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Theoretical Population Biology 73 (2008) 11–23

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Truthful signalling, the heritability paradox,and the Malthusian equi-marginal principle

Kjell Hauskena,�, Jack Hirshleiferb

aFaculty of Social Sciences, University of Stavanger, N-4036 Stavanger, NorwaybDepartment of Economics, University of California, Bunche 2263, Los Angeles, CA 90095-1477, USA

Received 14 September 2006

Available online 19 September 2007

Abstract

The article shows that heritable quality differentials are consistent with the Zahavi Handicap Principle (the Truthful Signalling

Hypothesis (TSH)). Earlier analyses have assumed non-heritable quality. The crucial innovation is the Malthusian equi-marginal

principle: under selection pressures the relative numbers of higher- and lower-quality organisms will change until, in equilibrium, not the

average but the marginal levels of quality will be equalized. Assuming kin selection, each male maximizes his own reproductive success

and signals until the marginal value of more signalling is zero. We further require evolutionary stability; displacements to higher or lower

population sizes must be restored to equilibrium. The article proposes an alternative to Fisher’s [1958. The Genetical Theory of Natural

Selection. Dover Publications, Inc., New York [Original publication 1929]] and Hamilton and Zuk’s [1982. Heritable true fitness and

bright birds: a role for parasites? Science 218, 384–387] suggestions. The model is solvable for ranges of parameters that constitute the

stable region. We particularly consider the unit signalling costs of the high- and low-quality males, where it has been widely believed that

for a TSH equilibrium the former must be lower than the latter. This article confirms our earlier result that this is not a necessary

condition for a truthful signalling equilibrium, though the unit signalling costs of the high-quality males cannot be too much larger.

r 2007 Elsevier Inc. All rights reserved.

Keywords: Heredity; Signalling; Mating competition; General equilibrium; Handicap principle; Malthusian equi-marginal principle; Kin selection

1Getty (2006) argues that the resulting internally consistent system does

not correspond to a sports handicapping system, and that the metaphor is

not externally valid because better quality individuals do not necessarily

1. Introduction

In an early passage of The Handicap Principle (1997,p. xv), Amotz Zahavi and Avishag Zahavi describe theinitially adverse reception of their idea: ‘‘When we firstsuggested this handicap principle in 1975, it was almostunanimously rejected.’’ The peacock’s tail has become oneprime example of the principle. So colorful and magnificenton the one hand, likely to impress peahen. But socumbersome to carry around when fighting with otherpeacocks and compiling food. Placing oneself at such acompetitive disadvantage may on the one hand sound self-destructive, but may on the other hand signal to female’ssuperiority despite a substantial burden. The tide in theearly reception turned with the publication of two logicallycomplete articles by Grafen (1990a, b). Actually, owing to

e front matter r 2007 Elsevier Inc. All rights reserved.

b.2007.09.002

ing author.

ess: kjell.hausken@uis.no (J. Hirshleifer).

their high degree of mathematical abstraction, most readersmust have taken the Grafen articles on faith as demon-strating the internal logical consistency of the ZahaviHandicap Principle. (Or, as we will usually term it, theTruthful Signalling Hypothesis (TSH).) But, building uponGrafen’s work, the present authors (Hausken and Hirsh-leifer, 2004) have generated a logically complete analyticalsystem and solution consistent with the TSH. This methodgenerates easily reproducible results. So the question of theinternal logical consistency of the Handicap Principle hasbeen resolved.1 However, this ‘‘happy ending’’ is notcomplete. The Grafen and the Hausken and Hirshleifer

carry a bigger handicap. Consequently, there is not necessarily a

correlation between signaler quality and the ‘‘handicap’’ nor is there

necessarily a correlation between signal size and signal cost, across

signalers of different quality.

ARTICLE IN PRESSK. Hausken, J. Hirshleifer / Theoretical Population Biology 73 (2008) 11–2312

analyses were both based upon an artificial premise, thatthe quality differentials forming the basis for handicapsignalling were non-heritable. To wit, that the qualitydifferentials forming the basis for handicap signalling werenon-heritable. ‘‘In the first stage, males are randomlyassigned a quality independently of their genotype y’’—Grafen (1990a, p. 475).

Grafen’s analysis considered only with non-heritable

quality. However, many readers seem to believe that healso considered heritable quality. First, Dawkins (1989,p. 310) writes that ‘‘quality, for Grafen, means that y

females would benefit genetically if they mated with goodmales y’’ Second, Zahavi and Zahavi (1997, p. 58), claimsupport from Grafen’s demonstration, and relate handicapsignals with heritable male quality traits. In contrast,Noldeke and Samuelson (2001) observe that Grafenconfines attention to non-heritable quality.

So, since the main concern of biologists is surely withheritable quality rather than or in addition to non-heritablequality, the question is really still open. Is the HandicapPrinciple, or TSH, valid—in the sense of internal logicalconsistency—when the quality differentials forming thebasis of signalling are heritable? The object of the presentarticle is to provide a demonstration that the TSH doesindeed—within certain parameter ranges—extend to thecase of heritable quality.

Applied to mating choices, the paradox of signalling is asfollows. High signalling and thus high-quality males enjoysuperior mating success if females mate according tosignalling. Consequently, high-signalling males increase inrepresentation in the population in the next generation ifquality is heritable. Through successive generations thisdrives low-signalling and thus low-quality males toextinction, aside from some representation due to mutationpressure. Evidently, if signalling is to persist throughsuccessive generations, both low- and high-quality malesmust remain jointly present in the population. We believethat this contradiction is what led Fisher (1958 [1929],p. 152) to suggest a runaway disequilibrium process.Another alternative, the so-called ‘‘Hamilton–Zuk hypoth-esis’’ (Hamilton and Zuk, 1982), intended as a biologicallyplausible solution to the paradox of lost genetic variation,is that parasite–host coevolutionary cycles may result in anever-ending chase after changing optima.

Our resolution of the paradox involves an economic-evolutionary principle—we call it the Malthusian equi-marginal principle. To wit, that under selection pressuresthe relative numbers of higher and lower-quality organismswill change until, in equilibrium, not the average but the

marginal levels of quality will be equalized. Each typesignals until the marginal value of more signalling is zero.That results in both types growing until density-dependentcompetition within types results in reproductive successwhere the proportions of both types stabilize. Since themarginal value of signalling falls with increasing numbersfor each type regardless of the quality of that type, inequilibrium the marginal values of signalling are equal for

both types despite the fact that the reproductive success islarger for higher quality than for lower quality organisms.(This will be spelled out in detail later on.) The analogouseconomic equilibrium condition is that, in a competitiveindustry, no entering marginal firm can do better than earnzero profit, while no exiting firm can do any better insidethan outside the industry. In this analogy, the size of anindustry corresponds to the number of males of a giventype. That one additional firm enters the industry meansthat the number of males of this type increases with one.That one firm leaves the industry means that the number ofmales of this type decreases with one.The present discussion follows the procedural sequence

of our preceding article that dealt with the non-heritablecase. We set forth an explicit general equilibrium system ofequations, an analytic system that weaves togetheroptimization conditions on the individual choice leveltogether with the balancing conditions on the global levelneeded to make all the individual decisions consistent withone another. This system of equations is then solved,showing that—subject to certain parametric limitations—heritable quality differentials are indeed consistent withthe TSH.All females are assumed identical. Male quality consists

of a heritable endowment, which we call endowed quality,which is in part dissipated to the extent that the maleexpends resources generating signals. That is, the indivi-dual male dissipates some of his endowment in order togenerate signals. Think of the peacock’s tail whichconstitutes a signal, but is costly to produce, maintain,carry around, and a hindrance when compiling food andcompeting for scarce resources. More energy devoted tosignalling, for example, may mean less available forpaternal care. The male’s net quality, which is the trait ofinterest to females, is what remains after deducting themale’s incurred signalling cost. A male’s reproductivesuccess is the product of his per-capita mating success, hisnet quality, and the female’s mating capacity.As a modeling strategy we divide activities into a

resource-acquisition and a mating competition stage.The first stage, resource-acquisition, involves com-petition that is strictly within quality types. If there aremany of the high-type male, for example, this reducesthe benefit to being a high-type male. This reflects thenotion that high-type males have access to some resourcethat is not available to low types. As a result, high- andlow-type males may to some extent have different niches,so that males of each type compete with each othermore than with members of the other type. For example,the high type male may be able to fight and hold territory,whereas low-type males scrounge for resources in areasthat are not suitable for territorial defense. Or, the high-type male may have sufficient speed, strength andendurance to forage in different areas from the lowtypes, or to seek resistant prey that low types do notattempt to acquire. In the second stage, a single ‘global’mating market opens. This reflects the fact that a female

ARTICLE IN PRESSK. Hausken, J. Hirshleifer / Theoretical Population Biology 73 (2008) 11–23 13

can be impregnated by either a high- or a low-qualityconspecific male.

One objective of this article is to find an equilibrium,which validates the TSH under conditions of heritablequality. Once a solution is found for the equation system, itremains necessary to test for evolutionary stability. For aparticular solution to be valid, an arbitrary displacementmust generate evolutionary forces—in this case, changes inrelative population numbers of the H and L males—tending to restore the system toward, rather than force itaway from, the candidate equilibrium.

As is common in most analyses in the signallingliterature, and also common in economics and biologymore generally, an infinitely large population is assumed.Many or most analyses show that this assumption isusually innocuous, except for very small numbers. As thenumbers increase, the solutions usually improve, andbecome exact at the limit for infinitely large populationsizes. Kimura (1964), Gillespie (1974) and Demetrius andGundlach (2000) are examples of authors who haveanalyzed the impact of finite population size.

It might be interjected at this point that a genetic modelwould be needed to make claims about genetically heritabletraits. We do not think so. We think that it is valid toconsider the logic by which the relative proportions ofhigh- and low-quality males are present in successivegenerations without specifying the genetics that drivesthese relative proportions. By equating the marginal levelsof quality for high- and low-quality males, the absolutelevels of quality can remain different in an equilibrium thatwe demonstrate is evolutionarily stable, and females matewith both types of males because of congestion or searchcosts for choosiness. This ensures that both high- and low-quality males remain present through successive genera-tions. Specifically, the low-quality males do not go extinct.Going one level down to the genetic level is not necessaryfor such an analysis.

The article abstracts from introducing the time dimen-sion, which would greatly complicate the model. Alldecisions are made simultaneously at a given point in time.We can refer to this as stationarity through time. Themodel accounts for all the characteristics that are necessaryto account for, and there is nothing that distinguishes onepoint in time from another point in time when a male or afemale makes a decision.

The equilibrium solutions satisfy the strong Nashcondition, which is such that when all other members ofthe population play their equilibrium strategies, then anydeviating individual will suffer a loss in reproductivesuccess. That the strong Nash condition is satisfied impliesthat the weaker ESS condition of Maynard Smith (1976) isalso satisfied.

Section 2 that follows sets forth and explains the model,showing how it generates equilibria consistent with theTSH. Sections 2.1–2.3 are similar to that of Hausken andHirshleifer (2004), and are included to ensure that thisarticle is self-contained. In Section 2.4 starts the main

contribution of this article. Section 3 provides a baselinenumerical solution. Section 4 discusses system breakdown,and the conditions of its occurrence. Section 5 discussessimulations for the existence of equilibrium, consideringpaired-parameter variations. Section 6 extends to single-parameter variations. Section 7 concludes.

2. The signalling model with heritable quality—evolutionary

equilibrium

In this section, the elements of the male choice situationare first summarized, followed by a similar consideration offemale choices. After that the male and female optimiza-tion conditions are outlined. Next, we turn to the crucialMalthusian equi-marginal principle, which resolves theparadox described above—that the high-quality high-signalling males are always doing better (earning higherreproductive success) than their low-quality low-signallingrivals, yet need not force the latter to extinction.A congestion effect prevents mating exclusively withhigh-quality males, and we will see that an equilibrium ispossible where both types enjoy non-negative but differentreproductive success, though equal marginal values. Ineach case, we emphasize, what is provided is not anabstract general proposition but an exact instantiation withfunctional forms, with an explicit solution consistent withthe TSH under conditions of heritable quality.The modelling aim is to create a complete and consistent

set of equations that generates the two crucial features ofthe TSH: (1) That the male signals are ‘‘truthful’’. Thecondition is s14 s2, where big vs. small signals aresymbolized by the subscripts 1 and 2. (2) That the signalsare ‘‘believed’’ by the females. This means that the femalesdevote a more-than-proportionate fraction m of theirmating capacity to the high-quality high-signalling males.(Yet, since the L-type males do not go extinct, the fractionremains short of unity: pomo1.)

2.1. Male choices—the elements

The high- and low-quality male types (indexed as H, L)are, respectively, by hypothesis, also those emitting biggerand smaller signals (indexed as 1, 2). Each male’s endowedquality ei (i ¼ 1,2)—his available resources, external orinternal—can be employed either to generate signals ofmagnitude si or else to generate net quality ni as a mate. Wewrite this as a conventional linear cost trade-off, i.e.

ni ¼ ei � bisi for i ¼ 1; 2; male endowment constraint:

(1)

The bi are conversion coefficients (assumed constant)between male resources and levels of signalling, i.e., theyrepresent the per-unit or marginal costs of signalling—ingeneral differing as between the two male types. Since thecoefficient of ni is unity, endowed quality ei and net qualityni are denominated in the same ‘‘currency’’. Signalling is

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needed to mate, but reduces net quality which is needed forreproductive success.

Let m be the proportion of a typical female’s matingcapacity going to the high-quality (H) male type, theremainder of course going to the low-quality (L) type.When the TSH is valid, these will also be the proportionsgoing, respectively, to the big- and small signallers. Andsince p, 1�p are the population proportions of the twotypes, the H and L per-capita mating success rates will bem/p and (1�m)/(1�p), respectively.

As a third essential condition of a valid equilibrium solutionconsistent with the TSH, it must be the case that not only ise14e2 but also n14n2. In other words, the H-type males notonly have initially higher endowed quality; their net qualitymust also remain higher even after expending more resourcesupon signalling than the L-type males.

A male of either type will choose (that is, will be selected‘‘as if’’ he were choosing) the signalling level that maximizeshis fitness or reproductive success RSi. Consider a high-quality male. First, we assume that his reproductive successdepends positively on his net quality, qRS1/qn140. That is,net quality, and not endowed quality, is what counts forreproduction. A peacock’s tail constitutes a burden limitingenergy that can be devoted to paternal care. But second,reproductive success also depends positively on the propor-tion of a typical female’s mating capacity, per populationproportion of H males, going to H males, qRS1=qðm=pÞ40.Here’s where the peacock’s tail comes in since signalling isneeded to generate high m, as modeled below. The male thusfaces a tradeoff between on the one hand maximimizing netquality combined with limited signalling which causes lowerm, and on the other hand increasing signalling to generatefemale mating attention which means accepting lower netquality. Third, reproductive depends positively on thefemale’s mating capacity K defined below, qRS1=qK40.As a representative example of these three assumptions,assumed representative across a variety of biological situa-tions, we assume that male reproductive success is theproduct of his per-capita mating success, his net quality ni,and the female’s mating capacity K (to be described below):

RS1 ¼ Km

p

� �n1 and RS2 ¼ K

1�m

1� p

� �n2

male reproductive success: ð2Þ

To generate a high m, the male needs to signal, whichreduces ni. The male performs this tradeoff and is alsoaffected by the female’s mating capacity K. If the females‘‘believe’’ the signal, then m/p41. So the H males will havemore than one mating per capita, and the L males less thanone. The RS measures cannot be negative. First, K, m, pare between zero and one. Second, if ei in (1) is very small,the male of type i will adjust his signalling downwards, ifnecessary to si ¼ 0 which gives ei ¼ ni. Consequently, RS1

and RS2 are positive or equal to zero.Let us position Eq. (2) relative to other models in the

literature. First, Grafen (1990b, p. 522) models reproductive

success as w(a,p,q) ¼ pr qa, where qa is net viabilitycorresponding to net quality ni in our model, q is qualitycorresponding to endowed quality ei in our model, a isadvertizing corresponding to signalling si in our model,pr is his number of mates corresponding to Km/p in ourmodel, and r is a positive constant. Grafen thus considersreproductive success to be a product of a male’s net viabilityand his number of mates. A male needs high viability, as wellas a high number of mates, to generate high reproductivesuccess. This illustrates the same tradeoff as between netquality and signaling to generate high m in Eq. (2). Observethe close resemblance between Grafen’s conception andEq. (2). Second, Getty (1998a, pp. 127–128) modelsreproductive success explicitly as w ¼W(V(q, a), P(a)) andimplicitly as w ¼W(p, a) where q is quality, a is advertizing,V(q, a) is viability, and P(a) is fecundity. Getty (1998a,p. 128) discusses the explicit model and argues, together withJohn (1997), that ‘‘as with any phenotypic trait, the fitnesseffects of advertizing depend on viability costs and fecunditybenefits.’’ Again this demonstrates the trade-off in Eq. (2).

2.2. Female choices—the elements

With the aim of maximizing her reproductive success,each female will be choosing the fractions m and 1�m ofher mating capacity going, respectively, to the high- andlow-signalling males. However, to prevent her fromdevoting 100% of her mating capacity to the bigger-signalling males (since she ‘‘believes’’ them to be also themale type of higher net quality), some kind of diminishingreturns must be operative. The model here postulates aform of congestion or search costs for choosiness. Femalesmay have to wait in line for the preferred males, or spendtime and effort searching for them. The consequence isthat, in attempting to mate disproportionately with suchmales, i.e., to raise her chosen m above p, a female mustaccept some reduction of her mating capacity K.

A specific instantiation is

K ¼ 1� hðm� pÞ2 female mating capacity: (3)

The ‘‘congestion coefficient’’ h, another of the fixedparameters of the model, scales the extent to whichattempting to make m exceed p impacts negatively uponfemale mating capacity. Somewhat akin to congestion wouldbe dilution of paternal care on the part of preferred males.Let M denote the average quality of the typical female’s

mates, weighted by the mating fractions m and 1�m:

M ¼ mn1 þ ð1�mÞn2

average net quality of female’s mates: ð4Þ

Then the female’s reproductive success is

RSF ¼ KM ¼ pRS1 þ ð1� pÞRS2

female reproductive success; ð5Þ

where the rightmost equation is developed applying (2).Female per-capita reproductive success equals mating

ARTICLE IN PRESS

2We thank an anonymous referee of this journal for the reference to

Lande (1980, 1987).3This can be interpreted as a ‘‘decreasing-proportional-marginal-cost’’

criterion for truthful signalling, which appears to have been independently

discovered by Eshel et al. (2002), Hausken and Hirshleifer (2004),

Houston (2003), Proulx et al. (2002). We thank Getty (2006) for pointing

this out to us.

K. Hausken, J. Hirshleifer / Theoretical Population Biology 73 (2008) 11–23 15

capacity times the weighted average of the two malequalities. Our Eqs. (2) and (5) make our instantiation amultiplicative signalling model. See Getty (1998a, b, 2002)for a discussion of the distinction between additive andmultiplicative models within the context of Grafen’s(1990a, b) model and other models.

2.3. Optimization and equilibrium

The female will be selected to maximize her individualreproductive success by choosing the fraction of her matingcapacity to devote to the high-signalling males. In this model,all females are identical, and each individual female’s chosenmating fraction is independent of the choices of otherfemales. The mating fraction m depends on the congestion orsearch costs for choosiness experienced individually by eachindividual female as expressed by K in Eq. (3), which plays arole in her reproductive success. Although experiencedindividually, in equilibrium all females behave identicallycausing congestion or search costs for choosiness to dependon the behavior of all females. So the ‘‘typical’’ orrepresentative female optimizes by differentiating RSF withrespect to m, leading to the first-order condition:

Kðn1 � n2Þ ¼ �DM female F:O:C:; (6)

where D ¼ dK/dm is the marginal congestion effect:

D ¼ �2hðm� pÞ marginal congestion effect: (7)

The representative female balances the marginal costagainst the marginal benefit of shifting a unit of her matingcapacity from the smaller-signalling type-2 males to thebigger-signalling type-1 males. The marginal benefit ofshifting, the left-hand side of (6), is the difference in netqualities n1–n2 weighted by her mating capacity K. Themarginal cost, the right-hand side, is the loss of matingcapacity due to congestion, multiplied by the average netquality of the males she would otherwise have mated with.Without congestion or search costs for choosiness, femaleswould mate exclusively with the high-signalling males,which would lead the low-signalling males to go extinct.Such an approach would not have empirical support. (It isstraightforward to verify that the second-order conditionfor a maximum is satisfied.)

Following a novel concept introduced in our earlier article,a mating success function (MSF) is employed to expressfemale sensitivity to male signalling differentials. If type-Hand type-L males are signalling at the respective levels s1 ands2, the MSF tells us how the signals translate into matingsuccess. Put another way, the MSF specifies the degree towhich the representative female attends to the signals inchoosing her mating fractions m and 1�m. Instantiating witha specific functional form, we postulate that females willrespond to the male signals in accordance with

m

p¼

1�m

1� ps1

s2

� �r

3m ¼psr

1

psr1 þ ð1� pÞsr

2

mating success function: ð8Þ

Eq. (8) is reminiscent of Lande’s (1980, 1987) psycho-biological mating system, where the frequency of eachmale is weighted by a females preference.2 Thus, the ratiom/(1�m) in which a female distributes her mating capacitybetween the high- and low-signalling male types reflectstheir proportionate representation in the population,p/(1�p), augmented by a signalling factor—the ratio ofthe signalling efforts raised to a power r 40. Here r, whichscales the extent to which differing signalling intensitiestranslate into mating success, is the ‘‘signalling decisivenessparameter’’. If r=0 there is no effect at all: the females payno attention to signals and mate at random, so that m=p.The higher is r the more decisive are the signals in bringingabout mating success. The MSF is an analog of what hasbeen called the contest success function in economicconflict theory (Tullock, 1980; Hirshleifer, 1989, 1991;Skaperdas, 1996; and see also Grafen, 1990b, p. 522).Although r is parametric from the point of view of each

and every male and female acting singly, for the analyticsystem as a whole the magnitude of r is an endogenousvariable. More specifically, r is the equilibriating variablethat glues together the male and female decisions. It reflectsthe overall ‘‘market conditions’’ in the signalling process—the willingness of the different types of males to emitsignals, and of the females to attend to them.Given the females’ choice of the mating fraction m, on

the male side there will be in equilibrium a chosen pair ofsignalling levels s1 and s2 for the H and L types,respectively. Given that, within each type, all males areidentical, each individual male’s chosen signalling level isindependent of the choices of the other males of his type.So for each male type, we can think of a typical orrepresentative individual as optimizing by differentiatingRS1 or RS2, as the case may be, with respect to therespective signalling levels s1 or s2. The differentiation isshown in Appendix A and gives the first-order conditions:

b1s1 ¼ rðe1 � b1s1Þ and b2s2 ¼ rðe2 � b2s2Þ male F:O:C:s:

(9)

So, apart from the fixed parameters, the equilibriummale signalling levels respond only to the intermediating‘‘decisiveness’’ parameter r that signifies the extent to whichthe females attend to the signals. Furthermore, s1 isindependent of s2 and N2, except through r.Eliminating r from (9) leads to the important result3:

s14s23b1=e1ob2=e2 truthful signalling condition:

(10)

This crucial proposition is the exact relationship betweenendowment levels and unit signalling costs required, in this

ARTICLE IN PRESSK. Hausken, J. Hirshleifer / Theoretical Population Biology 73 (2008) 11–2316

formulation, for the TSH to hold. As the simulations willconfirm, the TSH can be valid even if the H-type maleshave higher unit signalling costs than the L-type males(b1/b2 41)—provided this disadvantage is sufficientlycounterbalanced by the H-type males’ superiority inendowed quality (e1/e2 is sufficiently large).

2.4. Malthusian considerations and evolutionary equilibrium

This section turns to the conditions for evolutionaryequilibrium when quality is heritable. To recall theparadox, if females are attending to truthful signals, thenhigher-quality (hence higher-signalling) male contendersmust necessarily win disproportionate mating success. Soin the next generation the high-quality high-signallingmales will increase in representation in the population.Thus p would steadily rise. This process, continuing overthe generations would seem to imply inevitable extinctionof the low-quality males. But if signalling is to persist, bothlow- and high-quality males must remain substantiallypresent in the population. We believe that this offers apotentially more general solution than Hamilton and Zuk’s(1982) popular solution that parasite–host coevolutionsolves the paradox.

Allowing for diminishing returns, so that as p rises RS1

falls relative to RS2, does not itself suffice to resolve theparadox. So long as any positive difference between RS1

and RS2 persists, p will remain subject to upward pressure.But RS1 ¼ RS2 is not a possible equilibrium condition. In(2), if m exceeds p then necessarily n1 must be less than n2.But as a condition of equilibrium, the H-type males mustremain of higher net quality (n14 n2) despite expendingmore resources in signalling.

As suggested above, the escape from this paradox restsupon realizing that the true criterion, for balancing therelative numbers of high-quality vs. low-quality males,consists of equating not average fitness (RS1 or RS2) butmarginal fitness—the Malthusian equi-marginal principle.

The Malthusian equi-marginal principle is the essentialfeature permitting an evolutionary equilibrium withheritable quality that also meets the essential conditionsof the TSH: to wit, sH ¼ s1 4sL ¼ s2, and pomo1.The upshot is that, although the H-type males are alwaysdoing better on average (RS14 RS2), and consequentlyare gaining a disproportionate fraction of the femalemating capacity (m4p), there is a limit to how much theycan increase in numbers relative to the low-qualitymales. Once the marginal fitness marginal reproductivesuccess (MRS1) of an H-type male is no higher thanthe marginal fitness MRS2 of an L-type male, the numbersof the two types stabilize. The numbers of the two typesstabilize according to kin selection. Each individual maleof each type maximizes his fitness until his marginalfitness equals the individual marginal fitness of theother type. In equilibrium, all individual males of thesame type behave identically since they have equivalentcharacteristics.

Returning to the equation system, the total femalepopulation N, also considered as a variable, is byassumption equal to the sum of the two male types:

N ¼ N1 þN2. (11)

Changes in the fractions p, 1�p of the two male typeswill affect the degrees to which each is pressing upon theenvironments. As p grows, each individual of the H-type islikely to run into a kind of Malthusian diminishing returns,tending to reduce its endowed quality e1. Correspondingly,of course, the reduced environmental pressure is likely toincrease e2, the endowed quality of each individual of theL-type males. We instantiate the relation between endowedqualities and the respective population sizes with thefollowing specific functional form:

e1 ¼ E1 expf�q1N1Þ,

e2 ¼ E2 expf�q2N2Þ Malthusian diminishing returns:

ð12Þ

The constants E1 and E2 are the underlying qualityparameters. The endowed qualities e1 and e2, previouslyinterpreted as constants, now become variables respondingto population size. The constants q1 and q2 are ‘‘theMalthusian crowding parameters’’. They are measures ofdiminishing returns. As these parameters rise in numericalvalue, for each male type endowed quality falls off morerapidly as its numbers increase. Eq. (12) introduces theimportant idea of resource competition. For simplicity, weassume that it is strictly within types, implying that there isno resource competition across the high- and low-qualitytypes. That is, the model focuses on signalling competition,and the important characteristic of (12) is that the endowedqualities e1 and e2 both fall off as population sizes N1 andN2 increase. If Ni approaches infinity, which causes N toapproach infinity, ei approaches zero, which according to(2) causes the reproductive success RSi to approach zero,i ¼ H, L. If both N1 and N2 approach infinity, RSF

approaches zero according to (5). Eq. (12) implicitlyaccounts for both births and deaths, so if Ni increases,there must be more births than deaths. In a steady-stateequilibrium, all variables level out, and we get qNi=qt ¼ 0for Ni40, and similarly for all the other variables, where t

is time.Following the usual derivation provided in economic

textbooks for the algebraic relation between average andmarginal magnitudes, the MRS for the two male types canbe determined. Let us define the ‘‘total’’ RS measures forthe two male types as kin groups, as TRSi ¼ NiRSi. The‘‘marginal’’ measures, as new members of the respectivepopulations enter, are defined as in standard economictexts, by differentiating with respect to the populationsizes. This gives MRSi ¼ q(TRSi)/qNi ¼ RSi+NiqRSi/qNi.Since qRSi/qNi is negative, as expected the ‘‘marginal’’ RS

measures will be smaller than the ‘‘average’’ measures.Inserting (1) into (2) for the high-quality males givesRS1 ¼ K(m/p)(e1�b1s1). When differentiating RS1 with

ARTICLE IN PRESSK. Hausken, J. Hirshleifer / Theoretical Population Biology 73 (2008) 11–23 17

respect to N1 we take K and (m/p) as given (which meansthat they are interpreted as constants in the differentiation)since these are determined in the ‘‘female module’’. And wetake s1 as given since it is determined in the ‘‘male module’’.This gives qRS1/qN1 ¼ K(m/p)qe1/qN1, where qs1/qN1 ¼ 0.The reason for this different treatment of e1 and s1 is thatthe endowed quality e1 is a variable responding directly topopulation size N1 as specified in (12). In contrast, thesignal s1 is determined in the ‘‘male module’’ bydifferentiating RS1 with respect to s1 taking K, r, and e1as given, which means that the population size N1 is alsotaken as given. Hence although s1 in (9) can be expressed interms of e1 which depends on N1, s1 is determined takingN1 as given. Each male thus chooses his signal takingpopulation sizes as given when optimizing. This does notmean that signalling is independent of population sizes, butthe dependence is indirect through the other variables suchas the mating fraction m, the glue variable r, and the othervariables. In an equation system, of course, all variablesdepend on all other variables unless we deal with two orseveral independent equation systems. Using (12) andapplying analogous reasoning for the L-type males give

MRS1 ¼ RS1 þN1qRS1=qN1

¼ RS1ð1� q1E1N1 expf�q1N1g=n1Þ,

MRS2 ¼ RS2 þN2qRS2=qN2

¼ RS2ð1� q2E2N2 expf�q2N2g=n1Þ

definition of MRSi. ð13Þ

The population fraction represented by the H-typemales, p, is

p ¼N1

N1 þN2definition of p. (14)

Finally, the Malthusian equi-marginal principle dictatesthat, in evolutionary equilibrium:

MRS1 ¼ 1 ¼MRS2,

conditions of evolutionary equilibrium: ð15Þ

Let us assume that all males take the number N offemales as given when maximizing reproductive success.Since individuals that are related by kinship share commongenes, a randomly selected high-quality male has a higherprobability of being related by descent to another selectedhigh-quality male than to a low-quality male. Thus, kinselection (Hamilton, 1964) implies that, to the extent thatindividuals of a given type are related by descent, theyshould be altruistic toward each other. As a strategicmodeling simplification, we reflect this within-type kin-altruism in a specific form. Although males of a given typecompete for resources and mates, we assume thatindividuals of a given type adjust their numbers (N1 orN2) to maximize the joint reproduction of that type. Inother words, each type is maximizing the RS of the type.Normally, for group selection we would expect that each‘‘group’’ might contain both types. But for kin selectionthere is a reason why the KIN group would contain just

one of the types—they are genetically related. In otherwords, in the extreme a kin group will be geneticallyidentical. This means that the Malthusian equi-marginalprinciple is consistent with kin selection. The evolutionaryequilibrium is such that mutant types do not crowd eachother, but instead compete against each other in kin groupsthat try to maximize total productivity.Eq. (15) stands in contrast to the common stability

criterion that RS1 ¼ RS2. However, in nature, and in ourmodel as well, stability over time occurs exactly whenRS14RS2X0. We showed in Section 2.1 that the RS

measures cannot be negative. The high-quality malesachieve higher reproductive success RS1 than RS2 for thelow-quality males. Because of congestion or search costsfor choosiness, females will not mate exclusively with high-quality males, regardless how large is RS1, as long as RS2 ismarginally positive. This means that neither the matingfraction m, nor the fraction p of high-quality males, canincrease to one, though we have m4p since females to alimited extent (constrained by congestion or search costsfor choosiness) prefer high-quality males over low-qualitymales. In this light, equating RS1 and RS2 does not makesense. High- and low-quality males cannot earn equalreproductive success. Instead, each male maximizes hisown reproductive success and signals until the marginalvalue of more signalling is zero, which gives an equilibriumwhere both male populations stabilize, which also stabilizesthe female population. This means that the frequency ofhigh-quality males, and the frequency of low-quality males,remain constant through time. Consequently, the numberof females also remain constant through time. Each maletakes his own endowment, other characteristics, the femalemating fraction, and congestion or search costs forchoosiness as experienced by females, into account. Thereis no comparison of reproductive success across the maletypes. A high-quality male cannot switch to become a low-quality male, and a low-quality male cannot switch tobecome a high-quality male. Consistent with economicreasoning, equilibrium numbers are determined by themarginal conditions. The equilibrium is determined by themarginal profitability of male entry of the two types,determined by birth and death processes where the relativeproportions of the two male types are determined byfemale choice. As the population size increases for eachtype, the marginal value of signalling decreases, regardlessof the quality of that type. Hence, in equilibrium themarginal values of signalling are equal for both typesdespite reproductive success being larger for high-qualitymales than for low-quality males.Inserting (15) into (13) and requiring RS14RS2 implies

that we cannot have N1 ¼ 0 and N240. If we hypotheti-cally assume N1 ¼ N2 ¼ 0, it follows from (12) that e1 ¼ E1

and e2 ¼ E2, from (13) applying (15) that MRS1 ¼

RS1 ¼MRS2 ¼ RS2 ¼ 1, which contradicts RS14RS2,from (14) that p ¼ 1, and from m4p, that m ¼ 1. Whenm ¼ p, (3) gives K ¼ 1, and (2) gives RS1 ¼ n1 andRS2 ¼ n2. Hence n1 ¼ n2 ¼ 1. Applying (1) and (9) implies

ARTICLE IN PRESSK. Hausken, J. Hirshleifer / Theoretical Population Biology 73 (2008) 11–2318

b1s1=r=b2 s2. Applying (1) gives E1 ¼ 1+b1 s1 ¼ 1+r andE2 ¼ 1+b2s2 ¼ 1+r, which is a contradiction since werequire E14E2. Consequently, in the evolutionary equili-brium the low-quality males cannot persist while the high-quality males go extinct, and both types of males cannot goextinct, i.e. N140 and N2X0.

Taking stock at this point, the equation system has 20variables: m, s1, s2, n1, n2, K, M, r, D, e1, e2, N, N1, N2, RS1,RS2, RSF, MRS1, MRS2, and p. Counting Eqs. (1), (2), (9),(12), (13) and (15) provide two each and (3)–(8) plus (11),(14) provide one each. (That the numbers of equations andvariables match up does not of itself demonstrate that thesystem is internally consistent and complete, but thenumerical simulations that follow will verify the formalvalidity of the equation system.)

For the simpler case of non-heritable quality dealt within our earlier article, it was possible to find an explicitanalytic solution for the equation system as a function ofthe underlying exogenous parameters. Owing to theincreased analytic complexity, this has not proved possiblefor the case of heritable quality. But our numerical baselinesolution will suffice to demonstrate, by direct example, thelogical completeness and consistency of the equationsystem. After exploring a number of the features of thebaseline solution, we will examine how the solutionchanges in response to exogenous variations of a numberof the important determining parameters.

3. A baseline solution

In Table 1, the columns to the left define the variablesand display the numerical solution values, while thecolumns to the right show the parameter settings for thebaseline solution.

Table 1

Variable definitions and baseline settings

Variables and baseline for signalling solutions

m 0.408803 Proportion of female mating capacity going to H-ty

s1 3.526643 Signalling level of H-type males

s2 1.097933 Ditto for L-type males

n1 3.936009 Net quality of H-type males

n2 2.450758 Ditto for L-type males

K 0.972132 Female mating capacity

M 3.057933 Average net quality of mated males

r 0.4479974 Signalling decisiveness parameter

D �0.472169 Marginal congestion effect

e1 5.699331 Resource endowment of H-type males

e2 3.548691 Ditto for L-type males

N 4.834181 Number of females

N1 1.405591 Ditto for H-type males

N2 3.428591 Ditto for L-type males

RS1 5.379719 Reproductive success of H-type males

RS2 1.985936 Ditto for L-type males

RSF 2.972715 Ditto for females

MRS1 1 Marginal reproductive success of H-type males

MRS2 1 Ditto for L-type males

p 0.290761 Proportion of H-type males

3.1. Features of the baseline solution

At the chosen baseline parameter values, the TSH isvalidated:

1.

pe m

The H-type males, characterized by higher underlyingquality (E1 ¼ 104E2 ¼ 5) and endowed quality(e1 ¼ 5.6994e2 ¼ 3.549) do indeed signal more heavilythan the L-type males (s1 ¼ 3.5274s2 ¼ 1.098). Despitethis heavier signalling burden, the signalling remains‘‘truthful’’ (that is, the high signalling males remain ofsuperior quality) in terms of net quality after takingsignalling costs into account: n1 ¼ 3.9364n2 ¼ 2.451.

2.

The females ‘‘believe’’ the signals, that is, they grant thehigher-quality males a disproportionate share of thematings (m ¼ 0.4094p ¼ 0.291).

3.

Since p remains well below unity, the high-quality malesfall well short of forcing the low-quality males toextinction.

4.

The inferior-quality males remain in the population,even though their average reproductive success RS2

is far lower than the RS1 of the H-type males(1.986o5.380), because, at equilibrium, the marginalmeasures of reproductive success for both types equalunity (the Malthusian equi-marginal principle).

3.2. Evolutionary stability of the baseline solution

An acceptable solution must not only satisfy the systemof equations but must also be evolutionarily stable. Thetest is, if the number Ni of either type of male is arbitrarilydisplaced from the candidate equilibrium value, doMalthusian forces tend to restore the equilibrium—or dothey lead to further divergences from it? Thus, for

Settings of the fixed parameters

ales b1 ¼ 0.5 Signalling costs per unit for H-type males

b2 ¼ 1.0 Ditto for L-type males

h ¼ 2.0 Congestion effect coefficient

q1 ¼ 0.4 Malthusian crowding parameter of H-type males

q2 ¼ 0.1 Ditto for L-type males

E1 ¼ 10 Underlying quality of H-type males

E2 ¼ 5 Ditto for L-type males.

ARTICLE IN PRESS

Fig. 1. (a) Evolutionary stability of the baseline solution. MRS1 as a

function of N1 when MRS1 ¼ 1 is disabled. (b) Evolutionary stability of

the baseline solution. MRS2 as a function of N2 when MRS2 ¼ 1 is

disabled.

Fig. 2. MRS1 as functions of N1, MRS1 ¼ 1 is disabled.

K. Hausken, J. Hirshleifer / Theoretical Population Biology 73 (2008) 11–23 19

evolutionary stability, qMRS1/qN1 and qMRS2/qN2 mustboth be negative as either N1 or N2 is displaced awayfrom its equilibrium value. Differentiating and applyingqRS1/qN1 ¼ qRS1/qe1 leads to

qMRS1

qN1¼

RS1ðq1N1 � 2Þq1E1 expf�q1N1g

n1

o0 when q1N1o2 evolutionary stability: ð16Þ

And similarly for displacements of N2.(Owing to the complexity of the equation system, we

have not been able to reduce the stability conditions todirect functions of the exogenous parameters. So stabilitycannot be determined by analytic inspection, but has to betested by simulation.)

In Fig. 1a, the solution values for MRS1 do indeeddisplay negative slope as N1 increases or decreases awayfrom its equilibrium value. (The plotted values are foundby disabling the equation MRS1 ¼ 1 and then solving thereduced system for the values of MRS1 implied by values ofN1 in the neighborhood of the solution value.) Similarly,Fig. 1b shows that the curve for MRS2 also has negativeslope in the neighborhood of the equilibrium value of N2.This verifies that the baseline solution not only satisfies thesystem of equations, but is also evolutionarily stable.

4. System breakdown

System breakdown (SB) occurs when no solution exists.In Fig. 2, with N1 on the horizontal axis, the relevantcondition of equilibrium is MRS1 ¼ 1. (An analogousdiagram could be constructed for N2, using the conditionMRS2 ¼ 1.) If the equation MRS1 ¼ 1 is disabled and theremainder of the system solved for MRS1, for any given setof parameters a curve like those pictured is generated.Varying one of the parameters (in this case, b1) whileholding all the other parameters at their baseline values,the rightmost curve for b1 ¼ 1.312 shows that a solutionexists—since the generated MRS1 curve intersects the lineMRS1 ¼ 1. But the leftmost curve for b1 ¼ 1.315 neverintersects MRS1 ¼ 1, so no solution exists for this largervalue of b1. The middle curve gives the maximum valueb1 ¼ 1.3135 where a solution exists. This b1 value is farabove the baseline b1 ¼ 0.5, and the population size of H-type males decreases to N1 ¼ 0.5808. H males sufferbecause of the high unit cost of signalling. A higher costcannot be sustained while keeping the MRS as low asMRS1 ¼ 1.

5. Simulation tests for existence of equilibrium—paired-

parameter variations

By jointly varying related pairs of parameters, thissection explores the question of SB—or, equivalently, theexistence of equilibrium in our equation system. In Fig. 3a,the shaded stable region (SR) to the right and below thestraight solid linearly increasing line represents the set of

q1,q2 pairs—the Malthusian crowding parameters fromEq. (12)—that lead to (evolutionarily stable) solutions ofthe equation system. These parameters measure the respec-tive intensities of diminishing returns, that is, for each maletype how rapidly quality falls off as its population numbersincrease.

ARTICLE IN PRESS

Fig. 3. (a) Stable region SR in a q1 vs. q2 diagram. (b). Stable region SR in

a b1 vs. b2 diagram.

K. Hausken, J. Hirshleifer / Theoretical Population Biology 73 (2008) 11–2320

Inserting (1),(2),(8),(12),(14) into (15), and applyingfrom (1) and (9) that bisi=rni gives

n1ð1�N1q1ð1þ rÞÞ

n2ð1�N2q2ð1þ rÞÞ¼

s2

s1

� �r

. (17)

For the numerator and denominator on the left-handside to be positive we must have ro(1�N1q1)/(N1q1) andro(1�N2q2)/(N2q2). Requiring n14n2 and s14s2, (17)implies

fn14n2 and s14s2g

)N1q1

N2q2

41)b1

b2o

e1

e2¼

b1s1

b2s2¼

n1

n2o

E1

E2, ð18Þ

where the second implication follows from using(1),(9),(10),(12). At the end of Section 2, we proved thatwe must have N140 and N240 in the evolutionaryequilibrium. Eq. (18) specifies for finite values of N1 andN2 that q1 cannot be too small relative to q2. For each valueof q2, Fig. 3a shows the lowest value of q1 that gives a SR.Intuitively, for a solution to exist, diminishing returns mustoperate with greater intensity upon the superior-qualitypopulation, else it would eventually drive the lower-qualitypopulation to extinction. Hence N1q1 cannot be too small.

Fig. 3b is a corresponding diagram for the parameter-pair b1, b2—the respective marginal costs of signalling.There has been some discussion in the literature whether,consistent with the idea that signalling must reflect ahandicap, for a TSH equilibrium it must be that b14b2. SeeNur and Hasson (1984), Grafen (1990b, p. 522), Godfray(1991, p. 329), Hurd (1995, p. 219), Szamado (1999), Getty(1998a, b, 2002). Eshel et al. (2002) show that decreasingmarginal costs are unnecessary, while decreasing propor-tional marginal costs are necessary. Our earlier articleshowed that, assuming non-heritable quality, this was not astrictly necessary condition, although for an interiorequilibrium to exist it must be the case that b1 cannotexceed b2 by too wide a margin. Fig. 3b reveals thatessentially the same result applies also when quality isheritable. The entire shaded region above the 451 line ispart of the solution region, together with a shaded sectorbelow that indicates the extent to which solutions arepossible even when b14b2—that is, when the H-type malesmust incur heavier per-unit signalling costs.It might be wondered how it is that solutions exist even

when b1 and b2 rise without limit. The reason, as can beseen in Eq. (9), is that the signalling levels s1 and s2 aresimultaneously falling off toward zero. Indeed, as we willsee in the single-parameter simulations below that showmore detail, as the levels of per-unit signalling costs b1 andb2 rise, the corresponding total signalling costs b1s1 andb2s2 actually fall.

6. Simulation tests—equilibrium implications of single-

parameter variations

In order to explore some of the conditions bearing uponthe existence of solutions validating the TSH, the precedingsection studied joint variations of the parameter-pairs q1,q2and b1,b2. In order to generate more detailed implicationsfor the dependent variables—among them the signallinglevels s1 and s2, the population sizes N1, N2, and N, thedegrees of reproductive success RS1 and RS2, and the valueof the signalling decisiveness parameter r—this sectionexplores variations in a number of the crucial parameterstaken one at a time.

6.1. Parametric variation of relative underlying quality—

e1 vs. e2

Fig. 4 shows how equilibrium values of several of thecrucial variables respond to parametric variations of E1—the underlying quality of the H-type males—relative to afixed E2 ¼ 5. The underlying quality of the H-type males isallowed to vary both upward and downward from itsbaseline value E1 ¼ 10, all the other parameters being heldat their baseline values.In the top panel of Fig. 4, we see that as the H-type males

rise in quality, as expected both m and p—their fractionalmating success and their representation in the popula-tion—increase. In the lower panel especially s1—the

ARTICLE IN PRESS

Fig. 4. Endogenous variables as functions of the H male’s underlying

quality (E1).

Fig. 5. Endogenous variables as functions of the H male’s unit signalling

cost (b1).

K. Hausken, J. Hirshleifer / Theoretical Population Biology 73 (2008) 11–23 21

signalling level of the H-type males—increases dramaticallyin E1. The L-type males follow suit to their best ability withs2, which is costly, and s2 increases concavely. At the sametime, the population size N2 falls rapidly. (Division with 2for the population sizes N1 and N2 is for scalingconvenience.) The female population size N is not plottedsince it is given by N ¼ N1+N2, and is thus variable. As E1

increases above E2 ¼ 5, the decisiveness r first increaseswith strong concavity. The H-type males seek to separateor distinguish themselves from the L-type males throughincreased signalling, which gives increased decisiveness.This is costly when E1 is only marginally larger than E2, sothe population size N1 of H-type males first falls slightly,and thereafter increases slightly, i.e. is U formed. Thedecisiveness r is inverse U formed, reaches a maximum forE1 ¼ 15.35, and thereafter decreases very slightly. Asexpected, the reproductive success RS1 of H-type malesincreases in E1, while RS2 decreases, while femalereproductive success RSF is intermediate and increasesmoderately. Also the net quality n1 increases strongly,while n2 is slightly U formed. Finally, for the lowestunderlying quality for the H-type males, E1 ¼ E2 ¼ 5,

signalling ceases, s1 ¼ s2 ¼ 0, which also causes zerosignalling decisiveness r ¼ 0.

6.2. Parametric variation of relative underlying quality—b1

vs. b2

Fig. 5 shows response to parametric variations of b1—the unit signalling cost of the H-type males—relative to afixed b2 ¼ 5.The most intuitive result is that the reproductive success

RS1 of H-type males shows the largest decrease in b1. Also,intuitive is that the signalling of H-type males decreasesconvexly from infinity at zero cost b1 ¼ 0. When b1 is belowb2, L-type males realize their signalling inferiority, and s2 isvery low, which also causes decisiveness r to be low. L-typemales’ underlying quality is half that of H-type males’,E2 ¼ E1/2, and the low signalling allows for sustaining alarge population N2 of L-type males. L-type males have areproductive success RS2 around 1/3 that of H-type males.However, as b1 increases toward b2, male competitionbecomes more fierce and decisive. L-type males realize abenefit from signalling, and s2 increases. This causes the

ARTICLE IN PRESSK. Hausken, J. Hirshleifer / Theoretical Population Biology 73 (2008) 11–2322

population N2 of L-type males to fall off dramatically,which is remarkable. That is, higher unit cost for H-typemales causes higher decisiveness and a lower populationN2. The population size N1 of H-type males falls off too,but less dramatically.

As b1 increases above b2 ¼ 0.5, a new effect starts tooperate. To wit, N2 falls off more than N1, and s2 increases.Consequently, H-type males cannot allow s1 to decreasetoo much, so s1 levels out, and eventually increases, despitethe increased unit cost b1 of such signalling! s1 is thus U

formed. Although s1 increases, s2 increases more andconvexly since L-type males has the same fixed unit cost ofsignalling. r also increases convexly. This developmentnaturally gives SB at some upper value of b1, specified asb1 ¼ 1.3135 in Fig. 5, which also can be seen from Figs. 2and 3b. At this breakdown, the population sizes N1 and N2

is pushed very close to zero, though they cannot be pushedall the way to N1=N2=0 as shown in Section 5. Theincreased signalling s2, as b1 increases, causes also thereproductive success RS2 of L-type males to decrease,though less than RS1 decreases. RSF, n1, and n2 decreasetoo. Finally, the fractional mating success m, andrepresentation p in the population, of H-type males,are almost constant but increase slightly. Two oppositeeffects operate here. On the one hand, increasing b1 is adisadvantage for H-type males which should decrease m

and p. On the other hand, the rapidly decreasing N2 andeventual increase of s1 sustain female interest in H-typemales causing slight increase in m and p.

7. Conclusion

The Handicap Principle (Zahavi, 1975) has gainedincreased acceptance. Grafen (1990a, b) presented a for-malization of the principle, and its internal logicalconsistency has recently been demonstrated by Hauskenand Hirshleifer (2004). These analyses have assumed non-heritable quality. The main concern of biologists ishowever with heritable quality. Assuming heritable qualityleads however to a seeming paradox. On the one hand,high-quality high-signalling males will increase in repre-sentation in the population through the generations. Onthe other hand, for signalling to persist both low- and high-quality males must remain present. Fisher (1929) suggestedresolving the paradox with a runaway disequilibriumprocess, and Hamilton and Zuk (1982) proposed thatparasite–host coevolutionary cycles may cause never-end-ing chases after changing optima. This article takes a quitenovel approach to resolving the heritability paradox.Under selection pressures, the relative numbers of higher-and lower-quality organisms will change until, in equili-brium, not the average but the marginal levels of qualitywill be equalized. This Malthusian equi-marginal principleis the key analytic innovation of the article, and it is shownto be consistent with kin selection. In nature, and in ourmodel as well, high-quality males achieve higher reproduc-tive success than low-quality males in an equilibrium with

stable population sizes. This is possible since femalesbecause of congestion or search costs for choosiness willnot mate exclusively with high-quality males. Each malemaximizes his own reproductive success and signals untilthe marginal value of more signalling is zero.The equation system leading to solutions consistent with

the TSH contains, as key elements, a MSF which indicateshow female mating choices respond to higher and lowersignalling levels, and a congestion function which rules outcorner solutions in which females would mate exclusivelywith higher-quality males. Furthermore, for evolutionarystability, the derivative of the MRS for each male type withrespect to the population size of that type must be negative.A negative derivative implies that the population sizedecreases for that type. This means that an out-of-equili-brium displacement to a higher population size causeslower MRS, which reduces the population size and thusrestores the equilibrium, and similarly for a displacementto a lower population size. The absolute population sizesare determined by equating the MRS of the two male types,which gives an equilibrium with higher reproductivesuccess for high-quality than for low-quality males.The model developed is solvable for sets of parameters

that constitute the SR, in contrast to unstable regionswhere SB occurs. We test the SR and confirm evolutionarystability. That is, an arbitrary displacement must generateevolutionary forces, in this case, changes in relativepopulation numbers of the H and L males—tending torestore the system toward, rather than force it away from,the candidate equilibrium.We particularly consider the unit signalling costs of the

H- and L-type males, where it has been widely believed thatthe former must be lower than the latter. This articleconfirms our earlier result that this is not necessary, thoughfor a stable TSH equilibrium to persist the unit signallingcosts of the high-quality males cannot be too much larger.The article shows that, subject to certain parametric

limitations, heritable quality differentials are consistentwith the TSH.

Acknowledgments

We thank two anonymous referees of this journal,Thomas Getty, and David Hirshleifer for useful comments.

Appendix A. Male first-order conditions

The equilibrium signalling levels are s1 and s2, and thelevels chosen by possibly deviant males be s01 and s02. Inequilibrium the optimizing choices of possible type-1deviants are such that s01 is equal to s1, and similarly forpossibly deviating type-2 males. Even though deviation ispossible, it is always more profitable to conform. Considerfirst potentially deviant type-1 males who constitute aproportion A, where Aop, of the overall male population.Letting m0 and m denote the mating fractions assigned bythe typical female to the deviating and non-deviating type-1

ARTICLE IN PRESSK. Hausken, J. Hirshleifer / Theoretical Population Biology 73 (2008) 11–23 23

males, respectively, the corresponding RS functions are

RS01 ¼ Km0

�

� �n01 where n01 ¼ e1 � b1s

01,

RS1 ¼ Km

p� �

� �n1 where n1 ¼ e1 � b1s1. ðA:1Þ

Adapting the MSF equation (8) to the competitionbetween the deviating and non-deviating type-1 males:

m0

m¼

�

p� �

� � s01s1

� �r

. (A.2)

Inserting (A.2) into (A.1) and taking the derivative withrespect to s01 yields

RS01 ¼ Km

p� �

� � s01s1

� �r

ðe1 � b1s01Þ,

qRS01qs01¼

K

s01

m

p� �

� �½rðs01Þ

r�1ðe1 � b1s01Þ

þ ðs01Þrð�b1Þ� ¼ 0. ðA:3Þ

After cancellations, this first-order condition for thedeviant’s choice of s01 does not involve the non-deviant’s s1at all. So each and every type-1 male, regardless of what theothers are doing, will be choosing his signalling level inaccordance with Eq. (9). The optimality of s01 does notdepend upon the value of A, which obviates the need for Ato move toward zero as a limit. So no type-1 male will wantto deviate, and similarly for the type-2 males.

The second-order condition for the type-1 males is

q2U 01qðs01Þ

2¼

K

sr1

m

p� �

� �rðs01Þ

r�2½�e1 þ e1r� ðrþ 1Þb1s01�: (A.4)

Inserting e1r ¼ (r+1)b1s01 from the first-order condition

above ensures negativity and thus confirms existence of amaximum. The development for the type-2 males is entirelyparallel.

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