The ESS in an Asymmetric Generalized War of Attrition with Mistakes in Role Perception

J. theor. Biol. (2002) 214, 329}349 doi:10.1006/jtbi.2001.2454, available online at http://www.idealibrary.com on

The ESS in an Asymmetric Generalized War of Attrition with Mistakes in Role Perception PATSY HACCOU*-

AND

OLIVIER GLAIZOT?

*Institute of Evolutionary and Ecological Sciences, ¸eiden ;niversity, P.O. Box 9516, 2300 RA ¸eiden, ¹he Netherlands and ?Museum of Zoology, P.O. Box 448, CH-1000 ¸ausanne 17, Switzerland (Received on 26 March 2001, Accepted in revised form on 31 August 2001)

We derive the ESS for the generalized asymmetric war of attrition, where payo!s to contestants may vary in time and may depend on some characteristic, called the &&role'' of an individual. We use the same approach as Hammerstein & Parker (1982), who examined an asymmetric war of attrition. We consider two roles, A and B. Role A is assumed to be favoured with respect to payo!s. It is assumed that there is always a true asymmetry, so in each contest one individual has role A and the other has role B. It is assumed that roles are assigned to contestants at random and that they can make mistakes in role perception. It is shown that, under certain assumptions about shapes of payo! functions and probabilities of making mistakes, there is an ESS which can be characterized by two probability distributions with non-overlapping support. Individuals who perceive their role as A should choose larger persistence times. This ESS structure is similar to that in the asymmetric war of attrition. In that model, the resource values and the cost rates are constant. We consider situations where all these values may change in time and where rewards and costs may be equal after some "nite time. Such shapes of payo! functions arise naturally in competitive patch depletion (Sjerps & Haccou, 1994a, b). As a result, the probability density functions that specify the conditional strategies are no longer necessarily negative exponentials (as in the war of attrition), but may have very di!erent shapes. Furthermore, under some conditions there is a maximum persistence time, at which there can be an atom of probability. We give explicit expressions as well as numerical approximations for the ESS.  2002 Elsevier Science Ltd

Introduction The war of attrition is a model of contest behaviour, where two or more contestants compete for an indivisible resource (Maynard Smith, 1974). Prior to the contest, each competitor chooses a persistence time which is unknown to the others. The individual that chooses the longest persistence time obtains the reward. Each individual pays a cost which is equal to some positive -Author to whom correspondence should be addressed. 0022}5193/02/030329#21 $35.00/0

constant (the cost rate) times the duration of the game. Thus, the payo! of a &&winner'' of the game is the resource minus the cost and the payo! of a &&loser'' is minus the cost. This model has been applied in a variety of contexts, e.g. to study contests between male dung #ies (Parker, 1970a, b), or between larval damsel#ies (Crowley et al., 1988). In the original model, competitors are equivalent with respect to cost rate and reward. In many cases, however, there are di!erences between contestants that a!ect their expected reward and/or  2002 Elsevier Science Ltd

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cost rate, so-called &&payo!-relevant asymmetries''. For instance, food resources may be more valuable to hungry individuals than to sated ones, large animals may have smaller cost rates than small ones, etc. (Maynard Smith & Parker, 1976; Hammerstein, 1981; Parker & Rubinstein, 1981). Many empirical examples and references can be found in Riechert (1998). Bishop et al. (1978) studied a war of attrition where contestants may have di!erent &&roles'' that a!ect the resource value, but do not a!ect their cost rates. They assumed that there are a limited number of roles, and that each individual knows its own role, but not its opponent's. Contestants were paired at random. They showed that the ESS is characterized by a set of non-overlapping probability density functions, depending on the roles. Individuals with higher resource values should choose higher persistence times. Hammerstein & Parker (1982) derived the ESS for an asymmetric war of attrition, where the resource value as well as the cost rate depends on the role of the contestants. They considered two possible roles, which are called A and B. One of the roles is &&favoured''. This means that individuals with that role have a smaller cost rate and/or a higher reward. In contrast to the model considered by Bishop et al. (1978), they assumed that there is always a true asymmetry, i.e. two contestants can never have the same role. They considered situations with two contestants and argued that no ESS can exist if both have complete information about their role. Note that this implies that they also know their opponent's role. Usually however, individuals will not have complete information about their roles in a contest (Parker & Rubinstein, 1981). For instance, birds returning to their breeding sites from their winter range may not recall the value of their territory accurately (Walton & Nolan Jr, 1986). Ydenberg et al. (1988) argue that "ghts between territory owners and strange intruders are usually more vigorous than those between neighbours, due to lack of information on relative resource value and/or cost rates. Under the more realistic assumption of mistakes in role perception, Hammerstein & Parker (1982) showed that an ESS does exist and is characterized by two probability density functions, +p , p ,, where a and ? @ b denote the perceived roles. They derived ex-

plicit expressions for these functions and showed that if A is the favoured role, the support of p is ? the interval [s, R) and that of p [0, s) where s is @ a positive constant. Thus, the structure of the ESS is similar to the one found in the war of attrition with random rewards, that was examined by Bishop et al. (1978). Asymmetric contests with a continuum of possible roles have been examined by Kura & Kura (1998). They found that in such cases, pure conditional strategies do exist, i.e. contestants should choose a "xed persistence time that depends on their resource holding potential. Mesterton-Gibbons et al. (1996) examined e!ects of perception of di!erences in resource holding potential between contestants on the existence of a pure ESS. In the standard war of attrition model rewards as well as cost rates are "xed constants. This is not always true. For instance, competitors may deplete a resource during their contest, or changes in weather conditions may a!ect the expected future payo! of leaving a site (Sjerps & Haccou, 1994a, b). Cost rates of contests may also vary in time due to weariness (Payne & Pagel, 1996). To deal with such situations, Bishop & Cannings (1978) introduced the &&generalized war of attrition''. This is where the reward that the &&winner'' obtains at the end of the game is a function of the duration of the contest. Furthermore, the costs that both individuals pay are not necessarily proportional to the duration of the game. Sjerps & Haccou (1994a, b) further generalized their results to include cases where costs and rewards may be equal after a "nite time. They showed that the thus generalized model can be used to study patch leaving under competition when the payo!s correspond to expected net energy gain rates. They only considered symmetric situations. In this paper, we consider an asymmetric generalized war of attrition for contests between two individuals with two possible roles. We study situations where rewards are always higher than costs (as in the model of Bishop & Cannings, 1978) as well as the situations examined by Sjerps & Haccou (1994a, b), where rewards and costs may be equal after a "nite time. In the model of Kura & Kura (1998), the payo! of being the "rst to leave can be time-dependent, but it always decreases and resource value is assumed to be

AN ASYMMETRIC GENERALIZED WAR OF ATTRITION

constant. We consider situations where the payo! of being the "rst to leave may initially increase and where resource values may depend on roles and may change in time. The type of models discussed in this paper can be used to study contests when there are di!erences between contestants that a!ect their rewards and/or costs. For example, Sjerps & Haccou (1994a) studied migration strategies of moth larvae on a host plant with the generalized war of attrition. Age-di!erences between larvae or di!erences in growth rate may cause e.g. asymmetries in risks of migration. For instance, larval weight a!ects the searching e$ciency, since small larvae have more di$culty in passing obstacles (Rausher, 1979; Damman & Feeny, 1988; Damman, 1991). Larval size also a!ects predation risk by ground predators such as ants, during migration (Haccou & Hemerik, 1985). Using a similar approach to Hammerstein & Parker (1982), we derive the ESS for the situation with two competitors when there are mistakes in role perception. It turns out that for the generalized war of attrition, similar results can be obtained as for the war of attrition, i.e. the ESS is speci"ed by two probability distributions with non-overlapping support. The supports and forms of the distributions depend on the forms of the payo! functions. We give expressions for the distributions of the conditional strategies as well as numerical approximations. The Model: De5nitions and Notations We consider situations where two individuals compete for a resource. The value of the resource may change in time. Before the start of a contest, each individual is assigned an attribute which a!ects its payo! from the contest. This is called its &&role''. There are two possible roles, denoted by A and B. We assume that there is always a true asymmetry, so one individual has role A and the other role B. An individual's true role may, however, di!er from its perceived role. Perceived roles will be denoted by lower case letters, i.e. a and b. Whereas two contestants cannot have the same true role, they can have the same perceived role. Let x be the time from the start of the contest. We denote the payo! of being the "rst to leave at

331

time x for an individual with role R (R"A, B) by ¸ (x). The payo! to an individual with role R if 0 its opponent is the "rst to leave at time x is S (x). 0 If both contestants pick the same persistence time, they each get the average of the two payo!s, i.e.  (¸ (x)#S (x)). 0  0 For all x, S (x)*¸ (x) since the remaining 0 0 individual can follow the conditional optimal strategy, i.e. leave or stay according to which gives the highest payo! at that time. In the "rst instance we will assume that for each R there is a "nite time q after which payo!s for staying and 0 leaving are equal, i.e. S (x)'¸ (x) for x(s 0 0 0 and S (x)"¸ (x) for x*q . Later the model is 0 0 0 generalized to allow for situations where one or both q tend to in"nity. 0 We assume that all payo! functions are continuous functions of x with continuous and "nite "rst and second derivatives on the interval [0, q ). As in Sjerps & Haccou (1994a, b), we will 0 assume that S (x) decreases in x and that ¸ (x) 0 0 either increases initially and later decreases or decreases from the start, i.e. there is a time x* 3[0, q ) for which ¸ (x)'0 if x(x* and 0 0 0 0 ¸ (x)(0 if x'x*. Examples of the forms of the 0 0 payo! functions are shown in Fig. 1. In the asymmetric war of attrition model, ¸ (x) equals !C x, where C is a positive con0 0 0 stant representing the cost rate in role R. This cost rate is comparable to !¸ (x) in our model. 0 Note, however, that for small x-values !¸ (x) 0 may be negative, as in Fig. 1(a) and (c). The rewards in the asymmetric war of attrition are "xed constants that may di!er for the two roles. In the generalized war of attrition the analogue of the reward for role R is S (x)!¸ (x). This may 0 0 change in time and, when q is "nite, eventually 0 become zero. An example of a situation where the payo! ¸ (x) may increase initially [as in Fig. 1(a) and 0 (c)] is a contest between larvae on a host plant (see Sjerps & Haccou, 1994a). Since larger larvae have lower risks during migration, the payo! of leaving increases as larvae grow. Situations where the payo! functions become equal after a "nite time [as in Fig. 1(a) and (b)] occur e.g. in competitive patch depletion (see Sjerps & Haccou, 1994b). Finally, when there is a "xed resource value, as for instance in the war of attrition, the di!erence between the payo!

332

P. HACCOU AND O. GLAIZOT

FIG. 1. Examples of the forms of the payo! functions S (x) (**) and ¸ (x) (} } } ). ¸ (x) may increase initially (a, c) or 0 0 0 decrease from the start (b, d), in which case x*"0. Payo!s may be equal after a "nite time q (a, b) or S (x) may be larger than 0 0 0 ¸ (x) for all x (c, d). 0

functions remains positive for all x (as in Fig. 1(c) and (d)]. ROLE PERCEPTION

It can be shown that in this asymmetric game, no ESS exists if the contestants have complete information on their roles and the corresponding payo!s. We give a rough outline of the argument (see Hammerstein & Parker, 1982). As Selten (1980) showed, asymmetry implies that if an ESS exists, it is a pure strategy, i.e. it would be of the form &&choose persistence time t in role A and  persistence time t in role B''. However, such a strategy is not protected against drift in the direction of lower persistence times in the role

with the highest persistence time. Such a drift in turn triggers selection to increase the persistence time in the other role, etc. In this argument it is assumed that both contestants have complete information about their roles. This is usually not the case. Hammerstein & Parker (1982) therefore examined the asymmetric war of attrition when there are mistakes in role perception. We will apply their approach to the generalized war of attrition. Let g (r"R) be the conditional chance that an individual's perceived role is r (r"a, b) whereas its real role is R (R"A, B), for instance Pr [perceived role"B"true role"A] "g (b"A) ("1!g (a"A)).

(1)

AN ASYMMETRIC GENERALIZED WAR OF ATTRITION

Throughout, we make the reasonable assumption that for each role the probability of making a mistake in role perception is smaller than the probability of a correct role identi"cation, i.e.

The weighted payo! functions if the opponent leaves "rst, S (x), etc., are de"ned analogously. It ?? can easily be shown that ¸ (x)"h (aa; A) (¸ (x)#¸ (x)) ?? 

g (a"A)'g (b"A) and g (b"B)'g (a"B). (2) We focus on one of the individuals and denote the simultaneous chance that the focal individual perceives its role as r , its opponent perceives its  role as r , and the focal individual's real role is R,  by h (r r ; R), e.g.  Pr [both perceive their role as A and true role of focal ind. is B]"h(aa; B)

333

h (aa; A) " ¸ (x), h (bb; A) @@

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and the same relation exists between S (x) and ?? S (x). Note that the function S (x)!¸ (x) is @@ ?? ?? analogous to Hammerstein & Parker's (1982) w < (in their notation) and similar analogies   hold for the other weighted payo! functions. MAIN ASSUMPTIONS

focal ind. perceives its role as A,

We assume that role A is &&favoured'' with respect to payo!s (see Hammerstein & Parker, 1982). This is further speci"ed in the following assumptions.

Pr opponent perceives its role as B and true role of focal ind. is A "h (ab; A).

(3)

We assume that the roles A and B are distributed among contestants with equal chances. This implies for instance that h(aa; A)"g (a"A);g (a"B);"h (aa; B) 

(4)

h(ab; A)"g (a"A);g (b"B);"h (ba; B).  For ease of notation, we will use weighted payo! functions. Before giving a formal de"nition, the interpretation of such functions is illustrated by an example. Consider the situation where the focal individual's perceived role is a in a contest with an opponent who also has perceived role a. If the focal individual leaves at time x whereas the opponent selected a larger persistence time, its expected payo! is ¸ (x)"h (aa; A) ¸ (x)#h (aa; B) ¸ (x). (5) ??  More generally, the weighted payo! functions for perceived roles r (focal ind.) and r (opponent) if   the focal individual leaves "rst at time x is de"ned as ¸   (x)"h (r r ; A) ¸ (x)#h (r r ; B) ¸ (x).    PP (6)

A1: x**x* , 

q *q . 

Let m be the time at which ¸ (x) is maximal, ? ?@ m the time when ¸ (x) is maximal and m the @ @? time at which ¸ (x)#¸ (x) is maximal [i.e. this  is the place of the maximum of ¸ (x) as well as ?? ¸ (x)]. Assumption A1 implies that: x*)m ) @@ @ m)m )x* . ?  A2: For all m)x(z(q  !¸ (x) (S (z)!¸ (z))   '!¸ (x) (S (z)!¸ (z)).  Note that A2 is automatically satis"ed for x(x*, since in that region !¸ (x)(0 and   !¸ (x)'0. Furthermore, A1 implies that A2 is satis"ed for q )x)q , since in that region the  right-hand side of the inequality is zero and the left-hand side is positive. Since ¸ (x) as well as  ¸ (x) decrease in the remaining interval, a su$cient (though not necessary) condition for A2 to hold is: A2: for all x*)x(q :  !¸ (x* ) (S (x)!¸ (x))    '!¸ (x) (S (x)!¸ (x)). 

334

P. HACCOU AND O. GLAIZOT

(The validity of this condition is much simpler to verify than that of A2.) A3: for all m)x(q :  +h (ab; A)!h (aa; A), +!¸ (x), *+h(aa; A)!h(ba; A),+!¸ (x),. 

large. Assumptions A3}A5 imply that mistakes in role perception should be relatively scarce. In A5, it is furthermore assumed that the decrease of ¸ (x) should be su$ciently large on the interval where ¸ (x) still increases. The following numer ical example will serve as an illustration: payo! functions:

From eqn (2) it follows that h(ab; A)' h (aa; A)'h (ba; A), so the left-hand side of the inequality in A3 is positive for the speci"ed xvalues. Furthermore, since the right-hand side of the inequality is negative for x(x* , A3 is auto matically satis"ed for m)x)x* .  A4: for all m)x(q : 

20!5x for x)4, S (x)"  ¸ (x) for x'4, 

(i) +h(ab; A)!h (bb; A), +S (x)!¸ (x),  

S (x)"

¸ (x)"!x#3x#4,  ¸ (x)"!x!2x#8,

 

*+h(bb; B)!h (ab; B), +S (x)!¸ (x),, (ii) +h (aa; A)!h (ba; A), +S (x)!¸ (x),   *+h (ba; B)!h (aa; A), +S (x)!¸ (x),. A5: for all m )x(m: @ g (b"B) +!¸ (x),*g (b"A) +¸ (x),.  Assumption A1 concurs with the fact that A is the favoured role, as is illustrated in the example below (see Fig. 2). Assumptions A2}A4 imply that the relative advantage of role A is su$ciently

20!10x for x)2, ¸ (x)

for x'2,

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error probabilities: g (a"B)"g (b"A)"0.1. Figure 2 shows the four payo! functions. It can immediately be seen from this "gure that assumption A1 is satis"ed: The payo! in the unfavoured role decreases from the start whereas that in the favoured role initially increases (0"x*(x* ); Furthermore, the reward for the  favoured role is positive over a longer interval than that of the unfavoured role (q (q ). The  validity of assumptions A2}A5 can be veri"ed for this example by simple calculations. The ESS for Finite q

0 A strategy I is de"ned as a set of probability distributions +P , P , which specify the chance ? @ with which di!erent persistence times are chosen if the perceived role is, respectively, a and b. The expected payo! of persistence time t if the perceived role is r against a strategy I is: for r"a, b:

FIG. 2. The four payo! functions used in the numerical example. S (x) (**) and ¸ (x) (} } } ). It is indicated in the 0 0 "gures which payo!s belong to the di!erent roles A and B. After time q "2 the payo!s are equal in role B and after q "4 they are equal in role A. 



E (t, I)" P

R



S (z) dP (z)#¸ (t) P? ? P?



1 # (S (t)#¸ (t)) P (t) P? ? 2 P?



R

dP (z) ?

335

AN ASYMMETRIC GENERALIZED WAR OF ATTRITION



#

R



S (z) dP (z)#¸ (t) P@ @ P@

1 # (S (t)#¸ (t)) P (t). P@ @ 2 P@





R

dP (z) @ (9)

Note that the right-hand side consists of two main parts, one corresponding to the situation where the opponent's perceived role is a and the other part for the case where the opponent's perceived role is b. The payo!s of a strategy J"+Q , Q , against I can be calculated from ? @ this straightforwardly by taking the expectation over t. As Hammerstein & Parker (1982) showed, a strategy I"+P , P , is an ESS if and only if the ? @ following conditions hold: (1) I is a best reply to itself, so for r"a, b we have: (1a) E (x, I)"E (P , I) for all x3Support (P ) P P P P (1b) E (x, I))E (P , I) for all x , Support (P ), P P P P and (2) If JOI is an alternative best reply to I then E (I 3. J)'E (J, J)(r"a, b). (10) P P The ESS I is derived by proving eleven lemmas. These are listed below. The proofs of Lemmas 3}10 can be found in Appendix A. Lemma 1. ¹here can be no persistence times greater than q in the support of P or P .  ? @ This is easy to see, since after q all payo!s are  equal, so there is no advantage for individuals that stay the longest after that time. Lemma 2. ¹here are no atoms of probability in P or P except possibly at q . ? @  This lemma can be proved by using the same argument as in Bishop & Cannings (1978), with the provision that there can be an atom at s .  Lemma 3. ¹here are no x-values smaller than m in the support of P and no x-values smaller than m ? @ in the support of P . @

Lemma 4. All x3[m, q ] are included in the  union of the supports of P and P . ? @ Lemma 5. ¹here are no gaps in the support of P . ? Lemma 6. ¹here is no overlap between the supports of P and P . ? @ Lemma 7. High persistence times must be chosen if the perceived role is a. Lemma 8. All values of x3[m , m) are included in @ the support of P . @ Lemma 9. ¹here is a unique equilibrium strategy of the form speci,ed below. If the perceived role is a, persistence times should be chosen according to the following probability density: !¸ (x) ?? p (x)" ? S (x)!¸ (x) ?? ?? V !¸ (z) ?? ;exp ! dz S (z) !¸ (z) ?? ?? Q if x3(s, q ) (11) 

 



and the persistence time q is chosen with prob ability

 

q





!¸ (z) dz . S (z)!¸ (z) ?? ?? Q

P (q )"exp ! ? 

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If the perceived role is b, persistence times should have the probability density p (x), which @ is implicitly determined by the integral equation ¸ (x) !¸ (x)!¸ (x) @@ @? @@ # p (x)" @ S (x)!¸ (x) S (x)!¸ (x) @@ @@ @@ @@



;

V

m

@

p (z) dz, x3[m , s). @ @

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This is a Volterra type 2 integral equation with a unique solution (see e.g. Linz, 1985). The value of s is determined by



Q

m @

p (z) dz"1. @

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P. HACCOU AND O. GLAIZOT

Lemma 10. ¹he strategy I"+P , P , speci,ed in ? @ ¸emma 9 satis,es ESS condition (1b) in eqn (10) with strict inequality. Lemma 11. ¹he ESS condition (2) in eqn (10) is also satis,ed. The same reasoning as in Hammerstein & Parker (1982), making use of the results of Bishop & Cannings (1978) can be used to prove this lemma. From the last two lemmas, it follows that the strategy given in Lemma 9 is the ESS for the generalized asymmetric war of attrition with role-dependent payo! functions that satisfy assumptions A1}A5. Generalization: The ESS for In5nitely Large q  We now turn to the situation where q is in" nitely large and q may be either "nite or in"nite. The assumptions A1}A5 should be adjusted accordingly and Lemma 1 is omitted. Lemma 2 is replaced by Lemma 2. ¹here are no atoms of probability in P or P . ? @ Its proof remains the same. The proofs of Lemmas 3}5 can be easily adjusted to deal with the current situation. Lemma 6 must be replaced by Lemma 6. If the support of P is bounded, there is @ no overlap between the supports of P and P . ? @ The proof is similar to that of Lemma 6, but it is assumed that the support of P is bounded by @ some value. The rest of the proof remains essentially the same. The proof of Lemma 7 can be adjusted straightforwardly by letting q tend to in"nity. From  Lemmas 2 and 5 we can conclude that P is ? di!erentiable, whereas P contains no atoms @ (Lemma 2) but its support may contain gaps. We need an additional Lemma (see below) to deal with the situation where q is in"nitely large. In  the proof of that Lemma, which is given in Appendix A, it is assumed that P , too, is di!erentiable. @ Lemma 12. ¹he support of P has an upper bound. @

From Lemmas 2}8, plus Lemma 12 it follows that there is a value s'm such that the support of P is [s, R) and that of P is [m , s). The proofs ? @ @ of Lemmas 9 and 10 can be adjusted straightforwardly and Lemma 11 remains the same. The ESS is as given in eqns (11) and (13), with q re? placed by in"nity. In this case, of course, p does ? not have any atoms of probability. The Shape of the ESS From eqn (11) (with eqn (12) for "nite q ) ? together with eqn (13), we can derive some general features of the ESS. The probability density function p (x) speci"ed by eqn (11) is ? a so-called failure time distribution, with hazard rate !¸ (x) ?? . (15) j (x)" S (x)!¸ (x) ?? ?? The hazard rate is the instantaneous leaving rate at time x given that both contestants have stayed until this time (see e.g. Kalb#eisch & Prentice, 1980) and it can be interpreted as the leaving tendency of individuals with perceived role a. From the de"nitions of the weighted payo! functions [see eqn (7)] we "nd (!¸ (x))#(!¸ (x))  , j (x)" (S (x)!¸ (x))#(S (x)!¸ (x))  

(16)

so it can be concluded that the leaving tendency in perceived role a does not depend on the probabilities of mistakes in role perception. Equation (13) can be written as p (x)"f (x)#j (x)!j (x) @



V

m @

p (z) dz, x 3 [m , s), @ @ (17)

with !¸ (x) @? f (x)" S (x)!¸ (x) @@ @@ h (ba; A) (!¸ (x))#h(ba; B) (!¸ (x))  " h(bb; A) ((S (x)!¸ (x))#(S (x)!¸ (x)))   (18)

AN ASYMMETRIC GENERALIZED WAR OF ATTRITION

and j (x) as de"ned in eqn (16). In this case, however, j (x) cannot be interpreted as a leaving tendency in perceived role b, since eqn (17) does not de"ne the corresponding failure time distribution. To "nd either an explicit expression for p (x) ? and p (x) or a numerical solution, we must "rst @ solve eqn (17). Once p (x) is known, s can be @ determined from (14) and, once s is known, p (x) ? can be calculated from eqn (11) (with eqn (12) when q is "nite).  An explicit expression for p (x) can be found @ by means of the method of iterated kernels (see e.g. Kondo, 1991), which leads to p (x)"f (x)#j (x)#j(x) @



  



V

m

2

X

FIG. 3. The ESS probability densities p (x) (**) and ? p (x) (} } }) for the numerical example. The value of s is @ approximately 0.8.

( f (z)#j (z))

@

 V t ; 1# G X X t G\

337

G “ (!j (t )) dt H H H



.

(19)

However, unless j (x) and f (x) have very speci"c and relatively simple forms, eqn (19) is not so useful. For more complicated cases, it is more convenient to use a numerical method for solving Volterra type 2 equations. In this section, we apply one such method, described by Linz (1985). Equation (17) is approximated by I p (x )"f (x )#j (x )!j (x ) h w p (x ), @ I I I I IG @ G G k"2, 3, 2 ,

(20)

where x "m #kh (k"0, 1,2 ) and the weights I @ w are de"ned according to Simpson's 1/3 and IG 3/8 rule. p (x ) is equal to f (m ) and p (x ) is @  @ @  calculated by using a trapezoidal method with a step length h/2, to reduce the starting error (see e.g. Kondo, 1991). An estimate of the value of s is x , such that L VL p (x )+1/h. @ G G

(21)

After s has been determined, eqn (11) can be solved by numerical integration to "nd an approximation for p (x). ? As an illustration, we calculated the ESS for the numerical example given previously [cf. eqn (8)]. In that example the value of m equals @ zero and q is equal to 4. We used an h-value  of 0.1. The value of s is approximately 0.8. Figure 3 shows the ESS probability densities. Note that, although values of p (x) are very small for ? values of x near q , the support of p (x) includes  ? all values between s and q . In this case, there is  no atom of probability at q , since the integral of  p (x) from x"s to q equals one. ?  Discussion In this paper, we generalized the asymmetric war of attrition model, proposed by Hammerstein & Parker (1982) to take into account more general shapes of payo! functions. With our results it is possible to further examine e!ects of speci"c shapes of payo! functions. Precise e!ects of forms of payo!s and errors on the ESS will usually have to be examined numerically, even though an analytical expression for p (x) is available [see eqn (19)]. However, some @ general features of the ESS can be derived from our results. First of all, the expressions (16)}(18) show that not all changes in payo! functions will a!ect the forms of the conditional strategies p (x) ? and p (x). Only changes that a!ect the di!erence @

338

P. HACCOU AND O. GLAIZOT

in payo!s, S (x)!¸ (x), or the derivative of 0 0 ¸ (x) (for R"A or B), will have an e!ect. This 0 implies for instance that adding the same constant to S (x) and ¸ (x) will not a!ect   the ESS, even though at "rst sight it increases the asymmetry between the payo!s of the two roles. Since in the standard war of attrition the di!erence S(x)!¸(x) corresponds to the resource value and the derivative !¸(x) to the cost rate, this remark may seem trivial. It can, however, be a confusing issue. For instance, experienced individuals may be able to exploit a contested resource more e$ciently, so their value of S(x) is larger, but they may at the same time also be able to "nd alternative resources more e$ciently, so their cost rate is lower, and as a consequence their payo! of leaving, ¸(x), is higher. These effects may compensate each other, so the value of S(x)!¸(x) is not a!ected by experience. In contests, experienced and inexperienced individuals should then use the same strategy, even though experienced individuals have higher expected payo!s. From eqn (17) it can be inferred that p (x) @ increases when f (x) and/or j (x) is increased. An increase of either of these functions will therefore shift the point s to the left. Since j (x) is also the leaving tendency if the perceived role is a, transformations of payo! functions that increase j (x), like increases in !¸ (x) or decreases in 0 S (x)!¸ (x), will have the e!ect that expecta0 0 tions of both conditional strategies decrease. From eqns (16) and (18) it can be seen that any transformation of S (x)!¸ (x) will have exactly   the same e!ect as the identical transformation of S (x)!¸ (x), provided that the derivatives of the ¸ (x) and the values of q are not a!ected by 0 0 the transformation. This implies for instance that when the resource value for individuals in role B is decreased, even individuals that believe themselves to have role A should have an increased leaving tendency. Furthermore, as can be seen from eqn (16), the e!ect of such changes on the leaving tendency j (x) does not depend on the probabilities of making mistakes in role perception. This gives the counterintuitive result that changes in S (x)!¸ (x) a!ect the ESS strategy p (x) in exactly the same way as changes in ? S (x)!¸ (x) do, irrespective of the accuracy of   information on the true role (as long as there

is uncertainty). A similar conclusion holds for changes in the derivatives of either ¸ (x),  or ¸ (x) that do not a!ect the values of q 0 and S (x)!¸ (x). For instance, if the cost 0 0 rate of a contest in role B increases, the leaving tendency in role a should increase, irrespective of the probability of making mistakes. From eqn (18) it can be seen that probabilities of mistakes in role perception do a!ect the function f (x) and therefore p (x) and the value of s. @ Therefore, although the leaving tendency j (x) is not a!ected by these errors, the shape of the p.d.f. p (x) is. Shifting s to the right will generally ? increase the expectations of both strategies. (Exact e!ects have to be examined numerically.) After some calculations it can be derived that (if all else is being held equal) an increased probability of making mistakes in role B, g (a"B), leads to an increase in f (x) and this implies a decrease in the value of s. So, the expected persistence times will decrease. Similarly, it can be shown that as the probability of making mistakes in role A, g(b"A), increases, the value of s shifts to the right, and as a result the expected persistence times increase. Note that the strategy distributions +p (x), ? p (x),, such as given for example in Fig. 3, do not @ correspond to the distribution of contest durations. The realized duration of a contest is the minimum of the chosen persistence times of the contestants. There are three possible combinations of perceived roles in contests: (1) ab, (or, equivalently, ba), (2) bb and (3) aa. Since contests of type (1) are always terminated by the contestant who believes that its role is b, the durations of such contests are distributed according to p (x). Durations of contests of type (2) have the @ distribution of the minimum of two independent random variables that are identically distributed according to p (x). Durations of contests of type @ (3) have the distribution of the minimum of two random variables that are i.i.d. according to p (x). Thus, the distribution of contest durations ? can be calculated from the strategy distributions. One consequence of our results is, for instance, that if q is "nite, contests should never last  longer than that value. Furthermore, if there is an atom of probability at q , there is a positive  chance that contests between individuals with perceived role a will have exactly that duration

AN ASYMMETRIC GENERALIZED WAR OF ATTRITION

and that after that time, both contestants will give up simultaneously. We showed how the ESS can be calculated with numerical methods for solving Volterra equations and numerical integration. Now that we have proved that there is a unique ESS however, an alternative method can also be used. This consists of an iterative method that simulates an evolutionary process. This method is similar to the one used by Haccou & McNamara (1998) to "nd optimal mixed strategies. The iteration is started with arbitrary probability distributions +p , p ,. Then the expected payo! ?L @L of each persistence time x in the two di!erent I perceived roles is calculated. Subsequently, the probability distributions are adjusted, putting more weight on persistence times with the highest expected payo!s. This gives +p ,p ,. ?L> @L> The iteration is continued until the probability distributions are stable and the expected payo!s of all the persistence times that remain in the support of the two distributions are equal. This method may have more intuitive appeal. However, the numerical method that we presented is more e$cient with respect to computer time. Furthermore, the value of s may be di$cult to determine from such simulations. Also, in cases where values of p (x) are very small for x near q , ?  like in our example (see Fig. 3), an iterative method could give the wrong impression that p (x) is zero. However, the support of p (x) al? ? ways includes these values and at the ESS the expected payo!s of all x-values between s and q are equal. Therefore, it is essential to pay  attention to the theoretical characteristics of the strategies when results of simulation methods are interpreted. As mentioned, our approach is similar to the one used by Hammerstein & Parker (1982). Whereas they formulated their assumptions in terms of the weighted payo! functions [see e.g. eqn (6)], we prefer to use the true payo! functions ¸ (x) and S (x) (R"A, B), for reasons of clarity. 0 0 However, several of our assumptions are analogous (though not equal) to those of Hammerstein and Parker and can be cast in a similar form. Our assumption A2 is analogous to their inequality (5) and our assumptions A3 and A4 correspond to the &&weak asymmetry condition'' speci"ed in their inequalities (6) and (7). In addition, we need

339

assumptions A1 and A5 to deal with the more general case considered here. Assumption A1 is, in our opinion, a natural extension of the concept of a &&favoured role'' to the generalized war of attrition. Assumption A5 is only necessary in the proof of Lemma 8 and may be relaxed. In that case, the lower boundary of the support of P may no longer be equal to m but rather lie @ @ between m and m. This is a subject of further @ study. From our example (Fig. 3) it can be seen that shapes of the conditional strategies for generalized wars of attrition can be very di!erent from those of wars of attrition. In the latter case, the probability density functions are negative exponential functions, so they decrease in x (see examples in Bishop et al., 1978; Hammerstein & Parker, 1982). This implies that contest durations are mixtures of exponential distributions. In the generalized war of attrition discussed in this paper, the shape of the distribution of contest durations depends on the shape of the payo! functions and can be much more general. Furthermore, in the asymmetric war of attrition contest duration has no upper bound. In the generalized case that we consider, this is only true when the payo! S (x) is always larger than  ¸ (x). If, after a "nite time q these payo! func  tions become equal, contests should never last longer than that time. Whereas the probability distributions of conditional strategies in the war of attrition have no atoms, an atom of probability may occur at q in the generalized situation  (although in our numerical example this is not the case). Like Hammerstein and Parker, we only considered situations with two possible roles. In such cases no pure ESS exists, so leaving strategies should be stochastic. This conclusion holds as long as there are only a "nite numbers of roles. We expect that the ESS for a generalized war of attrition with a limited number of roles (larger than two) will have a similar structure as found by Bishop et al. (1978), consisting of several probability density functions with non-overlapping supports. As Kura & Kura (1998) showed, pure ESSs do exist when there is a continuum of possible roles. In their models the cost-rate may change, but it cannot be negative and also they assume that resource value is constant and

340

P. HACCOU AND O. GLAIZOT

positive. In our notation this implies that S (x)!¸ (x)"S (x)!¸ (x)"<, where < is   a positive constant. We expect, however, that their conclusions concerning the existence of pure ESSs continue to hold for more general shapes of payo! functions, such as those considered here. It would be interesting to try and generalize their results in this respect. We thank Chris Cannings for his hospitality and inspiring discussions and the referees for helpful comments on a previous version. Elizabeth van Ast kindly checked the English. OG was supported by a grant of the Fonds National Suisse de la Recherche Scienti"que. REFERENCES BISHOP, D. T. & CANNINGS, C. (1978). A generalised war of attrition. J. theor. Biol. 70, 85}124. BISHOP, D. T., CANNINGS, C. & MAYNARD SMITH, J. (1978). The war of attrition with random rewards. J. theor. Biol. 74, 377}388. CROWLEY, P. H., GILLETT, S. & LAWTON, J. H. (1988). Contests between larval damsel#ies: empirical steps towards a better ESS model. Anim. Behav. 5, 1496}1510. DAMMAN, H. (1991). Oviposition behaviour and clutch size in a group-feeding pyralid moth Omphalocera munroei. J. Anim. Ecol. 60, 193}204. DAMMAN, H. & FEENY, P. (1990). Dispersal rates under variable patch density. Am. Nat. 135, 48}62. HACCOU, P. & HEMERIK, L. (1985). The in#uence of larval dispersal in the cinnabar moth (¹yria jacobaeae) on predation by the red wood ant (Formica polyctena): an analysis based on the proportional hazards model. J. Anim. Ecol. 54, 755}769. HACCOU, P. & MCNAMARA, J. M. (1998). E!ects of parental survival on clutch size decisions in #uctuating environments. Evol. Ecol. 12, 459}475. HAMMERSTEIN, P. (1981). The role of asymmetry in animal con#icts. Anim. Behav. 29, 193}205. HAMMERSTEIN, P. & PARKER, G. A. (1982). The asymmetric war of attrition. J. theor. Biol. 96, 647}682. KALBFLEISCH, J. D. & PRENTICE, R. L. (1980). ¹he Statistical Analysis of Failure ¹ime Data. New York: Wiley and Sons. KURA, T. & KURA, K. (1998). War of attrition with individual di!erences on RHP. J. theor. Biol. 193, 335}344. doi:10.1006/jtbi.1998.0705. KONDO, J. (1991). Integral Equations. Oxford: Clarendon Press. LINZ, P. (1985). Analytical and Numerical Methods for
(Diptera: Scatophagidae). II. The fertilization rate and the spatial and temporal relationships of each sex around the site of mating and oviposition. J. Anim. Ecol. 39, 205}228. PARKER, G. A. (1970b). The reproductive behaviour and the nature of sexual selection in Scathophaga stercoraria L. (Diptera: Scatophagidae). IV. Epigamic recognition and competition between males for the possession of females. Behaviour 37, 113}139. PARKER, G. A. & RUBINSTEIN, D. I. (1981). Role assessment, reserve strategy, and acquisition of information in asymmetric animal con#icts. Anim. Behav. 29, 221}240. PAYNE, R. J. H. & PAGEL, M. (1996). Escalation and time costs in displays of endurance. J. theor. Biol. 183, 185}193. doi:10.1006/jtbi.1996.0212. RAUSHER, M. D. (1979). Egg recognition: its advantage to a butter#y. Anim. Behav. 27, 1034}1040. RIECHERT, S. E. (1998). Game theory and animal contests. In: Game ¹heory and Animal Behaviour (Dugatkin, L. A. & Reeve, H. K., eds), pp. 64}93. Oxford: Oxford University Press. SELTEN, R. (1980). A note on evolutionarily stable strategies in asymmetric animal con#icts. J. theor. Biol. 84, 93}101. SJERPS, M. & HACCOU, P. (1994a). A war of attrition between larvae on the same host plant: stay and starve or leave and be eaten? Evol. Ecol. 8, 269}287. SJERPS, M. & HACCOU, P. (1994b). E!ects of competition on optimal patch leaving: a war of attrition. ¹heor. Pop. Biol. 3, 300}318. doi:10.1006/tpbi.1994.1029. WALTON, R. & NOLAN JR, V. (1986). Imperfect information and the persistence of pretenders: male prairie warblers contesting for territory. Am. Nat. 128, 427}432. YDENBERG, R. C., GIRALDEAU, L. A. & FALLS, J. B. (1988). Neighbours, strangers, and the asymmetric war of attrition. Anim. Behav. 36, 343}347.

APPENDIX A Proofs of Lemmas 3+10, and Lemma 12 Lemma 3. ¹here are no x-values smaller than m in the support of P? and no x-values smaller than m@ in the support of P@ . We only give the proof for P? , since the proof for P@ is almost identical. Let Q and Q be probability distributions, where Q has support [0, m) and Q has support [m, q ], and let j be a constant between zero and one. If x-values smaller than m are included in the support of P? the ESS can be formulated as follows: I: If the perceived role is a choose persistence times according to Q with probability j and persistence times according to Q with probability 1!j. If the perceived role is b choose persistence times according to a probability distribution P@ . Now consider the alternative strategy:

341

AN ASYMMETRIC GENERALIZED WAR OF ATTRITION

J: If the perceived role is a choose persistence time m with probability j and persistence times according to Q with probability 1!j. If the perceived role is b choose persistence times according to a probability distribution P@ . We will show that I cannot be an ESS, by proving that J does better against I than I does against itself. For convenience, we use the notation u0 (a,b) to denote the expected payo! of a against b to an individual with role R, where a and b can be either a probability distribution or a "xed time, so for instance



u (m, P )" 0 @

K



S (z) dP (z)#¸ (m) 0 @ 0



#(1!j) +h (ab; A) u (Q , P )   @ #h (ab; B) u (Q , P ),,  @

and if the perceived role is a the expected payo! of I against I equals E (I, I)"j +h (aa; A) u (Q , Q ) ?    #h (aa; B) u (Q , Q ),   #j(1!j)



K





K

(A.3)

¸ (z) dQ (z) ?? 

#j (1!j) +h (aa; A) u (Q , Q )   

dP (z), @

#h (aa; B) u (Q , Q ),  

(A.1)

#(1!j) +h (aa; A) u (Q , Q )   

and

 

u (Q , P )" 0  @

K X



#

 K



#h (aa; B) u (Q , Q ),  

S (y) dP (y) dQ (z) 0 @  ¸ (z) 0





X

#j +h(ab; A) u (Q , P )   @ #h (ab; B) u (Q , P ),  @

dP (y) dQ (z). @  (A.2)

#(1!j) +h (ab; A) u (Q , P )   @

With this notation, if the perceived role is a the expected payo! of J against I equals E ( J, I)"j ?



K



#h (ab; B) u (Q , P ),,  @

(A.4)

so the di!erence in payo!s is S (z) dQ (z) ?? 

#j(1!j) ¸ (m) ?? #j (1!j) +h (aa; A ) u (Q , Q )    #h (aa; B) u (Q , Q ),   #(1!j) +h (aa; A) u (Q , Q )    #h (aa; B) u (Q , Q ),   #j +h (ab; A) u (m, P )  @ #h (ab; B) u (m, P ), @

E (J, I)!E (I, I) ? ? "j



K



S (z) dQ (z)!+h (aa; A) u (Q , Q ) ??    

 

#h (aa; B) u (Q , Q ),  



#j (1!j) ¸ (m)! ??

K





¸ (z) dQ (z) ?? 

#j [h (ab; A) +u (m, P )!u (Q , P ),  @   @ #h (ab; B) +u (m, P )!u (Q , P ),]. @  @ (A.5)

342

P. HACCOU AND O. GLAIZOT

The "rst expression on the right-hand side of this equation equals



j h (aa; A)

 

K





S (z) dQ (z)!u (Q , Q )      K

#h (aa; B)



S (z) dQ (z)!u (Q , Q )   



,

(A.6) and this is positive since S (z) and S (z) are the  highest possible gains that can be obtained in each of the roles. The second part is positive since ¸ (z) increases for z(m. For the remaining ?? part, "rst consider eqn (A.2). From this it follows that



u (Q , P )" 0  @

K

¸ (z) 0





#

K



¸ (z) 0

K

  K

K



X

#



#

X





"

dP (y) dQ (z) @ 

S (y) dP (y) 0 @



S (y) dP (y) dQ (z)  0 @

¸ (z) dQ (z); 0 



#

K





X



K



"





dQ (z); 

K



¸ (z) dQ (z); 0 





#

K



S (z) dP (z). 0 @

q 

K

dP (y) @

S (y) dP (y) 0 @



K



K



q 

!



"

K



+h (ab; A) ¸ (z)#h(ab; B) ¸ (z), dQ (z)  





dP (y) ¸ (m)! @ ?@

K





¸ (z) dQ (z) , ?@  (A.8)

Lemma 4. All x3[m, q ] are included in the  union of the supports of P and P . ? @

dP (y) dQ (z) @ 





q





K

¸ (z) 0

K

K

dP (y) h (ab; A) ¸ (m)#h (ab; B) ¸ (m) @ 

dP (y) dQ (z) @ 







q 

and this is larger than zero since ¸ (z) increases ?@ for z(m. We can conclude that the right-hand side of eqn (A.5) must be positive and thus I cannot be an ESS.

S (y) dP (y) dQ (z) 0 @ 







K

K X

#

(



q

[Since the support of Q is [0, m), its integral over  that interval equals one.] Therefore, the third part of eqn (A.5) is larger than j times:

q 

K

dP (y) @ (A.7)

It is easy to see that there is no upper bound less than q on the union of supports. Let m be  a value in the interval [m, q ) and suppose that  there are no values higher than m in the union of supports of P and P . Then, individuals that ? @ choose a persistence time y'm always get a higher payo! than those choosing persistence times smaller than m. Furthermore, there is no lower bound higher than m on the union of supports. Let f be a value in the interval (m, q ] and suppose that there are  no values lower than f in the union of supports of P and P . Then individuals choosing persistence ? @ time m if their perceived role is b always get a higher payo! than those choosing f, in that situation, since both ¸ (x) and ¸ (x) are de@? @@ creasing for x-values larger than m. Now we will show that there are no gaps in the union of supports. Consider a value x *m and  a value x (x )q . Suppose that the interval    (x , x ) is not included in the support of P or   ? P then @



E (x , I)" @ 

x





S (z) dP #¸ (x ) @? ? @? 



q 

x 

dP ?

343

AN ASYMMETRIC GENERALIZED WAR OF ATTRITION



x 

#



S (z) dP #¸ (x ) @@ @ @@ 



q 

x



dP , @ (A.9)

We "nd from eqns (A.11) and (A.12) DE (y, x, I )"+¸ (y)!¸ (x), P P? P?

whereas for y3(x , x )  



E (y, I)" @

x





S (z) dP #¸ (y) @? ? @?



x



#







# q 

x



S (z) dP #¸ (y) @@ @ @@

V

dP ?



q 

W

dP (z) ?

+S (z)!¸ (x), dP (z) P@ P@ @

#+¸ (y)!¸ (x), P@ P@

dP . @

x 

W



q 



q



W

dP (z). @ (A.14)

(A.10) Since y is smaller than x and since ¸ (x) and  @? ¸ (x) are both decreasing for x'm, the ex@@ pected payo! of persistence time y is higher than that of x . This provides a contradiction, since  I is the ESS. Lemma 5. ¹here are no gaps in the support of P . ? Consider values x and y, such that m)x (y)q , which are both included in the sup port of P , whereas the interval (x, y) is not ? included. From Lemma 4 it follows that (x, y) must then be included in the support of P . Thus, @ we have, for r"a, b



E (x, I)" P

V

S (z) dP (z)#¸ (x) P? ? P?





#

V





q 

W

S (z) dP (z)#¸ (x) P@ @ P@

dP (z) ?



q 



S (z) dP (z)#¸ (y) P? ? P?





#

W





q



W

S (z) dP (z)#¸ (y) P@ @ P@

V

+S (z)!¸ (x), dP (z) P@ P@ @

#+¸ (y)!¸ (x), P@ P@



"

V

dP (z), @

dP (z) ?



q 

W

dP (z). @ (A.12)

De"ne the di!erence in payo!s as (A.13)



q 

W

dP (w) @

+S (z)!¸ (z)#¸ (z)!¸ (x), dP (z) P@ P@ P@ P@ @



W

V



¸ (z) dz; P@

q 

W

dP (w) @

 W

V

+S (z)!¸ (z),#+¸ (z)!¸ (x), P@ P@ P@ P@

#¸ (z) P@



"

W



q





dz dP (w) dP (z) @ @ dP (z) W @ 



¸ (z)!¸ (x) P@ +S (z)!¸ (z), 1# P@ P@ P@ S (z)!¸ (z) V P@ P@



dz ¸ (z) P@ # dP (z) @ S (z)!¸ (z) dP (z) @ P@ P@



"

 E (y, I)!E (x, I). DE (y, x, I) " P P P

W

V

"

and V



W

#

(A.11)

E (y, I)" P

Since I is an ESS, both DE (y, x, I ) and ? DE (y, x, I ) must be zero. We now prove that @ DE (y, x, I ) is larger than DE (y, x, I ), so there is ? @ a contradiction. First, note that

W



 X ¸ (y) dy +S (z)!¸ (z), 1# V P@ P@ P@ S (z)!¸ (z) P@ P@ V



dz ¸ (z) P@ # dP (z). @ S (z)!¸ (z) dP (z) @ P@ P@

(A.15)

344

P. HACCOU AND O. GLAIZOT

From eqns (A.14) and (A.15) it can be seen that it is su$cient to prove that for all x and z, with m)x(z)q the following inequalities hold:  ¸ (z)!¸ (x)'¸ (z)!¸ (x), ?? ?? @? @?

(A.16)

S (z)!¸ (z)'S (z)!¸ (z), ?@ ?@ @@ @@

(A.17)

 X ¸ (y) dy  X ¸ (y) dy V ?@ ' V @@ . S (z)!¸ (z) S (z)!¸ (z) ?@ ?@ @@ @@

(A.18)



V

+!¸ (y) dy, (S (z)!¸ (z)) 



(

X

V

+!¸ (y) dy, (S (z)!¸ (z)),  

(A.19)

and this follows directly from assumption A2. Lemma 6. ¹here is no overlap between the supports of P and P . ? @ We already know that the support of P is @ bounded by q . Let y)q be the maximum   value of the support of P and suppose that there @ is an overlap with the support of P , so ? the interval [x, y] is included in both supports. The expected payo!s of persistence time x are (r"a, b)



E (x, I )" P

V

S (z) dP (z)#¸ (x) P? ? P?





#

V





q 

V

S (z) dP (z)#¸ (x) P@ @ P@

dP (z) ?



W

V

dP (z), @

and for persistence time y



W



S (z) dP (z)#¸ (y) P? ? P?



#

W



S (z) dP (z). P@ @



q 

W

W

V

+S (z)!¸ (x), dP (z) P? P? ?

#(¸ (y)!¸ (x)) P? P?



#

W

dP (z) ? (A.21)



q



W

dP (z) ?

+S (z)!¸ (x), dP (z). P@ P@ @ (A.22)

Below, we show that DE (y, x, I) is larger than ? DE (y, x, I ). This contradicts I being an ESS, @ since then both di!erences should be zero. By using similar methods as in the proof of Lemma 5 it can be derived that



DE (y, x, I)" P

W

V

+S (z)!¸ (z), P? P?



¸ (z) P? ; 1# S (z)!¸ (z) P? P?



q





dz dP (w) dP (z) ? ? dP (z) W ?

;



W



W

#

V

#

V

(A.20)

E (y, I )" P



DE (y, x, I)" P

V

Inequalities (A.16) and (A.17) follow in a straightforward manner from assumptions A3 and A4(i). It can be derived that inequality (A.18) holds if and only if X

So the di!erence in payo!s is





+¸ (z)!¸ (x), dP (z) P? P? ? +S (z)!¸ (z), P@ P@



¸ (z)!¸ (x) P@ dP (z). ; 1# P@ @ S (z)!¸ (z) P@ P@ (A.23) From inequalities (A.17) and (A.18), it follows that the last expression in eqn (A.23) is larger for DE (y, x, I) than for DE (y, x, I ). It remains to ? @ prove that for m)x(z)q  ¸ (z)!¸ (x)'¸ (z)!¸ (x), (A.24) ?? ?? @? @? ¸ (z) ¸ (z) @? ?? ' . S (z)!¸ (z) S (z)!¸ (z) @? @? ?? ??

(A.25)

345

AN ASYMMETRIC GENERALIZED WAR OF ATTRITION

Inequality (A.24) is the same as inequality (A.16) and inequality (A.25) follows from A2. Lemma 7. High persistence times must be chosen if the perceived role is a. Suppose that m(q is the upper bound of the  support of P . From the above, it follows that the ? interval (m, q ] is then included in the support of  P . So for y'm @



E (y, I)" P



K

S (z) dP (z)# P? ?



#¸ (y) P@



q 

W

W

K

S (z) dP (z) P@ @

dP (z), @

(A.26)



V

S (z) dP (z)#¸ (x) P? ? P?



#¸ (x) P@



q

K





K

V

dP (z) ?

dP (z), @

(A.27)

so



DE (y, x, I)" P

K

V

DE (y, m, I)"¸ (y)!¸ (m)#¸ (y)!¸ (m) @ @? @? @@ @@ "h (ba; A) +¸ (y)!¸ (m),   #h (ba; B) +¸ (y)!¸ (m), #h (bb; A) +¸ (y)!¸ (m),   #h (bb; B) +¸ (y)!¸ (m), "+h(ba; A)#h (bb; A), ;+¸ (y)!¸ (m),  

whereas for x)m E (x, I)" P

support of P . Suppose that they are not included ? in the support of P , and let y be a value in that @ interval, then

#+h (ba; B)#h (bb; B), ;+¸ (y)!¸ (m),,

(A.29)

so, from the de"nition of the chances h (r r ; R)  we have DE (y, m, I)" [g (b"A) (g(a"B)#g (b"B)) @ 

+S (z)!¸ (x), dP (z) P? P? ?

;+¸ (y)!¸ (m),  



#g (b"B) (g(a"A)#g (b"A))

#

W

K

+S (z)!¸ (x), dP (z) P@ P@ @

#+¸ (y)!¸ (x), P@ P@



q

W



dP (z). @ (A.28)

By the same methods as used in the proof of Lemmas 6 and 5, it can be shown that DE (y, x, I)'DE (y, x, I). If I is an ESS, ? @ DE (y, x, I)"0. However, this implies that @ DE (y, x, I)'0, which contradicts the fact that ? I is an ESS. Lemma 8. All values of x3[m , m) are included in @ the support of P . @ As we have already proved (see Lemma 3), values in the considered interval are not in the

;+¸ (y)!¸ (m),] " [g(b"A) +¸ (y)!¸ (m),    #g(b"B) +¸ (y)!¸ (m),], (A.30) and it follows from A5 that for y3[m , m) this is @ positive. This contradicts the fact that I is an ESS. Lemma 9. ¹here is a unique equilibrium strategy. We can conclude from Lemmas 1}8 that there is a value s3[m, q ) such that the support of P is  ? [s, q ] and the support of P is [m , s). Further @ @ more, there are no atoms of probability except possibly at q . Therefore, there is a probability 

346

P. HACCOU AND O. GLAIZOT

density function p corresponding to the distribu@ tion P and for x3(s, q ) we can write: @ 

Di!erentiation with respect to x gives

∀x3(s, q ): 

S (x) p (x)#¸ (x) @@ @ @@



E (x, I)" ?

V

Q

S (z) p (z) dz#¸ (x) ?? ? ??

#S (s) ?@



Q

m @



q 

V

p (z) dz, @

where p denotes the derivative of P . It follows ? ? from the ESS conditions in eqn (10) that this should be equal to E (s, I)"¸ (s)#S (s) ? ?? ?@



Q

m @

p (z) dz. @

(A.32)

#¸ (x) ??

 

V

Q q

S (z) p (z) dz ?? ?



V

p (z) dz"¸ (s). ? ??

(A.33)

Di!erentiation with respect to x gives S (x) p (x)#¸ (x) ?? ? ??



q 

p (z) dz!¸ (x) p (x)"0 ? ?? ? V q  !¸ (x) ?? p (z) dz, Np (x)" ? ? S (x)!¸ (x) V ?? ?? x3(s, q ). (A.34) 



The solution of this integral equation is the probability density given in eqn (11). We will now consider P . From the ESS condi@ tions (1a) in eqn (10) we can derive that

!¸ (x)!¸ (x)  Q p (z) dz @? @@ V @ , p (x)" @ S (x)!¸ (x) @@ @@ x3[m , s). @

We will "rst prove that when the perceived role is b, the payo! of all x that are not in the support of P is less than that of P . First note that this @ @ has already been proved for all x outside the union of supports of P and P . Furthermore, @ ? since it has been proved that the payo! of all strategies in the support of P have the same @ payo!, it su$ces to show that E (x, I)(E (s, I) @ @ for all x in the support of P . Filling in the payo!s ? of x and s in role r against the equilibrium strategy gives



E (x, I)" P

V

Q

S (z) dP (z) P? ?

#¸ (x) P?

DE (x, s, I)" P

and we can write

m @

S (z) p (z) dz#¸ (x) @@ @ @@

"¸ (m )#¸ (m ). @@ @ @? @



V

p (z) dz#¸ (x) @ @? (A.36)



q 

dP (z)#S (s), ? P@

V

E (s, I )"¸ (s)#S (s), P P? P@

E (x, I)"E (m , I)"¸ (m )#¸ (m ), (A.35) @ @ @ @@ @ @? @



(A.38)

Equation (13) is easily derived from eqn (A.38).

so

Q

(A.37)

so p (x) satis"es the integral equation @

∀x3[m , s): @

V

p (z) dz @

V

Lemma 10. ¹he strategy I"+P , P , speci,ed in ? @ ¸emma 9 satis,es ESS condition (1b) in eqn (10) with strict inequality.

Equating eqn (A.31) to eqn (A.32) gives ∀x3(s, q ): 

Q

!¸ (x) p (x)#¸ (x)"0, @@ @ @?

p (z) dz ? (A.31)





V

Q

S (z) dP (z)#(¸ (x) P? ? P?



!¸ (s)) P?



"

V

Q

(A.39)

q 



V dP (z)! ¸ (s) dP (z) ? P? ? V Q

(S (z)!¸ (z)) dP (z) P? P? ?

347

AN ASYMMETRIC GENERALIZED WAR OF ATTRITION

#(¸ (x)!¸ (s)) P? P?



q



V

#¸ (x) ?@

dP (z). ? (A.40)

Since x is in the support of P , it is known that ? DE (x, s, I)"0. Therefore it is su$cient to prove ? that DE (x, s, I)(DE (x, s, I ). The proof is sim@ ? ilar to that of Lemma 5. Proceeding in the same way we can derive from eqn (A.40) that



DE (x, s, I)" P

V

Q

+S (z)!¸ (z), P? P?



 X ¸ (y) dy ; 1# Q P? S (z)!¸ (z) P? P?



dz ¸ (z) P? # dP (z). ? S (z)!¸ (z) dP (z) ? P? P? (A.41)

#¸ (x) ?@

 

E (x, I)"¸ (x)# ? ?? #¸ (x) ?@

V

m

@

Q

#¸ (x) ?@

V

Taking the derivative gives *E (x, I) ? "¸ (x)#S (x) p (x) ?? ?@ @ *x

(A.45)



Q

V



Q

V



dP (z) @

dP (z) @

S (x)!¸ (x) ?@ "¸ (x)!¸ (x) ?@ ?? @? S (x)!¸ (x) @@ @@



Q

V

(A.43)

(A.44)

dP (z), @







dP (z) ¸ (x)!¸ (x) @ ?@ @@



S (x)!¸ (x) ?@ . ; ?@ S (x)!¸ (x) @@ @@

(A.46)

Since ¸ (x) decreases in x for x'm and @? S (x)!¸ (x)'S (x)!¸ (x) [this follows ?@ ?@ @@ @@ from A4(i)] this gives *E (x, I) ? '¸ (x)!¸ (x) ?? @? *x



# dP (z). @

V

; !¸ (x)!¸ (x) @? @@

#

S (z) dP (z) ?@ @



Q

(S (x)!¸ (x)) *E (x, I) ?@ ? "¸ (x)# ?@ ?? S (x)!¸ (x) *x @@ @@

(A.42)

Together this proves that DE (x, s, I)' ? DE (x, s, I ). @ By the same reasoning as before, we can argue that for the payo!s in role a, it su$ces to show that E (x, I)(E (s, I) for all x in the support of ? ? P . The payo! of x in role a equals @

dP (z)!¸ (x) p (x) @ ?@ @

and for x3[m , m] this is positive since m is the @ place of the maximum of ¸ (x) and m , the place ?? ? of the maximum of ¸ (x) is larger than m. Thus, ?@ the payo! increases in x on this interval. We now consider x values in the interval [m, s). From eqns (A.38) and (A.45) we "nd:

and from assumption A2 it can be derived that X ¸ (y) dy  X ¸ (y) dy Q ?? ' Q @? . S (z)!¸ (z) S (z)!¸ (z) ?? ?? @? @?

V

"¸ (x)#(S (x)!¸ (x)) p (x) ?? ?@ ?@ @

It follows straightforwardly from assumption A4(ii) that S (z)!¸ (z)*S (z)!¸ (z), ?? ?? @? @?



Q

Q






S (x)!¸ (x) ?@ . dP (z) ¸ (x)!¸ (x) ?@ ?@ @@ S (x)!¸ (x) @ V @@ @@ (A.47)

From A3 it can be derived that ¸ (x)'¸ (x) ?? @? for these x-values. Furthermore, it can be derived from A2 that the remaining part is also positive. We can, therefore, conclude that the payo! of

348

P. HACCOU AND O. GLAIZOT




x increases over the interval (m , s) and therefore @ s gives a higher payo! than any value of x in that interval.

#p (z) (S (x)!¸ (x)) @ @@ @@

Lemma 12. ¹he support of P has an upper bound. @

(S (x)!¸ (x)) (S (x)!¸ (x)) ?@ @? @? ! ?@ (S (x)!¸ (x)) ?? ??



We prove this Lemma by contradiction, so suppose that P has no upper bound. This means @ that there is some m such that all x'm are included in the union of supports of P and P . ? @ According to the ESS conditions, this means that for x'm



E (x, I)" P

V

S (z) p (z) dz#¸ (x) P? ? P?





#

W







p (z) dz ?

V

S (z) p (z) dz#¸ (x) P@ @ P@





V

p (z) dz @

"c , P

(A.48)

where c and c are constants. Taking the deriva? @ tive gives (S (x)!¸ (x)) p (x)#¸ (x) P? P? ? P?





V

p (z) dz ?





V

p (z) dz"0. @

(A.49)

Now it is proved that all three expressions on the left-hand side of this equation are negative, which gives a contradiction. Equation (A.51) states that





V

p (z) dz (¸ (x) (S (x)!¸ (x)) ? @? ?? ??

!(S (x)!¸ (x)) ¸ (x)) @? @? ?? #p (x) ((S (x)!¸ (x)) (S (x)!¸ (x)) @ @@ @@ ?? ??



p (z) dz (¸ (x) (S (x)!¸ (x)) @ @@ ?? ??

!¸ (x) (S (x)!¸ (x)))"0. ?@ @? @?

< (x)"S (x)!¸ (x), R"A, B. 0 0 0 (A.50)

Substituting this for p (x) into eqn (A.49) with ? r"b gives




(A.51)

(A.52)

Obviously, the values of p and p cannot be ? @ negative. We will prove that the three factors with which these probabilities are multiplied are all negative. It follows straightforwardly from inequality (A.25) that the "rst factor is negative. For ease of notation we de"ne

S (x)!¸ (x) ?@ p (x) ! ?@ S (x)!¸ (x) @ ?? ??





V








p (z) dz ¸ (x) @ @@

(S (x)!¸ (x)) @? ! @? ¸ (x) "0. (S (x)!¸ (x)) ?@ ?? ??



 ¸ (x) ?? p (z) dz p (x)"! ? ? S (x)!¸ (x) ?? ?? V



V

#

This implies that

 ¸ (x) ?@ ! p (z) dz. @ S (x)!¸ (x) V ?? ??



!(S (x)!¸ (x)) (S (x)!¸ (x))) ?@ ?@ @? @?

#(S (x)!¸ (x)) p (x) P@ P@ @ #¸ (x) P@



#



(S (x)!¸ (x)) @? ¸ (x) p (z) dz ¸ (x)! @? @? ? (S (x)!¸ (x)) ?? V ?? ??

(A.53)

The second factor equals h (bb; A) h (aa; A) (< (x)#< (x))  !(h (ab; A) < (x)#h (ab; B) < (x))  ;(h(ba; A) < (x)#h (ba; B) < (x)). 

(A.54)

349

AN ASYMMETRIC GENERALIZED WAR OF ATTRITION

From the de"nitions of the probabilities [see examples in eqn (4)] it follows that

h (ab; A) h (ba; A) (< (x)#< (x)) (¸ (x)#¸ (x))  

h (bb; A) h (aa; A)"h (ab; A) h (ba; A), h (ab; B)"h (ba; A), h(ba; B)"h (ab; A). (A.55) Substituting these into eqn (A.54) gives

!(h (ab; A) ¸ (x)#h (ba; A) ¸ (x)) (h (ba; A)  ;< (x)#h (ab; A) m< (x))  "< (x) ¸ (x) h (ba; A) (h(ab; A)!h (ba; A)) 

h (ab; A) h (ba; A) (< (x)#< (x)) 

!< (x) ¸ (x) h (ab; A) (h (ab; A)!h (ba; A)) 

!(h (ab; A) < (x)#h (ba; A) < (x)) 

"(h (ab; A)!h(ba; A)) (h (ba; A) < (x) ¸ (x) 

;(h(ba; A) < (x)#h(ab; A) < (x)) 

!h (ab; A) < (x) ¸ (x)). 

"< (x) < (x) (2h (ab; A) h (ba; A)!h(ab; A)  !h (ba; A)),

and, from eqn (A.55) it follows that this equals

Since proper role identi"cation is assumed to have a higher chance than making mistakes

(A.56) (h (ab; A)!h (ba; A))'0.

and this is negative since < (x) and < (x) are  both positive. The third factor equals h (bb; A) h(aa; A) (< (x)#< (x)) (¸ (x)#¸ (x))   !(h(ab; A) ¸ (x)#h (ab; B) ¸ (x))  ;(h (ba; A) < (x)#h (ba; B) < (x)), 

(A.58)

(A.57)

(A.59)

Furthermore, from this and A2 it follows that h (ba; A) < (x) ¸ (x)(h (ab; A) < (x) ¸ (x).   (A.60) Inequalities (A.59) and (A.60) prove that the right-hand side of eqn (A.58) is negative.