Costly Signaling and Cooperation

J. theor. Biol. (2001) 213, 103}119doi:10.1006/jtbi.2001.2406, available online at http://www.idealibrary.com on

Costly Signaling and Cooperation

HERBERT GINTIS*-, ERIC ALDEN SMITH? AND SAMUEL BOWLESA

*Department of Economics, ;niversity of Massachusetts, Amherst, MA, ;.S.A., ?Department ofAnthropology, ;niversity of=ashington, Box 353100, Seattle,=A 98195-3100, ;.S.A. and

ASanta Fe Institute and Department of Economics, ;niversity of Massachusetts, Amherst, MA, ;.S.A.

(Received on 20 November 2000, Accepted in revised form on 20 July 2001)

We propose an explanation of cooperation among unrelated members of a social group inwhich cooperation evolves because it constitutes an honest signal of the member's quality asa mate, coalition partner or competitor, and therefore results in advantageous alliances forthose signaling in this manner. Our model is framed as a multi-player public goods game thatinvolves no repeated or assortative interactions, so that non-cooperation would be a dominantstrategy if there were no signaling bene"ts. We show that honest signaling of underlyingquality by providing a public good to group members can be evolutionarily stable, and canproliferate in a population in which it is initially rare, provided that certain plausibleconditions hold, including a link between group-bene"cial signaling and underlying qualitiesof the signaler that would be of bene"t to a potential mate or alliance partner. Our modelapplies to a range of cooperative interactions, including unconditionally sharing individuallyconsumable resources, participating in group raiding or defense, and punishing free-riding orother violations of social norms.

( 2001 Academic Press

1. Introduction

Cooperation among unrelated individuals hasgenerally been explained by some form of condi-tional reciprocity (Trivers, 1971; Axelrod &Hamilton, 1981). However, there is an increasinginterest in examining alternative mechanisms forthe evolution of cooperation (Dugatkin, 1997;Pusey & Packer, 1997), including mutualism(Brown, 1983; Connor, 1995), indirect reciprocity(Alexander, 1987; Nowak & Sigmund, 1998;Wedekind & Milinski, 2000; Leimar & Hammer-stein, 2001), and multilevel selection (Wilson,1977; Bowles, 2001; Sober & Wilson; 1998;

-Author to whom correspondence should be addressed.E-mails: [email protected]; [email protected];[email protected]

0022}5193/01/210103#17 $35.00/0

Gintis, 2000b). Several authors (Zahavi, 1977b,1995; Roberts, 1998; Wright, 1999; Smith &Bliege Bird, 2000; Leimar & Hammerstein, 2001)have suggested that costly signaling couldprovide an explanation for cooperation andgroup-bene"cial behavior, but this proposal hasnot been formally modeled. Following argumentsoutlined in Smith & Bliege Bird (2000), here wepresent a game-theoretic model in whichcooperative behavior indicates the underlying(dichotomous) quality of the signaler, elicitinga (dichotomous) response from observers thatcan be mutually bene"cial. The model is framedas a multi-player game that involves no repeatedor assortative interactions, and assumes a payo!structure that would conform to a multi-playerpublic goods game (and hence universal defec-tion) if there were no signaling bene"ts. We show

( 2001 Academic Press

104 H. GINTIS E¹ A¸.

that honest signaling of underlying quality byproviding a bene"t to group members can beevolutionarily stable, and may proliferate whenrare as long as high-quality individuals areneither too common nor too rare, and the cost ofsignaling is su$ciently greater for low than forhigh-quality players.

Our signaling model is distinctive in applyingto group (n signalers and n observers) rather thandyadic or many-signaler}one-observer inter-actions. It is also novel in determining endo-genously the fraction of the group that signalshigh quality in equilibrium. Finally, our modeloperates at the phenotypic level, and abstractsfrom the speci"c genetic mechanisms involvedin trait-transmission. We supply novel andplausible conditions for the spread of prosocialsignals.

We study the case where cooperation involvesproviding a bene"t to all members of the groupunconditionally (i.e. without any necessary recip-rocation in kind). Given the resulting publicgoods game payo! structure and the one-shotnature of the interactions, the unique equilibriumof this game under standard assumptions is uni-versal defection, so no player supplies the groupbene"t. Hence, individually costly cooperationcould not evolve unless one postulated the groupselection of altruistic behavior. Even if interac-tions among group members were repeated,multi-player cooperation requires implausibleforms of coordination, particularly if the groupcontains more than a few individuals (Boyd& Richerson, 1988). The model presented here ismeant to apply to such cases, where reciprocity isunlikely to emerge and is vulnerable to free-riding.

We propose that cooperating by providinga bene"t to group members may be a reliablesignal of signaler quality, whereby &&quality''we mean genetic or phenotypic attributes thatare di$cult for others to assess directly, yethave important e!ects on the payo!s from socialinteractions with the signaler. Those who providethe bene"t to others, or who provide more of thebene"t (thus signaling more intensively), honestlyadvertise their quality as allies, mates, or com-petitors. This information alters the behavior ofother group members to act, for purely sel"shmotives, in ways that provide positive payo!s tosignalers* for example, preferring them as allies

or mates, or deferring to them in competitivesituations (Smith & Bliege Bird, 2000).

The paper is organized as follows. In Section 2,we develop an multi-player model showing thatthere exists an equilibrium in which quality ishonestly signaled over a wide range of parametervalues in which low-quality types must pay moreto signal than high-quality types. This model ismeant to capture the key features of verbal argu-ments about cooperation as costly signaling ingroup-living species, but applies to costly signal-ing of any kind in multi-player settings. In Sec-tion 3, we show that under plausible conditionsthe costly signaling equilibrium is dynamicallystable. In Section 4, we present conditions underwhich a non-signaling equilibrium will be dis-placed by a signaling equilibrium, and in Sec-tion 5, we develop a dynamic model determiningthe movement towards an equilibrium fraction ofcostly signalers. Given that these results are notspeci"c to cooperation, Section 6 analyses whencostly signaling will take the form of providingbene"ts to others. Section 7 draws some con-clusions and implications for further research.

2. A Multi-player Model of Costly Signaling

Consider a group consisting of n members (seeAppendix A for a list of symbols). We think ofn as ranging between 10 and 100, to re#ect thesize of a foraging group or residential band. Oncein each period each member of the group canperform an action at personal cost c'0 thatconfers a bene"t g'0 on each other member ofthe group. We assume that the individual provid-ing g to other group members either does notshare in the public good, or else c is the net cost ofproviding this good, incorporating the provider'sshare. Since there is a strictly positive cost c toproviding the bene"t, self-interested individualswill not do so, and those who do provide thebene"t will be eliminated by any evolutionaryprocess in which di!erential replication is mono-tonic in payo!s. We consider this bene"t to bepublic good in the sense that

g(n!1)'c. (1)

Our task is to show that providing the bene"tmay be stable and may evolve. To do this, we add

BSince in some cases a Signaler may always be of high-quality or low-quality, it may seem unnatural to assign toa Signaler a strategy part of which is never used (e.g. a high-quality Signaler does not need an option for the case where

COSTLY SIGNALING AND COOPERATION 105

two elements to the above structure of socialinteractions.

Find, suppose group members have a personalcharacteristic, which we will call &&quality'', thatcan either be high or low. We assume membersknow their own quality but not that of others inthe group. We assume that the expected costc per period of providing the bene"t to the groupfor the high-quality type is less than the corre-sponding expected cost c@ for the low-qualitytype, so 0(c(c@ (we consider the cases wherec"0, c(0, and c"c@ later, in Section 2.2).

Second, suppose during a given period eachindividual has occasion to enter into a pro"tablealliance (e.g. mating or political coalition) withany one of the other n!1 group members. Thisother member, whom we will call the Partner,derives a bene"t h"h

0#ag'0, where a*0,

from choosing a high-quality ally, and a bene"t(or possibly a loss) l(h from choosing a low-quality ally, and has payo! zero if no ally ischosen. The parameter a represents the possibili-ty that a Partner may receive greater privatebene"ts from allying with a member who suppliesa higher level of the group bene"t.

We assume it costs a partner a small amountl'0 to monitor signals. For a given Partner, letp be the frequency of high-quality membersamong the group of remaining n!1 members.We assume the Partner knows p but not thequality of individuals among the other n!1members. A group member receives a payo!s'0 from each of the n!1 Partners whochooses to ally with him.E Finally, we treat eachperiod as a one-period game, as would be the caseif periods represent generations, or an agent'squality in one period cannot be inferred from hisquality in previous periods.

We construct an n-player game in which theplayers, whom we will call Signalers, choose inde-pendently whether or not to signal by providingthe bene"t to group members. There are fourpossible Signaler strategies, which we label

E In an alternative treatment, we would allow the bene"th to a Partner and the bene"t s to the group member to befunctions of the number of alliances into which the memberhas entered. It will be clear from the following analysis thatour results will continue to hold in this more general setting.

Mss, sn, ns, nnN. Here, ss means &&always signal,regardless of quality'', sn means &&signal if high-quality and do not signal if low-quality'', nsmeans &&do not signal if high-quality and signal iflow-quality'', and "nally nn means &&never signal,regardless of quality''. We can abbreviate these asMAlways Signal, Signal Truthfully, Signal Un-truthfully, Never SignalN.B

Each of the n players also plays the role ofPartner who must choose an ally (we do notassume that if player i choose player j as an ally,then player j must choose player i as an ally).A Partner has four possible strategies, which welabel Maa, ar, ra, rrN. Here, using the same conven-tion as with the Signaler, aa means &&AlwaysAccept (whether or not the Signaler signals)'', armeans &&Accept if Signaler Signals, Reject if Sig-naler does not Signal'', ra means &&Reject if Sig-naler Signals, Accept if Signaler does not Signal'',rr means &&Always Reject''. We also specify that ifno Signaler signals high-quality, an ar Partnerchooses an ally randomly from the group.**

We assume that all Signalers on the one hand,and all Partners on the other, follow the samepure strategy, since it is well known that a mixedstrategy equilibrium in an asymmetric game ofthis type is always unstable (Selten, 1980), so itmay be ignored.

A distinctive aspect of this model is that indi-viduals signal their quality to all other membersof the group, prior to interacting with particularPartners within the group. This assumption re-#ects our interest in the role signaling might playin favoring collective action (multi-player co-operation) in social species, including humans.But it is often the case that biological signals inother domains such as mate choice, resource

he or she is low-quality). However, Harsanyi (1967) hasshown that this formality is harmless. It has the advantage ofconsiderably simplifying the analysis.

**This speci"cation is sensible only if l*0, or else a sig-nal-monitoring Partner prefers no ally to a low-quality ally.We will assume, unless otherwise stated, that l'0, thusavoiding a set of parallel derivations that hold when l(0.Our analysis applies equally to both cases.


competition, and even predator}prey interac-tions are not private to an intended receiver, butare emitted without the signaler knowing exactlywhich among a population of possible observersit might in#uence. Our model is general enoughto apply to any domain where costly signalingoccurs in a multi-player context.

We represent the probabilistic character ofa Signaler's quality by assigning high-qualitywith probability p and low-quality with probabil-ity 1!p. The Signaler, knowing his high or low-quality, then chooses either to signal or not, andthe other players (in their Partner roles) choosewhether or not to consider the Signaler in thepool of potential allies.

It is clear that as long as ph#(1!p)l'0,there is a non-signaling equilibrium (nn, aa) inwhich no one signals and Partners choose alliesrandomly from all other group members. Sim-ilarly, if ph#(1!p)l(0, there is a non-signal-ing equilibrium (nn, rr) in which no agent signalsand Partners never choose allies. The honest sig-naling equilibrium is denoted by (sn, ar), indicat-ing that Signalers signal high-quality if and onlyif they are high-quality, and Partners choose ran-domly among those who signaled high-quality.

Clearly, it is worthwhile for a Partner to playar only if the expected bene"t from allying witha high-quality agent exceeds the cost of moni-toring the signal. The payo! from alliances if aPartner chooses randomly from the group is(hp#l (1!p)). If we de"ne

d (x)"1!(1!p)x(n~1), (2)

then, assuming all Signalers are honest (i.e. playar), the expected payo! to a Partner from

FIG 1. The matrix of payo!s to a Signaler S and a Partner P

monitoring is hd (1)#l(1!d(1))!l, since withprobability d(1) the Partner "nds a high-qualityally, and with probability [1!d(1)] the Partnermust resort to a low-quality ally. Comparingthese two payo!s, we see that monitoring isworthwhile for a Partner only if

(h!l)(d(1)!p)'l. (3)

We assume l is su$ciently small so that thise.ciency of monitoring condition holds. Note thatin most cases d (1)+1 (e.g. if p"0.3 and n"20,d(1)+0.999; moreover, d(1) is increasing in p andn, so the approximation holds for all larger valuesof these variables), in which case the e$ciencyof monitoring condition is approximately(h!l) (1!p)'l.

2.1. THE HONEST SIGNALING EQUILIBRIUM

To "nd the conditions under which there is anhonest signaling equilibrium, it is su$cient toderive the conditions under which honest signal-ing is the best response for one agent, assumingall other members signal honestly (we will pres-ently explore the local stability and global acces-sibility of such an equilibrium). This gives rise tothe matrix shown in Fig. 1. Multi-player gamesare notoriously unwieldy, so we have introducedseveral notational simpli"cations to reduce theclutter of symbols in Fig. 1 and the subsequentanalysis dependent upon Fig. 1.

(a) If the fraction of high-quality types in thegroup is p, the fraction of high-quality typesremaining if the Partner is of high-quality is(pn!1)/(n!1) and is pn/(n!1) if the Partner is

, assuming all other Signalers play the same strategy as S.

--The assumption that a Partner can ally with only onesignaler is somewhat arbitrary, but it is completely straight-forward to extend our analysis to the case where Partnerscan ally with multiple Signalers. It is somewhat more chal-lenging to allow the Partner's alliance bene"t h to depend onthe number of allies possessed by the Signaler with whom heallies. On grounds of simplicity, we will avoid this moregeneral treatment.


of low-quality. We assume n is su$ciently largeso that both these numbers can be approximatedby p.

(b) We have multiplied the payo! to Signalersfrom alliances by n to re#ect the fact that a Sig-naler pays the cost c only once per period, but hasn potential Partners (actually n!1 but again weassume this is n to reduce notational overhead),each of whom can independently ally with theSignaler.

(c) We have not included the bene"t to othersthat accrues to both players from the actions ofthe other n!2 players, since this payo! merelyadds a constant to each row of the game matrixfor the Partner, and a constant to each column ofthe matrix for the Signaler. To see this, supposek of the n!2 remaining players provide thebene"t to group members. Then both Signalerand Partner receive payo! kg from the signalers'provision of bene"ts, no matter what strategiesthey choose. Moreover, whatever strategythe Signaler chooses, the Partner receives thesame bene"t from the Signaler. For instance, ifthe Signaler chooses sn, then the Partner receivesa payo! of pg from the Signaler's provision ofbene"ts, whether the Partner chooses aa, ar, ra,or rr. Since adding a constant to the payo!s ofa player, given the choices of the other players,cannot change the player's best response strategy,we omit these payo!s.

(d). We assume that n is su$ciently large andp is su$ciently far from zero, that the probabilitythat there is at least one high-quality player isunity (i.e. we assume d(1)"1).

To illustrate how the entries in Fig. 1 are cal-culated, we will derive them for the honest signal-ing equilibrium, which is the highlighted (sn, ar)box. In this case, a high-quality Signaler providesthe bene"t at cost c and receives expected bene"ts/pn from each Partner, since each Partner nowchooses randomly from among the group of pnSignalers who provided the bene"t. Multiplyingthe alliance payo! by n, because there are n Part-ners, and multiplying the net payo! by p, sincethe Signaler is of high-quality with probability p,the expected payo! to the Signaler isp(s/p!c)"s!pc. Similarly, since a Signalercan have multiple allies, a Partner certainly "ndsan ally among the high-quality members, giving

payo! h!v.-- A similar argument is used to "llin the other entries in Fig. 1.

From the construction of the matrix in Fig. 1,we know that a pair of best responses for Signalerand Partner determine a Nash equilibrium of thegame, since if honest signaling is the best re-sponse for one Signaler when all other signalersuse honest signaling, then the same is true forany signaler. It follows that the conditionspc@'s'pc, and h'l are necessary and su$-cient for honest signaling (sn, ar) to be a strictNash equilibrium. The conditions for this signal-ing equilibrium to exist are easily interpreted, andminimally what one would expect prima facie ina costly signaling framework. The "rst, s'pc,requires that the bene"ts of signaling exceed theexpected cost of signaling for the high-qualitytype. The second, s(pc@, requires exactly theopposite for low-quality types. Finally, h'lstates that the bene"t from allying with a high-quality type is greater than that with a low-quality type.

2.2 FREQUENCY DEPENDENCE OF THE

HONEST SIGNALING EQUILIBRIUM

We have the following theorem.

Theorem 1. Suppose there is a positive payo+ toan alliance for the Signaler (s'0), and Partnersprefer to ally with high-quality types (h'l). ¹henthere is a range of frequencies p of high-qualitytypes for which there is an honest signaling equilib-rium if and only if signaling is more costly forlow-quality than high-quality types (c@'c), andmore costly than the payo+ to a single alliance(c@'s).

To prove the theorem, we rewrite the conditionpc@'s'pc as

c@'sp'c. (4)


If eqn (4) holds, then clearly c@'c and sincep(1, we must have s(c@. Conversely, if theseparameter inequalities hold, then eqn (4) holdsfor any p'0 in the interval (s/c@, s/c). The e$-cient monitoring condition (3) is su$cient to en-sure that ar is the best response for the Partner.

Note that this theorem does not require c'0.If c"0, so signaling is not costly for the high-quality signaler, there still exists an honest signal-ing equilibrium for some p, so long as c@'s.Indeed, it is easy to see that we can have c(0and the signaling equilibrium will still exist underthe same conditions. This is an important obser-vation because it implies that the cost of signalingfor the high-quality type cannot be signed, soeven signals that are intrinsically bene"cial tothe Signaler can be part of an honest signalingequilibrium.

The reader will note that the equilibrium con-ditions for honest signaling that we have justderived are dependent upon the frequency p ofhigh-quality types in the group. This aspect ofcostly signaling as of course been noted in verbaldescriptions of costly signaling, but did not ap-pear in the Grafen's pioneering contribution(Grafen, 1990a, b) nor in most of the more recentpapers on the topic [but see Lachmann &Bergstrom (1988), and Siller (1998)].

2.3. HONEST SIGNALING BY PROVIDING

A PUBLIC GOOD ENHANCES AVERAGE FITNESS

To see that honest signaling by providinga public good is "tness enhancing for the groupas a whole, suppose that every member of thepopulation is both a Partner and a Signaler ineach period. Then average payo!s for the twostable equilibria are just the sum of the entries inthe reduced normal form matrix for the relevantstrategy pro"le, plus the gains to all parties fromthe provision of Signaler bene"ts. Thus, the hon-est signaling equilibrium has higher averagepayo!s if h#(n!1)pg#p(s!c)'hp#(1!p)l,which reduces to

pA1!(n!1)g#s!c

h!l B(1, (5)

which, by eqn (1), is true if the bene"t g is apublic good. Note that for the honest signaling

equilibrium to have higher average payo!s,g need not represent a public good. Even confer-ring harm (g(0) will entail higher averagepayo!s if a su$ciently large number of allianceswith high-quality individuals result.

2.4. COST, BENEFITS, AND PUNISHMENT

Our model assumes low-quality types havehigher signaling costs. An alternative assumptionthat is sometimes more accurate (Godfray, 1991;Maynard Smith, 1991; Johnstone, 1997; Getty,1998) is that both types face the same marginalsignaling costs, but high-quality types reap high-er marginal bene"ts per unit of signaling than dolow-quality types. In terms of our parameters,this means c"c@ and s's@, where s@ is the valueof an alliance to a low-quality signaler. If we solvefor Nash equilibria using the parameters c, c@, s,and s@ with c, c@'0, we "nd that the conditionsfor an honest signaling equilibrium are simplys'pc, and s@(pc@. These conditions of coursereduce to the above conditions when s"s@, andwe will not pursue this variant of the modelfurther in this paper.

Among the forms of signaling described by themodel is the punishment of those who violategroup-bene"cial norms. We o!er the followingbrief account to illustrate how our model cap-tures this form of signaling. Suppose that a groupof n members can cooperate to provide somegroup bene"t. By cooperating, each member con-tributes a total bene"t of b to others at a "tnesscost of c to himself. Thus, the gain from defectingis c!b/n and to induce cooperation, membersmust be punished at least c!b/n for defecting.Now suppose that a high-quality individual canimpose c!b/n on defectors at a personal cost ofc, whereas a low-quality individual must incurcost c@'c to achieve the same e!ect. Followingthe model presented above, there will be an equi-librium in which high-quality individuals willpunish and low-quality ones will not, providedcertain parameter values obtain. We will discussthe signi"cance of the punishment form of costlysignaling in Section 7.

3. Dynamics

We make this model dynamic by assumingthat the fraction of honest Signalers increases


when the payo! to honest signaling exceeds thatof Never Signal, and the fraction of ar Partnersincreases when the payo! to ar exceeds that of aa.We model these as a &&replicator dynamic''(Taylor & Jonker, 1978), which means that therate of growth of a Signaler strategy, representedas a fraction of Signalers in the population usingthat strategy, equals the di!erence between thepayo! to that strategy and the average payo! toall Signaler strategies, and similarly for Partnerstrategies. This dynamic can re#ect either cul-tural change, in which members are prone toswitch from inferior to superior strategies, orgenetic change, in which those who pursue suc-cessful strategies have more o!spring, who tendto follow their parents' strategies.??

In analysing the dynamics of our system, sincewe are now considering the whole range of valuesof a3[0, 1], we can no longer abstract from theprobability of "nding an ally schedule d(x). Wemust calculate the expected payo!s when thefraction of sn Signalers is a and the fraction of arPartners is b. We derive these payo!s as follows.

With probability p, an honest signaler indi-cates that he is of high-quality. The number ofPartners of type ar is b (n!1). Since the expectednumber of signalers who signaled high-quality isap(n!1), the expected payo! for each such sig-naler is b (n!1)s/ap(n!1). We multiply this byp, the probability that an honest signaler is ofhigh-quality to arrive at the "rst term for n

snbelow. With probability 1!p, an honest signaleris not of high-quality and does not signal. Thissignaler may be chosen as an ally of an ar Partnerwhen there are zero high-quality signalers. Thereare b (n!1) Partners of type ar, and the prob-ability that there are no high signalers is(1!d(a)), so the expected number of alliances ofthis type is b (n!1)(1!d (a)), which must beshared by the n!1 signalers. This accounts forthe second term in n

sn. The expected number of

alliances with aa types is (1!b)(n!1), sharedamong the n!1 signalers, giving the third termin n

sn. The "nal term in the expression is the

expected cost of signaling high-quality.Similarly, the "rst term for the expected payo!

to Never Signal (nnn

in the equations below) is

??For various derivations of the replicator dynamicequations, see Gintis (2000a, Chapter 9).

s times the probability that a non-signaler allieswith an ar-type when no agent signaled high-quality, and the second term is s times the prob-ability that a signaler makes an alliance with anaa-type Partner. The payo! to Always Accept isthe expected payo! to an alliance, ph#(1!p)l.Finally, the expected payo! to an ar-type ish times d (a), which is the probability of makingan alliance with someone who signaled high-quality, plus the probability of an alliance withsomeone who did not signal high-quality,1!d(a), times the average quality of such anagent, who is of average quality, hp#1(!p)lwith probability (1!a), and is a honest signalerof low-quality with probability a. We thus havethe following expected payo!s.

Signal honestly:

nsn"

bsa#(1!p)bs (1!d(a))#(1!b)s!pc.

Never signal:

nnn"bs(1!d (a))#(1!b)s.

Always accept:

naa"ph#(1!p)l.

Accept if Signaler signals:

nar"d(a)h#(1!d(a))(al#(1!a) (ph#ql))!l.

Average Signaler payo+:

nN 1"ansn#(1!a)n

nn.

Average Partner payo+:

nN 2"bnar#(1!b)n

aa.

The replicator equations are then

a5 "a(nsn!n6 1), (6)

b0"b (nar!n6 2), (7)

which reduce to

a5 "a(1!a)(sb (1!a (1!d(a)))/a!cp),

bQ "b (1!b ) ((h!l) (d(a)!p#(1!a)

]p(1!d(a)))!l).

These equations express the familiar result thatthe rate of change of the frequency of a trait in


a population varies with the variance of "tness,or equivalently, the variance of the trait times thee!ect of the trait on "tness. The "rst indicatesthat the rate of increase of honest signalingequals the variance of the frequency of honestsignaling, which is a (1!a) times the net gainfrom honest signaling. The second indicates thatthe rate of increase in ar equals the variance ofthe frequency of ar, which is b(1!b), times thenet gain from ar.

The replicator equations (6) and (7) have "veequilibria. Four correspond to a"0, 1 andb"0, 1, and the "fth is a mixed strategy equilib-rium in which all four types of agents are present.The Jacobian matrix of the replicator equationsis a quite complicated expression, but its eigen-values for a"b"0 are M!cp, !lN, which arestrictly negative assuming the cost of signaling, c,is positive. Thus, the no-signaling equilibrium isstable under this condition. At the equilibriuma"0, b"1 the Jacobian has eigenvalues

M!cp!ps, lN, (8)

indicating instability. At the equilibrium a"1,b"0 the Jacobian has eigenvalues

Mcp, (h!l) (d(1)!p)!lN. (9)

FIG 2. The phase diagram. Points below the ridge line ABC a

Thus, this equilibrium is also unstable for c'0,or when the e$cient monitoring condition holds.At the truthful signaling equilibrium, corre-sponding to a"b"1, the Jacobian has eigen-values

Mcp!(1!(1!d(1))p)s, l!(h!l) (d(1)!p)N,(10)

both of which must be strictly negative for honestsignaling to be a stable equilibrium. The secondexpression is negative by the e$cient monitoringcondition (3). The "rst expression is negativewhen

cs((1!(1!d(1))p), (11)

which becomes the criterion s'cp for an honestsignaling equilibrium, given the approximationd(1)+1. We have seen that this is the case aslong as p is not too small and n is su$cientlylarge. Assuming this, both eigenvalues in eqn (10)are negative, and the honest signaling equilib-rium is stable. We thus have the followingtheorem.

Theorem 2. Suppose the conditions for an honestsignaling equilibrium hold (see ¹heorem 1) with

re in the basin of attraction of the non-signaling equilibrium.


frequency p of high-quality types. ¹hen if eqns (11)and (3) hold and c'0, both the non-signaling andthe honest signaling equilibria are stable. More-over, there are three unstable equilibria and a ridgeline connecting them that separates the basins ofattraction of the two stable equilibria.

Figure 2 shows the phase diagram for the dy-namical system. The equilibrium indicated byB in Fig. 2 is unstable, as can be con"rmed by thevector "eld derived from the replicator equa-tions (6) and (7) indicated by the arrows.

4. The Evolution of Signaling

The analysis thus far has revealed that whilean honest signaling equilibrium will exist undera broad range of parameter values, so will anon-signaling equilibrium. Are there plausiblemechanisms leading from the non-signaling tothe signaling equilibrium? We can o!er threecomplementary mechanisms of this type.

4.1. STOCHASTIC SHOCKS

A population at the non-signaling equilibriummight be displaced by a series of stochastic eventsinto the basin of attraction of the signaling equi-librium, should the underlying parameters besuch that the honest signaling equilibrium exists.Since a small number of Signalers and signal-monitoring Partners can invade a non-signalinggroup, the movement from a non-signaling toa signaling equilibrium is much more likely thanthe reverse movement, which requires the simul-taneous extinguishing of signaling and signalmonitoring of most group members. Having thusproliferated within a single group or a fewgroups, signaling equilibria may proliferate ina larger population through multilevel selection.

4.2. MUTUALISM

Throughout this paper we have assumed high-quality signaling is costly to the high-qualityagent, i.e. c'0. But as Theorem 1 shows, theexistence of an honest signaling equilibrium doesnot depend on this fact. The condition c(0indicates that providing the bene"t g is bene"cialto the high-quality Signaler, and hence g is aform of by-product mutualism. Studies of animal

behavior indicate the importance of by-productmutualism (Dugatkin, 1996; Milinski, 1996;Dugatkin & Mesterton-Gibbons, 1996; Mester-ton-Gibbons & Dugatkin, 1997), which doubtlessplays a role in human societies as well.

When c(0, an inspection of the replicatorequations (6) shows that only the honest signal-ing equilibrium (a"b"1) is stable. Indeed, inthis case a mutant honest signaler has higher"tness than a non-signaler, even in the absence ofPartners who monitor and respond to the signal.Selection will then favor Partner strategy ar, andthe honest signaling equilibrium will obtain inthe long run.

Once a signaling equilibrium is attained, pro-vided the conditions for prosociality developedbelow in Section 6 obtain, there will be a tend-ency for signals that provided high social bene"tsto displace signals providing low social bene"ts,even when the costs associated with these bene"tsare strictly positive. Thus, even if the highly pros-ocial signals involve positive signaling costs, suchsignals will evolve in the long run.

4.3. INCLUSIVE FITNESS

Another mechanism leading from non-signal-ing to signaling equilibria involves the generaliz-ation of behavior from highly genetically relatedgroups to groups of unrelated individuals.

Suppose our group of n members has degree ofrelatedness r'0. Then the inclusive "tness costof honest signaling for a Signaler is c"c

0!

(n!1)gr. If signaling is prosocial and c0

is nottoo large, we will have c(0, so the analysis ofthe previous section applies: positive relatednessfacilitates the evolution of prosocial honestsignaling.

Once the honest signaling equilibrium is estab-lished in a group of related individuals, migrationin and out of the group will not destroy thehonest signaling equilibrium, provided p remainsin the appropriate range and the other conditionsspeci"ed in Theorem 1 obtain. Hence, the honestsignaling equilibrium will persist even underconditions of zero relatedness.

5. Heritability of Quality

We have shown that signaling by providingbene"ts to members could proliferate when rare,


and would under plausible conditions be sus-tained in a population in which behaviors evol-ved according to the payo!-monotonic updatingdescribed in eqns (6) and (7). But this is not su$-cient to ensure the evolutionary success of suchsignaling. Recall that the existence of the signal-ing equilibrium requires that the high-qualitytypes not be excessively prevalent in the popula-tion. But if high-quality types have higher "tnessthan low-quality types in the signaling equilib-rium, and if quality is heritable, their frequencyp may increase over time, thus undermining thesignaling equilibrium.

The payo! di!erence between high- and low-quality types in the signaling equilibrium iss/p!c'0, so in the absence of any other in#u-ence on p, high-quality would evolve to "xation.But we have modeled only a subset of the in#uen-ces on p, and we may suppose other in#uences tobe at work. The relevant di!erential equation is

pR "p(1!p)(s/p!c)!zp#w(1!p), (12)

where the "rst term expresses the rate at whichthe di!erential "tness of high-quality types istranslated into o!spring, z'0 is the proportionof o!spring produced by high-quality parentsthat are low-quality, and w'0 is the proportionof o!spring produced by low-quality parents thatare of high-quality. Equation (12) can be simpli"-ed to

pR "cp2!(c#s#w#z)p#s#w. (13)

Since the left-hand side of eqn (13) is s#w'0 atp"0 and !z(0 at p"1, there is surelya stable equilibrium p*, where 0(p*(1. Wehave

p*"

c#s#w#z!J(c#s#w#z)2!4c(s#w)2c

.

This expression is complicated, but it does havesome intuitive implications. Setting the right-hand side of eqn (12) to zero and totally di!eren-tiating to see how p* varies in response tochanges in our parameters, we "nd that (a) an

increase in the cost c to the high-quality type ofsignaling leads to a lower equilibrium value of p*;(b) an increase in the bene"t s of an alliance to theSignaler raises the equilibrium value of p*; (c) anincrease in rate w at which low-quality agentsproduce high-quality o!spring raises the equilib-rium value of p*; and (d) an increase in rate z atwhich high-quality agents produce low-qualityo!spring lowers the equilibrium value of p*.

There are, of course, various social and biolo-gical mechanisms that could lead to positivelevels for either z or w. For example, the charac-teristics that confer high-quality or low-qualitycould be purely or substantially environmentallydetermined, or determined by cultural factorsthat are passed on through biased or horizontalrather than vertical cultural transmission(Cavalli-Sforza & Feldman, 1981; Boyd &Richerson, 1985). Or, the "tness bene"ts asso-ciated with prosocial signaling of high-qualitymay be con"ned to some period of the life cycle,and be partially o!set by disadvantageous e!ectsat other ages.

We have explored the global dynamics of ourpopulation, showing "rst, that honest signalingmay proliferate when rare, and second that anhonest signaling equilibrium once attained neednot set in motion a rise in the fraction of high-quality types that would eventually destroy theconditions for an honest signaling equilibrium. Itremains to be shown that in an honest signalingequilibrium group-bene"cial signals would be fa-vored over other honest signals of quality.

6. Why Signal by Providing Bene5ts to Others?

We have shown that when signaling takesplace in an n-player social context, and providingbene"ts to other serves as an honest signal ofunderlying quality, the payo!s to signaling cansolve the problem of maintaining unilateral co-operation in a group when self-interested agentswould not otherwise provide the bene"ts. Inother words, under conditions standard to thecostly signaling framework and speci"ed in Sec-tion 2, if the social bene"ts that signalers receivefrom signaling exceed the costs of signaling whenthey are high-quality (s'pc), these can providea net advantage for what would otherwise bealtruistic contributions to the group. However,


such group-bene"cial signaling is only a specialcase of the model presented in Section 2. Indeed,in our model, the existence of a costly signalingequilibrium does not require that there be anygroup-bene"cial e!ects of the signal. To see this,note that the per member bene"t g could equallywell be positive, negative, or zero without alter-ing the conditions for existence of stability ofa costly signaling equilibrium. Moreover, ifph#(1!p )l'0, whenever there is a stablecostly signaling equilibrium, there is anotherstable equilibrium in which that signal is notused. Therefore, when there are a variety of pos-sible signals, our analysis to this point does nottell us which among them will in fact be favored.

Let us call a costly signal feasible if it satis"esthe conditions for a costly signaling equilibriumdeveloped in Section 2. We have shown thatprosocial signals can meet these conditions, butwe have not shown that there is any particularreason to expect feasible costly signals deployedwithin a group to be prosocial. We can specifyseveral conditions under which prosocial signal-ing will be favored over alternative signals.

6.1. SIGNALING PROSOCIAL QUALITY

First, suppose there are several feasible signalsand consider an honest signaling equilibrium inwhich the most prosocial signal (i.e. the signalthat provides the highest level of bene"t, g*, togroup members) is not used. Then as long asa'0, the bene"t h*"h

0#ag* to a mutant

Partner who allies with a mutant Signaler whoprovides the public bene"t at level g* will behigher than that of Partners who recognize onlysignals with public good value g(g*. This beingthe case, the Partner strategy of allying withSignalers providing g* will increase in frequencyaccording to a straightforward adaptive dynam-ics.AA This in turn increases the payo! to g*signalers, since they now achieve more alliances.Thus, as long as the marginal gain to the sig-nalers from increased alliances is greater than theadded cost of signaling by providing g* rather

AABy an adaptive dynamic we mean a situation in which, ifa parameter positively a!ects "tness, over time strategiesexhibiting higher values of this parameter will displace strat-egies lower values (Dieckmann, 1997; Hofbauer & Sigmund,1998).

than g, the equilibrium involving the less pros-ocial signal is not stable against invasion by moreprosocial Signalers and Partners who respond totheir signals.

The assumption that a'0, on which this ar-gument hinges, asserts that the level of publicbene"t provided is positively correlated with theexpected level of bene"t the Signaler provides tothe Partner. This argument is analogous to the&&direct bene"ts'', or &&good parent'', explanationfor female preference of males who signal su-perior ability to provide parental care or otherresources (Johnstone, 1995; Iwasa & Pomian-kowski, 1999). In both cases, a high-quality indi-vidual is more likely to provide social bene"tbecause the cost of doing so is lower than fora low-quality individual, and the individual whoprovides the most prosocial bene"t will also beequipped to supply the most private bene"t tothe observer of the signal. In this case, the groupbene"ts produced by the Signaler are incidentalto Partner's preference, but because they producedirect bene"ts to the Partner they are valued inthemselves, not simply as indicator traits.

One example of a positive correlation betweenthe social value of a signal and its value to a po-tential ally is the case where a high value ofg re#ects the Signaler's willingness to act in a gen-erous manner in bargaining over the distributionof the bene"ts of his e!orts. This behavioralpropensity will likely generalize to the signaler'sprivate alliances. Alternatively, a high value ofg might re#ect the signaler's capacity to honorcommitments by demonstrating the ability toshare bene"ts that would not be worth providingif the Signaler did not intend to continue thealliance in the future (Schelling, 1978; Zahavi& Zahavi, 1997).

Another example of a signal that is both pros-ocial (providing group bene"ts) and directlybene"cial to allies is signaling extraordinaryforaging ability through unconditional sharing ofsurplus resources. This will provide direct bene-"ts to Partners if Signalers also use this ability toprovision their allies (e.g. mates with dependento!spring). A "nal example is defending the groupagainst enemy attack. This signals qualities suchas strength and "ghting ability, which will pro-vide direct bene"ts to Partners if Signalers withsuch attributes are more likely to defend allies in


internal disputes. Again, we are assuming thata high-quality individual signals by providingthese collective and individual bene"ts becausethe cost of doing so, c, is lower than it would befor a low-quality individual, c@.

It is of course possible to construct scenarioswhere high-quality individuals signal in sociallyneutral or even antisocial ways. For example, onecould signal extraordinary foraging ability byfocusing on resources that require great skill tocapture but provide little food value (e.g. thespear"shing pursued by some Meriam men, asdescribed in Bliege Bird et al., 2001), or by con-spicuously consuming surplus resources (Veblen,1899), or even by destroying them (as in somecompetitive potlatches among the Kwakiutl In-dians; Ruyle, 1973). Similarly, strength and "ght-ing ability could be signaled by bullying membersof one's group, or engaging in repeated brawls.We do not discount these possibilities, but simplyargue that when prosocial acts are e!ective atsignaling underlying qualities, and also providedirect bene"ts to potential allies (enhancing h),Partners will prefer to ally with Signalers usingsuch signals, thus preferentially bene"ting them(via s).

6.2. RECIPROCITY

Suppose that a"0, so the above argumentdoes not hold, but suppose all else equal, Part-ners prefer to ally with the agents who haveconferred the highest bene"ts upon them. Notethat this behavior imposes no cost on the mutantPartner who responds to a more prosocial signal,so there is no "tness penalty associated withmutant Partners who show a preferences forSignalers who have provided them with greaterpublic bene"ts. In this case, once again the equi-librium with group bene"t g can be invaded bya mutant signaler that uses the most prosocialsignal g*. Mutant signalers of this type will thusincrease in frequency under the operation of anadaptive dynamic. This will continue until themost prosocial feasible signal displaces the anti-social or less prosocial signal.

6.3. BROADCAST EFFICIENCY

Another mechanism favoring prosocial signal-ing is that signaling by providing a bene"t to

group members may increase &&broadcast e$-ciency'' (Bliege Bird, 1999; Smith & Bliege Bird,2000), in the sense that it attracts a larger audi-ence to witness the signal. The broadcast-e$-ciency argument could be formalized in a varietyof ways. One simple approach that captures theessential point is as follows. We initially assumedthat all members of a Signaler's group &&see'' thesignal. But suppose there are several signalsMp

1,2, p

kN, each of which is observed by a frac-

tion M f1,2, f

kN of the group. We call f

ithe

&&broadcast strength'' of signal pi. Then the

payo!s to the Signaler using piwill be reduced by

(1!fi)s in a signaling equilibrium, because the

Signaler now receives an expected bene"t onlyfrom the f

i(n!1) members who &&viewed'' the

signal. Therefore, everything else being equal,Signaler strategies that maximize the fraction ofthe group that views the signal will gain higherpayo!s and will increase their share in the popu-lation. If a prosocial signal attracts a larger audi-ence, the higher broadcast strength per unitsignal cost ( f

i/c) will favor signaler strategies of

this type. Detailed study will be needed to deter-mine the degree to which this applies to particu-lar cases.

The broadcast e$ciency argument applieswhen prosocial signals attract a larger audiencethan alternative signals of equivalent cost. Sig-nals that consist of providing a consumable pub-lic good (e.g. foods of su$cient value or rarity toattract audiences) will plausibly have this e!ect,while prosocial acts such as punishing non-coop-erators may not. An example of the former caseinvolves chimpanzees (Pan troglodytes), who livein social groups that travel and forage in a &&"s-sion}fusion'' pattern (Wrangham, 1980), so thatpotential signal observers are often within closerange but not necessarily in visual contact. Whilemost foraging is non-cooperative, and most con-sumption is by the acquirer, an important excep-tion is hunting (Stanford, 1999). It is interestingto note that among wild chimpanzees, extra-kinfood sharing only involves group-hunted re-sources, and the hunters are always males. Stan-ford et al. (1994) found that the best predictor ofhunting frequency is the number of estrous fe-males present in the social group, even thoughfemales are not the primary recipients of meatshares, whereas Mitani & Watts (2001), working


with a di!erent population, found that in a multi-variate analysis only the number of adult maleswas a good predictor. Stanford interprets hisresults as supporting meat-for-sex exchange,while Mitani and Watts favor an explanationinvolving meat-for-male coalition support ex-change. However, the available evidence is fullyconsistent with a signaling/broadcast-e$ciencyargument, which does not require the direct tradeof Stanford's scenario and avoids the problems ofenforcing reciprocity that arise in that of Mitaniand Watt.

Costly signaling has been proposed as an ex-planation for certain types of food-sharing inhuman societies, such as providing large and/ordi$cult-to-harvest game, or large quantities offood for consumption at ritual feasts (Boone,1998; Gurven et al., 2000; Hawkes et al., 2001;Smith & Bliege Bird, 2000; Sosis, 2000). In mostsuch cases, as in the chimpanzee case, there is notsu$cient information available to judge if keyconditions for costly signaling, such as quality-dependent signaling, are present. In the casestudied by Bliege Bird and colleagues (BliegeBird et al., 2001), such data are now available,and agree with costly signaling predictions. Thisstudy also shows that the most prosocial formof signaling*unconditionally sharing largegame*has high broadcast e$ciency, and ismore likely to be done through feasts thanthrough household-to-household sharing. It isclear that young men are more likely to establishreputations for foraging ability by providinglarge game (marine turtles) for feasts attened byupwards of 200 people (mean"174.9, n"54feasts) than by other means, such as conspicuousconsumption or minimizing foraging time tosupply domestic needs, which would only beobserved by immediate neighbors, and evenfor them be less conspicuous than feastcontributions.

6.4. GROUP SELECTION OF ALTERNATIVE EQUILIBRIA

A plausible process favoring the selection ofprosocial costly signaling, though not modeled inthis paper, is a process of group selection amongalternative local equilibria modeled by Boyd& Richerson (1990) and others. The model pre-sented in Section 2 has both an honest signaling

and a non-signaling equilibrium, both stable inthe replicator dynamic. Which equilibriumwould occur in a particular case will depend onlocal circumstances, including social and envi-ronmental factors determining the level of keyparameters, as discussed in Section 3, and per-haps on initial composition of the population.Given this variation in the metapopulation, itfollows that local groups will vary in the level ofprosocial costly signaling, and that groups witha high level will have members who, on average,are most "t than groups in which such behavioris absent. Such groups, by withstanding extinc-tion and dispersion, and by having superiorstrength in hostile interactions with other groups,can spread the prosocial practices beyond theiroriginal boundaries (Gintis, 2000b). Coupled tosuch a framework, our model provides a possiblebasis for a more general understanding of whichamong the multiplicity of signals are likely toevolve and persist, namely, those characterizedby a large basin of attraction for the associatedequilibria.

6.5. SUMMARY OF FORCES FAVORING

PROSOCIAL SIGNALING

We have o!ered several distinct and com-plementary reasons why prosocial signaling maybe favored over other forms of signaling. First,Partners may receive higher private bene"ts fromsignalers who use more prosocial signals. Second,potential Partners may simply prefer agents whohave conferred the highest group bene"t g uponthem*a preference with no individual-level "t-ness-reducing consequences and strong group-level "tness-increasing consequences. Third,some prosocial signals may attract larger audien-ces than other signals of equal cost, and hencehave higher broadcast strength. Fourth, groupselection among alternative equilibria can favoroutcomes with higher group "tness, which cangive an evolutionary advantage to equilibra withmore prosocial e!ects.

7. Conclusion

Costly signaling of quality for purposes ofmating, alliance formation, and warning o!potential enemies has been reported in many


species, including humans (Zahavi, 1977a;Grafen, 1990a; Maynard Smith, 1991; Johnstone,1995; Wright, 1999). We have proposed a multi-player game-theoretic model of costly signalingand shown that under plausible parametervalues, a class of signals that themselves contrib-ute to group bene"ts may proliferate in a popula-tion when rare, and constitute evolutionarystable strategies. Costly signaling may thus pro-vide a mechanism for the evolution of coopera-tive and other group-bene"cial practices capableof working independently of repeated interac-tions, positive assortment, and multilevel section,though these latter factors may act to reinforcesuch evolution.

While existing game-theoretic models of sig-naling are framed as dyadic or many-signal}one-observer interactions (e.g. Johnstone, 1999), oursignaling model is distinctive in applying togroup interactions. It is also novel in determiningendogenously the fraction of the group that sig-nals high-quality in equilibrium. We show thathonest signaling of underlying quality by provid-ing a bene"t to group members can be evolu-tionarily stable, and may proliferate when rare aslong as high-quality individuals are neither toocommon nor too rare, and the cost of signaling issu$ciently greater for low-quality than for high-quality players.

Our model is general enough to apply toa range of social interactions. First, the bene"twhose provision signals high-quality may takethe form of individually consumable resources.For example, the widespread practice amonghunter}gatherers of sharing individually har-vested resources unconditionally with all mem-bers of the community, has presented a puzzle formodels based on reciprocity and risk reduction(Hawkes, 1993). Some hunters consistently pro-vide more than others while sharing equally inthe catch. These &&altruistic'' providers in fact reaphigher social status and reproductive successthan their less-productive peers, despite the ab-sence of any conditional exchange of &&meat formates'' (Kaplan & Hill, 1985; Marlowe, 1999;Bliege Bird et al., 2001). Our model formalizes theconditions under which a costly signaling ex-planation might account for such a pattern.Catching large game, the most commonly sharedresource, requires skill and endurance, and

readily attracts an audience to consume it, allow-ing the signals sent by hunting success and gener-osity to be broadcast e$ciently. Whether huntingsuccess is a condition-dependent signal of qualityis not yet "rmly established, though some currentwork matches this prediction (Bliege Bird et al.,2001).

The model can also be applied to the provis-ioning of public goods that are not individuallyconsumable. For instance, participating in groupraiding or defense*an individually costly behav-ior that is common among chimpanzees as wellas human societies (Boehm, 1992)*provides be-ne"ts that are available to all group members andcannot be hoarded. Similarly, punishment ofthose who free-ride or otherwise engage in anti-social behavior is a critical mechanism for enforc-ing cooperation in many social contexts (Frank,1995). The role that costly signaling might play inenforcement of prosocial behavior is as yet untes-ted, but deserves further investigation.

Our model applies as well to situations involv-ing punishing those who free-ride on the group'scooperative activities. It is well known that, whileenforcing cooperation by punishing defectors cansolve collective action problems, such enforce-ment itself provides a personally costly publicgood, and thus poses a second-order collectiveaction problem (Hardin, 1982). Boyd & Richer-son (1992) demonstrated that if enforcementtakes the form of punishing both non-cooper-ators and non-punishers, then cooperation (oranything else) can be evolutionary stable, even inlarge groups. Such punishment may be a potentelement in stabilizing cooperation in many typesof social systems (Boyd & Richerson, 1992;Clutton-Brock & Parker, 1995; Frank, 1995;Michod, 1997; Fehr & GaK chter, 2000). The modelpresented in Section 2 provides one mechanismfor the evolution of such a system. In this version,enforcement*punishment of non-cooperators*itself is the bene"t to others that signals high-quality. Our model easily allows such punish-ment or enforcement to serve as the costly signal,and hence to be maintained when the conditionsfor evolutionary stability speci"ed in the modelare met, as discussed in Section 2.4.

Honest signaling of quality need not be groupbene"cial, of course, and our signaling modelapplies equally well to socially neutral or harmful


forms of costly signaling, such as conspicuousprivate consumption, brawling and dueling,#outing social norms with impunity, and the like.We suggested several factors that might makesignals with prosocial consequences (such as pub-lic generosity) more likely to evolve than equallycondition-dependent neutral or harmful signals:the greater likelihood that observers will allywith the signaler (due to present and expectedfuture direct bene"ts), the likelihood that pros-ocial signals that take the form of individuallyconsumable public goods will attract a largeraudience for the signaler, and the operation ofgroup selection on multiple alternative equilibria.We related the "rst two of these to parameters ofour model. However, we do not claim on thebasis of this model (nor any existing theory) thatprosocial signals will always have an evolution-ary advantage over signals without group bene-"ts. Indeed, such a claim would #y in the face ofthe ample empirical evidence for individuallyadaptive but collectively harmful displays amonghumans and many other species. Rather, we sug-gest that several factors ( just summarized) maymake the evolution of prosocial signals morelikely, if the relevant conditions pertain in par-ticular cases. Our model speci"es the conditionsthat can produce an honest signaling equilibriumin multi-player social contexts, and illuminatesthe additional factors that could tip the balancetoward or away from prosocial signals per se.Further theoretical and empirical work is war-ranted to determine the fruitfulness of thisapproach to the evolution of cooperation.

We would like to thank Michael Alvard, CarlBergstrom, Rebecca Bliege Bird, Robert Boyd, StevenFrank, Michael Lachmann, Aimee Plourde, PeterRicherson, Steven Siller, and Richard Sosis for usefulcomments, and the MacArthur Foundation for "nan-cial support.

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APPENDIX A

List of Symbols

This is a list of symbols used in places otherthan where they were "rst de"ned.

a the weight of the Signaler bene"t in a Part-ner's alliance payo!s: h"h

0#ag, l"

l0#ag

c signaling cost for a high-quality typec@ signaling cost for a low-quality typef probability that a signal is perceived (broad-

cast strength)


g individual bene"t conferred on each groupmember by Signaler

h Partner's payo! from allying with a high-quality type

l Partner's payo! from allying with alow-quality type

n group sizep fraction of population who are high-quality

typess Signaler's payo! from allying with a Partnerw fraction of o!spring of low-quality parents

who are of high-quality

z fraction of o!spring of high-quality parentswho are of low-quality

a fraction of Signalers who signal honestlyb fraction of Partners who prefer to ally with

Signaler who signalsd probability that a Partner will successfully

allyl "tness cost to Partner from monitoring

signaln expected payo! for a given (subscripted)

strategypi

signal i

Costly Signaling and Cooperation

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