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J. theor. Biol. (2001) 209, 87}95 doi:10.1006/jtbi.2000.2248, available online at http://www.idealibrary.com on Heterogeneity Stabilizes Reciprocal Altruism Interactions MICHAEL A. FISHMAN*, ARNON LOTEM AND LEWI STONE Department of Zoology, Faculty of ¸ife Sciences, ¹el Aviv ;niversity, ¹el Aviv 69978, Israel (Received on 25 October 1999, Accepted in revised form on 9 December 2000) In considering the phenomena of reciprocal altruism few would dispute that there are di!erences in individual quality*in particular, that for some individuals, at least on occasion, the cost of doing favors will exceed the potential of future bene"ts. That is, at any given time, a typical population is heterogeneous with respect to the a+ordability of reciprocal altruism. However, methodological limitations of the traditional analytical framework*Single ¹ype (symmetric) Evolutionary Game ¹heory2have restricted previous analytical e!orts to address- ing populations idealized in terms of their averages. Here we use the methods of Multitype Evolutionary Game ¹heory to analyse the role of individual di!erences in direct reciprocity interactions. Multitype analysis shows that non-idealized populations possess an ESS pro"le wherein individuals who cannot a!ord reciprocity (low-quality) defect, while individuals who derive net bene"ts from reciprocity (high-quality) cooperate. Furthermore, this cooperation is implemented via unmodi"ed tit-for-tat (TfT) strategy. Hence, our results may help resolve a long-standing problem concerning the evolutionary stability of TfT in direct reciprocal altruism. Finally, this di!erence between idealized and real populations is not restricted to direct reciprocal cooperation. Previously (Lotem et al., 1999) we have demonstrated evolu- tionarily stable indirect reciprocal cooperation among high-quality individuals in hetero- geneous populations. 2001 Academic Press 0. Introduction scious inheritable trait maintained by Darwinian "tness advantages, was "rst advanced by Altruistic behavior is commonly attributed to Williams (1966), and given rigor by Trivers inclusive "tness, or reciprocity (Krebs & Davies, (1971). 1993). The idea of reciprocal altruism, albeit one By using game theoretical analysis Trivers has motivated by conscious calculation, was "rst pro- shown that an individual may help an unrelated posed by Darwin: &&as the reasoning powers and conspeci"c whenever: (i) the cost to the donor is foresight2become improved, each man would less than the bene"t to the recipient, and (ii) the soon learn from experience that if he aided his favor is likely to be returned at a latter date. fellow-men, he would commonly receive aid in Hence, in order for reciprocal altruism (coopera- return''. (Darwin, 1871). The idea of reciprocal tion) to persist, cooperators must protect altruism in its modern form, i.e. as an uncon- themselves from exploitation by individuals who accept favors, but do not reciprocate*defectors. Trivers, aided by W. D. Hamilton (Trivers, 1971, * Author to whom correspondence should be addressed. p. 39), resolved this issue by postulating that E-mail: ma"sh@post.tau.ac.il cooperation is conditional, i.e. a cooperator will 0022}5193/01/050087#09 $35.00/0 2001 Academic Press 88 M. A. FISHMAN E¹ A¸. keep helping an unrelated individual, unless the (a) The costs of reciprocity are less than its latter refuses to reciprocate.- These theoretical bene"ts for all individuals involved. results soon gained empirical support from ob- (b) The costs of reciprocity exceed its bene"ts servations detailing reciprocal exchange of favors for all individuals involved. in group-living species (Fisher, 1980; Seyfarth & (c) The costs of reciprocity exceed its bene"ts Cheney, 1984; Wilkinson, 1984). for some individuals, and are less than The concept of helpfulness conditional upon these bene"ts for the rest. Hence, at any opponents' reciprocity was further developed by given time, the subject population consists Axelrod and co-workers (Axelrod & Hamilton, of two quality classes: low-quality indi- 1981; Axelrod, 1984; Trivers, 1985) resulting in viduals*who cannot &&a!ord'' reciprocity the formulation of tit-for-tat (TfT) strategy: a TfT vs. high-quality individuals*who derive player will punish defection by refusing help in net bene"ts by exchanging favors. turn, but otherwise will cooperate. In particular, these authors demonstrated that a population of By adopting a representation in terms of popu- TfT players cannot be invaded by defectors, i.e. lation averages, cases (a) and (b) can be analysed TfT is an Evolutionarily Stable Strategy (ESS) in in terms of symmetric (single-type) evolutionary the context of the TfT vs. defector contest. How- game theory (cf. Nowak et al., 1995; Brembs, ever, Selten & Hammerstein (1984) have shown 1996), which addresses situations where all con- that TfT is not an ESS under biologically plaus- testants have the same choice of game strategies ible conditions. Brie-y: mutants that lost the abil- and receive the same payo!s for any particular ity to make TfT-type evaluation of opponents, interaction (Maynard Smith, 1982). However, and thus help others unconditionally, can in- such a simpli"cation would reduce case (c) to crease through genetic drift in a population of either (a) or (b), and therefore is inappropriate. TfT players, and a population containing sub- Thus, to analyse case (c), we must use the stantial fraction of these unconditional altruists methods of Multitype Evolutionary Game ¹heory (UA) can be invaded by defectors. that allow analysis of contests between con- The demonstration that TfT is not an ESS led testants di!ering in the strategy choices available to the concerted e!ort to formulate cooperation to them, or having the same strategy choices but strategies superior to TfT in the sense of not di!erent payo!s for some of the possible interac- being subject to invasion by unconditional altru- tions (Cressman, 1992; Weibull, 1996). ists, reviewed by (Nowak et al., 1995; Brembs, Our multitype analysis shows that individuals 1996). The alternate approach of this paper is who cannot a!ord to reciprocate (e.g. young, based on the following considerations. Even if the sick, handicapped, or those that simply do not bene"ts of receiving help are the same for all have su$cient resources at a given time) will members of a population, there are di!erences defect by default. Thus, these phenotypic defectors (genetic or phenotypic) among individuals with (Lotem et al., 1999) are a special case of respect to the ability to donate help. Hence, for phenodeviation*a name for the disruptive e!ects any type of exchange of favors, there are three of the environment on genotype expression pro- possible situations. posed by Thornhill & M+ller (1997) in their semi- nal work on developmental stability. Because -Trivers used a metaphor, known as the Prisoner1s Dilem- defection by default is a phenodeviation, and thus ma (PD), from game theory. In this game, the two players have the choice of cooperating or defecting. The payo! to cannot be eliminated by natural selection, it is each when they cooperate is greater than the payo! for a persistent feature of real populations. mutual defection, but less than the payo! to a defector The persistence of defection confers an advant- playing against a cooperator. Finally, the payo! to a cooper- ator playing against a defector is the least of all. A single- age on TfT players vis-a-vis unconditional altru- stage PD game can be shown to have a unique stable ists. This advantage is absolute, i.e. TfT players solution*mutual defection (cf. Fudenberg & Tirole, 1996, always have higher "tness than unconditional Section 1.1.3). Trivers have shown that in an open-ended altruists, leading to the elimination of the latter series of PD games between two opponents*a Repeated Prisoner1s Dilemma (RPD), a conditional cooperator does from the population. As was shown previously better than a defector. (Axelrod & Hamilton, 1981), in the absence of HETEROGENEITY AND RECIPROCAL ALTRUISM 89 unconditional altruists, TfT is an ESS. Thus, we arrive at an ESS pro"le in which individuals who derive net bene"ts from reciprocity play TfT, whereas individuals that cannot a!ord reci- procity defect. Since an individual's ability to reciprocate (quality) changes with time, our results can be interpreted in two, not necessarily mutually ex- clusive, ways. (i) A population might be perma- nently divided into high-quality cooperators and low-quality defectors. (ii) Individuals can switch FIG. 1. Here 0(r(1 is the probability that an indi- behavior as their capacity for reciprocity (quality) vidual requesting help has been requested to help recently varies. In this latter interpretation, our ESS result enough for its response to be remembered: defection is represents cooperators who occasionally defect. punished with a refusal to help (!), and cooperation is Thus, our derived strategy pro"le is reminiscent rewarded with cooperation (#). Alternatively, with prob- ability 1!r, there is no de"nite memory of previous interac- of the evolutionarily stable &&mistake-making tion, and therefore the TfT is not motivated to refuse help. TfT'' strategy proposed by Boyd (1989). Thus, the value of r depends on the probability of repeated Finally, the stabilizing e!ect of phenotypic interactions and the "delity of memory/individual recogni- tion. Considered in a di!erent way, r is the probability that defection, is not restricted to direct-reciprocity the favor will be repaid, if the recipient is another TfT. interactions. Previously (Lotem et al., 1999), we have shown that the presence of phenotypic defectors, introduced as a modelling assumption, indiscriminately; defectors (DE) that solicit, but stabilizes cooperation in the analogous situations never donate help; and conditional altruists, or of indirect reciprocity interactions (Nowak & TfT players*who retaliate for each defection by Sigmund, 1998a, b). refusing help in the next interaction, but other- This paper is organized as follows. In Section 1, wise act as unconditional altruists. Thus, unlike we introduce a notation, one that we "nd conve- the invariant responses of the UA and DE nient for multitype analysis, and recapitulate the players, the response to a request for help by work of our predecessors in this notation. That is, a TfT player depends on its memory of previous we formulate a symmetric game theoretical interactions. That is, a TfT player always helps model addressing cases (a) and (b), and show that unconditional altruists and other TfT players, in these situations defector strategy is the unique but helps defectors only when it lacks informa- ESS. In Section 2, we extend the single-type tion to classify them. Denoting the probability model of Section 1 to a multitype model of the that an individual requesting help has been re- two quality classes of situation (c). To facilitate quested to help recently enough for its response presentation we con"ne some of the technical to be remembered by 0(r(1, we obtain the details of the analysis to the appendix. TfT response scheme summarized in Fig. 1. Let us denote the average (per capita) accumu- lated bene"ts of receiving help over a lifetime by 1. Symmetric Model (Idealized Populations) B, and the average lifelong costs of donating help We start the analysis by constructing a sym- by C; we use capital letters to distinguish these, metric game theoretical model for idealized per lifespan, payo!s from the per encounter populations. That is, we address the issue of payo!s more usual in the literature (cf. Nowak direct reciprocal altruism in terms of population & Sigmund, 1998b).? averages. This yields situations (a) or (b), i.e. the (average) costs of reciprocity are either less (a), or greater than (b) its bene"ts. As discussed in the ? If the per encounter bene"ts and costs are given by b and introduction, we consider three evolutionary c, respectively; the probability of t encounters per lifespan is given by w (0(w(1); and, on the average, individuals game strategies (heritable behavior phenotypes): take turns soliciting and being solicited for help: then B"b/ unconditional altruists (UA) that help others (2(1!w)) and C"c/(2(1!w)). 90 M. A. FISHMAN E¹ A¸. In these terms the payo! matrix for donating excluding the strictly dominated strategies*has help is given by the same ESS solutions as the original system (cf. Weibull, 1996, Chapter 3.2.1). Therefore, defec- UA TfT DE tion is again the unique ESS. !C !C !C UA . (1a) !C !C !(1!r)C TfT 2. A Multitype Model for Heterogeneous Populations 0 0 0 DE In this section, we analyse a situation where Note that the entry i}j represents the costs of help the costs of reciprocity exceed its bene"ts for given by (an average player of ) strategy-i to (an some individuals, and are less than these bene"ts average player of ) strategy-j. Thus, to calculate for the rest. We start by dividing the population the bene"ts of receiving help, we transpose matrix into two classes: low-quality individuals for (1a) and substitute #B for !C. This yields whom costs of reciprocity exceed its bene"ts vs. high-quality individuals for whom reciprocity UA TfT DE yields net bene"ts. The membership in a class is not necessarily hereditary*a reader might "nd it B B 0 UA . (1b) convenient to think of these quality classes as B B 0 TfT juveniles and mature individuals, respectively. B (1!r)B 0 DE We shall denote the frequency of low-quality individuals by 0(q(1 (cases q"0, 1 have The payo! matrix, P, for both giving and receiv- been addressed in the previous section). We re- ing help, the game matrix, is obtained by adding tain C as the average for the accumulated lifelong matrices (1a, b) to obtain costs of altruism in the high-quality class, and denote the corresponding value for the low-qual- UA TfT DE ity class by D. We retain the use of B for the accumulated lifelong bene"ts. As discussed B!C B!C !C UA P" . (2) above, C(B(D. Using r as in Section 1, and B!C B!C !(1!r)C TfT using the subscripts H and ¸ to denote the qual- B (1!r)B 0 DE ity classes, we have the following payo! matrices: P , P , P , P . Here the "rst subscript de- && &* *& ** As discussed above, in this section we consider "nes the focal (recipient of the payo!, the row two possibilities. strategy) and the second subscript de"nes the opponent: (a) ¹he costs of reciprocity are less than its bene,ts, i.e. C(B. In the appendix we show that, P && if C(B, then system (2) has a unique ESS solu- tion, DE, i.e. defectors displace individuals using UA TfT DE & & & alternative strategies, resulting in a population B!C B!C !C UA consisting of defectors only. "(1!q) &, B!C B!C !(1!r)C TfT & (b) ¹he costs of reciprocity are greater than its B (1!r)B 0 DE bene,ts, i.e. C'B. If C'B, then every element & of the third row of P is greater than the corre- (3) sponding elements of its "rst and second rows. UA TfT DE That is, at any composition of the population, * * * defectors have higher "tness than cooperators B!C B!C !C UA P "q & (UA or TfT), leading to the elimination of the &* B!C B!C !(1!r)C TfT , latter from the population. Formally, UA and & B (1!r)B 0 DE TfT are strictly dominated by DE, and can be & excluded, i.e. a reduced system*obtained by (4) HETEROGENEITY AND RECIPROCAL ALTRUISM 91 P UA TfT DE *& & & & UA TfT DE B!C B!C !C UA & & & (1!q) & B!D B!D UA B!C B!C !(1!r)C TfT , "(1!q) !D * & B!D B!D !(1!r)D TfT , B (1!r)B 0 DE & * B (1!r)B 0 DE DE * * (5) 1 UA !qC & (7) UA * TfT * DE * 1!r TfT , & B!D B!D !D UA 0 DE P "q * & ** B!D B!D !(1!r)D TfT . * UA TfT DE DE B (1!r)B 0 DE & & * * * ((1!q)B (1!r) (1!q)B 0) DE , (0) (6) * . DE * Note that the payo!s depend on the fre- Since all low-quality individuals are (pheno- quencies of the two quality types in the popula- typic) defectors, the ESS solutions of system (7), tion. For example, every element of the P && and hence system (3}6), have the form and P is multiplied by (1!q) because this *& is the probability to encounter a high-quality (x*, DE ), (8) opponent. * The general mathematical framework for where x* is an ESS solution of the system analysing evolutionary stability in games with obtained by combining the payo!s for interacting two types of players was developed by Cressman with DE to the payo!s for high-quality vs. high- * and co-workers (Cressman & Dash, 1991; Cress- quality interactions. That is, we add the row man, 1992). The speci"c case of system (3}6), elements of the reduced P to each element of &* however, can be analysed by taking advantage the appropriate row of P to obtain && UA TfT DE & & & (1!q)B!C (1!q)B!C !C UA & (9) (1!q)B!(1!qr)C (1!q)B!(1!qr)C !(1!r)C TfT . & (1!q)B (1!q) (1!r)B 0 DE & of the fact that, similar to the case for symmetric Since q, r'0, every element of the second games discussed above, strictly dominated row is greater than the corresponding element strategies in multitype games can be excluded of the "rst row. This is due to the fact that without a!ecting the ESS solutions of the the burden imposed by the presence of game (cf. Weibull, 1996, Section 5.6.1). To phenotypic defectors on unconditional altruists is wit, since B(D, every element of the third row greater than the corresponding burden, eqn (7), of P and P is greater than the correspond- on the TfT players. Formally, TfT strictly *& ** & ing elements of the "rst and second rows. Hence, dominates UA *and therefore, as discussed & UA and TfT are strictly dominated by DE above, we can exclude UA . This yields a * * * & and can be excluded. That is, we see that reduced system low-quality individuals &&must'' defect. Exclu- sion of UA and TfT yields a reduced p p * * P " RR RB system: & p p BR BB 92 M. A. FISHMAN E¹ A¸. TfT DE Condition rB'C is both necessary and su$- & & " (1!q)B!(1!qr)C !(1!r)C TfT cient when we consider competition between TfT &. players and defectors (Axelrod & Hamilton, (1!q) (1!r)B 0 DE & 1981). However, as detailed in Section 1 and in (10) the appendix: eqns (A5, A6), because in the ab- sence of defectors there is no di!erence in "tness We see that p 'p , i.e. defector playing BB RB between TfT and UA players*TfT playing against defector does better than a TfT playing populations can be invaded by unconditional against defector. Hence, a population of defectors altruists and a subsequent mixed population can cannot be invaded by TfT players. Consequently, be invaded by defectors. This result, however, is defection is an evolutionarily stable strategy of only obtained when we neglect the heterogeneity system (10). of real populations. In heterogeneous popula- If p 'p as well, then defector playing against BR RR tions, there are individuals who cannot a!ord TfT does better than a TfT playing against TfT reciprocity, eqns (3}6), and therefore defect by (exploitation pays better than cooperation). default*phenotypic defectors. In the presence of Hence, a population of TfT players can be in- these phenotypic defectors, unconditional altru- vaded and taken over by defectors. Thus, if ists have lower "tness than TfT players, eqn (9), p 'p , defection is the only ESS of system (10). BR RR and can be excluded. Thus, in heterogeneous However, if p 'p , then cooperation pays bet- RR BR populations, the situation reduces to the com- ter than exploitation. In particular, a population petition between (high-quality) TfT players and of TfT players cannot be invaded by defectors. (high-quality) defectors, modi"ed by the presence That is, as discussed in connection with eqn (8), of phenotypic defectors. system (3}6) has two ESS solutions (DE , DE ) & * Although phenotypic defectors prevent de- or (TfT , DE ). & * stabilization of cooperation by unconditional al- Now, truists, their presence is not an unmixed blessing. p !p "(rB!C)!qr(B!C) Because TfT players help defectors (in particular, RR BR phenotypic defectors) when in doubt (Fig. 1), the "rB(1! ) ( !q), (11) presence of phenotypic defectors imposes a bur- where den on TfT players, and decreases their ability to compete with defectors. Thus cooperation per- C rB!C 1 r! " and " " . sists only if the frequency of phenotypic defectors B r(B!C) r 1! (q) is less than , i.e. can be thought of as the tolerance capacity (in analogy with the carrying Since C(B, (1. Thus, for cooperation to capacity term of the logistic equation), for the be more productive than exploitation, we must burden of phenotypic defectors. We summarize have q( . Since q'0, we must have '0, these results in Fig. 2. which in turn requires r' , i.e. rB'C. As dis- cussed in Section 1, rB is the expectation of re- payment when dealing with another TfT player. Hence, we obtain the unsurprising conclusion that cooperation can persist only if the expected repayment exceeds the investment.A A This result is analogous to the Hamilton1s rule of kin altruism (Hamilton, 1964). According to Hamilton's rule help may be donated to a relative if the degree of relatedness times the bene"t to the recipient (the inclusive "tness bene- "ts to the donor) exceeds the donor's costs. That is, in both cases apparently altruistic acts are undertaken only when FIG. 2. Here the inequalities on the paths from the origin they yield net bene"ts to the &&altruist'' (Nowak & Sigmund, ( ) to endpoints represent the conditions that must be satis- 1998b). "ed if the strategy pro"le(s) at the endpoint to be ESS. HETEROGENEITY AND RECIPROCAL ALTRUISM 93 Thus, we see that cooperation is possible if formulated following the demonstration that TfT a population contains individuals who cannot playing populations can be invaded by uncondi- a!ord reciprocity, but the frequency of such, low- tional altruists and subsequently by defectors quality individuals is not too high. (Selten & Hammerstein, 1984). Our work demon- strates that TfT is evolutionarily stable in competition with unconditional strategies in het- 3. Discussion erogeneous populations. However, it is by no The analysis of reciprocity undertaken in this means certain that an analogous analysis of paper shows that reciprocal altruism can be the competition between TfT and some of stable when individual variation is taken into the more sophisticated conditional cooperation account. Genetic and/or phenotypic di!erences strategies will demonstrate domination by in ability among individuals, in particular in indi- the former. Thus, examination of the e!ects vidual costs for donating help, create three dis- of heterogeneity on the functioning of condi- tinct situations. When the costs of reciprocity are tional cooperation strategies, and of their either less than its bene"ts or exceed its bene"ts relative merits in heterogeneous context, though for all individuals involved, then the situation can beyond the scope of the present paper, is the next be analysed in terms of the single-type (symmet- logical step in game theoretical investigation of ric) evolutionary game theory, and reciprocity is reciprocity. not an ESS (Selten & Hammerstein, 1984). How- Above all else, the current study illustrates that ever, the most likely situation is when the costs individual variation is more than just a noise, and of reciprocity exceed the bene"ts for some thus the study of the evolution of behavior in individuals, but are less than the bene"ts for the terms of population averages may yield mislead- rest. That is, the subject population consists of ing results. In terms of mathematical methods, two quality classes: low-quality individuals*who our results highlight the usefulness of multitype cannot &&a!ord'' reciprocity vs. high-quality evolutionary game theory in analysing real (het- individuals*who derive net bene"ts from reci- erogeneous) populations. On the empirical level, procity, and must be analysed using the methods they illustrate the importance of studying vari- of the multitype evolutionary game theory ations in quality in relation to the behavioral (Cressman, 1992). Multitype analysis shows that, phenotypes. given the appropriate conditions, such a two- class population exhibits an ESS pro"le wherein We would like to use this opportunity to thank the individuals who cannot a!ord reciprocity defect, unknown referees for encouraging evaluation and while individuals who derive net bene"ts from constructive criticism. reciprocity cooperate by playing unmodi"ed TfT. 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Nature 308, 181}184. stable relative to other strategies in . WILLIAMS, G. C. (1966). Adaptation and Natural Selection: Obviously, we cannot check every strategy in A Critique of Some Current Evolutionary ¹hought. Prince- the continuous set X for being an ESS (let alone ton, NJ: Princeton University Press. an element of an ES set). Hence, we enumerate the ESS solutions as a two-step process. First, we Appendix "nd all the potential ESS solutions by using the fact that every ESS solution is a Nash Equilibrium Methods and Notation point (though not vice versa) and therefore if Let +e , e , e , be the standard basis of R, x*3X is an ESS, then u(e , x*)"u(x*, x*) or H and let us denote the strategy set for the game x*"0, ∀j. H given by eqn (2) (payo! matrix P) by Consequently, every solution of the system of equations X"+x e #x e #x e " x , x , x *0 [u(e !x, x)]x "0 ∀j (A4) H H and x #x #x "1,. (A1) on X is a potential ESS. Once we have derived all the potential ESS solutions, we apply the ESS criterion (A3). That is, e , e , and e represent the pure strategies In the speci"c case of system (2), there is one UA, TfT, and DE, and their convex combina- potential ES set solution and two potential ESS tions represent mixed strategies. In these terms, solutions. HETEROGENEITY AND RECIPROCAL ALTRUISM 95 The potential ES solution However, since x Oe but "+ e #(1! )e " 3[0, 1], (A5) (x , e )"0 and represents the state where defectors are absent, (x , e )"!r(B!C) (1! ). (A8) and therefore there is no di!erence between un- conditional altruists and TfT players. Since x is not an ESS. e 3 and e , Finally, we have a potential ESS solution wherein defectors displace both unconditional but altruists and TfT players (e , e )"!C. (A6) is not an evolutionary stable set. x "e and (x , x)"C[x #(1!r)x ]. Next, we have a potential ESS solution where- (A9, A10) in TfT players and defectors coexist Hence, since 0(r(1, x (DE) is an ESS for all (1!r)C x " e #(1! )e : " . (A7) parameter values. r(B!C)

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reciprocal altruism, Journal of Theoretical Biology, Evolution of cooperation, indirect reciprocity, Theor Biol, Bulletin of mathematical biology, research profile, the Dolphins, group polarization, Robert Trivers

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posted: | 4/20/2010 |

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