ (1), implying an equilibrium value
u(t∗ (1), 1). For if gradual play ends at t < t∗ (1) then quantile q = 1 benefits from deviating to
t∗ (1). And if it ends at t1 > t∗ (1), then ut (t, 1) < 0 for all t∗ (1) < t < t1 . But since uq < 0 this
7
violates the differential equation (2). Thus, during gradual play, Q must satisfy the pre-emption
gradual play locus ΓP :
u(t, ΓP (t)) = u(t∗ (1), 1)
(5)
When uq < 0, a rush at time t > 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. Since u(t∗ (1), 1) is the Nash equilibrium payoff, inequality (3) precisely rules
out a time zero rush. But the left side of (3) reduces to u(0, 0) when uq < 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 implies V0 (0, q) > u(t∗ (1), 1)
for all q; any equilibrium with gradual play has Nash payoff u(t∗ (1), 1). Given alarm, stopping
immediately for payoff u(0, 0) dominates stopping during gradual play for payoff u(t∗ (1), 1),
and thus a rush must occur. But given V0 (0, 1) < u(t∗ (1), 1), any such rush must be of size
q0 < 1, and the indifference condition V0 (0, q0 ) = u(t∗ (1), 1) then pins down the rush size. But
V0 (0, q0 ) > u(0, q0 ) given uq < 0, so that stopping just after the time zero rush affords a strictly
lower payoff than stopping in the rush. This forces an inaction phase — a time interval [t1 , t2 ]
with no entry, where 0 < Q(t1 ) = Q(t2 ) < 1 — as seen in the right panel of Figure 2.
Appendix D completes the proof of the following result when u is strictly monotone in q.
Proposition 1 Assume the stopping payoff is strictly monotone in quantile. There is a unique
Nash equilibrium, and it is safe. If uq > 0, a pure war of attrition starts at t∗ (0). If uq < 0, and:
(a) with neither alarm nor panic, there is a pre-emption game for all quantiles ending at t∗ (1);
(b) with alarm there is a time-0 rush of size q0 obeying V0 (0, q0 ) = u(t∗ (1), 1), followed by an
inaction phase, and then a pre-emption game ending at t∗ (1);
(c) with panic there is a unit mass rush at time t = 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 peak quantile, two other types of rushes
are possible, likewise engulfing the quantile peak: A terminal rush happens when 0 < Q(t−) =
q < 1 = Q(t), so that all quantiles in [q, 1] rush at the same time t, collecting terminal rush
R1
payoff V1 (q, t) ≡ (1 − q)−1 q u(t, x)dx. A unit mass rush occurs when 0 = Q(t−) < 1 = Q(t),
so that all quantiles [0, 1] rush at the same time t. The latter is a pure strategy Nash equilibrium.
Since the harvest time t∗ (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 of Nash equilibria occur:
(a) An initial rush followed by a pre-emption phase time interval ending at harvest time t∗ (1) iff
there is not greed at time t∗ (1) and no panic. The equilibrium payoff is u(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 time t∗ (0). The equilibrium payoff is u(t∗ (0), 0).
(c) A unit rush at any t in an open interval around [t∗ (1), t∗ (0)] with no greed at t∗ (1) and no
fear at t∗ (0). Unit mass rushes cannot occur at any positive time with strict greed or strict fear.
We define a pre-emption equilibrium and a war of attrition equilibrium as 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 with uq > 0 would stop when ut < 0, i.e., later in time; meanwhile,
later quantiles with uq < 0 would stop when ut > 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 have Vi (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 of Vi and the peak quantile of u.
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 payoff V0 is maximized at q = 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
late peak rush locus Πi (t) ∈ arg maxq Vi (t, q), namely, i = 0, 1. Since an average peaks when
the margin equals the average, each locus equates payoffs in rushes and immediately adjacent
gradual play (see Figure 3):
u(t, Πi (t)) = Vi (t, Πi (t))
(6)
Lemma B.2 shows that each locus Πi (t) is decreasing when u(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 time t. We now depict safe equilibria in Figure 4.
9
Peak Terminal Rush
Peak Initial Rush
u(t, q)
u(t, q)
V1 (t, q)
V0 (t, q)
q ∗ (t)
Π0 (t)
q
Π1 (t)
q ∗ (t)
q
Figure 3: Rushes Include the Quantile Peak. The time t peak rush maximizes the average rush
payoff Vi (t, q), and so equates the average and adjacent marginal payoffs Vi (t, q) and u(t, q).
Proposition 3 Absent fear at the harvest time t∗ (0), there exists a unique safe war of attrition
equilibrium. Absent greed at t∗ (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 at t = 0 followed by a period of inaction and then a pre-emption game; or
(c) with panic, a unit mass rush at t = 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 panic nor alarm
obtains, secure equilibria must have both a rush and a gradual play phase and no inaction: either
an initial rush at 0 < 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 at t1 . 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 in t.
In any gradual play phase, the stopping rate Q′ (t) is strictly increasing in time from zero during
a war of attrition phase, and decreasing down to zero during a pre-emption game phase.
Proof: Wars of attrition begin at t∗ (0) and pre-emption games end at t∗ (1), by Proposition 2.
Since ut (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, since uq 6= 0 at the two quantile extremes q = 0, 1.
Next, assume that Q′′ exists. Then differentiating the differential equation (2) in t yields:4
Q′′ = −
4
1 1 utt + 2uqt Q′ + uqq (Q′ )2 = 3 2uqt uq ut − utt u2q − uqq u2t
uq
uq
(7)
Conversely, Q′′ exists whenever Q′ locally does precisely because the right side of (7) must be the slope of Q′ .
10
Safe War of Attrition
Safe Pre-Emption Game
No Alarm
1✻
1
✻
1
Π1
0
t∗ (0)
q0
✲
t1
t
✻
ΓP
ΓP
ΓW
q1
Safe Pre-Emption Game
Alarm
q0
Π0
0
Π0
✲
t∗ (1) 0
t0
✲
t∗ (1)
Figure 4: Safe Equilibria with Non-Monotone Payoffs. In the safe war of attrition equilibrium
(left), gradual play begins at t∗ (0), following the upward sloping gradual play locus (4), and
ends in a terminal rush of quantiles [q1 , 1] at time t1 where the loci cross. In the pre-emption
equilibrium without alarm (middle), an initial rush q0 at time t0 occurs where the upward sloping
gradual play locus (5) intersects the downward sloping peak rush locus (6). Gradual play in the
pre-emption phase then follows the gradual play locus ΓP . With alarm (right), the initial rush
occurs at t = 0 followed by an inaction phase, and then a pre-emption game follows ΓP .
after replacing the marginal rate of substitution Q′ = −ut /uq implied by (2). Finally, the above
right bracketed expression is positive when u is log-concave in t.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 nonmonotone 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 equilibrium, and (ii) convex prior to the terminal rush during any war of attrition equilibrium.
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 by ϕ ∈ R. When u(t, q|ϕ) is strictly log-supermodular in
(t, ϕ) and log-modular in (q, ϕ), greater ϕ 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 in ϕ, by log-supermodularity in (t, ϕ).
5
Since uq ut < 0, the log-submodularity inequality uqt u ≤ uq ut can be reformulated as 2uqt uq ut ≥ 2u2q u2t /u,
or (⋆). But log-concavity in t and q implies u2t ≥ utt u and u2q ≥ uqq u, and so 2u2q u2t /u ≥ utt u2q + uqq u2t .
Combining this with (⋆), we find 2uqt uq ut ≥ utt u2q + uqq u2t .
11
q✻
1
Pre-Emption Equilibrium
q✻
1
War of Attrition Equilibrium
ΓP
ΓW
❄
❄
❄
❄
0
Π1
Π0
✲
✲
0
t
✲
✲
t
Figure 5: Harvest Time Delay. The gradual play locus shifts down in ϕ. In a pre-emption game
(left): A smaller initial rush occurs later and stopping rates rise during gradual play. In a war of
attrition: 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 stochastically 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 payoff u is log-modular in (t, ϕ), so too is the average. The peak rush
locus Π0 (t) ∈ arg maxq V0 (t, q|ϕ) is then constant in ϕ. Now, rewrite the pre-emption gradual
play locus (5) as:
u(t∗ (1|ϕ), 1|ϕ)
u(t, ΓP (t)|ϕ)
=
(8)
u(t, 1|ϕ)
u(t, 1|ϕ)
The LHS of (8) falls in ΓP , since uq < 0 during a pre-emption game, and is constant in ϕ, by
log-modularity of u in (q, ϕ). Log-differentiating the RHS in ϕ, and using the Envelope Theorem:
uϕ (t∗ (1|ϕ), 1|ϕ) uϕ (t, 1|ϕ)
−
>0
u(t∗ (1|ϕ), 1|ϕ)
u(t, 1|ϕ)
since u is log-supermodular in (t, ϕ) and t < t∗ (1|ϕ) during a pre-emption game. Since the
RHS of (8) increases in ϕ and the LHS decreases in ΓP , the gradual play locus ΓP (t) obeys
12
Pre-Emption Equilibrium
q✻
1
q✻
1
ΓP
✻
✻
ΓW
❄
❄
✻
✻
0
War of Attrition Equilibrium
Π1
Π0
✲
✲
0
✲
✲
t
t
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 the payoff in the rush to the adjacent
gradual play payoff up. In the safe pre-emption equilibrium (left): Larger rushes occur later and
stopping rates rise on shorter pre-emption games. In the safe war of attrition equilibrium: Smaller
rushes occur later and stopping rates fall during longer wars of attrition.
∂ΓP /∂ϕ < 0. Next, differentiate the gradual play locus in (5) in t and ϕ, to get:
∂Γ′P (t)
=−
∂ϕ
∂[ut /u] ∂[ut /u] ∂ΓP
+
∂ϕ
∂ΓP ∂ϕ
u
ut
+
uq
u
∂[u/uq ] ∂ΓP
∂[u/uq ]
+
∂ΓP ∂ϕ
∂ϕ
>0
(9)
The first parenthesized term is negative. Indeed, ∂[ut /u]/∂ϕ > 0 since u is log-supermodular
in (t, ϕ), and ∂[ut /u]/∂ΓP < 0 since u is log-submodular in (t, q), and ∂ΓP /∂ϕ < 0 (as shown
above), and finally uq < 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 of u(t, q) in q.
Next consider pure changes in quantile preferences, by assuming the stopping payoff u(t, q|ϕ)
is log-supermodular in (q, ϕ) and log-modular in (t, ϕ). So greater ϕ inflates the relative return
to a quantile delay, but leaves unchanged the relative return to a time delay. Hence, the peak
quantile q ∗ (t|ϕ) rises in ϕ. We say that greed increases when ϕ rises, since payoffs shift towards
later ranks as ϕ rises; this relatively diminishes the potential losses of pre-emption, and relatively
inflates the potential gains from later ranks. Also, if there is greed at time t, then this remains
true if greed increases. Likewise, we say fear increases when ϕ 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 stopping 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
Greed Increases / Fear Decreases
✻
Greed
Fear
0
5
10
15
20
Time
Figure 7: Rush Size and Timing with Increased Greed. Circles at rush times are proportional
to rush sizes. As fear falls, the unique safe pre-emption equilibrium has a larger initial rush, closer
to the harvest time t∗ = 10, and a shorter pre-emption phase. As greed rises, the unique safe war
of attrition equilibrium has a longer war of attrition, and a smaller terminal rush (Proposition 6)
Proof: Since u is log-modular in (t, ϕ), the harvest times t∗ (0) and t∗ (1) are both constant in ϕ.
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.
R1
Define I(q, x) ≡ q −1 for x ≤ q and 0 otherwise, and thus V0 (t, q|ϕ) = 0 I(q, x)u(t, x|ϕ)dx.
Easily, I is log-supermodular in (q, x), and so the product I(·)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 locus Π0 (t) = arg maxq V0 (t, q|ϕ) rises in ϕ.
Now consider the gradual play locus (8). Its RHS is constant in ϕ since u(t, q|ϕ) is logmodular in (t, ϕ), and t∗ (q) is constant in ϕ. The LHS falls in ϕ since u is log-supermodular in
(q, ϕ), and falls in ΓP since uq < 0 during a pre-emption game. All told, the gradual play locus
obeys ∂ΓP /∂ϕ < 0. To see how the slope Γ′P changes, consider (9). The first term in brackets is
negative. For ∂[ut /u]/∂ϕ = 0 since u is log-modular in (t, ϕ), and ∂[ut /u]/∂ΓP < 0 since u is
log-submodular in (t, q), and ∂ΓP /∂ϕ < 0 (as shown above), and finally uq < 0 in a pre-emption
game. The second term is also negative: ∂[u/uq ]/∂ϕ < 0 as u is log-supermodular in (q, ϕ), and
∂[u/uq ]/∂ΓP > 0 as u is log-concave in q, and ∂ΓP /∂ϕ < 0, and ut > 0 in a pre-emption game.
All told, an increase in ϕ: (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 determines whether 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 in ϕ a co-monotone delay in this case, since the
harvest time t∗ (q|ϕ) and the peak quantile q ∗ (t|ϕ) both increase in ϕ. Intuitively, greater ϕ
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 temporal gains from waiting, and so the panic or alarm cases in Proposition 1 applied.
Next consider the sociology notion of a “tipping point” — the moment 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
But in our model, a rush is unavoidable when preferences are smooth, provided they are singlepeaked 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 preference 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 flexibility”, 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 abilities α ∈ [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
Rt
chooses the right type if it waits until time t to hire is e−δt /2 + 0 δe−δs ds = 1 − e−δt /2.6 Assume
that an ability α worker of the right type yields flow payoff α, discounted at rate r. Thus, the
present value of hiring the right type of ability α worker at time t is (α/r)e−rt .
Consider the quantile effect. Assume an initial ratio 2θ ∈ (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 quantile q of firms has already chosen is 1 − θq. We
also assume a stigma σ(q), with payoffs from early matching multiplicatively scaled by 1 − σ(q),
where 1 > σ(0) ≥ σ(1) = 0, and σ ′ < 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 in t, and initially increasing provided the learning effect is strong
enough (δ > r). This stopping payoff is concave in quantile q if σ is convex.
The match payoff (10) is log-modular in t and q, and so always exhibits greed, or fear, or
6
We assume firms unilaterally choose the start date t. One can model worker preferences over start dates by
simply assuming the actual start date T is stochastic with a distribution F (T |t).
16
R1
neither. Specifically, there is fear whenever 0 (1 − σ(x))(1 − θx)dx ≤ 1 − σ(0), i.e. when
the stigma σ 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,
R1
there is greed iff 0 (1−σ(x))(1−θx)dx ≤ 1−θ. This holds when the stigma σ of early matching
is high relative to the firm demand θ. 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 gastroenterology market. 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 advances in 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 tightness θ. So the rush for workers occurs earlier by Proposition 6, and is
followed by a longer gradual play phase (left panel of Figure 6). This predicted shift is consistent
with the observed change in match timing reported in Figure 1 of NR2004.7
Next consider comparative statics in the interest rate r. Since the match payoff is logsubmodular 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-emption equilibrium, 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 loci Πi are constant in t; therefore, rush sizes are unaffected by the interest rate. The sorority
rush environment of Mongell and Roth (1991) is one of extreme urgency, and so corresponds to
a high interest rate. Given a low stigma of early matching and a tight market (for the best sororities), 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.
7
One can reconcile a tatonnement process playing out over several years, by assuming that early matching in the
current year leads to lower stigma in the next year. Specifically, if the ratio (1 − σ(x))/(1 − σ(y)) for x < y, falls in
response to earlier matching in the previous year, then a natural feedback mechanism emerges. The initial increase
in θ stochastically advances match timing, further increasing fear; 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 the asset and chooses the time t to
sell. Let Q(t) be the fraction of investors that have sold by time t. There is common knowledge
among these investors that the asset price is a bubble. As long as the bubble persists, the asset
price p(t|ξ) rises deterministically and smoothly in time t, but once the bubble bursts, the price
drops to the fundamental value, which we normalize to 0.
The bubble explodes once Q(t) exceeds a threshold κ(t + t0 ), where t0 is a random variable
with log-concave cdf F 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 that κ is log-concave, with κ′ (t + t0 ) < 0 and limt→∞ κ(t) = 0.8 In other words, the
burst chance is the probability 1 − F (τ (q, t)) that κ(t + t0 ) ≤ q, where τ (q, t) uniquely satisfies
κ(t + τ (q, t)) ≡ q. The expected stopping price, F (τ (q, t))p(t|ξ), is decreasing in q.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 performance relative to their peers. This imposes an extra cost to leaving a growing bubble early
relative to other investors. For a simple model of this peer effect, scale stopping payoffs by
1 + ρq, where ρ ≥ 0 measures relative performance concern.10 All told, the payoff from stopping
at time t as quantile q 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 in t
and q. With ρ = 0, the stopping payoff (11) is then monotonically decreasing in the quantile,
and Proposition 1 predicts either a pre-emption game for all quantiles, or a pre-emption game
preceded by a time t = 0 rush, or a unit mass rush at t = 0. But with a relatively strong concern
for performance, the stopping payoff initially rises in q, and thus the peak quantile q ∗ is interior.
In this case, there is fear at time t∗ (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 high ρ, the stopping
payoff exhibits greed at time t∗ (1), and so we predict an intensifying war of attrition, climaxing
8
By contrast, AB2003 assume a constant function κ, but that the bubble eventually bursts exogenously even with
no investor sales. Moreover, absent AB2003’s asymmetric information of t0 , if the threshold κ were constant in
time, players could perfectly infer the burst time Q(tκ ) = κ, and so strictly gain by stopping before tκ .
9
A rising price is tempered by the bursting chance in financial bubble models (Brunnermeier and Nagel, 2004).
10
When a fund does well relative to its peers, it often experiences cash inflows (Berk and Green, 2004). In particular, Brunnermeier and Nagel (2004) document that during the tech bubble of 1998-2000, funds that rode the bubble
longer 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 survives, the price is p(t|ξ). Since it is log-supermodular in (t, ξ), if ξ rises, then so does the rate pt /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 as p(t|ξ) = eξt p̂(t), i.e. let ξ = −r
and let p̂ 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 bubble deterministically grows until the
rational trader sales exceed a fixed threshold κ > 0. They show that if κ increases, then bubbles
last stochastically longer, and the price crashes are larger. Consider this exercise in our model.
Assume any two quantiles q2 > q1 . We found in §7.3 that the bubble survival chance F (τ (q, t))
is log-submodular in (q, t), so that F (τ (q2 , t))/F (τ (q1 , t)) falls in t. Since the threshold κ(t)
is decreasing in time, lower t is equivalent to an upward shift in the κ function. Altogether, an
upward shift in our κ function increases the bubble survival odds ratio F (τ (q2 , t))/F (τ (q1 , t)).
In other words, the stopping payoff (11) is log-supermodular in q and κ — so that greater κ corresponds to more greed. Proposition 6 then asserts that sales stochastically delay when κ 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 in q
and relative performance concerns ρ, greater ρ is qualitatively similar to greater κ.
7.4 Bank Runs
Bank runs are among the most fabled of rushes in economics. In the benchmark model of
Diamond and Dybvig (1983) [DD1983], these arise because banks make illiquid loans or investments, 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
11
Schleifer and Vishny (1997) find a related result in a model with noise traders. Their prices diverge from true
values, and this divergence increases in the level of noise trade. This acts like greater κ in our model, since prices
grow 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 plus an 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 bank run 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 fraction R be held in the safe asset as reserves.
The safe asset has log-concave discounted expected value p(t), satisfying p(0) = 1, p′ (0) > 0
and limt→∞ p(t) = 0. The present value of the risky asset is p(t)(1 − ζ), where the shock ζ ≤ 1
has twice differentiable cdf H(ζ|t) that is log-concave in ζ and t 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 withdraw αp(t), where the payout rate α < 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 fraction R in the
safe asset, while fraction 1 − R will be invested in the risky project. Altogether, the bank will pay
depositors as long as total withdrawals αqp(t) fall short of total bank assets p(t)(1 − ζ(1 − R)),
i.e. as long as ζ ≤ (1 − αq)/(1 − R). The stopping payoff to withdrawal at time t as quantile q
is:
u(t, q) = H ((1 − αq)/(1 − R)|t) αp(t)
(12)
Clearly, u(t, q) is decreasing in q, log-concave in t, and log-submodular (since H(ζ|t) is).
Since the stopping payoff (12) is weakly falling in q, 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 avoided iff fundamentals p(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 payoff u(t∗ (1), 1). When
12
13
As DD1983 admit, with a first period deposit choice, if depositors rationally anticipate a run, they avoid it.
Thadden (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 size q0 of the initial run
via:
q0−1
Z
q0
H((1 − αx)/(1 − R)|0)dx = H((1 − α)/(1 − R)|t∗ (1))p(t∗ (1))
(14)
0
Since the left hand side of (14) falls in q0 , the run shrinks in the peak asset value p(t∗ (1)) or in
the hazard rate H ′ /H of risky returns.
Appendix H establishes a log-submodular payoff interaction between the payout α 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 in the reserve ratio. The stopping
payoff is log-supermodular in (t, R), since H(ζ|t) log-supermodular, and log-supermodular in
(q, R) provided the elasticity ζH ′(ζ|t)/H(ζ|t) is weakly falling in ζ (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, players must be indifferent between stopping in the rush and during gradual play. Thus, we introduce the associated the initial rush locus
RP and the terminal rush locus RW , 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 rush Π0 (t) in the safe pre-emption
equilibrium, nor from stopping immediately after the peak rush Π1 (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 time t rush is locally
optimal if V0 (t, q) ≥ u(t, 0), and a terminal time t rush is locally optimal if V1 (t, q) ≥ u(t, 1).
If players can strictly gain from pre-empting any initial rush at t > 0, there is at most one
pre-emption equilibrium. Since u(t∗ (1), 1) equals the initial rush payoff by (15) and ut (0, t) > 0
14
Equivalently, the stochastic return 1 − ζ has an increasing generalized failure rate, a property satisfied by most
commonly used distributions (see Table 1 in Banciu and Mirchandani (2013)).
15
An increase in the reserve ratio increases the probability of being paid at the harvest time, but it also increases
the probability of being paid in any early run. Log-concavity of H is necessary, but not sufficient, for the former
effect 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 time t∗ (0). Let
its rush include the terminal quantiles [q̄W , 1] and occur at time t̄1 . Likewise, let q P and t0 be the
initial rush size and time in the unique safe pre-emption equilibrium, when it exists.
Lemma 1 (Rush Loci) Given no fear at t∗ (0), there exist t1 ≤ t, both in (t∗ (0), t̄1 ), such that
RW is a continuously increasing map from [t1 , t̄1 ] onto [0, q̄W ], with RW (t) < ΓW (t) on [t1 , t̄1 ),
and RW locally optimal exactly on [t, t̄1 ] ⊆ [t1 , t̄1 ]. With no greed at t∗ (1), no panic, and (16),
there exist t̄ ≤ t̄0 both in (t0 , t∗ (1)), such that RP is a continuously increasing map from [t0 , t̄0 ]
onto [q P , 1], with RP (t) > ΓP (t) on (t0 , t̄0 ], and RP (t) locally optimal exactly on [t0 , t̄] ⊆ [t0 , t̄0 ].
Figure 8 graphically depicts the message of this result, with rush loci starting at t0 and t̄1 .
We now construct two sets of candidate quantile functions: QW and QP . The set QW is
empty given fear at t∗ (0). Without fear at t∗ (0), QW contains all quantile functions Q such that
(i) Q(t) = 0 for t < t∗ (0), and for any t1 ∈ [t, t̄1 ]: (ii) Q(t) = ΓW (t) ∀t ∈ [t∗ (0), tW ] where tW
uniquely solves ΓW (tW ) = RW (t1 ); (iii) Q(t) = RW (t1 ) on (tW , t1 ); and (iv) Q(t) = 1 for all
t ≥ t1 . The set QP is empty if there is greed at t∗ (1) or panic. Given greed at t∗ (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 at t∗ (1), no panic, and inequality (16), then QP
contains all quantile functions Q such that (i) Q(t) = 0 for t < t0 , and for some t0 ∈ [t0 , t̄]:
(ii) Q(t) = RP (t0 ) ∀t ∈ [t0 , tP ) where tP solves ΓP (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 at t∗ (0), while QP is non-empty iff there is not greed at t∗ (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 of RW (RP ) to quantile functions in the candidate sets
QW (QP ); and (b) all quantile functions in QW (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 set QW .
As the rush time postpones, the rush shrinks, and the gradual play phase lengthens. The set of
pre-emption equilibria is the candidate set QP . 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
✻
✻
1
Pre-Emption Equilibria
No Alarm
1
✻
Pre-Emption Equilibria
Alarm
RP
RP
ΓP
ΓP
q0
Π1
ΓW
q1
t∗ (0)
RW
t1 t t1 t̄1
Π0
✲
t
Π0
✲
0
t0 t0 t̄ t̄0
t∗ (1) 0
✲
t̄
t̄0
t∗ (1)
Figure 8: All Nash Equilibria with Gradual Play. Left: All wars of attrition start at time t∗ (0),
and are given by Q(t) = ΓW (t): Any terminal rush time t1 ∈ [t, t̄1 ] determines a rush size
q1 = 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 time t0 ∈ [t0 , t̄],
followed by inaction on (t0 , Γ−1
P (q0 )), and then a slow pre-emption phase given by Q(t) = ΓP (t),
∗
ending at time t (1). Right: The set of pre-emption equilibria with alarm is constructed similarly,
but using the interval of allowable rush times [0, t̄].
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 the smallest 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 a problematic 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 payoffs u(t, q|ϕ) introduced in §6.
Proposition 5∗ Consider a harvest delay with ϕH > ϕL and no panic at ϕL . Then
(a) For all QL ∈ QW (ϕL ), there exists QH ∈ QW (ϕH ) such that: (i) QL (t) ≥ QH (t); (ii) The
rush for QH is later and no smaller than for QL ; (iii) Gradual play for QH starts later and ends
later than that for QL ; and (iv) Q′H (t) < Q′L (t) in the intersection of the gradual play intervals.
(b) For all QH ∈ QP (ϕH ), there exists QL ∈ QP (ϕL ) such that: (i) QL (t) ≥ QH (t); (ii) The
rush for QH is later and no larger than for QL ; (iii) Gradual play for QH starts later and ends
later than that for QL ; and (iv) Q′H (t) > Q′L (t) in the intersection of the gradual play intervals.
23
1
✻
War of Attrition Equilibria
1
✻
Pre-Emption Equilibria
No Alarm
RL
P
ΓL
P
ΓH
P
RH
P
Π1
RL
W
ΓL
W
ΓH
W
RH
W
Π0
✲
✲
0
t
Figure 9: Harvest Delay Revisited. With a harvest delay, from low L to high H, the gradual
play loci Γji = Γi (·|ϕj ) and rush loci Rji ≡ Ri (·|ϕj ) shift down, where i = W, P and j = L, H.
As argued in the proof of Proposition 5, ΓP shifts down and steepens in ϕ, ΓW shifts down and
flattens in ϕ, while the initial peak rush loci Π0 and Π1 are unchanged. Meanwhile, as shown in
Lemma G.2, the rush loci Ri (·|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 Figure 9.
We now extend our comparative statics for quantile changes to Nash equilibria.
Proposition 6∗ Consider an increase in greed with ϕH > ϕL and no panic at ϕL , then
(a) ∀QL ∈ QW (ϕL ), there exists QH ∈ QW (ϕH ) such that: (i) QL ≥ QH ; (ii) The rush for QH
is smaller and occurs later than that for QL ; (iii) Gradual play for QH starts later than that for
QL ; and (iv) Q′H (t) < Q′L (t) in the intersection of the gradual play intervals.
(b) No alarm at ϕL : ∀QH ∈ QP (ϕH ), there exists QL ∈ QP (ϕL ) such that: (i) QL ≥ QH ; (ii)
The rush for QH is no smaller and occurs later than that for QL ; (iii) Gradual play for QH starts
later than that for QL ; and (iv) Q′H (t) > Q′L (t) in the intersection of the gradual play intervals.
(c) Alarm at ϕL : ∀QH ∈ QP (ϕH ), there exists QL ∈ QP (ϕL ) such that: (i) QL ≥ QH ; (ii) The
rush for QH is no larger and occurs no later than that for QL ; (iii) Gradual play for QH starts
later than that for QL ; and (iv) Q′H (t) > Q′L (t) in the intersection of the gradual play intervals.
The gradual play and peak rush loci shift as in Figure 6, while the rush loci shift down as in
Figure 9. Altogether, Figure 10 illustrates the effect of an increase in greed (or decrease in fear).
9 Conclusion
We have developed a novel and unifying theory of large timing games that subsumes pre-emption
games and wars of attrition. Under a common assumption, that individuals have hump-shaped
24
1
✻
War of Attrition Equilibria
1
✻
Pre-Emption Equilibria
No Alarm
RL
P
ΓH
P
ΠH
1
ΓL
P
ΠL
1
ΓL
W
ΓH
W
RH
W
ΠH
0
ΠL
0
✲
✲
0
t
Figure 10: Quantile Changes Revisited. With an increase in greed or decrease in fear, from low
L to high H, the gradual play loci Γji = Γi (·|ϕj ) and rush loci Rji ≡ Ri (·|ϕj ) shift down, where
i = W, P and j = L, H, while the peak rush loci Π0 and Π1 shift up.
preferences over their stopping quantile, a rush is inevitable. When the game tilts toward rewarding 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 this rush 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 stopping 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 muse that 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 observationally distinct from risk preference.
25
q✻
1
War of Attrition Equilibrium
ΓP
ΓW
✻
❄
ΠH
1
✻
0
Pre-Emption Equilibrium
q✻
1
✲
ΠL
0
❄
ΠL
1
ΠH
0
✲
0
t
✲
✛
t
Figure 11: Geometric Payoff Transformations. Assume a payoff transformation u(t, q)β . The
(thick) gradual play locus is constant in β, while the (thin) peak rush locus shifts up in β for a
war of attrition equilibrium (right) and down in β for a pre-emption equilibrium (left).
To prove (♣), consider any C 2 transformation v(t, q) ≡ f (u(t, q)) with f ′ > 0. Then vt =
f ′ ut and vq = f ′ uq and vtq = f ′′ ut uq + f ′ utq . So vt vq − vvtq = [(f ′ )2 − f f ′′ ]ut uq − f f ′ utq yields
vt vq − vvtq = [(f ′ )2 − f f ′′ − f f ′ /u]utuq − f f ′ [utq − ut uq /u]
(17)
Since the term ut uq changes sign, given ut uq ≥ uutq , expression (17) is always nonnegative when
(f ′ )2 − f f ′′ − f f ′ /u = 0, which requires our geometric form f (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 transformation f , Jensen’s inequality implies:
Z
−1
f (u(t, Π0 )) = f (V0 (Π0 , t)) ≡ f Π0
Π0
0
Z
−1
u(t, x)dx ≤ Π0
Π0
f (u(t, x))dx
0
So to restore equality, the peak rush locus Π0 (t) must decrease. Finally, any two geometric
transformations with βH > βL are also related by a geometric transformation uβH = (uβL )βH /βL .
B Characterization of the Gradual Play and Peak Rush Loci
Lemma B.1 (Gradual Play Loci) If q ∗ > 0, there exists finite t̄W > t∗ (0) with ΓW : [t∗ (0), t̄W ] 7→
[0, q ∗ (t̄W )] well-defined, continuous, and increasing. If q ∗ < 1, there exists tP ∈ [0, t∗ (1)) with
ΓP well-defined, continuous and increasing on [tP , t∗ (1)], where ΓP ([tP , t∗ (1)]) = [q ∗ (tP ), 1]
when u(0, q ∗(0)) ≤ u(t∗ (1), 1), and otherwise ΓP ([0, t∗ (1)]) = [q̄, 1] for some q̄ ∈ (q ∗ (0), 1].
26
S TEP 1: ΓW C ONSTRUCTION . First, there exists finite t̄W > t∗ (0) such that u(t̄W , q ∗ (t̄W )) =
u(t∗ (0), 0). For q ∗ > 0 implies u(t∗ (0), q ∗ (t∗ (0))) > u(t∗ (0), 0), while (1) asserts the opposite
inequality for t sufficiently large: existence of t̄W then follows from continuity of u(t, q ∗ (t)).
Next, since ut < 0 for all t > t∗ (0), we have u(t, 0) < u(t∗ (0), 0) and u(t, q ∗ (t̄W )) > u(t∗ (0), 0),
there exists a unique ΓW (t) ∈ [0, q ∗ (t̄W )] satisfying (4) for all t ∈ [t∗ (0), t̄W ]. Since uq > 0,
ut < 0 on (t∗ (0), t̄W ] × [0, q ∗ (t̄W )], and u is c2 , Γ′W (t) > 0 by the Implicit Function Theorem.
S TEP 2: ΓP C ONSTRUCTION . First assume u(0, q ∗ (0)) ≤ u(t∗ (1), 1). Then q ∗ (t) < 1 ⇒
u(t∗ (1), q ∗ (t∗ (1))) > u(t∗ (1), 1), while the continuous function u(t, q ∗(t)) is strictly increasing
in t ≤ t∗ (1). So there exists a unique tP ∈ [0, t∗ (1)) such that u(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) and ut (t, q) > 0 for all
t < t∗ (1) ≤ t∗ (q). In sum, there is a unique ΓP (t) ∈ (q ∗ (t), 1) solving (5), for t ∈ (tP , t∗ (1)). If
instead, the reverse inequality u(0, q ∗ (0)) > u(t∗ (1), 1) holds, then u(t, q ∗(t)) ≥ u(0, q ∗ (0)) >
u(t∗ (1), 1) > u(t, 1), for all t ≤ t∗ (1). Again by ut > 0, there is a unique ΓP (t) ∈ (q ∗ (t), 1]
satisfying (5) for all t ∈ [0, t∗ (1)], i.e. tP = 0. All told, ΓP (t) ≥ q ∗ (t), so that uq (t, ΓP (t)) <
0 < ut (t, ΓP (t)), while u is C 2 , so that Γ′P (t) > 0 by the Implicit Function Theorem.
Lemma B.2 (Peak Rush Loci) Absent greed at time t∗ (1), the peak rush locus Π0 : [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 time t∗ (0), the peak rush locus Π1 : [t∗ (0), ∞) 7→ (q ∗ (t), 1) is well-defined,
continuous, and non-increasing. Moreover, ∂V1 (t, q)/∂q ≷ 0 as q ≶ Π1 (t).
Proof: We focus on Π0 ; the peak terminal rush Π1 admits similar reasoning.
S TEP 1: G REED
AND
F EAR
OBEY
S INGLE C ROSSING . Since u(t, q) is log-submodular, the
ratio u(t, y)/u(t, x) is non-increasing in t for all y ≥ x. So if there is no greed at t∗ (1), then there
is no greed at any t ≤ t∗ (1), and if there is no fear at t∗ (0), then there is no fear at any t ≥ t∗ (0).
S TEP 2: Π0 IS WELL - DEFINED AND CONTINUOUS . Since u is log-concave with peak quantile
q ∗ (t) ∈ (0, 1), we have uq (t, q) ≷ 0 as q ≶ q ∗ (t). Thus, for sufficiently small q > 0, the stopping
payoff u exceeds the average V0 . So by continuity, a solution Π0 (t) ∈ (0, 1) to (6) exists iff
u(t, 1) < V0 (t, 1) (i.e. no greed at t). But since uq (t, q) ≷ 0 as q ≶ q ∗ (t), there can be at
most one q ∈ (0, 1) where the payoff u and associated average V0 coincide, necessarily at the
unique global maximizer V0 , i.e. Π0 (t) = arg maxq V0 (t, q) with ∂V0 (t, q)/∂q ≷ 0 as q ≶ Π0 (t).
Continuity of Π0 (t) follows from the Implicit Function Theorem.
Define I(q, x) ≡ q −1 for x ≤ q and 0 otherwise, and ℓ ≡
R1
t∗ (1) − t, and thus V0 (t∗ (1) − ℓ, q) = 0 I(q, x)u(t∗ (1) − ℓ, x)dx. Easily, I is log-supermodular in
(q, x), and so the product I(·)u(·) is log-supermodular in (q, x, ℓ). Thus, V0 is log-supermodular
S TEP 3: Π0
IS NON - INCREASING .
27
in (q, ℓ) since log-supermodularity is preserved by integration by Karlin and Rinott (1980). So
the peak rush locus Π0 (t∗ (1) − ℓ) = arg maxq V0 (t∗ (1) − t, q) rises in ℓ, i.e. falls in t.
C A Safe Equilibrium with Alarm
Assuming no greed at t∗ (1) and alarm, we construct a quantile function with a rush at t = 0
and a pre-emption phase. To this end, let quantile qA ∈ (Π(0), 1) be the largest solution to
V0 (0, qA ) = u(t∗ (1), 1). First, Π(0) = arg maxq 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 from V0 (0, q) continuously decreasing in q > Π(0) (Lemma B.2). Then,
given qA , define ΓP (tA ) = qA . To see that such a time tA ∈ (0, t∗ (1)) uniquely exists, observe
that u(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 endpoints ΓP (t∗ (1)) = 1
and ΓP (0) < qA , where this latter inequality follows from qA > Π(0) ⇒ u(0, qA ) < V0 (0, qA ) =
u(t∗ (1), 1) = u(0, ΓP (0)). Finally, define the candidate quantile function QA 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 at t∗ (1). Then QA is a safe Nash equilibrium. It is the
unique equilibrium with a t = 0 rush, one gradual play phase ending at t∗ (1), and no other rush.
P ROOF : The quantile function QA is safe since its support is {0} ∪ [tA , t∗ (1)]. By construction,
the stopping payoff is u(t∗ (1), 1) on the support of QA . The payoff on the inaction region (0, tA )
is strictly lower, since u(tA , qA ) = u(t∗ (1), 1) by construction and u(0, qA ) < u(t∗ (1), 1) by
ut (t, qA ) > 0 for t < t∗ (1) ≤ t∗ (qA ). Finally, since ut (t, 1) < 0 for t > t∗ (1), no player can gain
from stopping after t∗ (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 is u(t∗ (1), 1), and the unique time t = 0 rush size with this stopping value
is qA . Given qA , the time tA at which the pre-emption game begins follows uniquely from ΓP ,
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 at t∗ (0), satisfying (4), which uniquely exists by Lemma B.1 (and q ∗ = 1).
To see that this is an equilibrium, observe that no agent can gain by pre-empting gradual play,
28
since t∗ (0) maximizes u(t, 0). Further, since t∗ (q) is decreasing, we have ut (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 at t > 0 and that all equilibria
involving gradual play have expected payoff u(t∗ (1), 1).
S TEP 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 rush at t = 0. Now assume
panic, but gradual play, necessarily with expected payoff u(t∗ (1), 1). The payoff for stopping at
t = 0 is either u(0, 0) (if no rush occurs at t = 0) or V0 (0, q), given a rush of size q < 1. But
since uq < 0, we have u(0, 0) > V0 (0, q) > V0 (0, 1) ≥ u(t∗ (1), 1) (by panic), a contradiction.
S TEP 2: E QUILIBRIUM WITH A LARM . First we claim that alarm implies a t = 0 rush. Instead,
assume alarm and gradual play for all q. As in Step 1, the expected value of the proposed equilibrium must be u(t∗ (1), 1). Given uq < 0 we have u(0, 0) = maxq V0 (0, q), which strictly exceeds
u(t∗ (1), 1) by alarm; therefore deviating to t = 0 results in a strictly higher payoff, contradicting
gradual play for all quantiles. But by Step 1, any equilibrium with alarm must also include gradual play, ending at t∗ (1). Altogether, with alarm we must have a t = 0 rush, followed by gradual
play ending at t∗ (1). By Lemma C.1, there is a unique such equilibrium: QA .
S TEP 3: E QUILIBRIUM WITH N O A LARM 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 payoff u(t∗ (1), 1). But then given uq < 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 payoff u(t∗ (1), 1), a contradiction. Given
gradual play for all quantiles ending at t∗ (1), there exists a unique equilibrium. Indeed, absent
alarm u(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 locus ΓP is uniquely defined by (5). Thus, the unique candidate
equilibrium is Q(t) = 0 on [0, tP ); Q(t) = ΓP (t) on [tP , t∗ (1)]; and, Q(t) = 1 for t > t∗ (1).
Since ut < 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 phase ending at t∗ (1), a terminal rush
29
preceded by an uninterrupted war of attrition phase starting at t∗ (0), or a unit mass rush.
S TEP 1: RUSH N ECESSITY. Assume gradual play for all Q, starting at t0 . Since q ∗ (t0 ) > 0,
we have uq (t0 , 0) > 0 ⇒ ut (t0 , 0) < 0 (by (2)), i.e. t0 > t∗ (0). But then by t∗ non increasing,
ut (t, Q(t)) < 0 on the support of Q, and by (2) uq (t, Q(t)) > 0, i.e. Q(t) < q ∗ (t) for all t. Since
q ∗ non increasing, we have Q(t) < q ∗ (t) ≤ q ∗ (1) < 1 for all t ∈ supp(Q), which is impossible.
S TEP 2: AT M OST O NE RUSH . Assume rushes at t1 and t2 > t1 . Since u strictly falls after the
peak quantile, we must have Q(t2 −) < q ∗ (t2 ) else players can strictly gain from preempting the
rush. Likewise, to avoid a strict gain from post-empting the rush at t1 , we need Q(t1 ) > q ∗ (t1 ).
Altogether, q ∗ (t1 ) < Q(t1 ) ≤ Q(t2 −) < q ∗ (t2 ), which violates q ∗ weakly decreasing.
S TEP 3: S TOPPING ENDS IN A RUSH OR WITH GRADUAL PLAY AT t∗ (1). Assume gradual
play ends at t 6= t∗ (1). If t < t∗ (1) then quantile 1 benefits from deviating to t∗ (1). If instead,
t > t∗ (1), then we have ut (t, 1) < 0 ⇒ uq (t, 1) > 0, which violates q ∗ < 1.
S TEP 4: S TOPPING BEGINS IN A RUSH OR WITH GRADUAL PLAY AT t∗ (0). On the contrary,
assume gradual play begins at t 6= t∗ (0). If t > t∗ (0) then quantile 0 profits by deviating to t∗ (0).
If instead, t < t∗ (0), then ut (t, 0) > 0 ⇒ uq (t, 0) < 0, violating q ∗ > 0.
S TEP 5: N O I NTERIOR Q UANTILE RUSH . Assume a rush at t > 0, i.e. 0 < Q(t−) < Q(t) < 1.
By Step 2, all other quantiles must stop in gradual play. Then by Steps 3 and 4 we must have
Q(t∗ (0)) = 0 and Q(t∗ (1)) = 1, but this violates Q weakly increasing and t∗ weakly decreasing.
S TEP 6: O NLY O NE G RADUAL P LAY P HASE . Assume gradual play on [t1 , t2 ] < [t3 , t4 ], satisfying 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 since t∗ is non-increasing,
ut (t, Q(t)) < 0 for all t > t1 . But then u(t2 , Q(t2 )) > u(t3 , Q(t3 )), contradicting optimal stopping at t2 and t3 . If instead, stopping begins in a rush, then by step 3, t4 ≤ t∗ (1), and since t∗ is
non-increasing, we must have ut (t, Q(t)) > 0 for all t < t4 . But then u(t2 , Q(t2 )) < u(t3 , Q(t3 )),
again contradicting stopping at t2 and t3 .
Lemma E.2 Greed at t∗ (1) rules out pre-emption, while fear at t∗ (0) rules out wars of attrition.
By Lemma E.1 any pre-emption phase must end at t∗ (1), implying Nash payoff w̄ = u(t∗ (1), 1).
Also, since t∗ is non-increasing, ut (t, q) > 0 for all (t, q) < (t∗ (1), 1); and thus, w̄ is strictly beR1
R1
low the average payoff at t∗ (1) : 0 u(t∗ (1), x)dx. Altogether, w̄ = u(t∗ (1), 1) < 0 u(t∗ (1), x)dx,
which contradicts greed at t∗ (1). By similar logic, fear at t∗ (0) is inconsistent with the Nash payoff u(t∗ (0), 0) in any war of attrition starting at t∗ (0).
30
E.2 Existence and Uniqueness of Safe Equilibria: Proof of Proposition 3. We assume no
greed at t∗ (1). The case for wars of attrition with no fear at t∗ (0) follows the logic of Step 3.
S TEP 1: PANIC . By definition a t = 0 unit mass rush is safe, and is an equilibrium iff panic obtains. For without panic, V0 (0, 1) < u(t∗ (1), 1), and deviating to t∗ (1) offers a strict improvement
over stopping in the t = 0 rush, while panic implies V0 (0, 1) ≥ u(t∗ (1), 1) ≡ maxt u(t, 1).
We claim that with panic a unit rush at t = 0 is the unique safe equilibrium with an initial
rush. By definition, unit rushes at t > 0 are not safe, while by Lemma E.1 any equilibrium
with an initial rush of size qR < 1 ends with gradual play at t∗ (1), implying expected payoff
u(t∗ (1), 1). Players will only stop in the rush if the rush payoff V0 (t, qR ) weakly exceeds the
adjacent gradual play payoff u(t, qR ), i.e. that V0 (t, q) is non-increasing in q at qR , and thus
qR ≥ Π0 (t) = arg maxq V0 (t, q) by Lemma B.2. But since V0 (t, q) is falling in q > Π0 (t) (by
Lemma B.2) and increasing in t ≤ t∗ (1), we have V0 (t, qR ) > V0 (t, 1) ≥ V0 (0, 1) ≥ u(t∗ (1), 1).
Altogether, with panic V0 (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.
S TEP 2: A LARM . Assume alarm. Step 1 rules out a unit rush at t = 0, while a unit rush at t > 0
is not safe by definition. Then given Lemma E.1, any equilibrium with an initial rush must end
with gradual play at t∗ (1). By Lemma C.1, there exists one such safe equilibrium with an initial
rush at t = 0. We claim that any equilibrium with an initial rush at t > 0 is not safe. First,
given an initial rush at t > 0, safety requires that gradual play must immediately follow the rush.
Thus, any initial rush at time t of size q must be on both the gradual play locus (5) and peak rush
locus (6), i.e. q = ΓP (t) = Π0 (t), which implies u(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.
S TEP 3: N O A LARM OR PANIC . By Lemma E.1 pre-emption equilibria begin with an initial rush, followed by gradual play ending at t∗ (1), and thus have expected payoff u(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) yields ΓP (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 that t be on
both the gradual play (5) and peak rush locus (6), i.e. ΓP (t) = Π0 (t), but ΓP (t) is increasing and
Π0 (t) non-increasing on [0, t∗ (1)]: tP = 0 and q P = Π0 (0) is the only possible safe initial rush.
Now assume (3) is strict, which trivially rules out a t = 0 rush, since the maximum rush payoff
falls short of the gradual play payoff u(t∗ (1), 1). Given a rush at t > 0, safety again requires that
31
the rush time must satisfy ΓP (t) = Π0 (t), which we claim uniquely defines a rush time tP ∈
(0, t∗ (1)) and rush size q P ∈ (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
discover ΓP (tP ) = q ∗ (tP ) < Π0 (tP ), ΓP (t∗ (1)) = 1 > Π0 (t∗ (1)), while ΓP is increasing and Π0
is non-increasing on [tP , t∗ (1)): There exists a unique solution (tP , qP ) ∈ (tP , t∗ (1))×(q ∗ (tP ), 1)
satisfying q P = ΓP (tP ) = Π0 (tP ). For the second case, we assume the opposite u(0, q ∗(0)) >
u(t∗ (1), 1), then combine Lemma B.1 part (b) and Lemma B.2 to see that ΓP is increasing and Π0
non-increasing on [0, t∗ (1)], again with ΓP (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 both Π0 and ΓP exceed q ∗ (0), where uq < 0.
In all cases, there is a unique safe pre-emption equilibrium candidate: (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). Since ut < 0 for t > t∗ (1),
no player can gain from delaying until after the gradual play phase. To see that no player can
gain by pre-empting the rush, note that ut > 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).
S TEP 1: N O U NIT RUSHES WITH S TRICT F EAR OR G REED Assume a unit mass rush at time
tR > 0. Given strict greed at time tR , 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.
S TEP 2: T HE U NIT RUSH T IME I NTERVAL [T , T̄ ]. Since fear satisfies strict single crossing
(Lemma B.2 Step 1) no fear at t∗ (0) implies ∃T̄1 > t∗ (0) such that no fear obtains for all t ≤ T̄1 .
Likewise no greed at t∗ (1) implies ∃T 1 < t∗ (1) such that no greed holds for all t ≥ T 1 . Since
t∗ (1) ≤ t∗ (0), we have T 1 < T̄ ; thus, both no fear and no greed hold on [T 1 , T̄1 ]. Given t∗
non-increasing, V0 (t, 1) is strictly increasing for t < t∗ (1). Thus, no greed at t∗ (1) implies
∃T 2 ∈ [0, t∗ (1)), such that u(t∗ (1), 1) ≤ V0 (t, 1) for all t ∈ [T 2 , t∗ (1)]. Likewise, V0 (t, 1) is
strictly decreasing for all t ≥ t∗ (0) with limt→∞ V0 (t, 1) < u(t∗ (0), 0) (by inequality (1)). Thus,
if no fear obtains at t∗ (0), ∃T̄2 > t∗ (0), such that V0 (t, 1) ≥ u(t∗ (0), 0) for all t ∈ [t∗ (0), T̄2 ].
Finally, let T = max{T 1 , T 2 } and T̄ = min{T̄1 , T̄2 }.
S TEP 3: V ERIFICATION OF U NIT RUSH E QUILIBRIA . If a unit mass rush occurs at tR ∈
[T , t∗ (1)), then the stopping payoff rises until the rush, and no fear at tR , 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 rush iff 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 a flat payoff on an interval. In the
middle, 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 at tR , u(tR , 1) ≤
V0 (tR , 1) ensures post-empting the rush is not optimal. Finally, if tR > t∗ (0), the stopping payoff
rises until t∗ (0) and falls between t∗ (0) and the rush. Thus, u(t∗ (0), 0) ≤ V0 (tR , 1) rules out
pre-empting the rush, which holds for all t ∈ (t∗ (0), T̄ ] by Step 2. The stopping payoff falls after
the rush; and so, no greed at tR , u(tR , 1) ≤ V0 (tR , 1) rules out post-empting the rush.
F Safe is Equivalent to Secure: Proof of Theorem 1
S TEP 1: S AFE ⇒ S ECURE . Trivially, any rush at t = 0 is secure. Now, assume an interval [ta , tb ]
of gradual play with constant stopping payoff π̂. So for any ε′ < (tb − ta )/2 and any t ∈ [ta , tb ],
one of the two intervals [t, t + ε′ ) or (t − ε′ , t] will be contained in [ta , tb ] and thus obtain payoff
π̂. Security is maintained by adding a rush with payoff π̂ at ta or tb .
S TEP 2: S ECURE ⇒ S AFE . 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 Lemma E.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 q̂ ∈ (0, 1] at time t̂ ∈ (0, t∗ (1)], necessarily with Q(t) = 0 for all t < t̂. If this
equilibrium is not safe, then Q(t) = q̂ on an interval following t̂. Since this is an equilibrium,
V0 (t̂, q̂) ≥ u(t̂, 0). Altogether, inf s∈(t̂−ε,t̂] w(s; Q) = inf s∈(t̂−ε,t̂] u(s, 0) < V0 (t̂, q̂) = w(t̂; Q) for
all ε ∈ (0, t̂), where the strict inequality follows from ut (t, q) > 0 for all t < t̂ ≤ t∗ (1) ≤ t∗ (q).
Now consider an interval following the rush [t̂, t̂ + ε). If q̂ < 1, gradual play follows the rush
after delay ∆ > 0, and V0 (t̂, q̂) = u(t̂ + ∆, q̂). But, since t + ∆ < t∗ (1) we have ut (t, q̂) > 0 during the delay, and w(t; Q) = u(t, q̂) < V0 (t̂, q̂) for all t ∈ (t̂, t̂ + ∆). Thus, inf s∈[t̂,t̂+ε) w(s; Q) <
33
w(t̂; Q) for all ε ∈ (0, ∆). Now assume q̂ = 1 and consider the two cases t̂ < t∗ (1) and t̂ = t∗ (1).
If t̂ < t∗ (1), then V0 (t̂, 1) > u(t̂, 1), else stopping at t∗ (1) is strictly optimal. But then by continuity, there exists δ > 0 such that w(t̂; Q) = V0 (t̂, 1) > u(t, 1) = w(t; Q) for all t ∈ (t̂, t̂ + δ).
If t̂ = t∗ , equilibrium requires the weaker condition V0 (t∗ (1), 1) ≥ u(t∗ (1), 1), but then we have
ut (t, 1) < 0 for all t > t̂; and so, w(t̂; Q) = V0 (t̂, 1) > u(t, 1) = w(t; Q) for all t > t̂.
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.
S TEP 1: P RELIMINARIES . First we claim that:
Z q
−1
V0 (t, q|ϕ) ≥ u(t, q|ϕ) ⇒ q
ut (t, x|ϕ)dx ≥ ut (t, q|ϕ)
(18)
0
Indeed, using u(t, q|ϕ) log-submodular in (t, q):
1
q
Z
q
0
ut (t, x|ϕ)
1
dx =
u(t, q|ϕ)
q
Z
q
0
ut (t, x|ϕ) u(t, x|ϕ)
ut (t, q|ϕ)
dx ≥
u(t, x|ϕ) u(t, q|ϕ)
qu(t, q|ϕ)
Z
0
q
u(t, x|ϕ)
ut (t, q|ϕ)
dx ≥
u(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), and V(t, q, ϕ) =
Rq
q −1 0 ν(t, x, ϕ)dx. By u log-modular in (t, ϕ), νϕ∗ = 0, while νt∗ ≶ 0 as ut ≷ 0. By logsubmodularity in (t, q) and log-supermodularity in (q, ϕ), νt > 0 and νϕ < 0.
S TEP 2: Vt ≥ νt
Vt − νt
AND
−Vϕ > −νϕ
FOR ALL
(t, q) SATISFYING (6), I . E . q = Π0 (t).
Z q
ut (t, x|ϕ) u(t, x|ϕ)ut (t, 1|ϕ)
ut (t, q|ϕ) u(t, q|ϕ)ut(t, 1|ϕ)
= q
−
dx −
−
u(t, 1|ϕ)
u(t, 1|ϕ)2
u(t, 1|ϕ)
u(t, 1|ϕ)2
0
Z q
u(t, q|ϕ)ut(t, 1|ϕ)
u(t, x|ϕ)ut(t, 1|ϕ)
dx +
by (18)
≥ −q −1
2
u(t, 1|ϕ)
u(t, 1|ϕ)2
0
Z q
ut (t, 1|ϕ)
−1
=
u(t, x|ϕ)dx = 0
by (6)
u(t, q|ϕ) − q
u(t, 1|ϕ)2
0
−1
Since u is strictly log-supermodular in (q, ϕ) symmetric steps establish that −Vϕ > −νϕ .
S TEP 3: A D IFFERENCE
IN
D ERVIATIVES . Lemma B.1 proved Γ′P (t) > 0, Lemma B.2 es-
tablished Π′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 in ϕ by proving that starting from
34
any (t, q, ϕ), satisfying q = ΓP (t) = Π(t) and holding q fixed, the change dt/dϕ in the gradual
play locus (5) is smaller than the dt/dϕ in the peak rush locus (6). Evaluating both derivatives,
this entails:
−νϕ (t, q, ϕ)
νϕ (t, q, ϕ) − Vϕ (t, q, ϕ)
>
Vt (t, q, ϕ) − νt (t, q, ϕ)
νt (t, q, ϕ) − νt∗ (t)
(19)
Since ut > 0 during a pre-emption game, we have νt∗ < 0, νt > 0, and νϕ < 0 by Step 1; so that
inequality (19) is satisfied if νt (νϕ − Vϕ ) > −νϕ (Vt − νt ) ⇔ −Vϕ νt > −νϕ Vt , 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 alarm RP (0) falls.
P ROOF : Rewriting (15), we see that any initial rush is defined by the indifference equation:
RP (t)
−1
Z
0
RP (t)
u(t, x|ϕ)
u(t∗ (1|ϕ), 1|ϕ)
dx =
u(t, 1|ϕ)
u(t, 1|ϕ)
(20)
Since the initial rush includes the peak quantile, the LHS of (20) falls in RP (t), while the LHS
falls in ϕ by log-supermodularity of u in (q, ϕ). Now, (20) shares the RHS of (8), shown increasing 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 payoff u(t, q) ≡ (1 + ρq)F (τ (q, t))p(t|ξ) in (11) is log-submodular in (t, q), and
log-concave in t and q.
Proof: That κ(t + τ (q, t)) ≡ q yields κ′ (t+τ (q, t))(1+τt (q, t)) = 0 and κ′ (t+τ (q, t))τq (q, t) = 1.
So, τt ≡ −1 and τq < 0 given κ′ < 0. Hence, τtq = 0, τtt = 0, and τqq = −(κ′′ /κ′ )(τq )2 . Thus,
∂ 2 log(F (τ (q, t)))
F (τ (q, t))2 = F F ′′ − (F ′ )2 τt τq + F F ′τtq = F F ′′ − (F ′ )2 τt τq ≤ 0
∂t∂q
Twice differentiating log(F (τ (q, t))) in t likewise yields [F F ′′ − (F ′ )2 ]/F 2 ≤ 0. Similarly,
∂ 2 log(F (τ (q, t)))/∂q 2 = (τq )2 [F F ′′ − (F ′ )2 − (κ′′ /κ′ )F F ′]/F 2 ≤ 0
where −κ′′ /κ′ ≤ 0 follows since κ 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 elasticity ζH ′(ζ|t)/H(ζ|t) is weakly falling in ζ.
P ROOF : By Lemma 2.6.4 in Topkis (1998), u is log-submodular in (q, α), as H is monotone and
log-concave in ζ, and 1 − αq is monotone and submodular in (q, α). It is log-supermodular in
(q, R):
1 − αq
∂ 2 log(H(·))
2
2
(H ′ )2 − HH ′′ − HH ′ ≥ 0
(21)
H(·) (1 − R) =
∂q∂R
1−R
i.e. x(H ′ (x)2 − H(x)H ′′ (x)) − H(x)H ′ (x) ≥ 0, namely, with xH ′ (x)/H(x) weakly falling. I All Nash Equilibria: Omitted Proofs (§8)
Proof of Lemma 1.
We focus on RP , and thus assume no greed at t∗ (1) and no panic.
S TEP 1: RP ([t0 , t̄0 ]) = [qP , 1] IS
CONTINUOUS , INCREASING AND EXCEEDS
ΓP .
By Propositions 2 and 3, the unique safe initial rush (t0 , q P ) satisfies (15). And since any equilibrium initial rush includes the peak of V0 (Corollary 1) with V0 falling after this peak (Lemma B.2),
q P must be the largest such solution at t0 , i.e. RP (t0 ) = q P . Now, for the upper endpoint t̄0 , combine the inequalities for no greed at t∗ (1) and no panic: V0 (0, 1) < u(t∗ (1), 1) < V0 (t∗ (1), 1),
which along with V0 (t, 1) continuously increasing for t < t∗ (1) yields a unique t̄0 < t∗ (1) satisfying V0 (t̄0 , 1) = u(t∗ (1), 1). That is, RP (t̄0 ) = 1. Combining RP (t0 ) = q P < 1 = R(t̄0 ),
with V0 (t, q) smoothly increasing in t ≤ t∗ (1) and smoothly decreasing in q ≥ q P , we discover:
(i) t0 < t̄0 ; as well as, (ii) the two inequalities V0 (t, qP ) > u(t∗ (1), 1) and V0 (t, 1) < u(t∗ (1), 1)
for all t ∈ (t0 , t̄0 ). So, by V0 (t, q) smoothly increasing in t ≤ t∗ (1) and smoothly decreasing in
q ≥ q P , the largest solution RP to (15) uniquely exists for all t, and is continuously increasing
from [t0 , t̄0 ] onto [q P , 1] by the Implicit Function Theorem.
We claim that RP (t) > ΓP (t) on (t0 , t̄0 ]. First, V0 (t0 , q P ) ≥ u(t0 , q P ), else players would not
stop in the safe rush (t0 , q P ). Combining this inequality with V0 (t, q) ≥ u(t, q) for q ≥ Π0 (t) by
Lemma B.2, we find RP (t0 ) ≥ Π0 (t0 ). But then since RP is increasing and Π0 non-increasing
(Lemma B.2), we have RP (t) > Π0 (t) on (t0 , t̄0 ]; and thus u(t, RP (t)) < V0 (t, RP (t)) =
u(t∗ (1), 1) by Lemma B.2 and (15). Altogether, given (5), we have u(t, RP (t)) < u(t, ΓP (t));
and thus, RP (t) > ΓP (t) by uq (t, q) < 0 for q > q ∗ (t).
S TEP 2: L OCAL O PTIMALITY. Formally, the candidate initial rush RP (t) is locally optimal iff :
V0 (t, RP (t)) ≥ max{u(t, 0), u(t, RP (t))}
36
(22)
Step 1 established V0 (t, RP (t)) ≥ u(t, RP (t)) on [t0 , t̄0 ]. When inequality (16) holds, we trivially
have V0 (0, RP (0)) ≥ u(0, 0): a t = 0 rush of size RP (0) is locally optimal. Thus, we henceforth
assume that the lower bound on the domain of RP 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 obeys RP (t0 ) = arg maxq V0 (t0 , q), but then V0 (t0 , RP (t0 )) > u(t0 , 0). Now, since
V0 (t, RP (t)) is constant in t on [t0 , t̄0 ] by assumption (15) and u(t, 0) is increasing in t on this
domain, we either have V0 (t, RP (t)) ≥ u(t, 0) for all t ∈ [t0 , t̄0 ], in which case we set t̄ ≡ t̄0 or
there exists t̄ < t̄0 such that V0 (t, RP (t)) T u(t, 0) as t S t̄ for t ∈ [t0 , t̄0 ]. In either case RP (t)
satisfies (22) for all t ∈ [t0 , t̄], but for any t ∈ (t̄, t̄0 ] inequality (22) is violated.
Proof of Proposition 7.
Let QN P be the set of pre-emption equilibria. With greed at t∗ (1) or
panic, QN P , QP = ∅ by Proposition 2 and assumption. So assume no greed at t∗ (1) and no panic.
S TEP 1: QN P ⊆ QP . By Proposition 2, pre-emption equilibria share value u(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 size q = RP (t) at a time t ∈ [t0 , t̄0 ]. 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 defines ΓP . Finally, by Lemma 1, the interval [t0 , t̄] are
the only times for which RP (t) is locally optimal.
S TEP 2: QP ⊆ QN P . Recall that w(t; Q) is the payoff to stopping at time t ≥ 0 given quantile
function Q. Let QS ∈ QP be the unique safe equilibrium and consider an arbitrary Q ∈ QP
with an initial rush at t0 , inaction on (t0 , tP ) and gradual play following ΓP (t) on [tP , 1]. By
construction the stopping payoff is u(t∗ (1), 1) for all t ∈ supp(Q). Further, since QS is an
equilibrium and Q(t) = QS (t) on [0, t0 ) and [tP , ∞), we have w(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 thus V0 (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. Since u(t, 0) is increasing on this interval, no player can gain from pre-empting the
rush at t0 provided V0 (t0 , RP (t0 )) > u(t0 , 0), which is ensured by local optimality (22).
S TEP 3: C OVARIATE P REDICTIONS . Consider Q1 , Q2 ∈ QP , with rush times t1 < t2 . The rush
sizes obey RP (t1 ) < RP (t2 ) by RP increasing (Lemma 1). Gradual play start times are ordered
−1
−1
Γ−1
P (RP (t1 )) < ΓP (RP (t2 )) by ΓP increasing (Lemma B.1). Thus, gradual play durations obey
−1
∗
t∗ (1) − Γ−1
P (RP (t1 )) > t (1) − ΓP (RP (t2 )). Finally, by construction Q1 (t) = Q2 (t) = ΓP (t)
∗
on the intersection of the gradual play intervals [Γ−1
P (RP (t2 )), t (1)].
37
Lemma I.1 Assume a harvest delay or increase in greed ϕH > ϕL , with qH = RP (tH |ϕH )
locally optimal. If tL = R−1
P (qH |ϕL ) ≥ t0 (ϕL ), then RP (tL |ϕL ) is locally optimal.
S TEP 1: H ARVEST D ELAY. If qH = RP (tH |ϕH ) satisfies local optimality (22), then:
1≤
Z
0
qH
u(tH , x|ϕH )
dx =
qH u(tH , 0|ϕH )
Z
0
qH
u(tH , x|ϕL )
dx ≤
qH u(tH , 0|ϕL )
Z
0
qH
V0 (tL , qH |ϕL )
u(tL , x|ϕL )
dx =
qH u(tL , 0|ϕL )
u(tL , 0|ϕL )
where the first equality follows from log-modularity in (q, ϕ) and the inequality owes to u
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 of RP (tL |ϕL ), as shown in Lemma 1 Step 2.
S TEP 2: I NCREASE IN G REED . By Lemma 1 Step 2, RP (t|ϕ) is locally optimal for t ∈
[t0 (ϕ), t̄0 (ϕ)] iff u(t, 0|ϕ) ≤ V0 (t, RP (t|ϕ)|ϕ). Given u(t, 0|ϕ) increasing in t ≤ t̄0 (ϕ) and
V0 (t, RP (t|ϕ)|ϕ) constant in t by (15), if the largest locally optimal time t̄(ϕ) < t̄0 (ϕ)), it solves:
V̄ (t̄(ϕ), RP (t̄(ϕ)|ϕ), ϕ) = 1,
where V̄ (t, q, ϕ) ≡
V0 (t, q|ϕ)
u(t, 0|ϕ)
(23)
By assumption, tL ≥ t0 (ϕL ). For a contradiction, assume tL is not locally optimal: tL > t̄(ϕL ).
We claim that starting from any (t̄, q, ϕ) satisfying both (15), i.e. q = RP (t̄|ϕ) and (23), i.e.
V̄ (t̄, q, ϕ) = 1, the change in the rush locus dRP−1 (q|ϕ)/dϕ, holding q fixed, exceeds the change
along (23) dt̄/dϕ, holding q fixed. Indeed, defining h(t, q, ϕ) ≡ u(t∗ (1), 1|ϕ)/u(t, 0|ϕ) and
differentiating, we discover dRP−1 (q|ϕ)/dϕ − dt̄/dϕ = V̄ϕ /(ht − V̄t ) − V̄ϕ /(−V̄t ) > 0, where the
inequality follows from ht < 0 (by ut > 0 for t < t∗ (1|ϕ)), hϕ = 0 (by u log-modular in (t, ϕ)),
V̄t ≤ 0 (by u log-submodular in (t, q)), V̄ϕ > 0 (by log-supermodular in (q, ϕ)), and ht − V̄t ≥ 0
(else R(t|ϕ) falls in t contradicting Lemma 1). Altogether, given q̄L ≡ RP (t̄(ϕL )|ϕ), we have
shown t̄(ϕL ) = R−1
P (q̄L |ϕH ) ≥ t̄(ϕH ); and thus, tL > t̄(ϕH ), but this contradicts tH > tL (by
RP (·|ϕ) falling in ϕ) and tH ≤ t̄(ϕH ) (by qH = RP (tH |ϕ) locally optimal).
Proof of Propositions 5∗ and 6∗ . Consider the sets QP (ϕH ) and QP (ϕL ) for a co-monotone
delay ϕH > ϕL . The results vacuously hold if QP (ϕH ) is empty. Henceforth assume not. By
Proposition 2, QP (ϕH ) non-empty implies no greed at t∗ (1|ϕH ), which in turn implies no greed
R1
at t∗ (1|ϕL ) by 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)), and t∗ (1|ϕH ) ≥ t∗ (1|ϕL ) (Propositions 5 and 6). Then, since
we have assumed no panic at ϕL , QP (ϕL ) is non-empty, containing at least the safe pre-emption
equilibrium by Proposition 2. Two results follow. First, by Proposition 2, pre-emption games end
38
at t∗ (1), while Proposition 5 asserts t∗ (1|ϕH ) > t∗ (1|ϕL ) for a harvest delay and Proposition 6
claims t∗ (1|ϕH ) = t∗ (1|ϕL ) for an increase in greed. Thus, gradual play end times are ordered
as in part (iii) for any QH ∈ QP (ϕH ) and QL ∈ QP (ϕL ). Likewise, the exit rates obey part
(iv) for all QH ∈ QP (ϕH ) and QL ∈ QP (ϕL ), since Γ′P rises in ϕ by Propositions 5 and 6 and
Q ∈ QP (ϕ) share Q′ (t) = Γ′P (t) on any common gradual play interval by Proposition 7.
By construction, choosing QH ∈ Q(ϕH ) is equivalent to choosing some tH in the locally optimal interval [t0 (ϕH ), t̄(ϕH )] characterized by Lemma 1. Let qH ≡ RP (tH |ϕH ) be the associated
rush. Let t0 (ϕL ) and q L = RP (t0 (ϕL )|ϕL ) be the safe rush time and size for ϕL .
F INAL S TEPS
FOR 5∗ .
Assume a harvest delay ϕH > ϕL .
C ASE 1: q L > qH . Let QL be the safe pre-emption equilibrium. By Proposition 5 the safe
rush times obey t0 (ϕL ) ≤ t0 (ϕH ), while q L > qH by assumption: QL has a larger, earlier rush
than QH as in part (ii). Since ΓP (t|ϕ) is increasing in t by Lemma B.1 and decreasing in ϕ by
Proposition 5, the inverse function Γ−1
P (q|ϕ) is increasing in q and decreasing in ϕ. Thus, gradual
−1
play start times obey ΓP (qH |ϕH ) > Γ−1
P (q L |ϕL ), as in part (iii). Altogether, QL ≥ QH as in
part (i), since QL has a larger and earlier rush, an earlier start and end time to gradual play, and
the gradual play cdfs are ordered ΓP (t|ϕL ) > ΓP (t|ϕH ) on the common gradual play support.
C ASE 2: q L ≤ qH . Since QH is an equilibrium, qH = RP (tH |ϕH ) is locally optimal. And
by Lemma 1, RP (·|ϕL ) is continuously increasing with domain [q L , 1], which implies tL ≡
R−1
P (qH |ϕL ) ≥ t0 (ϕL ) exists. Thus, qH = RP (tL |ϕ) is locally optimal by Lemma I.1, and tL
defines an equilibrium QL ∈ QP (ϕL ). To see that QL satisfies part (ii) note that QL and QH have
the same size rush by construction, while rush times are ordered tL < tH ≡ R−1
P (qH |ϕH ) by RP
falling in ϕ (Lemma G.2). Now, since ΓP (t|ϕ) is increasing in t (Lemma B.1) and falling in ϕ
−1
(Proposition 5), gradual play start times obey Γ−1
P (RP (tL |ϕL )|ϕL ) < ΓP (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 end time to gradual play, and the gradual play cdfs are ordered ΓP (t|ϕL ) > ΓP (t|ϕH )
on any common gradual play interval.
F INAL S TEPS
FOR 6∗ .
Since QP (ϕH ) is empty with panic (Proposition 2), we WLOG assume no
panic at ϕH . The steps for part (c) (alarm at ϕL ) exactly parallel those for Proposition 5∗ above.
Thus, we henceforth assume no alarm at ϕL , and since the premise of Proposition 6∗ assumes
no panic at ϕL , inequality (3) obtains at ϕL , i.e. [maxq V0 (0, q|ϕL )]/u(t∗ (1|ϕL ), 1|ϕL ) ≤ 1. But
then, inequality (3) also obtains at ϕH , since V0 (0, q|ϕ))/u(t∗(1|ϕ), 1|ϕ) falls in ϕ by u logmodular in (t, ϕ) and log-supermodular in (q, ϕ). Altogether, neither alarm nor panic obtains at
ϕL and ϕH . Proposition 6 then states that safe rush times obey t0 (ϕL ) < t0 (ϕH ) ≤ tH with sizes
q L < RP (t0 (ϕH )|ϕH ) ≡ qH . By Lemma 1, RP (·|ϕL ) is continuously increasing onto domain
39
[q L , 1] ⊃ [q H , 1]; and thus, tL ≡ R−1
P (qH |ϕL ) > t0 (ϕL ) uniquely exists, is locally optimal by
Lemma I.1, and satisfies tL < tH by RP (·|ϕ) falling in ϕ (Proposition 6). Altogether, tL defines
QL ∈ QP (ϕL ) with an earlier rush of the same size as QH as in part (ii). The function ΓP (t|ϕ) is
increasing in t (Lemma B.1) and decreasing in ϕ (Proposition 6): Gradual play starts times obey
−1
Γ−1
P (RP (tL |ϕL )|ϕL ) < ΓP (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 obey ΓP (t|ϕL ) > ΓP (t|ϕH ) on any common gradual play interval.
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