Strong evolutionary equilibrium and the war of attrition

J. theor. Biol. (1980) 82, 383-400

Strong Evolutionary Equilibrium Attrition

and The War of

JOHN G. RILEY

Department

of Economics

UCLA, Los Angeles, California U.S.A.

90024,

(Received 7 March 1979, and in revised form 7 August 1979) In developing the concept of an evolutionarily stable strategy, Maynard Smith proposed formal conditions for stability. These conditions have since been shown to be neither necessary nor sufficient for evolutionary stability in finite populations. This paper provides a strong stability condition which is sensitive to the population size. It is then demonstrated that in the war of attrition with uncertain rewards there is a unique “strong evolutionary equilibrium” strategy. As the population becomes large this is shown to

approach the solution strategy proposed by Bishop, Cannings and Maynard Smith. The analysis is then extended to wars of attrition between different populations. It is concluded that for such contests there is a whole family of potential strong evolutionary equilibria.

1. Introduction

In the recent modelling of strategic behavior in conflicts between animals and insects one of the more remarkable conclusions is that “evolutionary equilibrium” may require the adoption of mixed strategies. For example, in the “war of attrition” analyzed by Maynard Smith (1974), Bishop & Cannings (1978), and Hines (1977), two members of a given species both desire the same food source or mate. Both use up valuable time in a ritualistic “display” of aggression. Eventually one of the contestants ends its display and departs. The other then collects the reward. Each, in effect, makes a sealed bid. The higher bidder wins and both must pay the low bid. In the simplest model of this conflict the value of reward to either contestant is the same and equal to p The above authors have argued that in such an environment the evolutionarily stable strategy is for each contestant to adopt the mixed strategy of bidding x or more with probability e-*‘“. However, there are two serious objections which can be levelled at the model and the inferences drawn from it. First of all, while adoption of the exponential mixed strategy yields, on average, a gain to the winner, there is 383

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@ 1980 Academic

Press Inc. (London)

Ltd.

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no social gain from contests: the full value of the reward is, on average, just offset by the value of the resources committed to the contest. This is readily demonstrated. Let p be the resource cost to the loser in any particular contest. In a symmetric equilibrium there is an equal chance of winning e -p and of losing p. Then the expected net gain is E{$(V-p)-ip}=:v-E{p}.

Moreover, a property of a mixed strategy equilibrium is that if it is adopted by one agent the expected net gain to the other agent is independent of his action. Since one of the actions in the mixed strategy equilibrium is to avoid conflict, it follows immediately that the equilibrium expected net gain must be zero. Thus the average value of the resources committed by the two contestants ( = 2,??(p)) is exactly equal to the value of the reward. Of course the absence of any social gain does not indicate that the model is wrong. However, it is hard to avoid reflecting that perhaps a richer model would yield less harsh conclusions about competitive gains in the natural world. The second objection to the simple model is that the evolutionary stability of the exponential mixed strategy is suspect. In Riley (1979) it is established that only for infinite populations are the equilibrium conditions proposed by Maynard Smith sufficient for a strategy to be evolutionarily stable. Furthermore, for any finite population it is shown that there are mutant strategies for which the expected return exceeds the return to the exponential mixed strategy regardless of the proportion of agents adopting the mutant strategy. Offsetting this apparently devastating result is the additional demonstration that successful mutant strategies are necessarily similar to the non-mutant. Since very different strategies have a higher expected return in a contest with an agent using the similar mutant strategy, instability would result were the latter to replace the exponential strategy completely. However, if the frequency of mutations were sufficiently high the similar mutant might itself be invaded by a second mutant before completing its invasion of the exponential strategy. As long as the second mutant were to have a lower expected return when used against the exponential strategy, both mutations might eventually disappear. Thus in such a world the exponential strategy might still survive in the long run. But this attempt to rescue the exponential mixed strategy is, at best, a plausible story. Again it is natural to inquire whether, in a richer model, the stability problems are compounded or whether they are reduced. In the following section the incorrectness of Maynard Smith’s equilibrium conditions for finite populations are illustrated and alternative conditions

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are derived. Then in section 3, strong stability results are obtained by taking into account a natural informational asymmetry between contestants. For a finite population of size N it is shown that there exists a bidding strategy v*(N) with the property that if any one agent switches to a different strategy CL,that agent necessarily has a strictly lower expected return than the rest of the population. Moreover, in the double limit as the population approaches infinity and the informational asymmetry is reduced, the strongly stable strategy approaches the exponential mixed strategy. The final section applies the strong evolutionary equilibrium concept to contests between different populations. It is shown that, in general, there is a whole family of equilibria, each with the strong stability property that adopting any mutant strategy yields a lower expected return than the equilibrium strategy. For a special case considered in more detail, the equilibria are shown to range from those in which the bidding strategies are almost indistinguishable to bidding strategies in which one population almost always gives up after only a brief encounter. While extending these results to more complex situations remains a task for the future, they do suggest that by careful modelling it is possible to avoid the instability problems emphasized in Riley (1979). 2. Strong Evolutionary

Equilibrium

We begin by illustrating the instability of Maynard Smith’s proposed equilibrium strategy when the population is finite. Let Y and p be any pair of feasible strategies, pure or mixed, and let r(v1p.c)be the expected return to adopting strategy Y when the opponent adopts strategy CL.Level curves for the expected returns of the two contestants are depicted in Fig. 1. Strategies are parametrized in such a way that, holding p constant, a higher value of Y lowers the expected return to the contestant adopting cc. A necessary condition for evolutionary stability as proposed by Maynard Smith is that if one contestant adopts the stable strategy then the other can do no better than adopt it also. Formally, Y is evolutionarily stable according to this condition, only if for all feasible alternatives J.L 444.M4.

(1)

This condition, weaker than Maynard Smith’s complete set of stability conditions, is the standard condition for a non-cooperative “CournotNash” equilibrium in symmetric games. In Fig. 1 if one agent adopts strategy yc the other’s best response is vc, hence condition (1) is satisfied. However, this strategy is not evolutionarily stable. To see this consider the alternative strategy ~1just to the right of yc. If adopted by one agent in a population of

386

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RILEY 45”

line

/

Strategy FIG.

1. Evolutionary

size N the probability the “mutant” is l/N agent is therefore

instability

of qent

I (v,)

of the Gournot-Nash

equilibrium.

that a “normal” agent will be randomly paired with The decline in the expected return to each normal

Also, since the mutant always competes with normal agents, its decline in expected return is However r(vlr+) takes on its maximum at v = vc. Therefore the mutant’s decline in expected return is zero to a first order of approximation. It follows that for p sufficiently close to vc the decline in expected return to each of the normal agents is greater than the decline in expected return to the mutant. That is, strategy k has a higher “fitness” than strategy vc and will therefore begin to spread in the population. The essence of the formal results in Riley (1979) is that for the war of attrition there are mutant strategies p which continue to have a higher fitness even when they are adopted by a large fraction of the population. Intuitively, if there is an eventual rest point of this dynamic process the equilibrium strategy vS will lie to the right of vc as illustrated. We therefore have r(~sl~s) < rbcl~c). However, if, as in the simple version of the war of attrition r(~~lv& = 0 it follows that ~(ZJS~VS)is strictly negative. Then vs has the undesirable

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property that it yeids a lower expected return than the strategy of avoiding all conflict. In section 3 we shall see that when the war of attrition is modified to take account of an informational asymmetry, there is a unique equilibrium strategy vs as illustrated in Fig. 1. Moreover in contrast to the simple war of attrition the expected return to each agent is strictly positive for all sufficiently large populations. First we provide a definition of strong evolutionary stability and then obtain conditions ensuring such stability. Consider a population of N agents identical except possibly in the strategies adopted in contests over food sources, mates, etc. Suppose one agent adopts a mutant strategy CLwhile all the rest adopt the non-mutant or “normal” strategy Y. For the agent adopting the mutant strategy any encounter is with an agent adopting V, therefore the expected return or “fitness” of the agent is given by RI(P)

= ~bl4.

(2)

Assuming random encounters, the probability that an agent adopting the normal strategy meets another using the same strategy is N -2/N - 1. The expected return is then (3) Suppose further than the environment will only support a fixed population and that the number of agents adopting the mutant strategy rises if and only if the fitness associated with this strategy is higher. Then the following definition of a strong evolutionary equilibrium is natural. DEFINmON

1 STRONG

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Strategy v is a strong evolutionary equilibrium strategy if it yields an expected return or “fitness” exceeding that of any feasible alternative ,u when the proportion adopting CCis small. It follows immediately

that if, for all feasible ,Q, Rlb)-Rl(~)
strategy Y is a strong evolutionary equilibrium conditions sutlicient for this strict inequality. Combining (2) and (3) we have

strategy. We now seek

RI(PL)-RI(~) =[r(~b)-~

rblv).

r(+)] -E

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Suppose r(p]v) - r(v]p)/n p # v,

RILEY

- 1) has a unique maximum

at p = Y. Then for all

= 0. We have therefore proved Proposition

equilibrium

1: For a population strategy if

has a unique global maximum

of size N, strategy Y is a strong evolutionary

at p = V.

3. The War of Attrition

with Uncertain Rewards

In the stark version of the war of attrition described in section 1 agents place an identical value on the contested object. Plausibly, however, the value of any such object is not fixed over time, but is strongly dependent upon the recent history of a given agent. For example in the competition for a food source, the food intake in the interval leading up to the contest will affect the payoff to further food consumption. Similarly, in the competition for a mate, experience gained from recent and presumably unsuccessful contests might be expected to alter an animal’s “perception” of the likelihood of victory and hence its strategic behavior. This introduces an element of informational asymmetry since each contestant is aware of its own valuation of the contested object but not that of its opponent. For simplicity we shall, in this section, retain symmetry at a more fundamental level by assuming that each contestant has the same underlying probability distribution of values. To be precise, it is assumed that for each contestant Pr {value of reward is no greater than x} = G(x), where G(0) = 0, G(a) = 1 and G(x) is differentiable and strictly increasing over [0, (~1. The following proposition is derived in the Appendix.

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Pruposition 2. If the value of the reward to each of N contestants is a random variable with distribution G(x) the strategy of bidding N

b&)=-

N-2

’ YGYY) J0 l-G(y)

dy

whenever the true value is x, is a strong evolutionary equilibrium yielding a positive expected return for all N greater than

strategy

2+J=(1-G(y))2dy/l’G(y)(l-G(y))dy. 0

0

As an example, suppose contestants valuations are distributed uniformly so that G(y) = y/a. Substituting this into the second expression the expected return is positive for N > 4 and substituting into the first the equilibrium bidding strategy is N b&r(x)=0 l--y/a N-2

Jk!!%dy

=&ln[e-X(l-~)-‘]

,

It is interesting to compare Proposition 2 with earlier results. Bishop, Cannings & Maynard Smith (1978), in the first discussion of the uncertain rewards model argued that the equilibrium bidding strategy would be

xl-Gcy)dy. yG’(x) b*(x)= Jo From Proposition 2 we see immediately that their solution is indeed the limiting case as N approaches infinity. We next compare the solution with that proposed in the original work by Maynard Smith (1974). It can be readily shown that there is no strategy satisfying Proposition 1 when, as Maynard Smith assumed, contestants always place identical values on rewards. However we can approximate the model by considering a case in which valuations must lie in some small interval (6 - E, 6 + E). Then to a first approximation the equilibrium bid function is given by N b(X)“G

J

’ fiG’(y) B-e(l-G(y))

dy = $$ln(l

Inverting and making the additional assumption large, we have G(n) z 1 -e-b(X)/Be

- G(x)). that the population

N is (4)

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In such an equilibrium the observed distributions characterized as follows:

of bids F(b) may be

F(b(x)) = Prob {bid does not exceed b(x)} = Prob (b(2) 5 b(x)} =Prob{2Ix}

(5)

= G(x)

Combining

(4) and (5) we have finally F(b)=

1 -cb’?

This is none other than the exponential probability distribution. Thus in this limiting case the observed bidding distribution approximates the solution proposed by Maynard Smith! However, it is important to keep in mind that here we have a pure strategy solution in contrast to Maynard Smith’s proposed mixed strategy equilibrium. This result, described in greater generality by Engelbrecht-Wiggans and Weber (1979) is summarized in the following proposition. Proposition 3: If the value of the reward to each contestant lies on a small interval (C-F, a+&) and the population size is large the observed distribution of bids is approximated by the exponential distribution with mean u. 4. Asymmetric

Reward

Distributions

A key assumption in the above analysis is that, while valuations differ in any particular contest, there is no more fundamental asymmetry in the sense that the distribution of values is identical for all contestants. While this assumption is a natural one for some examples of rivalry within a population, it certainly does not apply to contests between populations with different characteristics. Even within a population there may be recognizable different subgroups. As long as the distribution of valuations varies across these subgroups it is to be anticipated that mutations taking such differences into account will have an evolutionary advantage. For example, if the average value of a particular food source is higher for a full-grown male than for a young male, it would be surprising if the strategies of the young males were identical to those of the fully-grown males. In this section we explore the implications of generalizing the strong evolutionary equilibrium to take account of observable asymmetries. Consider two different (sub)populations. Suppose that agents from the first

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population adopt strategy ~1, when in a contest with agents from the second population. Suppose also that all but one of the latter adopt strategy vz in such contests. The remaining agent adopts the mutant strategy ,x. Modifying only slightly the notation established in section 2, the expected return to the mutant may be written as r~(p\vi) and the expected return to the nonmutant as r2(v&i). Since the strategies are only used in contest with a different population the finite population problems of section 3 do not arise and the mutant strategy, p, has a higher fitness if and only if ~*(cLI~1)--*(Y21Y*)~o. It follows that a sufficient condition for there to be no such mutant is that r& [vi) should take on its global maximum when, and only when, p = y2. In the same way we can compare the fitness of a mutant strategy in the first population with that of ZQ. A sufficient condition for there to be no mutant with higher fitness is that ti(p (~2) should take on its global maximum when, and only when, y = yl. These conclusions may be summarized as follows: 4: Strategies ul and u 2, used respectively by two observably different populations, are strong evolutionary equilibrium strategies if (I) ri(p(r~) has a unique global maximum at p = vl, (II) T&.L (vi) has a unique global maximum at ~1= z+

Proposition

In the language of game theory, a sufficient condition for the strategies yl, y2 used by two different populations to be strong evolutionary equilibrium strategies is that the two strategies form a “strong” Nash equilibrium. Returning to the war of attrition with uncertain rewards it is tempting to conclude that there will be two extreme equilibria. If the strategy of the first population, y1 is the aggressive strategy of staying in any contest until the opponent leaves, the optimal response of the second population ~2 is the pureIy passive strategy of avoiding all conflict. Given such behavior the first population then wins all contests at zero cost. However, this is not quite enough. While condition (II) is satisfied for these two strategies we have instead of condition (I) 0’)

h(ccl~2)

=

f~hlu2)

for all cL.

Any mutant strategy k appearing in the first population has the same fitness as the aggressive strategy ZQ. Thus eventually the proportion of the population adopting the latter could become small. But then the passive strategy is no longer the second population’s best response. Despite these problems the aggressive-passive equilibrium does have empirical appeal. Often closely related species specialize in their own

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RILEY

ecological niche rather than contest food sources. It is therefore tempting to conjecture that there might be an equilibrium strategy pair which is very similar to the aggressive-passive pair of strategies. Perhaps one population almost always wins particular types of contests, but the other population competes just often enough to eliminate the instability? In the Mathematical Appendix it is shown that this conjecture is correct. However it is also shown that there is a complete spectrum of strong evolutionary equilibria. Formally we have the following result. Proposition 5: Let the value of the reward to each member of population i, (i = 1,2), be a continuously distributed random variable. Then there exists a one parameter family of strong evolutionary equilibrium strategy pairs of the form 6i = 6:(x, k). Unfortunately there is no general analytical solution to the differential equations which define these equilibrium strategy pairs. We can therefore consider only special cases. One family of distribution functions that is very amenable to further analysis is the exponential family, Gi(x) = 1 -e-“‘,, where fii is the mean value of x. In the appendix it is shown that for this family the strong evolutionary equilibrium family of bid functions have the simple form b*(x) = (kX)(%+%m, b&) = (x/k)~~l+%J%~ (6) It is instructive to compare the family of equilibria distributions are similar so that

Substituting

for the case in which the

into (6) we have b,(x) = (kx J2-,

62(X)= (f)2+m. For k = 1 the bidding functions are similar enough to be empirically indistinguishable. However as k becomes large the bids of the second population are almost always small relative to those of the first population. Thus a member of the second population almost always retreats after only a brief encounter. At the opposite extreme with k small it is the members of the first population which almost always retreat very quickly leaving the prize to the first population.

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To conclude, we have shown that the strong evolutionary equilibrium concept generates a whole family of potential equilibria for contests between different species. Perhaps only by introducing an explicit dynamic model of mutations will it be possible to derive theoretical inferences about likely time paths of disequilibrium strategies and hence the more likely long run evolutionary equilibrium states. Helpful comments by Jack Hirshleifer, Robert J. Weber and a referee are gratefully acknowledged. Research on this paper was supported in part by National Science Foundation Grant S.O.C. 79-07573. REFERENCES BISHOP, D. T. & CANNINGS, C. (1978). I. them. Bid. 70, 85. BISHOP, D. T., CANNINGS, C. & MAYNARD SETH, J. (1978). J. rheor. Biol, 74,377. ENGELBRECHT-WIGGANS, R. & WEBER, R. J. (1979) Cowles Foundation, Yale University. HINES, W. G. S. (1977). J. theor. Biol. 67, 141. MAYNARD SMITH, J. (1974). J. rheor. Biol. 47,209. RILEY, J. G. (1979). J. theor. Biol. 76, 109.

APPENDIX Proposition 2: If the value of the reward to each of N contestants is a random

variable with distribution

G(x), the strategy of bidding N b&X) =N-2

x YGYY) dy I t, l-G(y)

whenever the true value is n, is a strong evolutionary equilibrium yielding a positive expected return for all N greater than

strategy

2+Ja(1-G(y))‘dy/J~G(y)(l-G(y))dy. 0

0

Proof Suppose that the strong evolutionary

equilibrium strategy, assuming it exists, is to make a maximum bid of b(x) whenever the value of the reward is x. Suppose furthermore, that b(0) = 0, II((Y) =OO and that b(x) is a differentiable strictly increasing function. Of course all these properties are later to be verified. Let the set of alternative feasible strategies be all those for which a specific bid b,(x) is assigned to each valuation of the reward X, where b,( . ) is assumed to be a piece-wise continuous function. Then the implied distribution of bids associated with a particular mutant strategy, cc, may be continuous or may have a finite number of mass points.

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Suppose an agent adopting the mutant strategy p places a valuation of x on the reward and hence makes a bid b,(x). Given the assumptions about the form of the function b(x) there is a unique y E [0, co] such that b(y) = b,(x), that is y = y(x) = b-‘(b,(x)).

(Al)

Since b( . ) is strictly increasing and continuous and b,( . ) is piece-wise continuous, y(x) is also piece-wise continuous. Given a bid b,(x), the expected return to the agent adopting the mutant strategy is = Pr {mutant wins}

r(b,(x)(v)

value of object -expected bid by non-mutant the mutant bid is higher [ - Pr{mutant loses}[bid by mutant] x

given that

1

=Pr{b,b,(x)}b,(x).

(A2) By construction b,(x) = b(y). Also b, = b(v) where v is the value placed on the contested object by the agent adopting the non-mutant strategy. Therefore Pr{b,
into (A2), we have

db,(x)b’)

Integrating

= G( y)[x -50’b(~t”,p’o)]

- b(y)(l

- G(y)).

the second term by parts this expression can be rewritten as:

r(b,(x)lv)=G(~)x-b(y)G(y)+

J0

‘b’(v)G(u)dv-b(y)(l-G(y))

=G(y)x-b(y)+[Ybl(V)G(v)dv. 0

Therefore on average, the return to the adoption of the mutant strategy CL

~(ILb)= Jr(b&+)dG(x)= J’0 r(b(y(x))jv) 0

=I[m

G(Y(x))x-b(y(x))+

0

dG(x)

Y(X)

J 0

b’(v)G(u) dv G’(x) dx.

I

(A3)

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To determine the return to the normal strategy Y we again begin by evaluating the average return to v for a given bid, b,, by the opponent. r(vlb,)

= Pr {non-mutant x

wins}

1

expected value of reward to the mutant given that its bid is higher-b, [

- Pr {non-mutant

loses} expected value of bid by non-mutant [ given that the bid is less than b,].

1

Since 6, = b(y) for some y, we may write Pr{non-mutant

wins}=Pr{b(u)>b(y)}=Pr{v>y}=

l-G(y)

Thus

= [‘vdG(u)-(1-G(y))b,-JYb(v)dG(u). y

0

Also, from (Al) b, =b,fx)=b(y(x)h Substituting this into the above expression and integrating parts we have r(v)b(y(x)))=

J‘L da(y)-b(y)+JYb’(v)G(v)dv. 0

Y

Since the opponent’s value of the object is distributed have finally 444

the last term by

according to G(x) we

dx Jur(~lb(yW)G’h) =JoQ[ JyIx, 0W4-bMd~+Joyix) b’WW]G’~~~ dx.

=

0

644)

We are now ready to make use of Proposition r$(p, v) we have

=

J0

1. From the definition

a I( y(x), x)G’(x) dx,

of

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where ICY, x>= E[

b’(u)G(u)

-b(y)+/"'

dv ] + G(y)x -A

Jymu dG(u),

(A5) is a continuous function of x and y. Selection of a piece-wise continuous function y(x) which maximizes 4(~, V) is a simple problem in the calculus of variations. The first-order necessary condition is simply

Partially

differentiating

(A5) with respect to y we have

g=N-[-bb’(y)+b’(y)G(y)]+G’(y)x+ ay N-l

j&

YWY).

(A7)

For 4 (CL,Y) to have a maximum at w = v we require that the mutant strategy of bidding b,(x) when the value of the reward is x be the same as the normal strategy. That is, condition (A6) must be satisfied for y(x) = b-‘(b,(x)) Substituting o=

Rearranging,

= b-‘(b(X))

= x.

(A8)

for x in (A7) and setting the expression equal to zero we have

~[-b’(y)+b’(y)G(y)l+G’(y)~

+A

yG’(

Y).

(A91

this reduces to N b’(y)=N-2

YG’(Y) l-G(y)’

Therefore, (AlO) Note that b(x) is a strictly increasing differentiable function with b(0) = 0 and b(a) = co. Thus, all the restrictions imposed on the form of the normal bid function are satisfied. Next we establish that the first-order necessary condition (A6) indeed defines the global maximum. Combining (A7) and (A9) we have ~$(Y(x),x)=

G’(y(n))(x-Y(X)).

(All)

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Consider any strategy p # v and let X+ be the set of values of x such that y(x) > x. Similarly, let X- be the set of values of x for which y(x) < x. For all XEX+ I(*, *) >I(Y (XL x), since, from (All), for all x E X-

I is decreasing in y over the interval (x, y(x)). Similarly, m,*)>nYb),*).

Therefore,

for any p f Y

It remains to establish that, for sufficiently large N, I(Y(V) is positive. To evaluate the expected return we first note that the winner is always the contestant with the higher valuation. The probability that both have a valuation less than x is G(x)‘. Then the expected reward from the contest is

J 20 1 a

x dG(x)2 =

(I J

xG’(x)G(x)

dx.

(A13

0

Also, since both agents must pay the lower bid and the latter has a density function of 2G’(x)(lG(x)), the average cost of entering a contest is

J

p b.&)2G’(x)(l-

G(x)) dx = [-&(x)(1

- G(x))*];

0

J

+ LIl&(x)(1 -G(x))*

dx

0

=-

Subtracting r(z+)=

N

N-2

J n

o

xG’(x)(l

-G(x))

dx. (A13)

(A12) from (A13) the expected return to a contestant is

JP =J

ui2*G’(x)G(x)-xC’(x)~dx-~J”(l-G(x))2dx

0

G(x)(l -G(x))

J

dx -A

p (1 - G(x))’ dx.

0

0

Rearranging the terms on the right-hand if and only if N>2+

0

J

side it follows that T(Y[Y) is positive

‘(I-G(x))‘dx/J-G(x)(I-G(x))dx.

0

0

Q.E.D.

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Proposition 5: Let the value of the reward to each member of population i (i = 1,2) be a continuously distributed random variable. Then there exists a one parameter family of strong evolutionary equilibrium strategy pairs of the form bi = bi(x, k). Proof. Let Gi(x) be the probability

that the reward is valued at x or less by a member of the ith population (i = 1,2), where G:(x)>0 on [0, CX] and G(a) = 1. Suppose that for each population the strong evolutionary strategy vi is to bid hi(x) when the reward is valued at x. Suppose further that bi(0) =O, b;(a) =a and that hi(x) is a strictly increasing differentiable function. Then b;(x) is invertible and we may write X = Zi(b),

where zi(b) is also a strictly increasing differentiable function. The distribution of bids by a member of the first population can therefore be expressed as follows Fl(b)=Pr{bl-=6}=Pr{zl(bl)~zzl(b)} =Pr{xl%z,(b)} = G,(zI@)).

(Al4)

Suppose a member of the second population switches to some mutant strategy. If it bids b when its valuation of the reward is x its expected return is r2(b(v1) = Pr {br < b}[x -E(bl(bl

Utilizing

(A14) and integrating

< b)]-Pr

{br > b}b.

by parts this can be rewritten as h

rdble)

= G~(z~(b))x

-b

+

I II

GI(zI(~I))

dbl.

(Al5)

The highest expected return attainable by the mutant is achieved by selecting b to maximize r2(b]V1). Note that r(Olvl) = 0. Moreover we shall see below [from equation (A19)] that limr;(b)=~. b+O

Then

5 rdbld = G;(zI(~))z;

(b)x -El - G~b~(b))l,

6416)

is strictly positive for sufficiently small values at b. It follows that there is an

STRONG

interior maximum

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satisfying the first-order necessary condition (A17)

Suppose that the optimal Proposition 5, that is

response b,(x) satisfies the requirements

of

b, b) = b&h Then we may invert and write x = Q(b) Combining

(A16)-(A18),

(A181

it follows that ~~(6) must satisfy

G; (z~(bNz; (bMb)-

Cl- GI(zI@))

= 0.

(Al91

Also, making an identical argument for a member of the first population have a second condition

G;(z*(b))z~(b)z1(6)-(1 -Gz(z,(bN

=O.

we

6420)

From (A20) we can solve for zI(b) as a function of z*(b) and z;(b), that is ZIP) = h(zz@), ZiuJ)). Substituting this into (A19) yields a second-order ordinary differential equation for z*(b). The requirement ~~(0) = 0 fixes one of the two constraints of integration. However, there is no other boundary condition. Thus the solution to (25) and (26) is a one parameter family of inverse bid functions 2 = Z&J, k). Substituting into (A20) then yields the corresponding for the first population

inverse bid function

2 = Zl(b, k). From (A19) and (A20) these are monotonically increasing functions. They can therefore be inverted to obtain the bid functions bi = bi(Z, k). Finally we note that by substituting (A19) and (A20) back into the expression for (a/ab)ri(blvi) and arguing exactly as in section 3 it can be confirmed that these two differential equations describe not only a local but also a global maximum. Q.E.D.

400

.I. G.

RILEY

For the special case in which Gi(x) = 1 -e-“‘“i, reduce to

equations (A19) and (A20)

r;(b).z*(b)-b~=O

L421)

z;(b)r~(b)-v2=0;

W2)

that is,

$ (Zl(b)Z2(6)) = 27,+ ?&. Integrating

and making use of the boundary condition

z 1(0) = 0 then yields

Zl(b)ZZ(b) = (01+ 212)b. Substituting for z,(b) in (A22) then yields the first-order ordinary differential equation z;(b)/zz(b) Integrating

= ti2/(?71+ &)b.

again yields *&)

= /&/(~l+v,

and, from (A22)

Finally, inverting these expressions we have the strong evolutionary rium bid functions

equilib-