Behavioral Ecology Vol. 13 No. 3: 393-400
© 2002 International Society for Behavioral Ecology
Food sharing: a model of manipulation by harassment
Department of Ecology, Evolution, and Behavior, University of Minnesota, 1987 Upper Buford Circle, St. Paul, MN 55108, USA
Address correspondence to J.R. Stevens. E-mail: jeff{at}nash.cbs.umn.edu .
Received 16 June 2001; revised 16 July 2001; accepted 7 August 2001.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
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Most analyses of food-sharing behavior invoke complex explanations such as indirect and delayed benefits for sharing via kin selection and reciprocal altruism. However, food sharing can be a more general phenomenon accounted for by more parsimonious, mutualistic explanations. We propose a game theoretical model of a general sharing situation in which food owners share because it is in their own self-interest—they avoid high costs associated with beggar harassment. When beggars harass, owners may benefit from sharing part of the food if their consumption rate is low relative to the rate of cost accrual. Our model predicts that harassment can be a profitable strategy for beggars if they reap some direct benefits from harassing other than shared food (such as picking up scraps). Therefore, beggars may manipulate the owner's fitness payoffs in such a way as to make sharing mutualistic.
Key words: food sharing, harassment, manipulation, mutualism, game theory.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
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Red colobus monkeys (Colobus badius) are an important and desirable source of protein for chimpanzees (Pan troglodytes; Goodall, 1986
). A monkey kill
attracts beggars that sidle up to the carcass owner, extend their hands,
vocalize, and sometimes grab a scrap of meat. In some situations, the carcass
owner may share with the beggar, breaking off a piece of meat and handing it
to a beggar. Why would the owner give away food that it could consume?
Food sharing, defined here as joint use of a monopolizable food source, can
be as active as this chimpanzee example or as passive as a lioness allowing
another lioness to feed on her gazelle carcass. Either way, sharing appears to
exemplify animal altruism because one individual accepts a fitness cost while
another receives a fitness benefit. Although sharing may be altruistic at the
time of the sharing event, mechanisms such as kin selection
(Hamilton, 1964) and
reciprocal altruism (Trivers,
1971) may provide indirect or delayed benefits to the sharer
(de Waal, 1989;
Mitani and Watts, 2001;
Perry and Rose, 1994). In the
case of reciprocal altruism, the food is recouped in the future; thereby
making sharing altruistic in the short term but selfish in the long term. The
complicated cognitive machinery necessary for reciprocal altruism and the
presence of relatives necessary for kin selection make these explanations
limited in the contexts in which they apply. If, however, sharing leads to an
immediate gain for the owner, the situation becomes much more general, and the
more complex, long-term accounts are unnecessary. Applying resource defense
theory (Brown, 1964;
Ydenberg et al., 1986) to our
general concept of sharing allows us to consider unexplored explanations of
sharing: non-food owners may harass or interfere with the owner's feeding,
thereby making it uneconomical to defend the food.
In our chimpanzee example, a beggar may harass the owner for food by screaming, grabbing at the carcass, or stealing the entire carcass. Often the owner defends its carcass, thus risking injury and incurring energetic costs, opportunity costs of slowing feeding rate, and other costs such as attracting more beggars. The beggar faces similar but probably reduced costs of its own when harassing. Therefore, the beggar can influence the net payoffs to the owner by inflicting or withholding the costs of harassment. If the costs of harassment are large, sharing might yield a higher net benefit to the owner than defending. If the beggar alters the owner's net benefits enough to change the owner's optimal strategy, the beggar has manipulated the payoffs in a way that makes sharing in the owner's immediate self-interest (i.e., sharing becomes mutualistic). This manipulative mutualism may occur commonly in situations in which an actor's behavior manipulates the net payoffs for another individual's cooperative behavior, making cooperation mutualistic rather than altruistic.
Harassment is a particularly interesting factor that may influence sharing across many situations and taxa because the beggar's actions rather than extrinsic forces (such as patch profitability or travel time) determine the owner's payoffs and optimal decision. To analyze the effect of harassment on food sharing explicitly, we consider an asymmetric game theoretical model. This model is one of only a few models of food sharing and, more important, one of the first to analyze the potential immediate fitness benefits associated with food sharing. This model examines the circumstances under which harassment and sharing should occur as well as optimal amounts of harassment and sharing.
Model
Elements of the sharing/harassment game
Consider two animals: a resource owner, who possesses a valuable food item
of size A (for amount), and a beggar, who has nothing but is aware of
the owner's food item. Both individuals may forage elsewhere, but this food
item offers a much more valuable fitness benefit per unit time. The owner may
choose to share a portion of its food, say As (for amount
shared). The beggar can choose to harass the owner or leave him alone. We
express the beggar's harassment intensity as a rate, c (e.g.,
measured in calories per second). If the beggar harasses the owner for time
t, this costs the beggar ct, and it costs the owner
ct (where represents a conversion factor that captures
how the beggar's harassment affects the owner). We assume that the beggar
harasses at intensity c when it is not busy eating; that is, while
the beggar consumes shared food, it does not or cannot harass.
Consumption and time available for harassment. We base our model
on time in the sense that we calculate the costs and benefits in terms of the
time engaged in various activities and corresponding rates of expenditure
during these activities. We assume that both players consume food at rate
r, so a single animal consuming the entire resource (amount
A) will spend time A/r eating. Next, we express the time
available for harassment (Th) in the form
 | (1) |
(p)
gives the proportion of the maximum consumption time (A/r) that is
available for harassment. If the owner shares nothing, the beggar has the
entire time available for harassment, so we expect that (0) = 1.
Similarly, if the owner gave everything away, the beggar would have no time
available to harass, so we expect that (1) = 0. More generally, we expect
that the time available for harassment will decrease as the proportion shared
increases. In this game, then, the owner influences the cost and duration of harassment by controlling As, while the beggar influences the cost of harassment by controlling the harassment intensity c.
The simplest game
Using these assumptions, we can write down the benefits to the owner
(Bo) as a function of the proportion of A shared
(denoted by p) and intensity of harassment c:
 | (2) |
The benefits obtained by the beggar, Bb, also depend on
the proportion shared and the intensity of harassment:
 | (3) |
In the next step in finding a solution to the game, we ask how the owner's choice of p affects the beggar's optimal c and vice-versa.
No share/no harass. The simple structure of
Bb(p, c) leads us to a simple conclusion.
"No sharing" (p = 0) and "no harassment"
(c = 0) is the only Nash equilibrium of the game as currently
constituted (a strong Nash equilibrium is equivalent to an evolutionarily
stable strategy, or ESS; Maynard Smith,
1982). The beggar's benefits can only decrease with increasing
harassment intensity (dBb/dc =
(p)/r), meaning that c = 0 is the beggar's best
option regardless of the owner's behavior. A nonzero harassment intensity
cannot persist because, according to Equation 3, a beggar that reduces its
harassment intensity always increases the benefits it obtains. In turn, this
means that p = 0 represents the owner's best choice because
Bo(p, 0) = A(1 - p) can only
decrease with increasing p. This result may seem disappointing
because no sharing and no harassment make for an uneventful interaction. We
believe, however, that it reflects a common natural situation: when an owner
possesses a completely defendable resource, a harasser only incurs costs by
harassing, and it only benefits by recognizing the possessor's ownership and
moving on to some other possibility.
Noncontingent benefits of harassment
This situation can change if harassment has some direct benefits that
accrue even if the owner does not share. A harasser may, for example, collect
scraps, cause a distracted owner to spill, or actively steal parts of the
resource. We call these gains the "noncontingent benefits" of
harassment because they do not depend on the owner's sharing.
Modeling noncontingent benefits. We suppose that these
non-contingent benefits should increase with the intensity of harassment,
c, and with the time available for harassment,
(A/r)(p). With these in mind, we can rewrite our benefit
functions as
 | (4) |
 | (5) |
A noncontingent benefit factor k can destabilize the no share/no
harass equilibrium by making harassment worth-while in its own right.
Generally speaking, if k > 1, the beggar can benefit from
harassment regardless of the owner's behavior, and the beggar's optimal
harassment level (
) should be the maximum intensity, say
c*. If, however, k < 1, the beggar should not
harass, and no share/no harass is the only equilibrium.
The owner's problem and the (p) function
We expect that the beggar should either not harass ( = 0)
or harass at the maximal intensity ( =
c*). To study whether the owner should share when
harassed, and if so, how much, we need to know more about the function
(p) that specifies the proportion of the maximum consumption time
available for harassment. In the Appendix we derive a (p)
function using a stochastic model of the sharing process. This model assumes
that the resource is subdivided into n discrete pieces, with the
owner deciding whether to share each chunk with probability p. Notice
that this subtle reinterpretation of p means that we should think of
p as the average proportion shared, rather than the realized
proportion shared—technically, we now have Ap =
E(As), instead of Ap =
As. If the resource is not divisible (n = 1),
(p) decreases linearly with p with a slope of
-1['(p) = -1; Figure
1]. For divisible resources (n > 1), the
(p) function has a slope of -2 at p = 0, increases in
slope around p =.5, and has a slope of 0 at p = 1
(Figure 1). Regardless of
divisibility, (p) always decreases with p.
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Optimal sharing when harassed. With the basic properties of
(p) in hand, we can now show how harassment should affect the
owner's willingness to share (p). To begin, we differentiate the
owner's benefit function (Equation 4):
 | (6) |
For indivisible resources (n = 1), '(p) = -1,
and
 | (7) |
We predict, therefore, that the owner should defend an indivisible resource
(set p = 0) when
 |
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The dimensionless term r/c( + k) plays an
important role in our model. The numerator includes the food consumption rate,
and the denominator expresses the rate at which costs accrue during
harassment. It represents, therefore, an efficiency—a quotient of rate
of benefit gain over rate of cost accrual, and we call it the
"efficiency of consumption when harassed" (ECH).
A divisible resource, however, complicates the analysis. Because
' (0) = -2, we know that the owner maximizes its benefits at
p = 0 (no sharing) when
 | (8) |
 | (9) |
That is, the owner should share a divisible resource only when the ECH is < 2.
Optimal amount to share. If 2 > r/c( +
k) 
0, then we expect the owner to share some portion of a
divisible resource (Figure 2). Although the complexity of (p) prevents a general algebraic
specification of the optimal p (or
), a graphical method gives a
relatively complete characterization. The
value is the solution of
 | (10) |
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0 (negative derivative at p =
0), the owner should not share
(
Because ' (p) is an increasing sigmoid function of
p, -' (p) is a sigmoid function that decreases
from 2 at p = 0 to 0 at p = 1.
Figure 3 shows how
is related to the term
r/c( + k). To find
corresponding to a given
r/c( + k), we locate r/c( +
k) on the vertical axis and trace a horizontal line to the sigmoid
-(p) curve, then we trace a vertical line to p axis to
the find . This graphical solution
gives a relatively complete picture of the economics of sharing: (1)
increases as r/c( +
k) decreases; (2) if r/c( + k) 2,
= 0; (3) if 2 >
r/c( + k) 1 then.5
> 0; (4) if 1 >
r/c( + k) 0 then 1
>.5.
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In addition, this graphical solution shows the effect of resource divisibility. For a more divisible resource (higher n), the sigmoid function will be more abrupt and steplike (Figure 3B), shifting optimal sharing proportions closer to one-half.
Assembling the pieces. Now that we have a reasonably complete
picture of the owner's and beggar's options, we assemble these pieces into a
game theoretical analysis. The simple form of the beggar's problem makes this
job easier because we only have to consider = 0 and
= c* (where c* is the
maximal intensity). This simplifies things because we only need to consider
two possibilities for the owner as well: the best reply to zero harassment and
the best reply to maximal harassment. As discussed above, the owner's best
reply to no harassment is no sharing
( = 0). We denote the best reply to
maximal harassment as p* and remark that this is given by:
 | (11) |
 | (12) |
Since we have two alternatives for each player ( = 0 or
= c* for the beggar and
= 0 or
= p* 
0 for
the owner), we can gain some intuition about the game using the familiar tool
of the two-by-two game matrix, as shown in
Figure 4.
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We can now characterize all possible Nash equilibria (share/no harass is never an equilibrium; Figure 5):
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- No share/no harass. If k < 1, the noncontingent benefits of
harassment are too small, so the beggar should not harass, and as a
consequence the owner should not share.
- No share/harass. If k > 1 and r/c* (
+ k) 2, noncontingent benefits make harassment worthwhile for
the beggar, but sharing does not benefit the owner because of the high ECH.
That is, harassment has little effect on the owner's consumption rate.
- Share/harass. If k > 1 and r/c* ( +
k) < 2, again, noncontingent benefits make harassment worthwhile
for the beggar, but now sharing benefits the owner because of high costs of
harassment relative to the rate of food consumption (low ECH).
Model discussion and conclusions
Now we review and highlight several key features and variables of the
model. First, notice that adopting the maximum harassment intensity,
c*, does not necessarily mean that the beggar will spend
much time harassing the owner. Our model assumes that beggars harass only when
not consuming shared food. Considering the three equilibria listed above,
then, we would expect the most harassment in the no share/harass case, the
least harassment in the no share/no harass case, and an intermediate amount in
the share/harass case.
The parameter k measures the noncontingent benefits of harassment
and is probably the most important variable in the model. The condition
k > 1 simply means that the benefits of harassment must outweigh
the costs even if the owner does not share. An animal that harasses when
k < 1 simply burns its own resources (and an owner's best strategy
is to let the harasser do so). Our model's second key parameter, the
efficiency of consumption when harassed, or ECH [r/c*
( + k)], measures the speed of food intake relative to the
cost rate of harassment. The role of intake rate (r) agrees with
intuition, for we do not expect sharing of small or easily processed
resources.
Given that the resource is divisible (n > 1), the amount of divisibility does not affect the equilibria outlined above, but it should affect the proportion shared when sharing occurs. Our model predicts that the amount shared should approach one-half as resource divisibility increases (Figure 3). One-half is special in our model because we assume that both players feed at the same rate. It follows that if the owner wants to eat the maximum amount in peace (keeping the beggar occupied while it eats), then a 50:50 split will achieve this goal.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
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Our model explores the effects of harassment on food sharing. For harassment to be profitable, the beggar must receive benefits for harassing (high k) whether or not the owner shares (e.g., gathering scraps, stealing small pieces). In the face of this harassment, an owner may share if harassment sufficiently reduces its feeding efficiency [r/c* (
+ k)].
Importance and implications
Harassment and manipulation
As one of the first models to explicitly examine immediate benefits of food
sharing, we set the stage for more general or parsimonious explanations of
sharing than kin selection and reciprocal altruism
(de Waal, 1989;
Perry and Rose, 1994).
Clutton-Brock and Parker (1995)
reviewed general forms of manipulation and punishment in animal societies, but
only two studies provide evidence that harassment influences animal sharing.
First, Wrangham (1975)
suggested that harassment may play a large role in chimpanzee sharing of
colobus monkey meat after kills. These kills often attracted beggars that
vocalized, used begging hand gestures, and even attacked the owner
(Goodall, 1986;
Wrangham, 1975). If the owner
shared part of the food, the recipient usually left, followed by a small band
of beggars. Wrangham (1975)
hypothesized that the owners "paid" the beggars with pieces of
food to avoid harassment. More recent evidence indicates that chimpanzees that
harass intensely receive more food than those that harass less intensely
(Gilby IC, unpublished data).
Hauser and Marler (Hauser,
1992; Hauser and Marler,
1993) described an extreme example of harassment affecting food
sharing. In experiments involving rhesus macaques (Macaca mulatta),
Hauser and colleagues provided food to individual monkeys that were out of
view of other monkeys. They found that, when detected by others, monkeys that
did not give food recruitment calls experienced more aggression than monkeys
that called. Calling females consumed more food than silent females because
silent females dropped food while being chased
(Hauser, 1992). The increase
in callers' consumption indicates that calling may be immediately
mutualistic.
Noncontingent benefits
Our model emphasizes benefits that beggars receive even if the owner
defends the food; that is, the owner cannot unilaterally defend the entire
food source, so harassing the owner to gather scraps or steal pieces of food
may benefit the beggar. Without these incentives to remain close to the owner,
harassment is not profitable, and in the absence of harassment, the owner has
no incentive to share.
Kummer and Cords (1991)
conducted experiments on captive long-tailed macaques (Macaca
fascicularis), varying non-contingent benefits for non-food owners. They
found that non-food owners tended to harass more (via stealing, stealing
attempts, and food manipulation) when the owner could not completely control
the food than when it could control the food. Unfortunately, the authors did
not present data on the non-food owner's success rate for obtaining food.
Resource divisibility
Our finding that resource divisibility does not affect the owner's decision
to share is a bit curious. Although it does not directly influence sharing,
divisibility may indirectly influence the decision to share if it affects the
noncontingent benefits of harassment. We did find, however, that divisibility
determines the proportion of the food source that the owner should share if it
does share: owners should share about one-half of highly divisible food.
Consider a lioness consuming a freshly killed gazelle. A single lioness can
defend an intact carcass, but when the carcass begins to disintegrate, the
lioness may have difficulty defending the entire carcass, and she may allow
others to take small pieces. Elgar
(1986) suggested that, upon
discovery of a food source, house sparrows (Passer domesticus)
sparrows gave "chirrup" calls, thereby attracting conspecifics and
decreasing individual predation risk. Interestingly, the sparrows called more
frequently after discovering a divisible food source than a solid source.
Hauser et al. (1993) reported
similar results with chimpanzees: individuals gave more food-associated calls
when consuming a divided watermelon than when consuming an intact watermelon.
Perhaps frequent sharing of a divisible food source is simply a question of
sheer monopolizability. Defending multiple food sources may prove much more
difficult than defending a single source.
Latent harassment
In natural situations in which owners defend food before sharing (such as
in chimpanzees), harassment is obvious. In situations in which the owner
shares immediately, however, blatant harassment may not appear even if the
ever-present threat of harassment maintains the sharing. Of course, latent
harassment may prove difficult to observe in nature, necessitating empirical
manipulations of sharing and potential for harassment. In a related vein, the
overt harassment and food defense may be an information-gathering ritual for
both the owner and beggar; each one gauging the other's motivation and resolve
(see Ydenberg et al., 1986,
for applications of the war of attrition to resource defense).
Related models
Although few models directly focus on the immediate benefits of food
sharing (but see Giraldeau and Caraco,
2000; Mesterton-Gibbons and
Dugatkin, 1999), several classes of models lay the foundation for
theoretically exploring the evolution of food sharing.
Resource defense
Whereas the term "sharing" often evokes thoughts of one
individual actively donating food to another, sharing can be much more
generally defined in terms of two or more individuals consuming a resource
that one can monopolize. This broader concept of sharing encompasses many
instances of resource defense. Brown
(1964) originally described the
"economic defendability" of territory defense as the circumstances
under which an individual should accept the costs involved in defending a
territory. Others have extended this work to the defense of food sources.
Ydenberg et al. (1986) modeled
the defense of food sources in a way relevant to this model by considering the
effects of interference (analogous to our harassment) on foraging decisions.
Their model predicted that interference will slow intake rate, thereby making
defense un-economical for individuals that are far from their home and for
those with richer food patches in their home range. The combined effect of
interference and the asymmetries in home-range distance and richness may allow
subordinate individuals to feed in patches with dominant individuals or even
interfere with and exclude dominant individuals from patches.
Tolerated theft
Blurton Jones (1984,
1986,
1987) argued that an asymmetry
of value based on satiety might be important in tolerated theft situations.
Although the next morsel of food is not very important to the sated owner, it
may be very valuable to the hungry beggar. Because of this decrease in the
marginal value of food during consumption, the owner should tolerate theft of
food by the beggar.
Although this idea provided a valuable foundation for modeling food
sharing, the important aspect of behavior in game theory is the relative value
of an individual's options, not the value difference between individuals.
Blurton Jones mentioned that owners must weigh the costs of defending food,
but never incorporated this idea into the model. Without this key inclusion,
the analysis ignores strategy stability. Winterhalder
(1996a,b)
continued the marginal analysis of tolerated theft, but still did not apply a
game theoretical approach.
Producer/scrounger games
Our model examines the conditions under which a non-food owner benefits by
harassing an owner and an owner benefits by sharing with the beggar. The
producer-scrounger game (Barnard and Sibly,
1981; Vickery et al.,
1991) addresses whether individuals specialize in either searching
for food individually (producing) or avoiding costs of foraging by
parasitizing the finds of the producers (scrounging); thereby assuming that
harassing and sharing occur. Keep in mind, however, that individuals can
choose between the two strategies; that is, for any given foraging bout, an
individual is either a producer or scrounger, but it can choose either
strategy in a future bout. Giraldeau and colleagues have experimentally
investigated theory-based predictions on the effects of dominance, resource
divisibility, role specialization, patch departure time, and competition
intensity in a producer-scrounger situation
(Beauchamp and Giraldeau, 1997;
Giraldeau et al., 1990).
Extensions
Destabilizing the no share/no harass equilibrium
In our model, harassment can only persist when harassing produces
noncontingent benefits for the harasser, and this result has led us to
hypothesize that phenomena such as stealing and scrap collecting are
prerequisites to harassment-induced sharing. We do not, of course, claim that
this is the only way to destabilize the no share/no harass equilibrium, but it
does seem to be a plausible and parsimonious approach. This result arises in
our model because we assume that the beggar's gains change linearly with
harassment intensity, so that the optimal harassment intensity must be either
the minimum level (0) or the maximum level (c*). Future
work, ideally guided by empirical results, might explore nonlinear benefit
functions which can (in theory) destabilize the no share/no harass equilibrium
without noncontingent benefits.
Food consumption rate
Food consumption rate (r) is an important parameter in our model.
We assume that the players have similar consumption rates, which means that
the amount required to keep a harasser busy is similar to the amount the owner
will be able to eat in peace. In natural situations, consumption rates may be
quite different because of differences in sex, age, or levels of satiety.
Systematic variation in individual consumption rates may provide an
interesting avenue to explore both theoretically and empirically. An informal
application of our model's logic suggests that differential consumption rates
may influence the beggar's decision to harass and the owner's decision to
share and how much to share. For example, owners may be more willing to share
with slow eaters (e.g., juveniles who have not learned efficient food handling
techniques) because they can be kept busy at minimal cost.
N-player game
Our model considers only two players to simplify the problem and to conform
to a standard two-by-two game matrix. In natural situations, however, multiple
beggars often surround an owner. We speculate that including multiple beggars
in our model will increase the overall intensity of harassment, thereby
forcing the owner to share more frequently (any parent knows it is easier to
tolerate harassment from one child than from several children). Chapman and
Kramer (1996) found
experimentally that as the number of food competitors increased, the owner's
intake rate decreased, guarding success decreased, and total number of chases
peaked at intermediate competitor numbers. The difficulty in analyzing the
effects of beggar number on sharing lies in how to distribute the food in such
a way to minimize harassment costs when facing multiple beggars. Further
analysis is necessary to explore optimal amounts of food that an owner should
share with multiple beggars: should the owner share one large piece to draw
some of the beggars away, or should it share small pieces with every
beggar?
The optimal strategy of the beggars offers a challenge as well. One can
imagine multiple beggars in a situation similar to that of a group of vigilant
prey. Like the concept of corporate vigilance
(Bertram, 1980), a beggar would
probably benefit more from having additional beggars around to increase
chances of sharing. Packer and Abrams
(1990) modeled vigilance
situations and found that Nash equilibrium vigilance levels were often lower
than Pareto equilibrium (or co-operative optimum) vigilance levels. Similarly,
food beggars are tempted to cheat or not harass by relying on harassment by
others, thereby avoiding their own costs of harassment.
Summary
Using a game theoretical approach, we modeled the effects of harassment on
food sharing. Our model predicts that a non-food owner should harass an owner
when the nonowner can gain benefits even in the absence of sharing. These
non-contingent benefits (such as gathering dropped scraps) can recoup
energetic costs of harassing. An owner should only share when a beggar
harasses, significantly reducing its consumption rate. Therefore, if an owner
consumes the food slowly, a beggar can harass for long periods of time, so the
owner pays high costs of defending. Experimentally manipulating parameters
such as feeding rate, noncontingent benefits, resource divisibility, and
number of beggars in a sharing context could provide rigorous tests of our
model.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
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The
(p) function and time available for harassmentHere we determine how the amount shared, As, influences the time available for harassment. Because we assume that both players feed at rate r, the owner consumes for time
 |
 |
The difference
 |
 |
Substituting this into Equation 1 suggests a (p) function of
the form
 | (A1) |
(p) as required.
Figure 1 uses Equation A1 to
plot (p) as a function of the proportion shared p.
Although one might construct a model based on the "kinked"
(p) function (Figure
1) discussed above, it is both inconvenient and implausible. It is
inconvenient because the discontinuity at p =.5 means that all
remaining calculations must also take account of this condition. It is
implausible because stochastic variation in consumption rates (r) and
the amount shared (As) will combine to create smooth
expected harassment time curve [E(Th)]. The next
few paragraphs discuss one simple way to incorporate this stochasticity.
Resource divisibility and binomial sharing
Some resources divide easily into parts, whereas others cannot. Suppose
that the resource in question can be divided into n equal parts of
size A/n. Now suppose that when the owner chooses the proportion to
share, p, it determines the probability of sharing each part. In this
scenario the number of parts shared is a random variable drawn from a binomial
distribution with parameters p and n, where p
represents the owner's willingness to share (p =
As/A), and the n represents the divisibility of
the resource.
The assumption that a binomial process governs sharing allows us to specify
completely the expected time available for harassment given the owner's
willingness to share, p. If the owner shares m of the
n parts, the owner retains amount
 |
 |
 |
n/2. To find the expected time available for
harassment [E(Th)], we calculate the product of
the time available (Th) and the probability of m
pieces being shared, summed over every possible m. We use
 |
).
 | (A2) |
(p)
function that we seek. Figure 1
shows this (p) function for a range of resource divisibilities
(n values). The figure compares this family of (p)
functions to the piecewise function (Equation A1) obtained when we assume
infinite divisibility and error-free sharing. For indivisible resources
(n = 1), (p) is a straight line [(p) = 1 -
p; Figure 1]. As
resource divisibility increases, (p) approaches the piecewise
function derived earlier (Equation A1; see
Figure 1).
Although we cannot express (p) in an algebraically convenient
closed form, we can easily state the important properties of (p).
The most important feature of (p) is its derivative at zero
[' (0)]. Direct differentiation shows
 | (A3) |
' (1) = 0]. In the indivisible case, (p) = 1 -
p, so ' (p) = -1 for all p. Finally, we
observe that (p) decreases with p (technically, it is
nonincreasing with p).
- Index of variables
- A
- entire resource amount
- As
- amount owner shares with beggar
- Bb
- fitness benefits received by beggar
- Bo
- fitness benefits received by owner
- c
- intensity of harassment
- c*
- maximum intensity of harassment
- optimal intensity of harassment
- ECH
- efficiency of consumption when harassed
- k
- noncontingent benefits factor
- m
- number of discrete parts of resource shared by owner
- n
- total number of discrete parts of resource
- p
- proportion of total amount shared
- p*
- proportion of total amount shared that is optimal reply to
c*
- optimal proportion of total amount shared
- r
- consumption rate
- t
- total harassment time
- Th
- time available for harassment
- effect of beggar's harassment intensity on owner's fitness
- (p)
- proportion of maximum consumption time that is available for harassment
when owner shares p proportion of food
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ACKNOWLEDGEMENTS |
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We thank Ian Gilby, Craig Packer, Alison Pearce, and two anonymous referees for reviewing the manuscript. We thank the Packer/Pusey/Stephens lab group at the University of Minnesota for stimulating discussions on this topic. This project was funded by the National Science Foundation grant IBN-9896102 (to D.W.S.).
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