FIGHT OR FLIGHT: ENDOGENOUS TIMING IN CONFLICTS

Boris van Leeuwen, Theo Offerman, and Jeroen van de Ven*

Abstract—We study a dynamic game in which players compete for a prize.
In a waiting game with two-sided private information about strength lev-
els, players choose ﬁghting, ﬂeeing, or waiting. Players earn a “deterrence
value” on top of the prize if their opponent escapes without a battle. Noi
show that this value is a key determinant of the type of equilibrium. For
intermediate values, sorting takes place, with weaker players ﬂeeing be-
fore others ﬁght. Time then helps to reduce battles. In an experiment, we
ﬁnd support for the key theoretical predictions and document suboptimal
predatory ﬁghting.

IO.

introduzione

FOLLOWING Maynard Smith’s (1974) seminal contribu-

zione, competition for a prize is often modeled as the war
of attrition. In this game, players choose the time at which
they intend to ﬂee. Time is costly, and players may differ
in their opportunity costs. The player who waits the longest
wins the prize and both players pay a cost proportional to
the time it takes for the losing player to ﬂee. Maynard Smith
refers to this type of interaction as a “display.” In a display,
no physical contact takes place, although if it does, it does
not settle the battle or convey information about which player
would win an escalated conﬂict.

The main contribution of our paper is that we develop and
analyze a game in which at any moment, players can not
only wait or ﬂee; they also have the option to start a ﬁght. In
case of a ﬁght, a battle ensues and the stronger player wins
the prize while the losing player incurs a loss. This dynamic
ﬁght-or-ﬂight game allows us to make sense of a much wider
variety of competitions. It captures the essence of many types
of interactions in which the timing of actions plays a crucial
role, such as R&D races, litigation, the launch of political
or advertising campaigns, and ﬁrm acquisitions. It also ﬁts
situations in the animal kingdom, where animals ﬁght over
territory or prey. In all these examples, players can “ﬂee”
(per esempio., reduce R&D spending, settle), wait to see if the other
gives in, or initiate a ﬁght (per esempio., sue the opponent, start a
hostile takeover), forcing the other into a battle.

Received for publication September 9, 2019. Revision accepted for pub-

lication June 18, 2020. Editor: Shachar Kariv.

∗van Leeuwen: Department of Economics, Tilburg University; Offerman:
CREED, University of Amsterdam and Tinbergen Institute; van de Ven:
ASE, University of Amsterdam and Tinbergen Institute.

We thank the editor, two anonymous referees, Jian Song, and audiences
at the University of Arizona, the University of Cologne, the University
of Lyon, the University of Manchester University, Middlesex University,
New York University, MPI Bonn, NHH Bergen, University of Oxford, UC
San Diego, Utrecht University, University of Vienna, WZB Berlin and at
IMEBESS Florence, M-BEES, NAG Toulouse, and TIBER for helpful sug-
gestions and comments. Financial support from the Research Priority Area
Behavioral Economics of the U. of Amsterdam, ANR–Labex IAST (Insti-
tute for Advanced Study in Toulouse), and CentER (Tilburg U.) is gratefully
acknowledged.

A supplemental appendix is available online at https://doi.org/10.1162/

rest_a_00961.

Our dynamic game helps to understand why in some situ-
ations, players want to wait and see if the other ﬂees without
a battle, while in other circumstances, both want to act as
quickly as possible. To illustrate the former type of situation,
consider two political candidates who may wait a long time
before they ofﬁcially announce that they are running for of-
ﬁce. If the other ﬂees without a battle, they avoid the costs
of a costly campaign that is required to win a ﬁght. Male
elephant seals that contest the right of exclusive access to a
harem usually wait a couple of minutes to allow the other to
ﬂee without a bloody ﬁght.

In other instances, players want to act as quickly as pos-
sible. A ﬁrm that wants to expand its market by acquiring
a competitor should act quickly to prevent the prospective
target from selling its valuable assets. Another possible in-
terpretation is that compared to letting the other escape, by
winning a ﬁght the player sends a stronger signal about its
strength to other players, thereby discouraging other players
from ever making a challenge. A ﬁrm that drives out another
ﬁrm by force will deter potential future competitors more than
if the other ﬁrm left voluntarily. In a lawless society without
a state monopoly of violence, people may want to rob each
other if they can. In an encounter, the stronger player prefers
to act as quickly as possible to avoid that the other ﬂees with-
out losing his money.

Notice that both types of examples are not well described
by the war of attrition. In the ﬁrst type of example, it may
happen that players ﬁght after a waiting period, which is not
a possibility in the war of attrition. The war of attrition also
does not capture the essence of the second type of interaction.
In particular, the war of attrition does not accommodate that
strong players decide to ﬁght in a split-second.

in questo documento, we analyze the ﬁght-or-ﬂight game theoreti-
cally and experimentally. Theoretically, we identify a key pa-
rameter, the deterrence value, that determines how the compe-
tition between two players will unfold. The deterrence value
is the amount that a player earns on top of the prize if the other
player manages to escape. Our theoretical analysis based on
standard preferences yields two main novel insights. Primo, if
the deterrence value is negative, all player types will rush and
act in a split-second. A negative deterrence value is illustrated
by the sale of valuable assets by a ﬂeeing prospective ﬁrm
in the takeover example. If the deterrence value is positive,
players prefer to avoid the costly ﬁght and wait before they
act. In the example where two political candidates engage in
a battle for ofﬁce, the costs to organize a campaign represent
a positive deterrence value.

The second insight is that if the deterrence value is positive
but not too large, sorting will occur in the dynamic ﬁght-or-
ﬂight game. Questo è, the weakest players will ﬂee just before
the end. Così, the dynamic structure helps players to avoid
costly ﬁghts, in comparison to a static version of the game that

The Review of Economics and Statistics, Marzo 2022, 104(2): 217–231
© 2020 The President and Fellows of Harvard College and the Massachusetts Institute of Technology. Published under a Creative Commons Attribution 4.0
Internazionale (CC BY 4.0) licenza.
https://doi.org/10.1162/rest_a_00961

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218

THE REVIEW OF ECONOMICS AND STATISTICS

is stripped of its time element. These two results cannot be
obtained in a standard war of attrition. In that game, players’
waiting correlates positively with their strength, and rushing
by all types is never observed in equilibrium. Inoltre, IL
dynamic standard war of attrition does not help players to
sort and avoid costly ﬁghts in comparison to the static version
(Hörisch & Kirchkamp, 2010).

We also investigate what happens in a behavioral model
in which players differ in their degree of risk aversion. Questo
model yields two additional testable implications. Primo, Esso
predicts that sorting will occur in a wider set of circumstances
than in the standard model. Secondo, it predicts that the more
risk-averse players ﬂee more frequently before the end.

We test the predictions in an experiment in which we sys-
tematically vary the deterrence value and the dynamic or
static nature of the game between treatments. Our experimen-
tal ﬁndings support some of the key features of the theory,
at least in terms of its comparative statics. With a negative
deterrence value, subjects quickly learn to decide in a split-
second. With a positive deterrence value, subjects tend to
wait much longer and indeed use time to sort. In agreement
with the model of heterogeneous risk aversion, we ﬁnd that
endogenous timing reduces the likelihood of costly battles
in a wider set of circumstances than predicted by standard
theory. Subjects classiﬁed as more risk averse on the basis
of an independent task are indeed the ones who tend to ﬂee
more often early in the game. Così, while not all results are
consistent with the point predictions of the model, in terms
of comparative statics, behavior often moves in the expected
direction.

An interesting ﬁnding that deviates from the predictions
is that a sizable minority of subjects ﬁght early when the
deterrence value is positive. This is the case even after am-
ple time to learn. This ﬁnding is in stark contrast with some
behavioral ﬁndings in related dynamic games. For instance,
Roth, Murnighan, and Schoumaker (1988) report that the
deadline effect, a striking concentration of agreements in
the ﬁnal seconds of the game, is the most robust behav-
ioral ﬁnding in a class of games designed to test axiomatic
models of Nash bargaining. Roth and Ockenfels (2002) E
Ockenfels and Roth (2006) identify substantial last-minute
bidding in second-price auctions. They attribute this phe-
nomenon of sniping to both strategic and naive considerations
of the bidders. We discuss some potential explanations for the
anomaly of early ﬁghting in our contest game at the end of the
section IV.

One feature of our experimental design is that time is dis-
crete but with very short time intervals. This makes it hard for
subjects to precisely time their actions and could be one of the
reasons behind the decrease in costly battles in the dynamic
games. In a follow-up experiment, we make it easier for sub-
jects to time their action by making the time intervals longer.
Consistent with the theoretical predictions, we no longer ob-
serve a decrease in battles compared to the static games when
the deterrence value is negative. In other respects, the results
closely resemble that of the original experiment.

Our paper contributes to the literature on dynamic games
in which players compete for a prize. Several studies compare
dynamic with static environments. Hörisch and Kirchkamp
(2010) investigate how experimental subjects behave in static
and dynamic versions of the war of attrition and some closely
related games. Theoretically, the dynamic version of a war
of attrition does not help players to sort, and indeed, they do
not observe such a difference in their experiments.1 Theo-
retically, in an auction with symmetric interdependent valu-
ations, Goeree and Offerman (2003UN) do not ﬁnd that the ef-
ﬁciency of a dynamic English auction is improved compared
to the static second-price auction. In contrasto, Kirchkamp and
Moldovanu (2004) investigate a setup where a bidder’s value
is determined by his own signal in combination with the sig-
nal of his right neighbor. In this setting, bidders can retrieve
valuable information in a dynamic auction process. In an ex-
periment, they ﬁnd that the efﬁciency of the English auction
is higher than in a second-price auction in which no such
information can be retrieved, which accords with theory.2

The remainder of the paper is organized as follows. Sezione
II introduces the ﬁght-or-ﬂight game and presents the theory.
Section III discusses the experimental design and procedures.
Section IV provides the experimental results, and section V
concludes.

II. Theory

UN. Dynamic Fight-or-Flight game

We ﬁrst describe the dynamic version of the ﬁght-or-ﬂight
game. In this section, we present a basic version of the game.
In section IIC and online appendix B, we discuss several
extensions.

Time is discrete, with a ﬁnite number of periods t =
0, 1, . . . , T . For each t < T , as long as the game has not ended, the two players independently decide to wait, ﬂee (R, 1There is a large literature on static contest games. Carrillo and Palfrey (2009) study a contest game that is quite close to our static benchmark. They ﬁnd that subjects compromise more often than in equilibrium, and they discuss some explanations based on cognitive limitations. De Dreu et al. (2016) investigate a game in which a group of attackers competes with a group of defenders. They ﬁnd that in-group defense is stronger and better coordinated than out-group aggression. Oprea, Henwood, and Friedman (2011) show how the matching protocol affects outcomes in continuous time Hawk-Dove games. Dechenaux, Kovenock, and Sheremeta (2015) provide a survey of the experimental literature on contest games. 2The war of attrition has been applied to various settings, including versions with private information (Fudenberg & Tirole, 1986; Ponsati & Sákovics, 1995) and applications to public good provision (Bliss & Nale- buff, 1984; Weesie, 1993). Oprea, Wilson, and Zillante (2013) experimen- tally study war of attrition games with two-sided private information (as in Fudenberg & Tirole, 1986) and observe behavior close to theoretical pre- dictions. More generally, the study of dynamic games reveals novel insights that signiﬁcantly surpass what we know from the study of static games. Re- cent contributions include Potters, Sefton, and Vesterlund (2005), Levin and Peck (2008), Ivanov, Levin, and Peck (2009), Kolb (2015), and Agranov and Elliott (2017). The recent experimental literature on continuous time ex- periments shows that outcomes in continuous time may substantially differ from outcomes in discrete time (Friedman & Oprea, 2012; Oprea, Charness, & Friedman, 2014; Bigoni et al., 2015; Calford & Oprea, 2017). l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / r e s t / l a r t i c e - p d f / / / / 1 0 4 2 2 1 7 1 9 9 6 1 4 0 / r e s t _ a _ 0 0 9 6 1 p d . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 FIGHT OR FLIGHT 219 for “retreat”), or ﬁght (F ). In the ﬁnal period, players can no longer wait and have to choose F or R. The game ends with at least one player choosing F or R, at which point the action set becomes null. At the start, each player i is privately informed of her ﬁghting ability ai. It is common knowledge that ai is in- dependently drawn from a uniform distribution over the unit interval. A player’s strategy lists for every ability the number of periods in which she chooses to wait and her choice if play reaches the period in which she wants to act. A player type’s strategy s(ai) is described as (t, A), where A ∈ {F, R}. This means that player i with ability ai will choose action A (ﬁght or ﬂee) in period t if the other player did not ﬁght or ﬂee earlier. The game ends as soon as one of the players decides to ﬁght or ﬂee. The outcome can be a battle or an escape. A battle occurs if the player with the shortest waiting time chooses to ﬁght or if both choose to ﬁght at the same time. An escape occurs if the player with the shorter waiting time chooses to ﬂee or if they both choose to ﬂee at the same time. If one of the players chooses to ﬁght and the other chooses to ﬂee at the same time, an escape occurs with probability p and a battle with probability 1 − p. Payoffs. In case of a battle, the player with the higher ability receives v h > 0 (the prize), and the other earns −v l ,
where v h, v l > 0. In case of an escape, the player who chose
to ﬂee earns 0 while the other earns v h + k, the prize plus
a deterrence payoff k. This deterrence value can be positive
or negative. A positive deterrence value captures situations
where ﬁghting is costly, so that players prefer to get the prize
without ﬁghting for it. A negative deterrence value captures
situations in which beating the other generates a higher value
compared to when the other escapes. We restrict the analysis
to k > −v h, so that if the other escapes, this always gives a
higher payoff than escaping. As tie-breaking rules, we assume
that if there is a battle between equally strong players, it is
randomly determined (with equal probability) which player
receives v h and which player receives −v l . If both players
decided to ﬂee at the same time, it is randomly determined
(with equal probability) who earns 0 and who earns v h + k.
Alternatively, players could be allowed to share the prize
equally in case both ﬂee. This would not affect the theoretical
analysis if players are risk neutral.

We assume that players maximize their expected utility and
do not discount the future. In online appendix B, we analyze
the case with discounting, but here our aim is to show how
time per se affects the ability of players to sort themselves
according to their strength. The case without discounting is
also relevant for many cases, such as when the cost of waiting
is small compared to the prize, the maximum duration of the
game is short, or the consumption of the prize happens at a
ﬁxed point in time.3

We allow for the possibility that players are risk averse. A
keep the model parsimonious, we assume that each player’s

utility function is piecewise linear in the payoff x and given
by

U (X) =

(cid:2)

if x ≥ 0
X
λx if x < 0. (1) Here, λ > 0 captures the degree of risk aversion (for λ > 1)
or risk seeking (0 < λ < 1). Naturally, when λ > 1, Questo
speciﬁcation is also consistent with loss aversion. Our ap-
proach does not distinguish between loss and risk aversion.

B. Equilibrium

We look for pure-strategy Bayesian Nash equilibria. In
this section, we derive equilibria under the assumption that
players have threshold strategies, where types below a certain
threshold ﬂee and types above that threshold ﬁght. Intuitively,
stronger types have more to gain from ﬁghting. We also as-
sume that no type acts after the period in which the stronger
type acts. In appendix A, we show that all equilibrium proﬁles
satisfy these properties.

Negative deterrence value. −v h < k < 0. For a negative deterrence value, the payoff of winning a battle exceeds that of allowing the other to escape. In this case, there is a unique equilibrium outcome in which all players ﬁght or ﬂee immedi- ately. The very strong types will want to ﬁght, and very weak types will want to ﬂee. If the weakest types would ﬂee after t = 0, the strongest types have an incentive to ﬁght before that so the opponent does not escape. But then the weakest types would deviate to ﬂeeing earlier. This implies that the strongest types ﬁght immediately, and the weakest types ﬂee immediately. Any other type will then act immediately as well. Acting later is costly, because it does not result in fewer battles with stronger types who ﬁght and gives weaker types the possibility of escaping. With all types acting immediately, let type ˜a be indifferent between ﬁghting and ﬂeeing. All stronger types ﬁght and all weaker types ﬂee. Suppose type ˜a ﬂees. If the opponent is weaker, the expected payoff is (v h + k)/2, and this happens with probability ˜a. If the opponent is stronger, a battle results with probability 1 − p, and this will always be lost by type ˜a, giving a payoff −λv l . The expected payoff of ﬂeeing is therefore given by ˜a 1 2 (v h + k) + (1 − ˜a)(1 − p)(−λv l ). (2) Suppose type ˜a ﬁghts. A weaker opponent escapes with probability p, giving a payoff v h + k, and otherwise there is a battle that will be won by type ˜a, giving a payoff v h. If the opponent is stronger, there will always be a battle that will be lost by ˜a. The expected utility of ﬁghting is then given by 3By design, discounting also cannot play a role in the experiment. ˜a[p(v h + k) + (1 − p)v h] + (1 − ˜a)(−λv l ). (3) l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / r e s t / l a r t i c e - p d f / / / / 1 0 4 2 2 1 7 1 9 9 6 1 4 0 / r e s t _ a _ 0 0 9 6 1 p d . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 220 THE REVIEW OF ECONOMICS AND STATISTICS Since type ˜a is indifferent between ﬂeeing and ﬁghting, it follows that ˜a = pλv l v h + pλv l + k 1 2 (cid:4) . (cid:3) p − 1 2 (4) The threshold ˜a is increasing in the probability of an es- cape p. As p increases, ﬁghting against weaker types becomes less attractive since they become more likely to escape. More types will then ﬂee in equilibrium. The effect of k on ˜a de- pends on the value of p. For p < 1 2 , an increase in k has a larger impact on the ﬂeeing payoff than on the ﬁghting pay- off. This means ﬂeeing becomes more attractive, and more types will ﬂee in equilibrium. For p > 1
2 , the reverse is true.

Positive deterrence value. k > 0. With a positive deter-
rence value, players are better off when the other manages
to escape than when they win a battle. In questo caso, all the ac-
tion will be concentrated in the ﬁnal two periods of the game.
Intuitively, sufﬁciently strong players will wait until the last
period to give other players the option to escape. Fighting
should take place only in the last period. Weaker types will
then also prefer to wait until at least the penultimate period,
since waiting until then gives opponents the option to escape
without the risk of ending up in a ﬁght.

Consequently, for k > 0, there is a fraction of types that
ﬂees at T − 1 and a fraction that ﬂees at T . The remaining
fraction ﬁghts at T . All types that ﬂee have the same payoff
independent of the moment that they ﬂee; they always lose a
battle with a type that ﬁghts, and their payoff when the oppo-
nent ﬂees is independent of their ﬁghting ability. The equilib-
rium therefore does not pin down which types ﬂee ﬁrst, only
the fraction. To determine the fraction of types that ﬂee, we
can assume without loss of generality that the weakest types
ﬂee at T − 1. The equilibrium can then be characterized by
two threshold levels, ˆa1 and ˆa2 > ˆa1. Type ˆa1 is indifferent
between ﬂeeing at T − 1 and ﬂeeing at T . Type ˆa2 is indif-
ferent between ﬂeeing at T and ﬁghting at T . A fraction of
types ˆa1 ﬂees at T − 1, and a fraction of types ˆa2 − ˆa1 ﬂees
at T . Types above ˆa2 ﬁght at T . The values of ˆa1 and ˆa2 are
given by

λv l [(v h − k)(1 − 2p) − 2k p2]

ˆa1 =

(v h + k + 2(1 − p)λv l )

(cid:3)

v h −

1
2

(cid:3)

1
2

(cid:4)
− p

ˆa2 =

2(1 − p)λv l
v h + k + 2(1 − p)λv l

(cid:4) ,

(5)

The fraction of types ﬂeeing at T − 1 is positive for values
of k below ˆk, Dove

ˆk =

1 − 2p
1 − 2p + 2p2

v h.

(6)

For larger values of k, all types wait until the ﬁnal period.
Intuitively, if k is large, it always pays off to wait and give

others the option to escape, even if that implies risking a
battle with stronger types. The same is true for larger values
of p. If the probability of an escape is large, it becomes more
attractive to wait, even if the opponent ﬁghts.

Thus there can be three types of equilibrium outcomes.
If k < 0, there is a rushing equilibrium in which all types immediately ﬁght or ﬂee. For intermediate positive values of k, there is a timing equilibrium in which some types wait until the penultimate period and then ﬂee, while all others wait until the ﬁnal period and then ﬁght or ﬂee. For high values of k, there is a waiting equilibrium in which all types wait until the last period and then ﬁght or ﬂee. While we derived these equilibria under the assumption that players have threshold strategies, in appendix A, we show that no other equilibria exist. The equilibrium outcome is generically unique, except for k = 0 or k = ˆk. Proposition 1 (Equilibrium). (i) If k < 0, the unique equilibrium outcome is a rushing equilibrium in which all players act immediately. Play- ers with abilities [0, ˜a] ﬂee at t = 0, and players with abilities ( ˜a, 1] ﬁght at t = 0. (ii) If 0 < k < ˆk, the unique equilibrium outcome is a tim- ing equilibrium in which a fraction ˆa1 of types ﬂee in period T − 1, a fraction ˆa2 − ˆa1 of types ﬂee in period T , and all types above ˆa2 ﬁght in period T . (iii) If k > max{ˆk, 0}, the unique equilibrium outcome is a
waiting equilibrium in which types [0, ˜a] ﬂee in period
T and types ( ˜a, 1] ﬁght in period T, and ˜a = 1 for any
v h < (1 − 2p)k. Proof. All proofs are in appendix A. Figure 1 illustrates the equilibrium outcomes. Figure 1a shows equilibrium outcomes for different combinations of the probability of an escape (p) and the deterrence value (k). Figure 1b shows how the threshold values change with k. For k < 0, fewer types ﬁght as k increases. A higher k makes let- ting the other escape relatively more attractive, and such an escape becomes less likely by ﬁghting. This reverses for pos- itive values of k, with more types ﬁghting as k increases. For higher values of k, fewer types ﬂee early. Fighting becomes relatively more attractive with more weaker types still around. The ﬁgure also illustrates how these thresholds change with an increase in p. To shed light on whether the dynamic time element of the ﬁght-or-ﬂight game decreases costly battles, we use a static version of the game as benchmark. In the static game, players choose simultaneously between ﬁght and ﬂee, and the same payoffs result as when players reach the ﬁnal period of the dynamic game. The Bayesian Nash equilibrium of the static game coincides with the equilibrium of the dynamic game for parameters where all players act in the same period (that is, either case i or case iii described in proposition 1). An interesting feature of the timing equilibrium of the dy- namic game is that sorting takes place over time, resulting in fewer battles compared to what happens in the static game. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / r e s t / l a r t i c e - p d f / / / / 1 0 4 2 2 1 7 1 9 9 6 1 4 0 / r e s t _ a _ 0 0 9 6 1 p d . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 FIGHT OR FLIGHT 221 FIGURE 1.—EQUILIBRIUM OUTCOMES WITH HOMOGENEOUS RISK AVERSION terrence values for which the timing equilibrium materializes does not depend on players’ risk aversion. This result changes when the population is heterogeneous in the degree of risk aversion. Intuitively, players who are relatively averse to risks will want to ﬂee earlier. Indeed, a population that is hetero- geneous in the degree of risk aversion can sustain a timing equilibrium for a larger set of deterrence values. We show this in a simple framework with two levels of risk aversion and outline the two main strategic features of this model. Suppose that a fraction 1 − q of the population has a risk aversion parameter λ1, and a fraction q has λ2 > λ1. UN
player’s value of λ is private information, but all players know
the distribution. Consider the case where q is very small. In
that case, the threshold levels derived assuming homogeneous
risk aversion in section IIA are not much affected for the less
risk-averse types. Fix an equilibrium in which k > ˆk, so that
all types with λ1 wait until period T .

If λ2 is such that

˜a

1
2

(v h + k) + (1 − ˜a)(1 − p)(−λ2v l ) < 0, (7) then types with λ2 and a ﬁghting ability less than or equal to ˜a prefer to ﬂee in period T − 1, while types with λ1 prefer to wait until T . Thus, for the same level of k, we now have a timing equilibrium instead of a waiting equilibrium. Another feature of this model is that the more risk-averse types will be the ones who ﬂee more frequently before the end. To see this, note that for the ability level for which the less risk-averse type is indifferent between ﬂeeing in period T − 1 and period T , the more risk-averse type still strictly prefers to ﬂee in period T − 1. The reason is that the expected payoff of ﬂeeing in period T − 1 is not affected by the degree of risk aversion (since there are no negative payoffs), while the expected payoff of ﬂeeing in period T decreases in a player’s risk aversion (since the negative payoff when a battle is lost weighs more heavily). In the experiment, we will test these two implications of the model with heterogeneous risk aversion. Other extensions. We also considered some natural exten- sions of the model. Here we describe the main qualitative fea- tures of these extensions. In appendix B, we provide further details of these extensions, as well as discussion of the pros and cons of discrete versus continuous modeling in waiting games. So far we simply assumed that the stronger player always wins a battle. A natural possibility is that stronger types are more likely to win but do not win with certainty. When relative strength correlates sufﬁciently strongly with winning a battle, the results are qualitatively the same. That is, with a positive deterrence value, all the action will be concentrated in the ﬁnal two periods. The strongest types still want to ﬁght in the ﬁnal period, while no type wants to ﬂee before the penultimate period. Likewise, with a negative deterrence value, all types will still act immediately. When the link between relative (a) The solid dots indicate the experimentally implemented values (with vh = vl = 10, p = 0.1, and k = {−6, 6, 12}). (b) Rushing occurs to the left of the vertical axis, timing occurs between the vertical axis and the shaded area, waiting occurs in the shaded area. The dashed lines show a decrease in the escape probability (p) (for p < 1 2 ). The dark shaded area shows the waiting equilibrium for the lower value of p. In the dynamic game, the strongest types remain in the game until the last period, while some weaker types ﬂee before any battle may take place. Moreover, a smaller fraction of types will ﬁght; ﬁghting becomes less attractive with fewer relatively weak players remaining. Proposition 2 (Battles and Sorting). Compared to a static (simultaneous-move) version of the game: (i) The frequency of battles is reduced in case of a tim- ing equilibrium and the same in case of a rushing or waiting equilibrium. (ii) The rate at which the weaker player in a pair man- ages to escape is increased in case of a timing equi- librium and the same in case of a rushing or waiting equilibrium. C. Extensions Heterogeneous risk aversion. A surprising feature of the analysis with a homogeneous population is that the set of de- l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / r e s t / l a r t i c e - p d f / / / / 1 0 4 2 2 1 7 1 9 9 6 1 4 0 / r e s t _ a _ 0 0 9 6 1 p d . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 222 THE REVIEW OF ECONOMICS AND STATISTICS strength and winning a battle becomes weak, other types of equilibria exist. In the extreme case, where each type has an almost equal chance of winning a battle against any other type, there can be equilibria where all types prefer to ﬁght, possibly at different periods. There can also be an equilibrium in which all types prefer to ﬂee in the last period. Another natural extension is discounting of future pay- offs. Conditional on a discount factor sufﬁciently close to 1, our main theoretical ﬁndings remain qualitatively similar. That is, we ﬁnd a rushing equilibrium when k < 0, a tim- ing equilibrium when 0 < k < ˆk, and a waiting equilibrium when k > max(0, ˆk). As in the timing equilibrium without
discounting, all the action happens in the penultimate and
last periods. The main difference with the model without dis-
counting is that the thresholds now also depend on the dis-
count factor. When discounting is important, the comparison
between the static and dynamic case becomes less clear-cut
in terms of welfare: the higher degree of sorting comes at the
cost of waiting longer.

A ﬁnal variant that yields somewhat different predictions
is the one where players face a known cost for time. Here, Esso
may happen that weak players decide to drop out earlier than
the penultimate period. In a recent paper, Song and Houser
(2021) study the interesting case of costly waiting in detail.
In the experiment, we focus on the variant where time is
not costly for two reasons. Primo, it allows us to investigate
in a meaningful way how the dynamic game helps players
avoid costly battles compared to the static game where time
plays no role. Secondo, we think that it is a stronger result if
players use time as a sorting device when time is not costly.

III. Experimental Design and Procedures

UN. Design

Subjects participated in a laboratory experiment in which
they played the ﬁght-or-ﬂight game. In all treatments, we
set the value of winning a battle to v h = 10 and losing a
battle to −v l = −10. The probability of an escape when
at the same time one player decided to ﬁght and the other
decided to ﬂee was set to p = 0.1. Each subject played
the game forty rounds, with random rematching after ev-
ery round within a matching group of eight subjects. A
the start of each round, the subjects were informed of their
ﬁghting ability for that round, which was an integer number
from {0, 1, 2, . . . , 1000}. They knew that each number was
equally likely, that each subject faced the same distribution,
and that draws were independent across subjects and rounds.
At the end of a round, each subject was informed of the out-
come, the paired subject’s ﬁghting ability, and the resulting
payoffs.

We implemented two treatment variations. The ﬁrst treat-
ment variable was the deterrence variable k, which was either
−6, 6, O 12. The second concerned the dynamic or static
nature of the ﬂight-or-ﬁght game. This gives a 3 × 2 progetto.
Every subject participated in only one of the treatments. In to-

Treatment
Version

Dynamic
Dynamic
Dynamic
Static
Static
Static

TABLE 1.—OVERVIEW OF TREATMENTS

Deterrence
Value (k)
−6
6
12
−6
6
12

N Subjects

N Matching
Groups

64
56
64
56
64
56

8
7
8
7
8
7

tal, 360 subjects participated, with seven or eight independent
matching groups per treatment. Tavolo 1 presents an overview.
In the dynamic ﬁght-or-ﬂight game, UN 5 second countdown
started after all subjects in the laboratory had indicated that
they were ready to start. This ensured that subjects knew
exactly when the game would start. During the game itself,
a clock started counting down from 10 seconds to 0. IL
program divided the 10 seconds in 50 periods of 200 mil-
liseconds each. Subjects implemented their strategies in real
time. For instance, a subject could decide to wait for 5 sec-
onds (cioè., for the ﬁrst 25 periods) and then choose to ﬁght,
which would determine the outcome of the game (unless the
other subject had already terminated the game earlier). Questo
way she would implement the strategy (25, F ). If subjects let
the time run down to 0, they entered the endgame, in which
they simultaneously decided between ﬁght and ﬂee (with no
time constraints, as they decided simultaneously anyway).

Our dynamic game has 50 periods, more than the minimum
required to test the theoretical predictions of the model, E
short time intervals of 200 ms. Our goal was to have a design
that is closer to the examples that motivated our research.
A disadvantage is that rational subjects might ﬁnd it hard
to exactly implement equilibrium strategies in our setup. UN
follow-up experiment with longer time intervals addresses
this concern (see section IVD).

The static version of the game abstracted from the time
element and only consisted of the endgame of the dynamic
version. Questo è, in this version of the game, subjects were
immediately put in the same position as the players of the
dynamic game who had both decided to wait until the end of
the game. So in the static game, both subjects simultaneously
chose between ﬁght and ﬂee.

After the main part, we obtained additional measurements.
We assessed subjects’ risk aversion using the method of
Gächter, Johnson, and Herrmann (2007). A subject chooses
whether to accept or reject six different lotteries. In a lottery,
the winning amount is 6 euros. The losing amount varies
across lotteries, from two to seven. In each lottery, the win-
ning and the losing amounts are equally likely. If a subject
rejects a lottery, she surely receives 0 euro. At the end of
the experiment, one of the six lotteries is selected at random
and played out for actual payment. The number of rejected
lotteries is our measure of a subject’s degree of risk aversion.
We also measured physical strength. We asked subjects to
press a hand dynamometer as hard as they could, following

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FIGHT OR FLIGHT

223

the procedure of Sell et al. (2009). This measurement was
obtained twice, and the best attempt was rewarded with 5
eurocents per kilo pushed. Finalmente, we obtained some self-
reported measurements on social dominance and prestige
(from Cheng, Tracy, & Henrich, 2010), perceived masculin-
ità, sex, and age.4

This design allows us to investigate the predictions summa-
rized in propositions 1 E 2. Inoltre, it makes it possible
to test the predictions from the behavioral model of hetero-
geneous risk aversion.

B. Procedures

The experiment was computerized and run at CREED
(University of Amsterdam). The instructions are in appendix
E. Subjects read the instructions at their own pace. They could
continue only after correctly answering test questions at the
end of the instructions. To ease understanding, we used non-
neutral labels such as “ﬁght” and “escape.” Subjects were
informed that there would be two parts, receiving new in-
structions at the start of each part.

During the experiment, subjects earned points, Dove 1
point = €0.70 (≈$0.84). To avoid a net loss at the end of
the experiment, they received a starting capital of 21 points,
and any proﬁts or losses would be added to or subtracted
from this. At the end of the experiment, one round of the
main part was randomly selected for payment. Total earnings
averaged €19.09, ranging from €5.30 to €38.20.5 A session
took approximately 65 A 75 minutes.6

IV. Results

In sections IVA and IVB, we consider the testable predic-
tions following from propositions 1 E 2, rispettivamente. Then,
in section IVC, we turn to decisions at the individual level. Tutto
statistical tests comparing treatment differences use match-
ing group averages as the independent unit of observation,
unless indicated otherwise.

UN. Timing of Actions

Following proposition 1, we address the comparative static
prediction that the timing of actions is inﬂuenced by the de-
terrence value. Speciﬁcally, we expect very quick decisions
if the deterrence value is negative and decisions in the ﬁnal
periods if the deterrence value is positive. Figura 2 shows
the average elapsed time before subjects made a decision

4Perceived masculinity is measured by the answer to the question: “On a
scale from 1 (very feminine) A 7 (very masculine), how would you describe
yourself?"

5The payment subjects received consisted of the starting capital and
their earnings in the ﬁght-or-ﬂight game, the lottery task, and the physi-
cal strength task.

6In addition to the forty decision rounds (which lasted around 20 minutes),
subjects spent time on the instructions and test questions (25 minutes), IL
lottery task, questionnaire and physical strength task (15 minutes), E
payment of subjects (10 minutes).

FIGURE 2.—AVERAGE WAITING TIME (IN MS) BEFORE SUBJECTS MAKE A
DECISION IN THE DYNAMIC GAME, BY TREATMENT AND ROUND

Lines are moving averages of three rounds.

in the dynamic games. As predicted, we observe a clear ef-
fect of the deterrence value on the timing of actions. Con
a negative deterrence value, subjects tend to ﬁght or ﬂee al-
most immediately. On average, subjects make a decision af-
ter 273 ms. When the deterrence value is positive, subjects
tend to wait much longer. For k = 6, the average elapsed
time before making a decision is 3,545 ms, and for k = 12,
questo è 3,973 ms. For both treatments with a positive deter-
rence value, the average waiting time is signiﬁcantly longer
than for k = −6 (Mann-Whitney tests, p = 0.001, N = 15
for k = −6 vs k = 6 and p < 0.001, N = 16 for k = −6 versus k = 12). While subjects wait slightly longer when k = 12 than with k = 6, the difference is not statistically sig- niﬁcant (Mann-Whitney test, p = 0.908, N = 15 for k = 6 vs k = 12). For all three treatments, we observe learning ef- fects. When the deterrence value is positive, subjects learn to wait, reﬂected by the strong, positive time trend over the rounds. The reverse holds for the negative deterrence value. In this case, subjects decide increasingly faster. The average elapsed time is 402 ms in the ﬁrst ten rounds and 200 ms in the ﬁnal ten rounds. When comparing the average wait- ing times in the ﬁrst ten rounds and ﬁnal ten rounds, all time trends are statistically signiﬁcant (Wilcoxon signed-rank tests, p = 0.017, N = 8 for k = −6, p = 0.018, N = 7 for k = 6 and p = 0.017, N = 8 for k = 12). Figure 3 gives a more detailed picture of the timing of de- cisions. The ﬁgure plots the distribution of actions for each of the 10 seconds plus the endgame (T). The left panels show this for the ﬁrst twenty rounds and the right panels for the ﬁnal twenty rounds. Several patterns emerge. First, with a negative deterrence value, we clearly observe rushing: sub- jects decide almost immediately. None of the matches make it to the endgame, and 99.5% of all matches end in the ﬁrst second. In fact, 90% of all matches end within the very ﬁrst 200 ms, that is, in the ﬁrst period.7 With a positive deterrence 7Figure A1 in appendix C shows the distribution of actions by 200 ms periods. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / r e s t / l a r t i c e - p d f / / / / 1 0 4 2 2 1 7 1 9 9 6 1 4 0 / r e s t _ a _ 0 0 9 6 1 p d . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 224 THE REVIEW OF ECONOMICS AND STATISTICS FIGURE 3.—DISTRIBUTION OF DECISIONS OVER TIME (SECONDS) BY DETERRENCE VALUE IN THE DYNAMIC GAME l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / r e s t / l a r t i c e - p d f / / / / 1 0 4 2 2 1 7 1 9 9 6 1 4 0 / r e s t _ a _ 0 0 9 6 1 p d . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Period T indicates the endgame. Left panels are for the ﬁrst twenty rounds and right panels for the ﬁnal twenty rounds. Only observations where a player made a decision to ﬁght or ﬂee are included in the graph, that is, observations where a player was waiting when the other moved are omitted. value, most action is at the very beginning and the very end: subjects tend to decide either relatively quickly or wait until the ﬁnal periods. In the ﬁnal twenty rounds, a larger fraction of subjects waits until the end. This fraction might be under- estimated because a subject who is willing to wait until the end will reach the end of the game only if the paired player is also willing to wait until then. Among those waiting, there are some subjects who ﬂee right before the endgame. Result 1. When the deterrence value is negative, players act immediately. When the deterrence value is positive, players are more likely to wait until the end of the game and they learn to wait longer. In contrast to the theoretical predictions, some subjects move at the very beginning of the game when the deterrence value is positive. This fraction decreases over time, but even in the ﬁnal twenty rounds (the right-hand panels of ﬁgure 3), we do observe such behavior. This behavior is not in line with the timing equilibrium or waiting equilibrium. We return to this anomaly when we discuss individual behavior (section IVC). The comparative static results of increasing k are in line with the theoretical predictions. B. Frequency of Battles and Sorting The second main testable prediction, following from proposition 2, is that endogenous timing helps to avoid costly battles. Speciﬁcally, we expect fewer battles in the dynamic games in case of a timing equilibrium, but not in case of a rushing or waiting equilibrium. The left panel of ﬁgure 4 shows the frequency of battles for each treatment (we discuss the results for experiment 2 in section IVD). We do indeed observe fewer battles in the dynamic treatments compared to the static treatments. The difference varies between 15 and 26 percentage points depending on the deterrence value and is always highly signiﬁcant (p < 0.003 in each case, two-sided Mann-Whitney tests). A regression analysis (table A1 in ap- pendix C, column 1) conﬁrms that there are fewer battles in the dynamic treatments, and this effect is slightly stronger when the deterrence value is positive. FIGURE 4.—FRACTION OF BATTLES (LEFT PANEL) AND FRACTION OF TIMES THAT THE WEAKER PLAYER IN A PAIR ESCAPES (RIGHT PANEL) FIGHT OR FLIGHT 225 Error bars indicate 95% conﬁdence intervals, based on matching groups as the independent unit of observation. The reduction of battles for k = 6 is in line with the com- parative static prediction following from proposition 2. For k = 6, the unique equilibrium outcome in the dynamic game is a timing equilibrium, resulting in fewer battles than in the equilibrium of the static game. Although we observe devia- tions from the timing equilibrium (in particular, some sub- jects move at the beginning of the game), we do ﬁnd that the number of battles is reduced compared to the static case. The observed lower frequency of battles for k = 12 is not ex- pected if players are homogeneous in their risk aversion, but it is consistent with the comparative static prediction of our version of the model in which players differ in their degree of risk aversion.8 In contrast to the theoretical predictions, we also observe a decrease in battles when the deterrence value is negative. This result is, however, partly mechanical; even if all subjects wanted to act immediately, some subjects might be a fraction of a second slower than others, resulting in more escapes.9 It is also a possibility that random noise reduces the fre- quency of battles in the dynamic game. For instance, if players in the dynamic game choose ﬁght, ﬂee, and wait in each pe- riod with equal probabilities while players in the static game choose between ﬂee and ﬁght with equal probabilities, fewer battles will be observed in the static game.10 However, as we will illustrate in section IVC, the behavior of our subjects is very remote from this random benchmark. Our subjects re- spond in a sensible way to their private strength parameters. Moreover, in agreement with theory but in contrast to the random benchmark, we ﬁnd that the dynamic nature matters most for reducing the frequency of battles when k > 0.

8As for k = 6, we also observe deviations from a timing equilibrium when
k = 12 as a number of subjects move early in the game. We discuss these
deviations in more detail in section IVC.

9Of the 15 percentage point difference in battles between static and dy-
namic games when k = −6, 6 percentage points can be attributed to escapes
that occur just because the subject who wanted to ﬁght is a fraction slower
than the subject who wanted to ﬂee. The remaining 9 percentage points can
be attributed to more subjects ﬁghting in the static games.

10We thank a referee for this insight.

Also following proposition 2, we expect that players sort
themselves according to their ﬁghting ability in case of a
timing equilibrium. The strongest players should wait longer
than weaker players, giving weaker players the opportunity
to escape. Hence, weaker players should manage to escape
more frequently in the dynamic games than the static games
if the deterrence value is positive. Our results are in line with
this prediction. The right panel of ﬁgure 4 shows how often
the weaker subject in a pair escapes. Subjects sort on ﬁghting
ability more often in the dynamic than the static game and the
increase is larger for dynamic games with a positive deter-
rence value. For k = −6, the weaker player escapes in 12%
of the matches in the static game and 26% of the matches in
the dynamic game. For k = 6 (k = 12), the weaker player es-
capes in 15% (18%) of the matches in the static game and 38%
(45%) of the matches in the dynamic game. The difference-
in-difference analysis reported in table A1 in appendix C
shows that the larger increase for positive deterrence values
is statistically signiﬁcant.11
Result 2. There are fewer battles in the dynamic game than
in the static game. The dynamic version of the game helps
players to sort themselves according to their ﬁghting abil-
ità, and this effect is stronger when the deterrence value is
positive.

The reduced number of battles in the dynamic games also
positively affects earnings. Figura 5 shows the mean earn-
ings for each treatment and for different levels of ﬁghting
ability. As expected, stronger types attain higher earnings.
Averaging across all ﬁghting abilities, earnings are higher in
the dynamic games than in the static games (Mann-Whitney
tests, P < 0.003 for all three comparisons). Note that the dif- ference for k = −6 is much smaller than the differences for the treatments with a positive deterrence value. Moreover, for 11Figure A2 in appendix C shows decision times for weak and strong players separately. It conﬁrms the comparative static prediction that stronger subjects wait longer than weaker subjects if the deterrence value is positive. Moreover, with experience, both weak and strong players learn to wait longer. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / r e s t / l a r t i c e - p d f / / / / 1 0 4 2 2 1 7 1 9 9 6 1 4 0 / r e s t _ a _ 0 0 9 6 1 p d . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 226 THE REVIEW OF ECONOMICS AND STATISTICS FIGURE 5.—MEAN EARNINGS BY TREATMENT AND FIGHTING ABILITY FIGURE 6.—BEHAVIOR BEFORE THE FINAL PERIOD IN THE DYNAMIC GAME, BY DETERRENCE VALUE k AND FIGHTING ABILITY a (IN TEN BINS OF EQUAL SIZE) l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / r e s t / l a r t i c e - p d f / / / / 1 0 4 2 2 1 7 1 9 9 6 1 4 0 / r e s t _ a _ 0 0 9 6 1 p d . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 In the category “wait” refers to subjects who made it to the endgame, and in “other moves” refers to subjects who did not make a move before the endgame but the other subject did. k = −6, the difference is driven by weaker subjects, whereas for the k > 0 treatments, all types on average beneﬁt from
endogenous timing.

Individual Behavior

We start this section by considering how actions in the
dynamic games depend on ﬁghting ability. Figura 6 plots the
fraction of subjects who ﬂee or ﬁght before the endgame,

those who were waiting while the other moved, and those
who wait until the endgame. We show this for the different
deterrence values and for different ﬁghting ability levels (In
ten bins of equal size). In line with the results on decision
times discussed in section IVA, no subject waits until the
ﬁnal period when the deterrence value is negative. Only a few
subjects (6%) are still waiting when the other moves. When
the deterrence value is positive, many subjects wait until the
endgame or are waiting when the other moves. Combining

FIGHT OR FLIGHT

227

TABLE 2.—FLEEING BEFORE ENDGAME

(1)
k = −6

0.037***
(0.007)
−0.025
(0.031)
0.007
(0.011)
−0.015
(0.019)
−0.980***
(0.003)
0.000
(0.001)

(2)
k = 6

−0.001
(0.018)
−0.012
(0.052)
−0.028
(0.023)
−0.004
(0.020)
−0.881***
(0.034)
−0.000
(0.001)

(3)
k = 12

0.041**
(0.018)
0.003
(0.060)
−0.001
(0.015)
0.003
(0.020)
−0.889***
(0.016)
−0.000
(0.001)

Risk aversion

Female

Dominance

Physical strength

Fighting ability

Round

k = 6

k = 12

Observations

2,520

2,080

2,520

(4)
All k

0.026***
(0.009)
−0.008
(0.030)
−0.006
(0.010)
−0.002
(0.011)
−0.935***
(0.013)
−0.000
(0.000)
−0.127***
(0.030)
−0.112***
(0.022)
7,120

Panel data probit regressions, with random effects at the subject level. Coefﬁcients are average marginal effects. Dependent variable is a dummy indicating whether the player decided to ﬂee before the endgame or
non. Risk aversion is measured as the number of rejected lotteries. Dominance and physical strength are normalized (mean 0 and SD 1). Fighting ability takes on values between 0 E 1. Standard errors (clustered at
the matching group level) in parentheses. Additional speciﬁcations with fewer or more controls are reported in table A4 in appendix C. ∗ p < 0.10, ∗∗ p < 0.05, and ∗∗∗ p < 0.01. those groups, we ﬁnd that 44% of subjects (intend to) wait for both k = 6 and k = 12. In line with theory, we ﬁnd in all treatments that weaker players are much more likely to ﬂee and stronger players are much more likely to wait or ﬁght. This pattern clearly shows that the behavior of subjects is far from a random benchmark. In the appendix, we provide further details on individual strategies. In appendix D, we estimate individual cutoff strate- gies. We ﬁnd that most behavior is consistent with the use of cutoff strategies: around 90% of all decisions are captured by individual cutoff strategies. There is substantial heterogene- ity in the type of cutoff strategies that individuals employ. Although the estimated cutoffs organize the data very well, for a substantial number of subjects, the estimated cutoffs are remote from the theoretical prediction. In section IVB, we reported that sorting was observed not only for k = 6 but also for k = 12. Although behavior in both treatments does not exactly follow the predictions from the timing equilibrium (notably, some subjects move early on in the game), the ﬁnding that subjects sort in k = 12 is consis- tent with the idea that heterogeneous risk aversion enlarges the set of environments for which the timing equilibrium ap- plies. A more direct implication of heterogeneous risk aver- sion is that the more risk-averse players should ﬂee early more often. Table 2 presents panel data probit regressions of how the probability of choosing to ﬂee before the endgame (T ) depends on a subject’s level of risk aversion, together with some controls. In agreement with the model of heterogeneous risk aversion, more risk-averse subjects are more likely to ﬂee before the endgame when k = −6 and when k = 12, and the effect survives when we combine all three treatments.12 An anomaly is the ﬁghting behavior early on in the game when there are beneﬁts of letting the other escape, that is, when k > 0. In questo caso, ﬁghting early is weakly dominated.
Given the observed actions in the experiment, the losses of
ﬁghting early are substantial. Consider the strongest possible
type who wins every ﬁght. This type would earn 14% higher
expected payoffs by waiting to ﬁght in the endgame if k = 6
E 42% higher expected payoffs if k = 12. Note that ﬁghting
early is even more costly for weaker types. One possible rea-
son for why we observe this anomalous behavior is that sub-
jects may need some time to learn. As ﬁgure 3 shows, we do
indeed observe less of this behavior in the ﬁnal twenty rounds
compared to the ﬁrst twenty rounds. Another, more psycho-
logical, explanation for ﬁghting early on in the game might
be a preference for social dominance. The evidence does not
support this. Tavolo 3 shows that the survey measure of social
dominance is not a predictor of ﬁghting early. We also do not
ﬁnd an association with physical strength, but we do ﬁnd that
women are more likely to ﬁght early than men.13

It may be that some of our subjects start playing the game
with a misguided behavioral rule that in contests, it generally
pays off to strike ﬁrst. Myerson (1991) proposes that behav-
ior that is apparently suboptimal behavior can sometimes be
understood by assuming that observed behavior is optimal
in a related but more familiar environment, which he calls a
“salient perturbation” (see Myerson, 1991; Samuelson, 2001;
Jehiel, 2005). Alternatively, it could be that intuition favors
ﬁghting behavior. According to the social heuristics hypothe-
sis (Rand, Greene, & Nowak, 2012; Rand et al., 2014) applied
to our setting, if ﬁghting is typically advantageous, it could
become the intuitive response. Note that subjects who ﬁght

12When we regress the estimated cutoff ﬁghting ability below which sub-
jects ﬂee before the endgame on risk aversion and other individual charac-
teristics, we obtain qualitatively similar results. The regressions are reported
in table A2 in appendix C.

13When we regress the estimated cutoff ﬁghting ability above which sub-
jects ﬁght before the endgame on risk aversion and other individual charac-
teristics, we obtain qualitatively similar results. The regressions are reported
in table A3 in appendix C.

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228

THE REVIEW OF ECONOMICS AND STATISTICS

TABLE 3.—FIGHTING IN THE FIRST SECOND

(1)
k = 6

−0.030**
(0.015)
0.130
(0.100)
0.002
(0.019)
0.009
(0.054)
0.415***
(0.067)
−0.004***
(0.001)

All Rounds

(2)
k = 12

−0.013
(0.018)
0.029
(0.058)
0.018
(0.021)
0.001
(0.036)
0.368***
(0.060)
−0.005***
(0.001)

Risk aversion

Female

Dominance

Physical strength

Fighting ability

Round

k = 12

Observations

2,080

2,520

(3)
all k > 0

−0.020**
(0.010)
0.066
(0.050)
0.015
(0.016)
−0.001
(0.029)
0.389***
(0.044)
−0.005***
(0.001)
−0.017
(0.042)
4,600

(4)
k = 6

−0.020
(0.017)
0.161
(0.111)
−0.025
(0.021)
0.025
(0.053)
0.272***
(0.072)
−0.002
(0.001)

Final 20 Rounds

(5)
k = 12

−0.015
(0.013)
0.046
(0.049)
0.011
(0.014)
0.022
(0.027)
0.218**
(0.090)
−0.003*
(0.001)

1,040

1,260

(6)
all k > 0

−0.018*
(0.011)
0.097**
(0.048)
0.006
(0.011)
0.022
(0.025)
0.242***
(0.048)
−0.002***
(0.001)
−0.026
(0.036)
2,300

Panel data prohibit regression with random effects at the subject level. Coefﬁcients are average marginal effects. Dependent variable is a dummy indicating whether the player decided to ﬁght in the ﬁrst second or
non. Risk aversion is measured as the number of rejected lotteries. Dominance and physical strength are normalized (mean 0 and SD 1). Fighting ability takes on values between 0 E 1. Standard errors (clustered at
the matching group level) in parentheses. Additional speciﬁcations with fewer or more controls are reported in table A5 in appendix C. ∗ p < 0.10, ∗∗ p < 0.05, and ∗∗∗ p < 0.01. early on have limited opportunities to learn, since they never experience the beneﬁts of waiting. This could explain why they do not converge fully to waiting until the end of the game. The fact that we observe an approximately equal frequency of early battles when k = 6 as when k = 12 suggests that this behavior is not due to a separate utility component reﬂect- ing (for instance) a desire to control the outcome or a joy of winning. If people have a preference to control the outcome, we would expect fewer early battles when it becomes more costly in k = 12.14 Still, when play has not yet converged to equilibrium, we cannot exclude that early ﬁghting is en- couraged by players who experience a joy of winning when they beat the other in a battle. In our follow-up experiments reported in section IVD, we include some measures of joy of winning to get direct evidence for this possibility.15 Result 3. A sizable minority of players acts immediately when the deterrence value is positive. This behavior decreases with experience. D. Experiment 2 In the dynamic treatments, a period lasted 200 ms. Such short periods can make it hard for participants to precisely time their actions. This could potentially explain why even for k = −6, we observe fewer battles and more escapes in 14The same argumentation would apply to a distaste for surprise or sus- pense. 15Sheremeta (2010), Price and Sheremeta (2011), and Cason, Masters, and Sheremeta (2018) all report evidence that joy of winning and risk aversion are important factors in driving subjects’ behavior in contest games. In a second price auction with value uncertainty, Goeree and Offerman (2003b) ﬁnd that bidders tend to submit bids below the expected value of the object, which suggests that risk aversion may be the stronger force. Sheremeta (2013) provides a survey. the dynamic game compared to the static game. We address this in a follow-up experiment.16 Experimental design and procedures. The design of exper- iment 2 closely follows that of the ﬁrst experiment. We col- lected data for all dynamic treatments, using periods of 5 seconds instead of 200 ms and four periods per round (with forty rounds in total). This gives subjects more scope to time their actions. We also added two items to the survey, measur- ing subjects’ joy of winning. The ﬁrst (incentivized) measure is taken from Sheremeta (2010, 2018). In this task, subjects can bid to win a contest with a prize of 0 points. For the second (nonincentivized) measure, subjects indicated how strongly they agreed with the statement: “I enjoy winning an amount by competing against another person more than I enjoy re- ceiving that same amount without having to compete for it” (rated on a 7-point Likert scale). The experiment was run online. Participants were recruited from the same subject pool as for the ﬁrst experiment (ex- cluding subjects who already participated). As in the ﬁrst experiment, we included test questions at the end of the in- structions. We showed the correct answers after two failed attempts on a question. We did this to prevent that subjects would log out if they had to wait too long. We kept track of the mistakes they made so that we can control for this in the analysis. In total, 168 subjects participated, with seven matching groups of eight subjects in each of the three dynamic treat- ments (k = −6, k = 6, k = 12).17 Sessions lasted around 60 minutes in total, and earnings varied between €4.20 and €35.70 (€19.00 on average). 16We thank a referee for this suggestion. 17We have some missing data for ﬁve subjects who lost the connection. If a subject could not be paired in a round because of this, he or she received the maximal payoff. l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / r e s t / l a r t i c e - p d f / / / / 1 0 4 2 2 1 7 1 9 9 6 1 4 0 / r e s t _ a _ 0 0 9 6 1 p d . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 FIGURE 7.—DISTRIBUTION OF DECISIONS OVER PERIODS BY DETERRENCE VALUE IN THE DYNAMIC GAME WITH FOUR PERIODS (EXPERIMENT 2) FIGHT OR FLIGHT 229 l D o w n o a d e d f r o m h t t p : / / d i r e c t . m i t . e d u / r e s t / l a r t i c e - p d f / / / / 1 0 4 2 2 1 7 1 9 9 6 1 4 0 / r e s t _ a _ 0 0 9 6 1 p d . f b y g u e s t t o n 0 8 S e p e m b e r 2 0 2 3 Left: First twenty rounds. Right: Final twenty rounds. Only observations where a player made a decision to ﬁght or ﬂee are included (omitting observations where a player was waiting when the other moved). Results. Figure 7 shows the timing of actions, which strongly resembles the results of experiment 1. With a nega- tive deterrence value, virtually all action happens in the ﬁrst period. With a positive deterrence value, many subjects wait until later periods. Compared to the ﬁrst twenty rounds (left panels), more subjects wait in the ﬁnal twenty rounds (right panels). The mean waiting time does not increase with ex- perience for k = −6 and does increase for positive deter- rence values (ﬁgure A3, appendix C). Moreover, we again observe that some participants act in the ﬁrst period when k > 0. In line with the theoretical predictions, some subjects
ﬂee just before the endgame, although some do so in period
2 rather than period 3, and very few ﬁght just before the
endgame.

Figura 4 plots the frequency of battles and escapes by the
weaker player in both experiments. The results are very com-
parable to those of experiment 1. In particular, for positive
values of k, the dynamic game leads to a reduction in battles
and an increase in escapes by the weaker player compared to
the static version. The difference between the static and dy-
namic game is signiﬁcant in all those cases (Mann-Whitney
test, P < 0.005 in all cases). The main difference with ex- periment 1 is that for a negative k, there is no reduction in battles or increase in escapes compared to the static games (p = 0.898 for battles, p = 0.368 for escapes). This supports the idea that in experiment 1, the decrease in battles and in- crease in escapes are driven by coordination failures: subjects may have attempted to immediately ﬁght but were not always able to precisely time their action.18 In experiment 2, we again observe anomalous early ﬁght- ing if k > 0. The two measures of joy of winning do not
explain this early ﬁghting, while the number of mistakes in
the test questions and the social dominance score do explain
(some) of the anomalous behavior (see table A7 in appendix
C). In experiment 2, we do not replicate the ﬁnding that risk
aversion correlates with ﬂeeing before the endgame (see table
A8 in appendix C).19

18A regression analysis conﬁrms these results. The interaction effects be-
tween dynamic timing and positive deterrence values are statistically sig-
niﬁcant, indicating that the effect of dynamic timing on battles and escapes
matters more for k > 0 (table A6 in appendix C).

19If we combine the data of both experiments, risk aversion is signif-
icantly correlated with ﬂeeing before the endgame and dominance with
early ﬁghting. See tables A9 and A10 in appendix C.

230

THE REVIEW OF ECONOMICS AND STATISTICS

V. Conclusione

in questo documento, we present a dynamic ﬁght-or-ﬂight game that
makes sense of a large range of conﬂicts observed in prac-
tice. We highlight the crucial role that the deterrence value
plays that players receive when the other player successfully
escapes. If it is negative, players will act in a split-second.
When it is positive, players will be patient and try to make
the other player ﬂee. An interesting feature of the analysis is
that if the deterrence value is positive but not too large, sorting
will occur. Questo è, the weakest players will ﬂee just before
the end, and thereby avoid costly battles. Così, this paper
clariﬁes how time can help people reach better outcomes in
dynamic games, even when time is not costly. The important
role of the deterrence value is conﬁrmed in our experiments.
Compared to a static version of the game, players are better
able to avoid costly battles.

In the experiment, we ﬁnd support for a behavioral version
of the model that allows for heterogeneous risk aversion. In
agreement with this model, sorting occurs for a wider range
of situations than predicted by the model with standard pref-
erences. Inoltre, subjects who appear to be more risk
averse in an independent task tend to be the ones who more
frequently ﬂee early, although we do not replicate this in
the follow-up experiment. We also observe an interesting
anomaly. A fraction of the players choose to ﬁght early even
in situations where the strategic incentive is to be patient. Nostro
conjecture is that some subjects come to the interaction with
a homegrown notion that it generally pays off to strike early
in contests. Over time, this costly behavior diminishes but
does not disappear.

We think that our setup provides a lower limit of the amount
of sorting that can be expected in practice. In our game, play-
ers manage to sort even though they do not receive any sen-
sory input about the ability of the opponent. In particular
when there is a strategic incentive to wait, sensory cues before
or during the contest may help players to avoid costly ﬁghts.
In an actual display, body odor or a high-pitched voice may re-
veal fear and help identify the weaker player (Mujica-Parodi
et al., 2009; Sobin & Alpert, 1999). A dominant performance
in a television show by a candidate running for presidential
ofﬁce may convince a weaker opponent that it is better to ﬂee
early. In the future, artiﬁcial intelligence may further help
players to agree on how they are ranked in terms of ability
before they engage in a costly battle. Relevant information
about the opponent’s ability will also affect players’ decisions
when the deterrence value is negative. Tuttavia, in such sit-
uations, a positive frequency of battles cannot be avoided.
Even when information about the opponent helps players to
perfectly forecast who will win the ﬁght, the stronger player
will still want to catch the weaker player in a battle. We think
that extending the analysis in this direction is an interesting
avenue for future research.

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