# The forager's dilemma: food sharing and food defense as risk-sensitive foraging options.

**ABSTRACT** Although many variants of the hawk-dove game predict the frequency at which group foraging animals should compete aggressively, none of them can explain why a large number of group foraging animals share food clumps without any overt aggression. One reason for this shortcoming is that hawk-dove games typically consider only a single contest, while most group foraging situations involve opponents that interact repeatedly over discovered food clumps. The present iterated hawk-dove game predicts that in situations that are analogous to a prisoner's dilemma, animals should share the resources without aggression, provided that the number of simultaneously available food clumps is sufficiently large and the number of competitors is relatively small. However, given that the expected gain of an aggressive animal is more variable than the gain expected by nonaggressive individuals, the predicted effect of the number of food items in a clump-clump richness-depends on whether only the mean or both the mean and variability associated with payoffs are considered. More precisely, the deterministic game predicts that aggression should increase with clump richness, whereas the stochastic risk-sensitive game predicts that the frequency of encounters resulting in aggression should peak at intermediate clump richnesses or decrease with increasing clump richness if animals show sensitivity to the variance or coefficient of variation, respectively.

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**ABSTRACT:**In many species, the intake rate of foraging individuals is negatively related to forager density due to agonistic interactions among foragers. Recently a variety of game theory models has been developed to address the question how such interference behaviour is shaped by nat- ural selection. These modelling approaches have not yet led to a com- prehensive understanding of interference competition; models that appear to be very similar yield strikingly different predictions regarding the evolutionary stability of various interference strategies. Here we attempt to unify approaches. To avoid model inconsistencies, we plead for a systematic, event-based description of the foraging process, the explicit account of feedback effects and the systematic derivation of a payoff function. To analyze the resulting evolutionary game, we use techniques from Adaptive Dynamics theory, since classical ESS tech- niques can be highly misleading when applied to the payoff functions resulting from interference competition. By means of this unified approach, we show that foraging animals can generally be expected to make their aggressive behaviour dependent on the role they play in interactions, that alternative evolutionarily stable interference strategies may evolve at the same ecological conditions, and that interference effects on intake rate cannot be taken for granted as the logical outcome of evolution. By critically discussing the setup, the assumptions and the way of analysis of some evolutionary models of interference competi- tion, we identify crucial assumptions and potential pitfalls in modelling the evolution of interference behaviour, and we demonstrate that the discrepancies between earlier model predictions often reflect seemingly subtle differences in the assumptions on behavioural flexibility. - SourceAvailable from: staff.city.ac.uk
##### Article: When should animals share food? Game theory applied to kleptoparasitic populations with food sharing

Behavioral Ecology 08/2012; 23(5). · 3.16 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Mechanisms promoting the evolution of cooperation in two players and two strategies (22) evolutionary games have been investigated in great detail over the past decades. Understanding the effects of repeated interactions in multiplayer spatial games, however, is a formidable challenge. In this paper, we present a multiplayer evolutionary game model in which agents play iterative games in spatial populations. -player versions of the well-known Prisoner's Dilemma and the Snowdrift games are used as the basis of the investigation. These games were chosen as they have emerged as the most promising mathematical metaphors for studying cooperative phenomena. Here, we have adopted an experimental approach to study the emergent behavior, exploring different parameter configurations via numerical simulations. Key model parameters include the cost-to-benefit ratio, the size of groups, the number of repeated encounters, and the interaction topology. Our simulation results reveal that, while the introduction of iterated interactions does promote higher levels of cooperative behavior across a wide range of parameter settings, the cost-to-benefit ratio and group size are important factors in determining the appropriate length of beneficial repeated interactions. In particular, increasing the number of iterated interactions may have a detrimental effect when the cost-to-benefit ratio and group size are small.IEEE Transactions on Evolutionary Computation 08/2012; 16(4):537-555. · 5.55 Impact Factor

Page 1

The Forager’s Dilemma: Food Sharing and Food Defense as Risk‐Sensitive Foraging Options.

Author(s): Frédérique Dubois and Luc‐Alain Giraldeau

Reviewed work(s):

Source: The American Naturalist, Vol. 162, No. 6 (December 2003), pp. 768-779

Published by: The University of Chicago Press for The American Society of Naturalists

Stable URL: http://www.jstor.org/stable/10.1086/379202 .

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Page 2

vol. 162, no. 6the american naturalistdecember 2003

The Forager’s Dilemma: Food Sharing and Food Defense

as Risk-Sensitive Foraging Options

Fre ´de ´rique Dubois*and Luc-Alain Giraldeau

De ´partement des Sciences Biologiques, Universite ´ du Que ´bec a `

Montre ´al, Case postale 8888, Succursale Centre-Ville, Montre ´al,

Que ´bec H3C 3P8, Canada

Submitted October 11, 2002; Accepted June 17, 2003;

Electronically published December 19, 2003

abstract: Although many variants of the hawk-dove game predict

the frequency at which group foraging animals should compete ag-

gressively, none of them can explain why a large number of group

foraging animals share food clumps without any overt aggression.

One reason for this shortcoming is that hawk-dove games typically

consider only a single contest, while most group foraging situations

involve opponents that interact repeatedly over discovered food

clumps. The present iterated hawk-dove game predicts that in sit-

uations that are analogous to a prisoner’s dilemma, animals should

share the resources without aggression, provided that the number of

simultaneously available food clumps is sufficiently large and the

number of competitors is relatively small. However, given that the

expected gain of an aggressive animal is more variable than the gain

expected by nonaggressive individuals, the predicted effect of the

number of food items in a clump—clump richness—depends on

whether only the mean or both the mean and variability associated

with payoffs are considered. More precisely, the deterministic game

predicts that aggression should increase withclumprichness,whereas

the stochastic risk-sensitive game predicts that the frequency of en-

counters resulting in aggression should peak at intermediate clump

richnesses or decrease with increasing clumprichnessifanimalsshow

sensitivity to the variance or coefficient of variation, respectively.

Keywords: aggression, foraging groups, ESS model, prisoner’s di-

lemma, risk-sensitive theory.

Group foraging animals frequently compete for resources

uncovered by others within the group (see Giraldeau and

Beauchamp 1999). This food discovered by others can be

consumed either exclusively by one forager that succeeds

in forcefully keeping the other competitors out of the

* Corresponding author; e-mail: fdubois@u-bourgogne.fr.

Am. Nat. 2003. Vol. 162, pp. 768–779. ? 2003 by The University of Chicago.

0003-0147/2003/16206-20379$15.00. All rights reserved.

clump or by several individuals that scramble for its con-

tent without any sign of overt aggression (Giraldeau and

Caraco 2000). From a theoretical point of view, the ques-

tion of which competitive tactic, fight versus scramble,

should be used has been addressed by game-theoretic

models (Sirot 2000; Dubois et al. 2003), most notably the

hawk-dove game (Maynard Smith and Price 1973). This

game considers a pair of opponents that compete for a

resource using either hawk, an escalating aggressive be-

havior, or dove, a nonescalating ritualized display, and

seeks the conditions under which one or the other strategy

is expected. Because a hawk uses force to gain exclusive

access to the food while a dove does not, an animal that

plays hawk always wins against one that plays dove. Con-

sequently, dove is never an evolutionarily stable strategy

(ESS) but can only exist as part of a mixed ESS at a

frequency that depends on the value of winning and the

cost of losing an aggressive encounter.

The basic hawk-dove game has been modified to gen-

erate more realistic predictions. In particular, games al-

lowing asymmetric players and assessment of these asym-

metries (e.g., Matsumura and Kobayashi 1998; Dubois et

al. 2003) as well as expanding the contests to include more

than two competitors (Dubois et al. 2003) have received

some empirical support (see Sirot 2000; Dubois et al.

2003). However, none of the games developed to date

predicts the generalized nonaggressive dovelike scrambles

for food clumps of the type reported in pigeons (Columba

livia; Giraldeau and Lefebvre 1986), water crickets (Velia

caprai; Erlandsson 1998), nutmeg mannikins (Lonchura

punctulata; Giraldeau et al. 1990), and many other organ-

isms argued to forage by local enhancement. Hawk-dove

games typically consider only a single contest and evaluate

the ESS for that contest over different sets of conditions.

Group foraging situations, however, likelyinvolvethesame

opponents that interact repeatedly over discovered food

clumps. If the players can modify their behavior from one

round to the next on the basis of previous experience, the

problem of fighting versus nonfighting over food clumps

within foraging groups can be addressed as an iterated

game where the solution depends not only on the expected

Page 3

The Forager’s Dilemma769

Table 1: Payoff matrix of the hawk-dove game

Choice of the finder

Choice of the joiner

HawkDoveTFT

Hawk(F?a)/2?C

(F?a)/2?C

a

F?a

a?w[(F?a)/2?C]

(F?a)?w[(F?a)/2?C]

F

0

(F?a)/2

(F?a)/2

(F?a)/2

(F?a)/2

F?w[(F?a)/2?C]

w[(F?a)/2?C]

(F?a)/2

(F?a)/2

(F?a)/2

(F?a)/2

Dove

TFT

Note: F is the total number of items contained in a food clump, a is the finder’s advantage, C is the

energetic cost induced by fighting, and w is the probability of interacting with the same individual that

the one encountered during the previous move of the game. Within each cell, the first line represents the

expected gain of the finder, whereas the second line represents the average gain expected by the joiner.

gain in the current contest but also on future gains in

subsequent contests. We therefore develop an iterated

hawk-dove game adapted to a group foraging contest and

ask, what conditions, if any, predict the adoption of non-

escalating scramble food exploitation strategies?

The Social Foraging Hawk-Dove Game

A foraging hawk-dove game must correspond to a number

of important characteristics of the group foraging process

(Dubois et al. 2003). The first individual to arrive at a

food clump can exploit it before the arrival of the con-

testant. This means that the finder can gain more from

the resource than the joiner, a correlated role asymmetry

that will be included in the foraging game (Giraldeau et

al. 1990; Vickery et al. 1991). While injury or even death

may be reasonable expressions of fighting costs for highly

valuable resources suchas mates (HoustonandMcNamara

1991), energy and time losses are more realistic depictions

of fighting costs for food clumps, and so the model will

incorporate time and energy costs rather than injury. Fi-

nally, the game must deal with foraging groups that are

of finite size and composed of individuals capable of al-

tering their behavior strategically according to the current

payoffs of the alternatives. If the parameters F and a rep-

resent the number of food items per clumpandthefinder’s

advantage, respectively, the ESS of a similar but noniter-

ated hawk-dove foraging game is (H, H) when the cost of

fighting (C) is smaller than(F ? a)/2

wise, where the first letter denotes the finder’s strategy and

the second the joiner’s (Dubois et al. 2003). The reason

why (H, D) is the only ESS when the costs of fighting are

relatively large comes from the assumption that the finder

and the joiner decide sequentially whether to compete

aggressively, since the finder arrives on the clump before

the joiner and hence has more informationabout thevalue

of the resource for which they compete when it is joined

and (H, D) other-

(Dubois et al. 2003). Here we explore the outcome of

allowing iterated play.

A Deterministic, Iterated, Social Foraging

Hawk-Dove Game

The payoffs of the game (table 1) are based on the fol-

lowing assumptions. A pair of foragers compete for a food

clump containing F indivisible items using either an ag-

gressive hawk behavior or a nonfighting dove behavior.

When two doves meet, they share the contested resource

equally at no cost. When a dove meets a hawk, the dove

leaves all the remaining food without a fight. When two

hawks meet, they engage in an escalated fight with ener-

getic cost C until one of them leaves the clump, with each

having a .5 probability of being the victor. The contestants

do not arrive at the clump simultaneously, so the finder

gets the finder’s advantage a (

arrives. The ESS is a strategy that, if adopted by all mem-

bers of a population, cannot be invaded by a mutant strat-

egy through the operation of natural selection (Maynard

Smith and Price 1973). Both the finder and the joiner can

use hawk or dove, resulting in four conditional strategies:

(H, H), (H, D), (D, H), and (D, D), where the first letter

denotes the finder’s strategy and the second the joiner’s.

We consider an individual that plays n consecutive

rounds of the foraging game in a group of

In every round, the individual is paired at random with

one of the G other group members. Because the foraging

success of an animal may depend on its age or experience

(Goss-Custard and Durell 1987; Sundstro ¨m and Johnsson

2001), we assume that individuals differ in their efficiency

p () at discovering food clumps. An animal with0 ! p ! 1

will be the finder of the clump over all of the np p 1

consecutive food discoveries. Inversely, when

join others’ food discoveries over all of the n consecutive

rounds. Finally, when an individual’s

finding efficiency does not differ from the average effi-

) before the joiner0 ≤ a ! F

foragers.G ? 1

, it willp p 0

, its relativep p 0.5

Page 4

770The American Naturalist

ciency of the other group members so that it will be the

finder in half the n rounds. Given that an individual’s

probability of discovering food clumps depends on its age

or experience, we assume that at a given moment each

group member has a fixed finding efficiency p and that

individuals have perfect information about their own rel-

ative finding efficiency.

When the Costs of Escalation Are Relatively Large

When

a single contest, the finder obtains a greater payoff if it

plays hawk rather thandoveagainstanopponentthatplays

dove, while the joiner obtains a greater payoff if both play

(D, D) rather than (H, D). Thus, if all the group members

play (H, D), the average expected gain per round of an

animal with finding efficiency p will be

, that is, when (H, D) is the ESS forC 1 (F ? a)/2

:E(H,D)

E

p pF. (1)

(H,D)

And if all groupmembersplay(D,D),theaverageexpected

gain per round of an animal with finding efficiency p will

be:E(D,D)

F ? a

2

E

p pa ?

.(2)

(D,D)

The peaceful sharing of food clumps by the finder and

its joiner can be an ESS only if both opponents do better

in every round when playing (D, D) rather than (H, D),

that is, if. Solving this inequality requiresE

1 E

(D,D)(H,D)

. Iffor one of the two opponents, then itp ! 0.5p ! 0.5

must follow that for the other. The consequencep 1 0.5

of this is that one opponent has no interest in competing

nonaggressively when it finds, and so (D, D) cannot be

an ESS.

When the Costs of Escalation Are Relatively Small

When

a single contest, both the joiner and the finder have a

higher payoff in every round if both play (D, D) rather

than (H, H), provided thatC 1 0

an animal that plays (H, H) against a (D, D) player is

greaterthan that of a (D, D)playingagainstitselfregardless

of its finder or joiner role. This means that (D, D) can

always be invaded by the (H, H) strategy and hence cannot

be an ESS. This foraging situation, when

analogous to the prisoner’s dilemma game, in which two

players choose between cooperation and defection (Ax-

elrod and Hamilton 1981). In its original formulation, the

prisoner’s dilemma game is characterized by a payoff ma-

trix in which players obtain a greater payoff from mutual

, that is, when (H, H) is the ESS forC ! (F ? a)/2

. However, the payoff of

, isC ! (F ? a)/2

cooperation than from mutual defection. However, de-

fecting against a cooperator provides the highest payoff,

while cooperating against a defector results in maximum

penalty, hence the dilemma. Expressed as a foraging di-

lemma, mutual dove provides a greater payoff thanmutual

hawk. However, a dilemma arises because playing hawk

against a dove leads to maximum payoff, whereas playing

dove against a hawk leads to maximum losses. As in the

prisoner’s dilemma, it is always best to play hawk in a

single round of play of the forager’s dilemma no matter

what the opponent does. As a consequence, mutual de-

fection is the expected outcome. Iterated games, however,

could generate stable mutual dovelike strategies when the

number of repetitions of the game is sufficiently large and

the opponents adopt a conditional strategy like tit-for-tat

(TFT), which consists of playing dove on the first round

and then doing whatever the opponent did on theprevious

round (Axelrod and Hamilton 1981). For now, we ignore

the many variants of conditional strategies (e.g., Nowak

and Sigmund 1993) in order to explore the outcome of

the simplest available conditional strategy.

We ask whether a TFT foraging strategy could resist an

(H, H) invasion by comparing the average expected gain

per round of an (H, H) and a TFT player over n consec-

utive rounds in a group where the G other individuals all

play TFT. We assume that a TFT player adopts the (D, D)

strategy on the first round and then copies its opponent’s

strategy on all subsequent rounds (Axelrod and Hamilton

1981). Table 1 shows the gains expected by a TFT player

in relation to the probability w of interacting again with

the same individual. Conventionally, evolutionaryanalyses

of the iterated prisoner’s dilemma game assume that the

game is played repeatedly by the same two individuals and

hence evaluate the mean payoff of eachstrategistaccording

to a constant probability w of interacting with the same

player on the next move of the game. We assume that

animals remember all opponents they encounter so that

they can modify their behavior according to their oppo-

nent’s play on the previous round. So if an animal plays

n consecutive rounds in a group of

have on each round i a probability

opponent and a probability1 ? ai

it has previously met, with

foragers, it will

of meeting a new

of facing an opponent

G ? 1

ai

i?1

)

G ? 1

(

G

a p

i

, (3)

where

already played.

When an (H, H) player encounters a TFT opponent for

thefirsttime,thelattercompetesnonaggressively,andhence

the aggressive competitor obtains either F or

denotes the number of rounds the animal hasi ? 1

foodF ? a

Page 5

The Forager’s Dilemma 771

items depending on whether it is the finder or the joiner,

respectively. When the two opponents have previously met,

however, they both play (H, H) and engage in an escalated

fight until one of them retreats, leaving all the remaining

resource to its opponent.

So, let denote the average expected gain per roundE(H,H)

of an animal with finding efficiency p that plays (H, H),

with

E

p

(H,H)

n

1

n

F ? a

2

#

a # (pa ? F ? a) ? (1 ? a) # pa ?

i

? C ,

?

ip1

i

[()]

(4a)

which simplifies to

n

F ? a

2

1

n

F ? a

(

2

E

p pa ?? C ?

#

a #

? C .

?

ip1

(H,H)i [)]

(4b)

If we replace by its value given in equation (3), we get

ai

F ? a

2

1

n

F ? a

(

2

E

p pa ?? C ?

#

? C

(H,H)

)

2n?1

)

G ? 1

(

G

G ? 1

G

G ? 1

(

G

…

# 1 ????

,

) ()

[]

(4c)

which can be simplified to

n

F ? a

2

G

n

F ? a

(

2

G ? 1

(

G

E

p pa ?? C ?

#

? C # 1 ?

.

(H,H)

))

[]

(4d)

When an animal plays TFT in a group of all TFT, the

two opponents compete nonaggressively in every round.

So, let denote the average expected gain per roundE(TFT)

of an animal with finding efficiency p that plays TFT, with

F ? a

2

E

p pa ?

. (5)

(TFT)

The TFT can resist the invasion of (H, H) if the average

expected gain per round of TFT is greater than that of (H,

H) in a group where all other group members play TFT,

that is, if. Solving this inequality requiresE

! E

(H,H)(TFT)

n

F ? a

2C

G ? 1

(

G

n 1 G 1 ?

# 1 ?

.(6)

())

[]

Moreover, when all group members play (H, H), the av-

erage expected gain of an animal playing (H, H) is

F ? a

2

E

p pa ?? C,(7)

(H,H)

whereas the gain expected by an animal playing TFT and

whose expression is given in table 1 is equal to

F ? a

(

2

E

p pa ? w

? C .(8)

(TFT)

)

The aggressive (H, H) strategy is an ESS if it can resist

the invasion of TFT, that is, if

inequality requires

. Solving thisE

1 E

(H,H)(TFT)

F ? a

(

2

(1 ? w) #

? C 1 0.(9)

)

Given that

C 1 0

C ! (F ? a)/2

, thenif0 ≤ w ≤ 1E

1 E(F ? a)/2 ?

(H,H)(TFT)

and hence, as in the case of a noniterated game, if

.

Predictions

The model predicts that TFT should resist the invasion of

(H, H) more effectively in small groups sizes and when

the number of consecutive rounds expected to be played

by each group member is large (fig. 1). This arises because

(H, H) does better than TFT if an animal faces a TFT

opponent for the first time but worse if it faces an op-

ponent it has previously met, because in this case the TFT

opponent does not retreat, as in the first round, but es-

calates. So, when the number of competitors within the

group is small or when an animal has already been in-

volved in a large number of contests, its probability of

facing an opponent it has previously met is high, which

is expected to prevent (H, H) from invading a TFT pop-

ulation. Inversely, in very large group sizes or when the

number of consecutive rounds expected to be played by

each group member is small, an (H, H) player has an

increased probability of obtaining thewholeresourcewith-

out a fight in every round, simply because it meets a series

of new TFT players that play dove.

Increasing the finder’s advantage reduces the remaining

amount of food for which animals compete. So, when the

finder’s advantage is very large, the quantity of food that

Page 6

772The American Naturalist

Figure 1: Average expected gain per round for an animal who plays

either (H, H) or TFT in a population of TFT players in relation to the

number of consecutive rounds;F p 6 a p 2 C p 1 p p 0.5

.G p 3

,,,, and

can be gained from joining a clump is likely to be insuf-

ficient to cover the energetic cost of a fight. Inversely,when

food clumps contain a large amount of food, the benefits

of fighting are increased. The probability that (H, H) in-

vades a TFT population is then predicted to increase with

the number of items in a clump—clump richness—but to

decrease as the finder’s advantage increases. Resource dis-

tribution in space and time is likely to affect both the

number of simultaneously available food clumps and their

richness. For instance, increasing the temporal clumping

of resources increases the number of food clumps that are

simultaneously available. Inversely, for a fixed total num-

ber of food items, increasing the spatial clumping reduces

the number of simultaneously available food clumps and

increases the number of items available per clump. So,

from these assumed effects of spatiotemporal food distri-

bution, we predict that the probability that TFT can resist

invasion from an (H, H) strategist will be higher when

resources are either clumped in time or dispersed in space.

Furthermore, if, as it is often assumed (Caraco and Gi-

raldeau 1991; Vickery et al. 1991), the finder’s advantage

results from the time required for a joiner to reach the

discovered food clump, then as the density of competitors

increases, the time required to reach a discovery and hence

the finder’s share may decline. If increasing group size

increases forager density and if increased density reduces

the finder’s advantage, then we predict that the proportion

of (H, H) players and hence the level of aggression should

increase with group size. Moreover, by definition, the

finder’s share decreases with increasing number of items

per clump.

The model predicts that an animal’s decisiontocompete

aggressively or not does not depend on its relative finding

efficiency. Indeed, when an animal has an increased prob-

ability of discovering food clumps, its average expected

gain increases regardless of whether it plays (H, H) or TFT,

and the difference in the average expected gain between

the two strategies remains constant.

So, in summary, the deterministic social foraging hawk-

dove game predicts that food should be shared without

aggression more frequently when the number of simul-

taneously available food clumps is large, a situation that

is most likely to occur when resources are clumped in time

or dispersed in space into many clumps of low richness.

It predicts, in addition, that food sharing will be common

only when the finder’s share (

decreases with increased clump richness and that we as-

sume should also decrease with increased group size.

) is large, a variable thata/F

A Stochastic, Iterated, Social Foraging

Hawk-Dove Game

In the previous deterministic game, the variability in the

amount of food associated with each foraging strategy was

ignored such that TFT is the ESS when its mean payoff

when playing against itself is greater than when playing

against any alternative strategy. However, an individual’s

payoffs likely vary from one round to the next as a result

of the strategy it adopts and the asymmetry in roles of

finder and joiner that influences the quantity of food that

can be obtained from a clump. We now consider the effect

that variability may have on the survival value of alter-

native foraging strategies for animals that are risk sensitive,

that is, that are sensitive to differences in payoff uncer-

tainty (Caraco 1981; for a review, see Kacelnik and Bateson

1996). Risk-sensitive foraging theory predicts that an an-

imal prefers the more certain option when its energy re-

quirements are low and the more uncertain or variable

option when they are high (Stephens 1981; Bateson and

Kacelnik 1997). Risk-sensitive behavior may therefore in-

fluence the expected outcome of the forager’s dilemma,

so we calculate not only the mean payoff of (H, H) and

TFT in a population of TFT players over n consecutive

rounds but also the variance in payoff associated with each

strategy.

Analysis

In the deterministic model, we assumed that a TFT player

adopts the (D, D) strategy on the first round and then

copies its opponent’s strategy on all subsequent rounds.

To simplify the analysis, here we assume that an animal

playing TFT competes nonaggressively if it encounters an

opponent for the first time or if its opponent has never

been observed to be aggressive in any previous play with

any other group members. Otherwise, it plays the same

Page 7

The Forager’s Dilemma 773

strategy the opponent adopted in the previous round.

There is some empirical realism to this assumption. For

instance, the agonistic displays exchanged during a fight

and the outcome of the encounter are often detectable by

neighbors who are not directly involved in the interaction

(see Johnstone 2001; McGregor et al. 2001). Empirical

studies also report that when individuals can gather in-

formation from interactions between others, they use this

information in subsequent encounterswiththecontestants

(e.g., Oliveira et al. 1998; McGregor et al. 2001). Now an

animal that plays (H, H) in a group of TFT players will

face a nonaggressive opponent during the first round but

an aggressive opponent in all subsequent (

for any group size. So, its average expected gain per round

will be:E(H,H)

) roundsn ? 1

1

n

F ? a

2

E

p

# pa ? F ? a ? (n ? 1) # pa ?? C ,

(H,H)

[( )]

(10a)

which can be simplified to

F ? a

2

1

n

F ? a

(

2

E

p pa ?? C ?

#

? C .(10b)

(H,H)

)

However, the average expected gain per round of TFT will

be equal to:E(TFT)

F ? a

2

E

p pa ?

.(5)

(TFT)

Variances associated with each strategy were used as a

measure of variability and calculated using the formula

. So let

V(X) p E(X ) ? E(X)

associated with the (H, H) strategy, with

denote the variance

22

V(H,H)

22

V

p pF ? (1 ? p)(F ? a) ? (n ? 1)

(H,H) {

(

22

F ? a

2

F ? a

(

2

# p a ?? C ? (1 ? p)

? Cn

))

[]}

U

?

pF ? (1 ? p)(F ? a) ? (n ? 1) (11a)

{ (

2

F ? a

2

F ? a

(

2

# p a ?? C ?(1 ? p)

? Cn ,

[ () )]}Z )

which can be simplified to

2

n ? 1

n

F ? a

(

2

2

V

p pa (1 ? p) ?

#

? C . (11b)

(H,H)

2

)

However, the variance associated with the TFT strategy is

equal to , withV(TFT)

222

F ? a

2

F ? a

(

2

F ? a

2

V

p p a ?? (1 ? p)

? pa ?

,

(TFT)

())()

(12a)

which can be simplified to

2

V

p pa (1 ? p).(12b)

(TFT)

As above, the average expectedgainperroundofplaying

(H, H) decreases as the number of consecutive rounds

increases, whereas it is independent of n for TFT players.

Thus, the average expected gain per round of a mutant

(H, H) outweighs playing TFT when the number of con-

secutive rounds is smaller than, with

∗

n

F ? a

2C

∗

n p 1 ?

.(13)

The strategic choices of risk-sensitive foragers will be

affected by differences in both the payoffs mean and var-

iance, and the relative importance of each depends on a

number of parameters (see Kacelnik and Bateson 1996).

To simplify our analysis, we formulate our predictions on

the basis of the following two assumptions: first, the dif-

ferences in the magnitude of variance in rewards will affect

the animals’ preferences only when the averagenetbenefits

do not differ too widely between the twostrategies;second,

when the two alternatives provide lowaverage payoffs,that

is, when animals are on negative energy budgets, individ-

uals will prefer the more variable strategy. A recent review

(Shafir 2000) indicates that the relative magnitude of var-

iability provides a better account of patterns of risk sen-

sitivity than measures of variability per se and suggests

that the coefficient of variation (CV) is a better measure

of variability for the purposes of risk sensitivity analyses.

Therefore, we also estimated the coefficient of variation

as(SD/mean) in order to explore the condi-CV p 100

tions, if any, under which the use of one or the other

variable may lead to different predictions.

Predictions

Neither the mean nor the variance in payoff associated

witheachstrategydependsongroupsize.Therefore,unlike

the deterministic game, the stochastic game predicts that

Page 8

774The American Naturalist

Figure 2: Payoff variance for (H, H) against TFT and TFT against TFT

in relation to the finder’s advantage;

.0.5

,,, andF p 10 C p 1 n p 5p p

Figure 3: Average expected gain per round for an animal that playseither

(H, H) or TFT in a population of TFT players in relation to clump

richness;,, , anda p 2 C p 1 n p 5.p p 0.5

the frequency of aggression will be unaffected by the num-

ber of competitors within the group. However, as for the

deterministic game, the probability that TFT resists an (H,

H) invasion increases with the number of expected con-

secutive rounds. Increasing n decreases the variance as-

sociated with the (H, H) strategy but has no effect on the

payoff variance for TFT. So, when few consecutive rounds

are expected to be played, the mean, variance, and coef-

ficient of variation of the (H, H) payoffs are all greater

than those for TFT. When the expected number of con-

secutive rounds increases, the mean expected gain of (H,

H) per round decreases. As the number of consecutive

rounds increases, players are increasingly likely to meet

their energy requirements and so should become risk

averse and prefer the less variable foraging TFT option.

Like the deterministic game, the stochastic game pre-

dicts that when the finder’s advantage declines, a greater

number of expected consecutive rounds is requiredtokeep

(H, H) from invading TFT. When the finder’s advantage

is small, the variance of the (H, H) payoff is larger than

the variance associated with TFT, but the difference be-

tween the two declines as the finder’s advantage increases

(fig. 2). As a consequence, for a given number of expected

consecutive encounters, (H, H) is more likely to be the

ESS when the finder’s advantage is small andthelikelihood

decreases as the finder’s advantage increases. Likewise, for

a given finder’s share, the likelihood that (H, H) is the

ESS declines with an increase in the expected number of

consecutive encounters. Using the coefficient of variation

as a measure of variability leads to the same predictions:

increasing the finder’s advantage increases the coefficient

of variation for both (H, H) and TFT but decreases the

difference between the two.

As above, increasing clump richness (F) increases the

mean payoff of both strategies, although at a faster rate

for the (H, H) strategy. So TFT expects more food than

(H, H) when the number of food items per clump is small

but less when F is large (fig. 3). The difference in the mean

rewards for each strategy, however, is weak, even for ex-

treme values of F (fig. 3), and so strategic choice more

likely depends on differences in the levels of variability.

The energy budget rule (Stephens 1981) can be used to

predict the effect of clump richness on risk sensitivity. In

environments where food clumps are rich(large F),players

are likely to be on positive energy budgets and so are

expected to be risk averse. When environments offer

clumps of low richness (small F), however, the animals

are more likely to be on negative energy budgets and so

are expected tobe risk proneinordertoreducethechances

of suffering an energetic shortfall. As clump richness in-

creases, so does the variance associated with (H, H), while

the variance associated with TFT remains constant (fig.

4A). When clump richness is low and individuals are un-

likely to meet their energetic requirements, the risk-prone

(H, H) strategy will therefore be preferred. This preference

will increase with clump richness because the difference

in variance betweenthetwostrategiesincreaseswithclump

richness (fig. 4A). At some critical clump richness, animals

will expect to meet their energy requirement and will

switch to a risk-averse foraging strategy and hence prefer

the TFT alternative. The preference for that alternativewill

increase with increasing clump richness as the (H, H) be-

comes increasingly risky compared with TFT. Conse-

quently, for risk-sensitive animals, the stochastic hawk-

Page 9

The Forager’s Dilemma 775

Figure 4: Effect of clump richness on the variability associated with

payoffs for an animal that plays either (H, H) or TFT in a population

of TFT, when the variability is expressed either by the variance (A) or

by the coefficient of variation (B);a p 2 C p 1 n p 5,, , and.p p 0.5

Figure 5: Average expected cumulative gain for an animal that plays

either (H, H) or TFT in a population of TFT players in relation to its

finding efficiency relative to that of the other group members;

, , and.a p 2 C p 1n p 10

,F p 10

dove foraging game predicts a dome-shaped relationship

between aggression and clump richness; it will increase

over a range of low clump richness values for which an-

imals do not expect to meet their energy requirements,

reach a maximum at some intermediate richness when

energy requirements are expected to be met, and then

decrease with further increases in clump richness. This

leads to a new prediction: the clump richness (F) at which

maximum aggression is observed will depend on total en-

ergy requirement. A different prediction is drawn if one

uses the CV as a measure of variability. Aggression is now

predicted to decrease in frequency with increasing clump

richness. This arises because as clump richness increases,

so do the mean payoffs of both strategies, leading to a

decrease in the coefficients of variation (fig. 4B). Thus, as

clump richness increases, not only do individuals face a

lower probability of starvation but their level of risk sen-

sitivity is also likely to decrease, leading to an increased

use of TFT.

Increasing p, an animal’s relative food finding efficiency,

affects neitherthedifferenceinthemeansnorthevariances

associated with each strategy (fig. 5). So the decision of

whether to compete aggressively or not remains indepen-

dent of an individual’s p. However, an individual with a

low p likely expects to encounter less food than one with

a high value of p, and as a consequence the average ex-

pected gain of an individual with a low p is smaller than

that of an individual that obtains the finder’s advantage

during each round, regardless of whether it plays (H, H)

or TFT (fig. 5). Animals with low p are therefore likely to

be risk prone, while those with high p are likely to be risk

averse. Given that both the variance and the coefficient of

variation associated with the (H, H) strategy are always

greater than those associated with the TFT strategy, ani-

mals with the smallest p should play the (H, H) strategy

more frequently than those with higher values of p. Spe-

cifically, we expect that all individuals will share the re-

source with other group members when they have similar

finding efficiency (i.e.,p p 0.5

because then the average cumulative gain of a TFT player

exceeds the food requirement (fig. 5). When individuals

differ in their efficiency at discovering food clumps, how-

for all group members)

Page 10

776The American Naturalist

ever, neither strategy provides sufficient average energy for

individuals whose finding efficiency is !0.25. Given that

the variance of the (H, H) payoff is much larger than the

variance associated with TFT, then their only chance of

meeting their energy requirement is to compete aggres-

sively in order to get the whole food clump. We therefore

predict that the frequency of encounters resulting in ag-

gression will increase with an increase in the asymmetry

in finding efficiency among group members.

To summarize, like the deterministic game, the sto-

chastic game predicts that food sharing will be common

when the number of simultaneously available food clumps

is large—a situation that, for reasons explained in “A De-

terministic, Iterated, Social Foraging Hawk-Dove Game,”

is more likely to occur when resources are clumped in

time or dispersed in space into many clumps of low rich-

ness—and when the finder can consume a large number

of food items before the joiner arrives. However, it predicts

that aggressive interactions will increase in frequency as

the asymmetry in finding efficiency among group mem-

bers increases and will reach a maximum for intermediate

clump richnesses.

Discussion

Foraging group members that exploit food clumps that

can be shared may be engaged in a forager’s dilemma.

Social foraging games provide one of the few biologically

relevant nonhuman situations for the application of the

logic of the prisoner’s dilemma (see Clements and Ste-

phens 1995; Brembs 1996). Both the iterated deterministic

and stochastic games predict that conditions exist under

which the ESS for group foraging animals is to compete

nonaggressively and hence scramble for resources (Gi-

raldeau and Caraco 2000). Moreover, the stochastic game

shows that the use of aggressive escalation (H, H) cor-

responds to a risky foraging alternative that may be valu-

able under conditions that call for the use of a risk-prone

foraging behavior. Therefore, when assessing the effects of

competitor number and resource distribution on the fre-

quency of aggression within a group, it may prove nec-

essary to consider the differences between the means as

well as the levels of variability in payoffs associated with

each strategy.

The Effect of Competitor Number on Aggression

within Foraging Groups

Both the deterministic and the stochastic games predict

that a decline in finder’s shares leads to higher rates of

aggression because the value of the remaining food in-

creases. If increasing competitor number reduces mean

interindividual distances, then a challenger likely reaches

a food finder sooner in large group sizes. If this is so, then

increasing group size may reduce the finder’s share, and

so aggression should increase with competitor number in-

dependently of competitor encounter frequency.

The deterministic game predicts that TFT is more re-

sistant to invasion from (H, H) as the likelihood of re-

peated play among players increases. When the competitor

number is large, so is the number of potential opponents,

and so a mutant (H, H) more likely meets a new TFT

opponent on each encounter, therefore obtaining each

clump without a fight (because TFT always plays dove on

first encounters). Thus, as competitor number increases,

the likelihood that (H, H) is the ESS increases, leading to

an increase in aggression levels once again independently

of encounter frequency of opponents. Unlike the deter-

ministic game, the stochastic game predicts that the cost

of defending and hence the likelihood that (H, H) is the

ESS are unaffected by competitor number. This prediction,

however, stems from the model’s assumption that indi-

viduals can gather information about potential opponents

from interactions in which they were not directly involved

(e.g., Oliveira et al. 1998; McGregor et al. 2001). Sirot’s

(2000) model also predicts that the number of escalated

fights should increase with competitor number. However,

the increase in the number of aggressive interactions with

competitor number predicted by Sirot is due to an increase

in the encounter rate with conspecifics, which leads to a

decrease in the proportion of unchallenged prey items.

Although many empirical studies in foraginggroupsreport

higher frequencies of aggression for large competitornum-

bers (e.g., Caraco 1979; see also Sirot 2000), tests need to

discriminate whether the observed increase in the number

of aggressive interactions with competitor number results

from an increase in the likelihood to engage in aggression

or in the encounter rate between group members. Fur-

thermore, one simulation model developed to predict the

altruistic sharing of food among vampire bats (Desmodus

rotundus; Wilkinson 1984) predicts that the frequency of

altruistic acts should increase withcompetitornumber,but

that prediction has yet to receive empirical support.

Food Distribution and Abundance

If increased numbers of simultaneously available food

clumps translate into larger expected numbers of encoun-

ters among players, then our game predicts that peaceful

food sharing will be more common in environments with

a larger number of available food clumps. That is because

increased expected repeated play prevents (H, H) from

invading a TFT population. The average expected gain per

round of (H, H) in a population of TFT decreases with

an increase in the expected number of consecutive rounds,

while that of TFT remains constant. The difference in the

Page 11

The Forager’s Dilemma 777

average expected gain between (H, H) and TFT therefore

decreases with the number of consecutive rounds. As the

difference between the two declines, the number of ag-

gressive competitors should also decline. Because the spa-

tial distribution of food in an environment likely affects

the number of simultaneously available food clumps, we

predictthatthefrequencyofaggressionwillalsobeaffected

by food distribution. We assume that the number of si-

multaneously available food clumps increases with in-

creased temporal clumpiness. In addition, for a given total

number of items, decreased spatial clumpiness should lead

to a larger number of poorer clumps. If our assumptions

are correct, then our games predict that the proportion of

encounters resulting in aggression increases with increased

spatial clumpiness and with decreased temporal clumpi-

ness. Several empirical studies support thepredictedeffects

of clumping on aggression levels (Grant and Kramer 1992;

Bryant and Grant 1995; Robb and Grant 1998). Moreover,

our predictions are consistent with a sexual selection game

that predicts that males whose probability of surviving

between contests is low should compete aggressively more

frequently than males whose life span is longer (Houston

and McNamara 1991). Houston and McNamara’s (1991)

model, however, assumes that fights may result in death,

so that the number of consecutive rounds expected to be

played by each male is not fixed as it is in the foraging

game but depends on whether the animal plays hawk or

dove. Therefore, although each escalated fight represents

a loss of opportunities for future contests, Houston and

McNamara’s (1991) model predicts that hawk is an ESS

when the male’s life span in the absence of fighting is

short, because in this case the value of future contests is

low.

Resource defense theory, which is based on the idea of

economic defendability, also predicts that aggression

should increase as resources become clumped in space but

less clumped in time (Brown 1964; Grant 1993), but for

different reasons. In particular, resource defense theory

predicts that aggressive interactions should occur more

frequently when resources are clumped in space, because

in this case only a small area needs to be defended to gain

access to a large portion of the resources. In contrast,when

resources are temporally clumped, any time spent on de-

fense is time away from resource exploitation.

The deterministic game predicts that the frequency of

aggression increases with clump richness as do other ESS

models on aggression (Sirot 2000; Dubois et al. 2003). In-

creasing the number of food items per clump increases the

difference between the average expected gain of an aggres-

sive versus nonaggressive animal that faces a nonaggressive

opponent. The stochastic game, however, predicts that the

effect of clump richness on aggression depends on whether

animals show sensitivity to the coefficient of variation or

the variance. Animals sensitive to the coefficient of vari-

ation should show decreasing aggression levels with in-

creasing clump richnesses. Shafir (2000) reports that sen-

sitivity to the coefficient of variation applies mostly to

nectarivores when variability concerns the volume of nec-

tar. The coefficient of variation, however, has nosignificant

effect on the strength of risk sensitivity for studies with

nonnectarivores, including birds and fishes, in which var-

iability is in the number of food items. For animals that

are risk sensitive in response to variance, our stochastic

game predicts an increase in aggression over an initially

low range of clump richness, with a peak at a clump rich-

ness that corresponds to a switch from negative to positive

energy budgets followed by a decline in aggression with a

further increase in clump richness. Because animals will

more likely meet their energy requirements when clumps

contain a large amount of food, peaceful food sharing

should occur more frequently under these conditions. Ac-

cordingly, several experimental studies report the absence

of agonistic interactions when animals forage in rich

clumps where they reach satiety before the clump is de-

pleted (Goss-Custard 1970; Kotrschal et al. 1993). More-

over, evidence for a dome-shaped relationship between

aggression and food abundance has recently been dem-

onstrated in convict cichlids (Archocentrus nigofasciatum;

Grant et al. 2002). Thus, Grant et al.’s results apparently

contradict predictions from previous game-theoreticmod-

els on aggression (Sirot 2000; Dubois et al. 2003) but give

some support to our model, emphasizing the need to take

into account differences in magnitude of variance in re-

wards when attempting to predict animals’ preferences.

Unlike the deterministic game, the stochastic game al-

lows us to consider temporal patterns of aggression over

the course of a day. Because both the animals’ energy

reserves and the amount of food they need to survive to

the next feeding event likely change over the course of the

day, the frequency of aggression will also change with the

time of day. In particular, late in the day just before the

nonforaging period, animals have to maintain their energy

reserves at a very high level in order to survive until the

next morning. Animals with low reserves that are unlikely

to survive the night should therefore adopt a risk-prone

strategy and play hawk. Early in the day, however, when

only a small amount of foodisrequiredtoavoidstarvation,

foragers should most frequently play dove in ordertomax-

imize their chances of obtaining a sufficient amount of

food to remain above the survival thresholdwhileavoiding

any loss of energy associated with fighting.

Competitor Asymmetry in Food Finding Efficiency

The stochastic game predicts that which of the competitive

tactics is used depends on p, the animal’s food discovering

Page 12

778The American Naturalist

efficiency relative to that of the other group members.

Specifically, animals whose finding efficiency is below the

population’s average should compete aggressively more

frequently than those whose efficiency is above average.

Inefficient food finders likely have low energetic reserves

and hence are more likely to prefer the aggressive risk-

prone strategy. In this study, we have considered that an

animal’s probability of discovering food clumps relative

to that of the other group members is fixed. Many ex-

perimental studies, however, report that an individual’s

propensity to discover food may vary in response to var-

iations in clump richness (Koops and Giraldeau 1996) or

spatial distribution of food (Coolen et al. 2001). An ani-

mal’s decision to compete aggressively or not is predicted

to depend on clump richness, since this parameter affects

the quantity of food that can be gained from a clump and

hence the costs and benefits of defending. However, if the

frequency at which group foragers join others’ food dis-

coveries is influenced by clump richness, the animals’ en-

ergy requirements will also vary with this parameter,which

should in turn affect the level of aggressiveness. Although

a large number of game-theoretic models have addressed

the issue of the frequency of the scrounger tacticinsystems

of scramble and aggressive kleptoparasitism (Broom and

Ruxton 1998; Ruxton and Broom 1999; Giraldeau and

Caraco 2000), no model has yet addressed the issue of the

factors that contribute to the frequency of aggression

within kleptoparasitic systems. Further extensions for our

model should then investigate the effects of competitor

number and resource availability on the level of aggres-

siveness as well as the use of producer and scrounger.

Conclusion

Our stochastic hawk-dove iterated foraging game model

is the first attempt to cast aggressive resource defense in

risk-sensitive foraging framework. The result is that the

model makes a number of predictions that are consistent

with both earlier game theoretic games (Sirot 2000; Dubois

et al. 2003) as well as resource defense models (Brown

1964; Grant 1993). While the predictions are qualitatively

similar, they may be quantitatively different. Moreover, it

is often the case that the predicted effect follows from

different causal chains so that it is imperative now that

the predictions of the iterated game be subjected to ex-

perimental investigation.

Acknowledgments

This research was financially supported by a postdoctoral

fellowship from Foundation Fyssen (France) and Foun-

dation Singer-Polignac (France) to F.D. and by a Natural

Sciences and Engineering Research Council (Canada) Dis-

covery Grant as well as a grant from Fonds Que ´be ´cois de

la Recherche sur la Nature et les Technologies to L.-A.G.

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Associate Editor: Joel S. Brown

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