Unifying the Theories of Inclusive Fitness and Reciprocal Altruism

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Jeffrey Alan Fletcher
Portland State University, [email protected]
Martin Zwick
Portland State University, [email protected]
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CCiittaattiioonn DDeettaaiillss
Fletcher, J.A., and Zwick, M., 2006. Unifying the Theories of Inclusive Fitness and Reciprocal Altruism. The
American Naturalist 168: 252-262.
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vol. 168, no. 2 the american naturalist august 2006 (cid:1)
Unifying the Theories of Inclusive Fitness
and Reciprocal Altruism
Jeffrey A. Fletcher1,2,* and Martin Zwick2,†
1.DepartmentofZoology,UniversityofBritishColumbia,#2370- More than 40 years ago, Hamilton developed an expla-
6270UniversityBoulevard,Vancouver,BritishColumbiaV6T1Z4, nationfortheevolutionofaltruismamongrelativesbased
Canada;
ontheideaofinclusivefitness(Hamilton1963,1964,1970,
2.SystemsSciencePhDProgram,PortlandStateUniversity,
1972,1975).Hismostfamousresult,knownasHamilton’s
Portland,Oregon97207
rule (HR), is usually interpreted as specifying the condi-
SubmittedDecember19,2005;AcceptedMay31,2006; tions under which the indirect fitness of altruists (due to
ElectronicallypublishedJuly14,2006 helping relatives have more offspring) sufficiently coun-
terbalancestheimmediateself-sacrificeofaltruists.Inthis
Onlineenhancement:appendix. way, the altruistic trait can increase overall. This mecha-
nismisalsoknownaskinselection(MaynardSmith1964).
Twenty-five years ago, Axelrod and Hamilton (1981)
launched a still vigorous areaofresearch(forreviews,see
Dugatkin 1997; Sachs et al. 2004; Doebeli and Hauert
abstract: Inclusive fitness and reciprocal altruism are widely
thought to be distinct explanationsforhowaltruismevolves.Here 2005) in which computer-based models of the iterated
we show that they rely on the same underlying mechanism. We prisoner’s dilemma (IPD) areusedtostudytheevolution
demonstratethiscommonalitybyapplyingHamilton’srule,normally of cooperation via reciprocal altruism (Trivers 1971). In
associatedwithinclusivefitness,totwosimplemodelsofreciprocal their article, Axelrod and Hamilton suggestthatthereare
altruism:one,aniteratedprisoner’sdilemmamodelwithconditional
thesetwoalternativeexplanationsfortheevolutionofco-
behavior;theother,amutualisticsymbiosismodelwheretwointer-
operative traits: when the benefits of altruism fall to rel-
acting species differ in conditional behaviors, fitness benefits, and
atives, cooperation can evolve by inclusive fitness, and
costs.WeemployQueller’sgeneralizationofHamilton’srulebecause
when benefits fall to nonrelatives, cooperationcanevolve
the traditional version of this rule does not apply when genotype
and phenotype frequencies differ or when fitness effects are non- by reciprocal altruism.1 A third theory for the evolution
additive, both of which are true in classic models of reciprocalal- ofaltruism,basedonmultilevel(orgroup)selection(Wil-
truism. Queller’s equation is more general in that it applies to all son1975;Wade1978),isnotaddresseddirectlybyAxelrod
situations covered by earlier versions of Hamilton’s rule but also and Hamilton (or in this article), but several researchers
handlesnonadditivity,conditionalbehavior,andlackofgeneticsim- havedemonstratedtheunderlyingunitybetweeninclusive
ilarity between altruists andrecipients.Ourresultssuggestchanges
fitness and multilevel selection theories (Wade 1980;Bre-
tostandardinterpretationsofHamilton’srulethatfocusonkinship
den 1990; Queller 1992b; Frank 1998; Sober and Wilson
andindirectfitness.Despitebeingmorethan20yearsold,Queller’s
1998). For reasons discussed below, reciprocal altruismis
generalizationofHamilton’sruleisnotsufficientlyappreciated,es-
peciallyitsimplicationsfortheunificationofthetheoriesofinclusive often left out of these unification efforts. Here we focus
fitnessandreciprocalaltruism. on this missing piece: the unification of the theories of
inclusive fitness and reciprocal altruism.
Keywords:conditionalbehavior,Hamilton’srule,iteratedprisoner’s It is understandable thatAxelrodandHamilton(1981)
dilemma,kinselection,mutualism,synergy. donotsuggestthatHRcouldapplytoreciprocalaltruism
andtheirIPDmodels.Inadditiontoassumingthatplayers
are unrelated, these models involve conditional strategies
(genotype/phenotype differences) and nonadditive fitness
* E-mail:fl[email protected].
† E-mail:[email protected]. 1 Weusecooperationandaltruismsynonymouslybecausecooperationinour
Am.Nat.2006.Vol.168,pp.252–262.(cid:1)2006byTheUniversityofChicago. modelsinvolvesanimmediatealtruisticsacrificeinfitnessthatprovidesrel-
0003-0147/2006/16802-41512$15.00Allrightsreserved. ativefitnessbenefitstorecipients.
Unifying Theories of Altruism 253
functionsthatHRcouldnotaccommodate.Yetjustafew they incur for their own helping behaviors. The required
yearslater,Queller(1985)developedaversionofHRthat combination of genotype-phenotype assortment, benefit
handles both of these issues, and in his article, Queller to cost ratio, and any nonadditive effects is given by
suggeststhathisversioncouldapplytoreciprocalaltruism. Queller’s generalization of HR. This rule applies whether
More recently, Sober and Wilson (1998) suggest a unifi- the source of positive assortment is interactions among
cationofinclusivefitnessandreciprocalaltruismtheories. relatives (the original application), conditional behaviors
They show how additive versions of the prisoner’s di- among nonrelatives, or reciprocalinteractionsacrossspe-
lemma (PD) correspond to fitness functions used in in- cies(mutualisticsymbiosis).Thissinglerequirementgov-
clusive fitness models, but they do not address the two erned by Queller’s version of HR brings unitytothesep-
critical issues mentioned above: genotype/phenotype dif- arate theories of inclusive fitness and reciprocal altruism.
ferenceswhenbehaviorsareconditionalandnonadditivity. We begin by briefly reviewing HR, Queller’s contribu-
Frank (1994, 1998) notes that regression coefficients be- tions, and the original IPD experiments (Axelrod and
tween species in his model of mutualism measure reci- Hamilton1981;Axelrod1984).WethenuseQueller’sver-
procity and are similar to coefficients of relatedness in sionofHRforgenotype/phenotypedifferencestoanalyze
inclusivefitnessmodels,buthealsodoesnotaddressthese an experimentinvolving an additiveIPD.UsingQueller’s
two issues. The emphasis of his analysis (Frank 1998) is nonadditive version of HR, we also show how additive
onpartitioningselectionandtransmissionandonunifying behavior that is iterated within generations gives fitness
quantitative genetic and population genetic approaches. consequences similar to those of synergistic behavior
Despite these suggestions that inclusive fitness and re- whereapairingisasingleinteractionpergeneration.(Syn-
ciprocal altruism theories are related, unification of these ergy is defined below as positive nonadditivity.) Finally,
two theories requires that HR be effectively applied to we extend this model so that there are conditional co-
reciprocal altruism. However, until now, there has been operator and defector types in each of two mutualistic
no directandsuccessfuldemonstrationofusingQueller’s speciesthatinteract.Heretheparticularconditionalstrat-
moregeneralversionofHRinreciprocalaltruismmodels. egy, benefit level provided, cost paid for cooperative be-
In fact, in an expansion of his original results, Queller haviors, and any nonadditive effects can be different in
(1992a, 1992b) drops any mention of its applicability to each of the species. For each species, we use Queller’s
reciprocal altruism. version of HR to accurately predict whether the cooper-
HerewedemonstratethatQueller’sequationsdoindeed ative trait will increase in that species. Finally, we discuss
provideafoundationfortheunificationofinclusivefitness the implications of these results for understanding and
and reciprocal altruism theories. Our approach differs unifying the theories of inclusive fitness and reciprocal
from Nee’s (1989) application of a version of HR to an altruism.
IPDmodelofreciprocalaltruism.InNee’swork,thegen-
erality of Queller’s equation was not utilized; instead,
The Progressive Generalization of Hamilton’s Rule
Queller’sversionwasbroughtbacktothesharedgenotype
level by adding an additional term that related the phe- Hamilton’srule(Hamilton1963,1964)givesthecondition
notype of others to their common genotype with focal necessaryforanaltruistictraittoincreaseinapopulation
altruists. Here we use Queller’s equation in its full gen- in the next generation and is deceptively simple:
eralitybysimplytakingitatfacevalue:itisthephenotype
of others that is crucial, not their genotype. This more rb1c, (1)
encompassing viewpoint allows us to include heterospe-
cific interactions in mutualistic symbiosis, where altruists wherebisusuallyinterpretedastheaveragefitnessbenefit
and recipients are clearly genetically unrelated and toarecipientofthealtruisticbehaviorandcistheaverage
nonidentical. cost to an altruist for this behavior. Complications arise
Ourmodelsandanalysissuggestthat,ratherthanbeing inthemeaningoftherterm,whichhasbeenprogressively
fundamentallydifferentmechanisms,inclusivefitnessand generalizedovertheyears.Originallythoughtofasasim-
reciprocal altruism are alternative ways to satisfy a com- ple measure of relatedness via descent (Hamilton 1963,
mon single requirement for self-sacrificing traits to in- 1964), Hamilton (after interacting with Price [1970])
crease in a population. This requirement can be statedas broadened the meaning of r to be a measure of the as-
follows: there must be sufficient positive assortment be- sortment of genetic types regardless of relatedness by de-
tween individuals with the altruisticgenotypeinquestion scent (Hamilton 1970, 1972, 1975):
and the helping phenotypes of others they interact with,
suchthatonaveragethosewiththefocalgenotypebenefit Cov(G ,G )
A O b1c, (2)
from the helping behaviors of others more than thecosts Var(G )
A
254 The American Naturalist
Table1:ProgressivegeneralizationofHamilton’sruleillustratedbysituationsforwhichdifferentversions
(eqq. [1]–[4])apply
Kin Nonkin genetic Genotype-phenotype Nonadditivefitness
Equation interactions similarity differences(G(P) functions(d(0)
(1) Yes … … …
(2) Yes Yes … …
(3) Yes Yes Yes …
(4) Yes Yes Yes Yes
whereG isthegenotype(orbreedingvalue)withrespect interpretedasthecontributiontothedirectfitnessofthose
A
tothealtruistictraitforeachpotentialactor(subscriptA) with the altruistic genotype from the behavior of others.
and G is the average genotype value of others(subscript Intheindirectfitnessconcept,bisseenasthecontribution
O
O) that interact with each potential actor. Queller (1985) by the actor to the fitness of others. This is a possible
further generalized Hamilton’s r term by explicitly in- interpretationofHRwhenthesetwobenefitsarethesame,
cludingtheconsequencesofthephenotype(behaviors)of but when the amount given and the amount received by
actors and others on selection for a genetic trait rather focal altruists differ, as in our model of symbiosis below,
than focusing on the effect of genotypes directly. This only the interpretation suggested by Queller’s version
yields works correctly.
Whenthereisadeviation(d)fromfitnessadditivityfor
Cov(G ,P ) mutual cooperation (as there is in many IPD models),
A O b1c, (3)
Cov(G ,P) thenanadditionaltermisneededthatspecifiesthedegree
A A
to which the focal genotype covaries with mutual coop-
whereP isthephenotypeoftheactorandP istheaverage eration,2scaledbytheamountofdeviation(Queller1985):
A O
phenotype of others interacting with each actor. In the
appendixintheonlineeditionoftheAmericanNaturalist,
weprovidemoredetailsaboutthegeneralizationofHam- Cov(G ,P )b(cid:1)Cov(G ,PP )d1Cov(G ,P)c. (4)
A O A A O A A
ilton’sruleaswellasthemathematicaldetailsofourmod-
els and analysis. Thisdeviationvaluecanbepositive(representingsynergy),
These covariance ratio expressions (eqq. [2], [3]) may negative (representing diminishing returns), or 0 (repre-
be convenient in comparing different versions of r, but sentingadditivity).Notethatdividingbothsidesofequa-
crossmultiplyingresultsinamoreeasilyinterpretedform tion (4) by Cov(G , P ) results in the same r term as in
A A
of Queller’s equation. For example, equation (3) can be equation (3), plus an additional covariance ratio related
written as to the deviation from additivity. There are other versions
of HR (for reviews, see Pepper 2000; West et al. 2002),
Cov(G ,P )b1Cov(G ,P)c. (3a) butequations(1)–(4)representsignificantstepsinapro-
A O A A
gressivegeneralizationofHRthataresummarizedintable
ThissaysthatthegenotyperepresentedbyG willincrease 1.
A
in the population if the covariance between its presence Queller’s versions (eqq. [3], [4]) apply to all situations
in each potential actor and the helping behaviors (phe- covered by Hamilton’s versions (eqq. [1], [2]), plus they
notypes) of others, scaled by the benefit of this help, is handleadditionalsituationsthatmaynotallowarecursive
more than the covariance between its presence and the analysis (Grafen 1985), such as when the frequency of
helping behaviors of actors themselves, scaled by thecost cooperative behavior depends on both genotype and en-
of these behaviors. Simply put, the genotypewillincrease vironmental factors (e.g., the behaviors of others). To see
if on average individuals carrying it receive more fitness thateachequationaboveismoregeneralthantheprevious
benefits than they pay out. Note that it is the phenotype ones, note that if fitness functions are additive (dp0),
or behaviors of others (P ) that is critical, not their ge- thenequation(4)becomesequation(3);ifphenotypefre-
O
notype; there is no G term in this equation. This has quencies equal genotype frequencies (G pP and
O O O
consequences for the usual indirect fitness interpretation G pP), then equation (3) becomes (2); andif thesim-
A A
of HR. As Frank (1998, p. 68) points out, “The direc-
tionalityofQueller’srelatednesscoefficient…isopposite 2 TheproductPP representsmutualcooperation,wherecooperate(C)be-
A O
to the directionality of Hamilton’s inclusive fitness coef- haviorshaveaphenotypeof1anddefect(D)behaviorsaphenotypeof0.
ficient.” The benefit (b) term in Queller’s version is best Thisisexplainedfurtherinthefollowingsections.
Unifying Theories of Altruism 255
ilarityingenotypebetweenactorsandothersissolelydue Table 2: Typical prisoner’s dilemma fitness values for an
to interactions within kin groups, then equation (2) can actor,givenitsbehavior(phenotypeP )anditsopponent’s
A
become equation (1), where r represents relatedness by behavior(phenotypePO)
descent. Queller’s version (eq. [4]) is the most general in Opponent’sbehavior
that it works without these restrictions or assumptions. C (P p1) D (P p0)
O O
Actor’s behavior contributesb contributes0
C (P p1) sacrificesc 3 0
The Iterated Prisoner’s Dilemma Model A
w (cid:1)b(cid:2)c(cid:1)d w (cid:2)c
of Reciprocal Altruism 0 0
D (P p0) sacrifices0 5 1
A
The PD captures a fundamental problem of social life: w (cid:1)b w
0 0
individually rational behavior may lead to a collectively
Note:Thesefitnessvaluescanberepresentedastheresultofadditive
irrationalanddeficientoutcome.Inn-playerversions,this benefit contributions (b) fromits partner, itsownsacrificeorcost
dilemma is also known as a “tragedy of the commons” (c),thebasefitnessvalueuncorrelatedwithCandDbehaviors(w),
0
(Hardin 1968) or a freeloader (free rider) problem (Mc- andthedeviationfromadditivityformutualcooperation(d).Forthe
shownfitnesspayoffvalues,bp4,cp1,w p1,anddp(cid:2)1.
Millan 1979; Avile´s 2002). Typical two-player PD fitness 0
values for the actor, given its own and its opponent’sbe-
was also one of the simplest. Called “tit for tat” (TFT),
haviors, are shown in table 2. Here behaviors are either
this strategy always cooperates with an opponent in the
cooperate(C)ordefect(D).Table2alsoshowsparameters
first interaction (PD game) and, in all subsequent inter-
thatdecomposethesefitnesspayoffsintermsofthebenefit
actions, simply plays whatever the opponent did in the
(b)providedtoanopponentbyaCbehavior,thecost(c)
lastgame.ThisconditionalbehaviorallowedTFTtomin-
paid by the cooperator, the base fitness (w) that is in-
0 imizeexploitationbydefectingopponents,suchas“always
dependentofcooperation,andthedeviation(d)fromad-
defect” (ALLD), while taking advantage of mutual coop-
ditivitywhencooperationismutual.InthePD,eachplayer
eration when it met other “nice” strategies. Since these
has a dominant strategy to defect (D), but if they both
original experiments more than 25 years ago, much re-
cooperate, both can receive more (in this case, three in-
search has been done on the IPD (Dugatkin 1997; Sachs
stead of one) than if theybothdefect.Itispresumedthat
et al. 2004; Doebeli and Hauert 2005).
players exhibit their behaviors simultaneously and that
From the perspective of Queller’s version of HR (eq.
there is no knowledge or guarantee about what the other
[3]), the combination of iterated games and conditional
player will do. The dilemma is that cooperation makes a
play can create positive assortment among the helping
player vulnerable to exploitation; in the case of mixed
behaviors (phenotypes) of others and the conditionally
behaviors,thedefectorgetsthehighestpayoff(five),while
cooperative genotype (e.g., TFT), even when there is no
the cooperator gets the lowest (zero).
positive assortment among genotypes. That is, if one cal-
Note that while a two-player fitnessmatrixcanberep-
culatesHamilton’srusingonlygenotypes(eq.[2]),itwill
resented in terms of these four parameters, other param-
be 0 in the case of random binomial pairing. Therefore,
eterizations are also possible. Also note that this typical
the traditional version of HR cannot be satisfied, and it
PD matrix, which is the one used by Axelrod (1984), is
appearsasiftheincreaseincooperation(e.g.,theincrease
nonadditive: it cannot be achieved without a nonzero d
in TFT types) observed in IPD models of reciprocal al-
term.Inthiscase,therearediminishingreturnsformutual
truism is not due to inclusive fitness as measuredbyHR.
cooperation (dp(cid:2)1), but synergistic (d10) matrices
Thisisnotinfactthecase;aswewillsee,Queller’sversion
that still define a PD are also possible. While Axelrod
of HR predicts exactly when conditional cooperationwill
purposely chose a nonadditive PD to ensure that tour-
increase in these models.
nament results did not depend on additivity (R.Axelrod,
personal communication, 2005), the nonadditive nature
of these common PD fitness values is not generally ap- Hamilton’s Rule Applied to a Classic
preciated.Becauseofnonadditivity,thisPDcannotbean- Reciprocal Altruism Model
alyzed with a pre-Queller HR.
Confirming Queller’s Version
AlthoughinaPDsituationitisindividuallyrationalto
defectineachsingleplayofthegame,AxelrodandHam- Here we offer a simple example where Queller’s version
ilton(1981)providedearlysupportforreciprocalaltruism of HR (eq. [3]) is applied to reciprocal altruism using a
theory (Trivers 1971) by showing that conditional coop- population consisting of the two classic types mentioned
erativestrategiescandowellwheninteractions(PDgames) above, TFT and ALLD. This kind of population hasbeen
areiterated.ThemostsuccessfulstrategyinAxelrod’stour- used previously to apply a modified HR to an IPD (as
naments (submitted by social scientist Anatol Rapoport) mentioned above; Nee 1989), to classify typesofaltruism
256 The American Naturalist
production as a function of the benefit value (b) for dif-
ferentvaluesoftheinitialTFTfrequency(Q)andnumber
of iterations (i). For convenience, other parameters are
held constant at cp1 and w p1. Notice that, all else
0
being equal, more iterations or higher starting Q make it
easier for DQ to increase. Arrows in figure 1 indicate the
predicted equilibrium points (DQp0) using Queller’s
(eq. [3]) version of HR and the same parameters used in
figure 1 (appendix). If instead of equation (3), which in-
volves P , we use the more restrictedequation(2),which
O
involvesG ,rp0forthisbinomialpopulationstructure,
O
and we do not correctly predict the increase in the pro-
portion of TFT (Q). In contrast, Queller’sversionexactly
predictsthese“tippingpoints,”thatis,thevalueofbenefit,
b, beyond which TFT increases in each population.
Figure 1: Change in the proportion of TFT players (DQ) after one
generationasafunctionofbenefit(b)levelinapopulationofTFTand
ALLDplayerswithbinomialpairing.Resultsshownforvariousstarting
Iterations and Synergy
TFT proportions (Q) and numbers of iterated games (i), where other
parameters for each game are held constant at cp1, w p1, and
0 Queller’s version of HR (eq. [4]) suggests two ways to
dp0. Arrows indicate where DQp0. These equilibrium points are
enhance the evolution of cooperation, given random
predictedbyQueller’sversionofHR(seetableA2intheonlineedition
oftheAmericanNaturalist). grouping: if cooperative behaviors toward altruists are
more frequent than the frequency with which they are
groupedwithotheraltruistsorifthereisnonadditivesyn-
(KerrandGodfrey-Smith2002),andtostudytheevolution
ergyformutualcooperation.Hereweshowthatthesetwo
of altruism in finite populations (Nowak etal.2004).Be-
effects can have equivalent fitness consequences. Coop-
cause one of the types (TFT) uses conditional behaviors,
eration evolves in the populations plotted in figure 1 be-
we must measure genotype and phenotype frequencies
cause conditional behavior positively assorts cooperation
separately.Tocalculatethecovariancesneededinequation
withTFTgenotypes,butanalternativeanalysisispossible.
(3), we take TFT as our focal genotype (G) and assign it
Assuming additive PD parameter values of bp4, cp
a value of 1 and ALLD a value of 0. For phenotype (P),
1, w p1, and dp0 and iterations of ip10, the cu-
thecooperate(C)behaviorhasavalueof1,andthedefect 0
mulative intergenerational fitness consequences of differ-
(D) behavior a value of 0. In a population of these two
entpairingsareshownintable3asiftheyweretheresult
types,therewillbethreepossibleparings(TFT-TFT,TFT-
of a single interaction between the players. This resulting
ALLD,ALLD-ALLD),eachwithpredictablevaluesforG ,
A game matrix, no longer a PD, can be decomposed into
P ,andP ,giventhenumberofiteratedgames(i)within
O A ourfourparameters.Weuseprimestodistinguishparam-
generations(tableA1intheonlineeditionoftheAmerican
etersoftheresultinggamefromthoseoftheoriginal.The
Naturalist).3 Again, subscript A indicates the focal actor,
andsubscriptOmeansothers(inthiscase,thefocalactor’s
opponent). We can now calculate the change in TFT fre- Table3:Cumulativefitnessvaluesforpairings
quency in the population (DQ), assuming fitness payoffs of tit-for-tat(TFT)andalwaysdefect(ALLD)
are proportional to offspring representation. Given the players thatlast for ip10 iteratedgames
population averages forthefrequencyofCbehaviorsand Opponent’sbehavior(s)
theinitialfrequencyofTFTtypes(Q),wecanalsocalculate
Actor’s behavior TFT ALLD
whether Queller’s version of HR is satisfied for any given
40 9
parameter settings (see appendix). TFT w(cid:1)(cid:1)b(cid:1)(cid:2)c(cid:1)(cid:1)d(cid:1) w(cid:1)(cid:2)c(cid:1)
We begin with an additive PD game (dp0). Figure 1 0 0
14 10
showsthechangeintheproportionoftheTFTtype(DQ) ALLD w(cid:1)(cid:1)b(cid:1) w(cid:1)
0 0
after one generation of random pairing and asexual re-
Note: For each game, bp4, cp1, w p1, and
0
dp0.Theshownfitnesspayoffvaluesinterpretedas
3 For mathematical convenience, i represents a fixed number of games in theresultofonlyasingleinteractioncanbedecomposed
eachinteraction.Similarresultsholdforgamesofaveragelengthi,andthe withb(cid:1)p4,c(cid:1)p1,w(cid:1)p10,andd(cid:1)p27.Thisgame
simpleplayersinthismodelarenotcapableofusingknowledgeofthenumber isnotaprisoner’sdilem0 ma;inthegametheorylitera-
ofgamesforbackwardinduction. ture,itiscalled“assurance”or“staghunt.”
Unifying Theories of Altruism 257
parameters for table 3 are b(cid:1)p4, c(cid:1)p1, w(cid:1)p10,
0
d(cid:1)p27 (appendix).
Now suppose that we do not know the fitness conse-
quences of each social interaction (game) or how often
interactions(iterations)occurwithineachgeneration.In-
stead, we see only who is paired with whom and the re-
sulting fitness consequences to each type at the end of
eachgeneration.Thefitnessconsequencesarethesameas
in the original situation, but from this “black box” per-
spective, there are no iterations and no difference in ge-
Figure 2: Fitness relationships between two interactingspecies(1and
notype versus phenotype frequencies. There are just bi-
2). The origin of each arrow correspondsto acooperatebehavior(C)
nomial single-interaction (ip1) pairings but a strong bythespeciesatitsorigin,andtheterminationofeacharrowindicates
fitness synergy when TFT meets TFT. Analyzing thissyn- whichspecies’fitnessisdirectlyaffectedbythisbehavior.Forinstance,
ergistic (d(cid:1)p27) situation with equation (4) and no ge- aCbehaviorexhibitedbyamemberofspecies1hastwodefiniteeffects—
decrementofc initsownfitnessandanincrementofb toitsspecies
notype/phenotype differences gives the exact same in- 1 1
2partner’sfitness—aswellasonepotential“interaction”effect(indicted
equality as assuming additivity (dp0) and genotype/
bydashedlines),achangeofd initsownfitnessonlyifthereisalsoa
phenotype differences due to iterations (ip10) and simultaneous C behavior by its1species 2 partner. We label benefitsby
conditional play (appendix). Thisperspectiveprovidesan theirsourcetoemphasizethathelpisgivenheterospecifically.
alternativeexplanationforhowconditionalstrategiessuch
as TFT evolve. From this point of view, itis not somuch species1exhibitingaCbehaviorareb (benefittospecies
1
that iterations and conditional play “solve” the PD itself 2) and c (its own cost) and similarly for species 2. We
1
(in which defection is favored) but that they effectively labelthe benefits bytheirsourcetohighlightthefactthat
change the game into one in which mutual cooperation the benefit received by a member of one species comes
(in table 3 labeled TFT) has the highest fitness payoff. fromacompletelyunrelatedmemberoftheotherspecies.
Note,however,thatthe“assurance”gamethatthePDhas Nonadditive effects, which have their source in bothspe-
been converted into is not itself dilemma free: ALLD still cies, are subscripted with the species whose fitness is di-
receivesmorethanTFTinallheterogeneouspairings(table rectly affected. The proportion of the cooperative typein
3), and therefore a maximin strategy results in a Nash eachspeciesisgivenbyQ andQ,respectively.Ingeneral,
1 2
equilibrium of mutual defection, which is Pareto whether the focal genotype (e.g., TFT) of species 1 in-
nonoptimal. creases in the next generation is predicted by Queller’s
version of HR (eq. [4]), where A (actor) is a member of
species 1 and O (other) is a member of species 2:
Hamilton’s Rule Applied to Cooperation across Species
Cov(G,P)b (cid:1)Cov(G,PP)d 1Cov(G,P)c . (5)
Mutualism Model 1 2 2 1 1 2 1 1 1 1
Here we use Queller’s version of HR to analyze a simple A symmetric equation (where all subscripts areswitched)
model of mutualistic symbiosis in which there are two predicts whether the focal type in species 2 increases or
interacting species (labeled 1 and 2), each with two dif- not. We will refer to these two instances of Hamilton’s
ferenttypes:ALLDand(usually)aconditionalcooperator rule as HR and HR, respectively (appendix).
1 2
type such as TFT. For convenience, the cooperative be- This model of symbiosis has some similarities to one
haviors of interest take place only heterospecifically. For developed by Frank (1994), but his model assumes that
instance,thebehaviorsbetweencleanerfishandtheirhosts phenotypeandgenotypefrequenciesarethesameandthat
(Bshary and Grutter 2002) are strictly heterospecific: fitnessfunctionsareadditive.Whilethissimplemodeldoes
cleaner fish do not clean conspecifics, and hosts are not not capture all types of mutualisms (Bronstein 2001), as
cleanedbyotherhosts.Wealsoassumerandom(binomial) far as we know, the analysis presented here is the first
pairings, but conditional behavior will provide the asym- example of the use of Queller’s version of HR to analyze
metry in benefits within species necessary for mutualists cooperation across species in which behavior canbecon-
to increase (Ferriere et al. 2001). Note that these inter- ditional (G(P).
actions, unlike our within-species cooperation examples
above, can be asymmetric between species in terms of
Dynamic Simulations
costs,benefits,anddeviationfromadditivity(Frank1994;
Sachsetal.2004).Figure2illustratesthesefitnessparam- Figure 3 illustrates the coupled dynamics in our model,
eters between the two species. The benefit and cost for where cooperation can reach saturation in both species
258 The American Naturalist
Figure3:Dynamicsbetweentwospecies,eachwithacooperativetypeandanALLDtypeforvariousparametersettings.A,Cooperationevolves
even though in species 1 cooperation costs more than the benefit it produces. Parameters areip4,b p1.5,c p2,d p0, initialQ p0.2,
1 1 1 1
b p5,c p0.1,d p0,andinitialQ p0.1;inbothspecies,thecooperativetypeisTFTandw p2.B,Allparametersarethesameforboth
2 2 2 2 0
species,exceptthatthecooperative type inspecies1 isTF2Tand in species2isPavlov.Parametersareip80,b pb p4,c pc p1,d p
1 2 1 2 1
d p0,initialQ pQ p0.1,andw p1.C,Cooperativetypeisunconditional(ALLC)inspecies1andTFTinspecies2.Otherparametersare
2 1 2 0
ip100,b p2,c p1,d p0,initialQ p0.01,b p2.2,c p0.1,d p1.3,initialQ p0.5,andinbothspeciesw p1.Thebarsaboveeach
1 1 1 1 2 2 2 2 0
graphindicateatwhichgenerationsQueller’sversionofHR(e.g.,eq.[5])issatisfied(solidbar)foreachspeciesrespectively(HR andHR).Note
1 2
thatsatisfyingHRcorrespondstowhenthecooperativetypeineachspeciesincreases.
evenundersomesurprisingconditions.Infigure3A,mu- an increase in altruism is not that the benefit given by
tual symbiotic cooperation evolves even though cooper- altruists sufficiently exceed their costs but rather that, on
ation costs members of species 1 more than the benefit average, the benefit received by those with the altruistic
they provide (c 1b ). This inefficient formof altruismis genotype exceed their costs.
1 1
not generally thought to evolve, but it can evolve here In the run shown in figure 3B, all parameters are the
because of the high benefits of cooperation provided by same for both species and additive, but the cooperative
the other species; that is, b is sufficiently greater than c strategies differ: in species 1, it is tit for two tats (TF2T),
2 1
(eq. [5]). In this case, fitness is additive,d pd p0.In which plays C unless the previous two plays by its op-
1 2
these dynamic numerical simulations, HR and HR are ponentwerebothD,andinspecies2,itisPavlov(Nowak
1 2
calculatedeachgeneration,andhorizontalbarsaboveeach and Sigmund 1993), which initially cooperates but
graphinfigure3indicatewhentheserespectiveinequalities switchesitsbehaviorifitdidnotgetoneofthetwohighest
are satisfied in each species. In all cases, Queller’s version payoffs in the last game. The terms Q and Q give the
1 2
of HR accurately predicts when the cooperative type will fraction of the more cooperative type in each species, re-
increase. For instance, at the start of the run depicted in spectively. Initially, when both species are dominated by
figure 3A, HR (eq. [5]) is not satisfied and the TFT type ALLD types (Q pQ p0.1), Pavlov looses ground in
1 1 2
decreasesinspecies1,butbecausethefractionoftheTFT species 2 as it alternates C and D behaviors when paired
type in species 2 (Q) is simultaneously increasing, Q is with the ALLD type in species 1; it never gets one of the
2 1
eventually pulled up in this coupled system. two highest payoffs and therefore keeps switching. The
Notethatifthebenefitprovidedbycooperativebehav- TF2T type in species 1 fares better because it cooperates
iors in species 1 (b) is used in equation (5) instead of b onlyinthefirsttwogameswhenitmeetsanALLD.Even-
1 2
for comparison with c, HR does not accurately predict tually, the fact that the TF2T type in species 1 increases
1 1
the fate of TFT in this species. To work, the b term must providesmoreopportunitiesforthePavlovtypeinspecies
be the benefit received, not the benefit provided, by car- 2 to end up in a mutually cooperative interaction, and it
riers of the focal genotype (e.g., TFT). The criterion for too eventually increases.
Unifying Theories of Altruism 259
Finally, in figure 3C, the cooperative type in species 1 procity (even if nonconditional): on average, carriers of
is unconditional always cooperate (ALLC) and in species the altruistic genotype must receive directbenefits.While
2 is TFT. In this case, even having conditional behaviors this reciprocated benefit can be asymmetric, on average,
in only one species can be enough for cooperation to it must sufficiently exceed focal carriers’ costs, where the
evolve in both. Here species 1 faces a PD in each inter- meaning of “sufficiently” is captured by Queller’sversion
action, experiences no synergy for mutual cooperation of HR. Inclusive fitness is also broadenedbeyondtheno-
(d p0),andstartswithonly1%ofitspopulationbeing tion of genes helping other copies of themselvesinrecip-
1
the ALLC type (Q p0.01). We would expect this naive ients (Williams 1966; Dawkins 1976). This “selfish gene”
1
cooperator to be selected out of species 1 under random interpretation of HR holds only in the special case where
pairing,butinsteaditsteadilyincreasestosaturation.Here helping behaviors are predicted by the common alleles
species 2 starts with an even mixture of TFT and ALLD betweendonorandrecipients.Abroadernotionofinclu-
types and experiences synergy for mutual cooperation sive fitness, as Queller (1985) argued for, is fitness aug-
(d p1.3). mented by help from others regardless of their genotype.
2
Of course, many other parameter settings are possible, Therearetwoimportantandrelatedideasherethatare
including where cooperation goes extinct. Here we just reflected in the most general form of HR (eq. [4]; table
illustrate that conditional behavior with iterationsand/or 1), which can encompass not only kin selection but also
nonadditivity can allow a cooperative symbiotic relation- reciprocal altruism and symbiosis. For its most general
shiptoevolveunderrandomgroupingevenifonespecies application, first, the direct cost of behaving altruistically
is inefficient (c1b) in its help (fig. 3A), less effective in shouldbecomparedwiththedirectbenefitgaintocarriers
avoiding exploitation by ALLD (fig. 3B), or even uncon- of the altruistic genotype from others (rather than to an
ditionallycooperative(fig.3C).Ourdifferentstrategytypes indirectbenefitviatheenhancedfitnessofothercarriers),
are not meant to represent any particular symbiosis ex- and second, the direct fitness benefit depends on the be-
amples in nature but to show that the evolution of mu- haviors (phenotypes) of others, not their genotypes. The
tualismsneedsnotdependonparticularstrategiesandthat firstpointisplainlyevidentinoursymbiosisresults,which
strategiescanvarybetweenmutualisticpartners.Notethat show that, in the generalcasewherebenefitprovideddif-
in this simple model with its obligatoryheterospecificin- fers from benefit received, only an interpretation of HR
teractions,thefatesofthecooperativetypesineachspecies basedondirectbenefitreceivedbycarriersgivesacorrect
are ultimately tied together, and the system reaches an result. Rather than employing multiple interpretations of
equilibriumofeitherallcooperation(Q pQ p1.0)or HR—one for relatives, one for nonrelatives having com-
1 2
all defection (Q pQ p0.0). At each generation along mon alleles, and one for different alleles (including in
1 2
the way, Queller’s version of HR accurately predicts the different species)—it is more parsimonious for a theory
direction of selection for the (conditionally) cooperative of altruism to be based simply on the most general in-
type in each species. terpretation of HR. From this perspective, the equation
(1) version (for relatives) of HR is just a special case of
the equation (2) version, which does not depend on re-
Discussion
latedness by descent, and the genotype of others (G ) in
O
A recent review article (Sachs et al. 2004), with the same the equation (2) version is just a stand-in for the phe-
titleasAxelrodandHamilton’s(1981)seminalarticleand notypeofothers(P )inthemostgeneralformoftherule,
O
Axelrod’s (1984) book, echoes the traditional view that that is, Queller’s equations (3) and (4). It is also more
inclusivefitnessandreciprocalaltruismarefundamentally parsimonious to see HR as measuring whetherthefitness
distinct explanations for the evolution of altruism. For gains to carriers are sufficiently greater than their costs
instance, these authors state that these theories differ be- rather than in terms of indirect fitness.
cause reciprocal altruism can operate “between nonrela- Note that explaining how, on average, benefits to car-
tives and between species” (Sachs et al. 2004, p. 139) and riersofthealtruisticgenotypeendupexceedingtheircosts
that inclusive fitness is unique because “the cooperative doesnotaffectthedefinitionofaltruismattheindividual
individualneednotbenefitfromitsact”(Sachsetal.2004, level (Kerr et al. 2004). The conventional perspective in
p. 143). which an individual altruist incurs cost and gives benefit
Inthisarticle,wedemonstratethat,onthecontrary,the remainsessentialtodefiningaltruism.Giventhatindivid-
distinction between inclusive fitness and reciprocal altru- uals have no guarantees abouttheirpartner’spresentand
ismisnotsharp.WeshowthatHamilton’sinclusivefitness future behaviors, cooperation (C) in any given PD inter-
rule (in Queller’s generalized form) applies to reciprocal action (game) is altruistic because, comparedwiththeal-
altruism. Analysis of Hamilton’s rule also reveals thatthe ternativebehavior(D),acooperatorgivesbenefittoothers
evolution of altruism by inclusive fitness involves reci- ata cost toitself.Evensummingoveriterations,TFTcan
260 The American Naturalist
beseenasaltruisticfromarelativefitnessperspective(Wil- At the same time, cooperative interaction may create
son 2004) in that TFT never does better than its paired new opportunities for defection (Michod and Nedelcu
opponent (Rapoport 1991; Sober and Wilson 1998) and 2003), for example, the free-riding hunter or swimming
its opponent does better than if it had been paired with spermthatexpendslessenergythanaveragebutstillreaps
ALLD. The fact that pairs of TFT do better than pairs of the benefit of others’ cooperation. The relationship be-
ALLD for particular parameters helps explain how altru- tween synergy and exploitation by defectors is difficultto
ism evolves overall, and this is captured in Queller’s ver- appreciate in the paired interactions modeled here. This
sion of HR and in the multilevel selection framework is because when there is one C behavior, there is no syn-
where groups are of size 2 (Sober and Wilson 1998). ergy, and when there are two C behaviors, there are no
Wehaveintentionallyusedsimplemodelsofreciprocal defectorstodotheexploiting.Moregenerally,wheresyn-
altruisminordertoillustrateourpointaboutunification, ergistic benefits are an increasing function of thepropor-
but Queller’s version of HR can be applied to a much tion of cooperators in a group (and benefits are shared
broader array of circumstances. This includesvaluesofb, among all group members), there are necessarily more
c, and d that fall outside the definition of a PD, group cooperators insituationswiththehighestsynergisticpay-
sizes greater than two, diploid genetics, other population offs,whiledefectorsareatarelativeadvantagewithineach
structures besides binomial random grouping, degrees of group because they do not pay the cost.
cooperation rather than just all C or D, and other forms Synergy may also be important in addressing recent
of conditional behavior beyond those based on just the argumentsthatreciprocalaltruismrarelyoccursinnature
past behavior of others. (Hammerstein2003).Someoftheseassessmentsarebased
on the low frequency of repeated interactions, but as we
have shown, fewer iterations are required in the presence
The Role of Synergy
of nonadditive synergy. Elsewhere, we showed that mul-
Models of the evolution of altruism typicallyassumevar- tiple generations within groups similarly result in non-
iouscombinationsofrandominteractions,additivefitness additivity(FletcherandZwick2004).Thoughnotexplored
functions, and a one-to-one correspondence between ge- here, nonadditivity may also be negative, as inthetypical
notypeandphenotypefrequencies.Droppingoneormore PD(table2)orothercasesofdiminishingreturns(Foster
of these assumptions can make the evolution of cooper- 2004; Hauert et al. 2006).
ationmorelikely.Traditionalinclusivefitnessmodelsfocus
on nonrandom interactions due to kinship while leaving
A General Theory with Many Specific Mechanisms
the other assumptions in place. Traditional reciprocal al-
truism models assume random encounters but use con- In a review on the evolution of mutualism, Herre et al.
ditional behavior to break the correspondence between (1999, p. 52) lament that “there is no general theory of
genotype and phenotype. mutualism that approaches the explanatory power that
What has been less explored but is explicitlyaddressed ‘Hamilton’s rule’ appears to hold for the understanding
by Queller’s version of HR (eq. [4]) is the role of non- of within-species interactions.” In this article, we have
additive synergy, something quite different from the po- shownthatinfactHRitself,inQueller’sgeneralizedform,
tentially additive benefit of mutual cooperation (see provides a general theoretical basis for understandingthe
Hauert et al. 2006). Its significance will of course depend evolution of cooperation across species. Fundamentally,
on fitness consequences in particular interactions. In the the evolution of altruism (within or between species) de-
accompanying commentary on Queller’s (1985) original pends on sufficient positive assortment between individ-
article, Grafen (1985, p. 311) argued that “for genes of ualswiththealtruisticgenotypeofinterestandthehelping
small effect, additivity is restored and the correctness of behaviors(i.e.,phenotypes)ofothersand/orsufficientsyn-
Hamilton’s rule is restored with it.” While HR in its ad- ergistic effects of mutual cooperation (eq. [4]).
ditive form may be a good approximation when fitness Sufficient association between cooperators and coop-
effects are small, cooperative traits may have strong syn- eration from others or synergistic effects can be created
ergisticeffects.Fromcooperativehuntersthatbringhome inavarietyofways.Theseincludeinteractionsinspatially
spoilsgreaterthantheycouldgetalone(PackerandRuttan structured populations among kin (Hamilton 1964) or
1988), to cooperatively swimming sperm that reach the across species (Doebeli andKnowlton1998),iteratedand
egg faster than individual swimmers (Moore et al. 2002), conditional behavior based on the pastbehaviors(Trivers
to the potential synergistic benefits of mutualisms(Herre 1971;AxelrodandHamilton1981;Axelrod1984;Dugatkin
etal.1999;Bronstein2001;Ferriereetal.2001),thenatural 1997) or reputations (Nowak and Sigmund 1998, 2005;
worldisfullofpotentiallysuperadditivesituations(Wright Panchanathan and Boyd 2003) of others, policing (Frank
2000; Michod and Nedelcu 2003). 1995, 2003), punishmentofnonaltruists(BoydandRich-