Proc. Natl. Acad. Sci. USA Vol. 92, pp. 3596-3600, April 1995 Evolution
Types of evolutionary stability and the problem of cooperation (evolution of cooperation/evolutionary game theory/evolutionarily stable strategies/iterated prisoner's dilemma/tit for tat)
JONATHAN BENDORt AND PIOTR SWISTAKt tGraduate School of Business, Stanford University, Stanford, CA 94305-5015; and *Department of Government and Politics, University of Maryland, College Park, MD 20742-8221 Communicated by Robert Axelrod, University of Michigan, Ann Arbor, MI, November 14, 1994
ABSTRACT The evolutionary stability ofcooperation is a choices, to cooperate or defect; the same payoff matrix de- problem of fundamental importance for the biological and scribes the outcomes of these choices in all periods. The players social sciences. Different claims have been made about this observe each others' choices. A pure strategy in a repeated issue: whereas Axelrod and Hamilton's [Axelrod, R. & Ham- game between two players specifies an action for one player in ilton, W. (1981) Science 211, 1390-1398] widely recognized a current period as a function of the two players' past sequence conclusion is that cooperative rules such as "tit for tat" are of moves. A mixed strategy is a probability distribution on a set evolutionarily stable strategies in the iterated prisoner's of pure strategies. In dynamical analyses, it is assumed that dilemma (IPD), Boyd and Lorberbaum [Boyd, R. & Lorber- strategies that obtain higher payoffs reproduce at a higher rate. baum, J. (1987) Nature (London) 327, 58-59] have claimed (See ref. 7 for a discussion of the standard model and of its that no pure strategy is evolutionarily stable in this game. modifications.) Here we explain why these claims are not contradictory by In the one-shot PD when one player cooperates while the showing in what sense strategies in the IPD can and cannot be other defects, denote the payoff to the defector as T and the stable and by creating a conceptual framework that yields the cooperator as S; their payoff is R when both cooperate and P type of evolutionary stability attainable in the IPD and in when both defect. The PD structure of the game is given by the repeated games in general. Having established the relevant assumption that T > R > P > S. In the repeated version we concept of stability, we report theorems on some basic prop- further assume that 2R > T + S. The essence of the dilemma erties of strategies that are stable in this sense. We first show in a one-shot PD is that defection yields a higher payoff that the IPD has "too many" such strategies, so that being regardless of the opponent's action (T > R and P > S), yet stable does not discriminate among behavioral rules. Stable mutual defection is less rewarding than mutual cooperation (P strategies differ, however, on a property that is crucial for < R). Thus, in a one-shot PD, defection is the unique their evolutionary survival-the size of the invasion they can equilibrium strategy. Likewise perpetual defection is the resist. This property can be interpreted as a strategy's evo- unique best strategy when the game is iterated but the future lutionary robustness. Conditionally cooperative strategies does not matter "too much"; i.e., when w, the probability of such as tit for tat are the most robust. Cooperative behavior playing one more round, is low. The emergence and stability supported by these strategies is the most robust evolutionary of cooperation become meaningful issues only when w is equilibrium: the easiest to attain, and the hardest to disrupt. sufficiently large. Evolutionary Stability as Defined by Payoff Conditions: In the last decade, one of the most famous theoretical prob- Conceptual Issues lems of the biological and social sciences-how humans and other animals can cooperate despite temptations to defect- In games where w is sufficiently large, the conventional wisdom has attracted much interest. The issue has been approached via had been that conditionally cooperative strategies such as TFIl a new framework, evolutionary game theory, which has pro- are evolutionarily stable. This point was first made by Axelrod vided one of the most interesting ways of thinking about this and Hamilton (4) in their celebrated article in Science. The problem. However, different claims have been made about the claim-a deductive one-was backed by a proof showing that evolutionary stability of cooperation, and these solutions seem for all strategies j, V(TF[, TFT) 2 V(j,TFT), where V(i, j) to contradict each other. Several well-known though appar- denotes strategy i's payoffwhen it plays strategyj. (We use this ently incongruent results have been reported in many re- framing of the result, since this payoff condition was later nowned books (1-3) and journals (4-8). Clearly, those who adopted by Axelrod to define collective stability; see Eq. 1.) claim tit for tat (TFT) to be evolutionarily stable and those who This, however, does not imply that TFT meets Maynard claim it is not cannot both be right-if,.of course, they are using Smith's condition (see Eq. 2) for an evolutionarily stable the same concept of stability. Our first objective, then, is to strategy. clarify these seemingly contradictory claims by showing in what Indeed, Axelrod later (2, 6) distinguished collective stability sense strategies are and are not stable in the iterated prisoner's from evolutionary stability, defining strategy i as collectively dilemma (IPD) and other repeated games. Our second objec- stable if for all strategies j, tive is to report theorems, within the clarified conceptual framework, that establish some basic properties of strategies V(i, i) V(j, i). [1] which are stable in the IPD. The problem of cooperation among unrelated animals is On the other hand, by Maynard Smith's condition, strategy i is typically (1, 2, 4-6, 9-11) modeled as a population of indi- evolutionarily stable if, for all strategies j 0 i, viduals engaged in random pairwise prisoner's dilemmas V(i, i) 2 i), and if V(i, i) = V(j, i), (PDs). The players have a constant probability, w, of meeting V(j, in the next period (0 < w < 1). In each period there are two then V(i,j) > V(j, j). [2]
The publication costs of this article were defrayed in part by page charge Abbreviations: PD, prisoner's dilemma; IPD, iterated PD; TFT, tit for payment. This article must therefore be hereby marked "advertisement" in tat; TF2T, tit for two tats; STFT, suspicious TFT; PFR, proportional accordance with 18 U.S.C. §1734 solely to indicate this fact. fitness rule. 3596 Downloaded by guest on October 1, 2021 Evolution: Bendor and Swistak Proc. Natl. Acad. Sci. USA 92 (1995) 3597
In fact, while Axelrod turned to distinguishing collective from three strategies-and a sufficiently high w, of course-TF2T evolutionary stability, others kept referring to TFT as evolu- has the highest fitness. tionarily stable in the standard sense of condition 2. Perhaps Yet what Boyd and Lorberbaum (5) have shown is not the most conspicuous example here is John Maynard Smith. In directly related to condition 1, 2, or 3. What they have proved his seminal work (1), Maynard Smith reported the Axelrod- is that any pure strategy, no matter how common in the Hamilton finding about TFT (ref. 1, pp. 167-168) and, having population, can be outscored by some strategy in an invading reproduced their proof, concluded that it shows that TFT is an group of mutants. (The assumption of heterogeneous mutants evolutionarily stable strategy. The idea that cooperation can be is essential for Boyd and Lorberbaum's result. If there is only made evolutionarily stable via reciprocity has been further one type of mutant strategy in the population, then TFT is disseminated by the exceptionally influential books of Axelrod indeed unbeatable for sufficiently large w.) Hence Boyd and (2) and Dawkins (3). Lorberbaum refer to an evolutionarily stable strategy as one Meanwhile, however, researchers started pointing out some which, once sufficiently frequent, is unbeatable. In general, problems. One problem related to condition 2, for instance, is however, satisfying Maynard Smith's condition 2 does not that neither TFT nor any other pure strategy in the IPD can imply unbeatability. This difference between Boyd and Lor- satisfy it. Selten and Hammerstein (12) pointed this out about berbaum's notion of stability and that implied by condition 2 TFT: if a neutral mutant strategy, such as tit for two tats becomes clear when we note that a strategy which is stable in (TF2T), invades a population of TFTs, all strategies will Boyd and Lorberbaum's sense-one that, when sufficiently cooperate with all; hence, all will score the same in all games. common, is maximally fit-can be equivalently described by This, of course, violates condition 2, since this condition's the following condition 4. (The easy proof is left to the reader.) second part requires strict inequality. And this problem is not We will say that strategy i meets condition 4 if for all strategies created by some arcane construction: every nice strategy-one J, that never defects first-is a neutral mutant of TFT. While Maynard Smith's condition 2 cannot be met by any V(i, i) 2 V(j, i), and if V(i, i) = V(j, i), pure strategy in the IPD, the following weaker version of condition 2 can. We will say that i meets condition 3 if for all then for allj*, V(i,j*) 2 V(j,j*). [4] strategies j, It is now easy to see how conditions 1-4 relate to one V(i, i) 2 V(j, i), and if V(i, i) = V(j, i), another. It is, for instance, obvious that 4 implies 3 and 3 implies 1, so that 1, 3, and 4 form a sequence of increasingly then V(i,j) 2 V(j,j). [3] stringent demands imposed on strategy i. Much less obvious are the dynamic consequences of 1-4, yet understanding these The proofs ofAxelrod and Hamilton (4) and of Maynard Smith consequences is necessary if we are to put a label of "stability" (1) imply, in fact, that TFT satisfies condition 3. Here, how- on any one of these conditions. ever, we should note that condition 3 is just a condition on Stability, naturally, refers to behavior in a dynamic system, payoffs; what type of stability, if any, is implied by condition and conditions 1-4 are useful for evolutionary analysis only as 3 is yet to be shown. We will address this problem later. For the long as they imply the dynamic stability of strategies that satisfy current argument, we focus solely on the logical interdepen- them. The only reason for analyzing stability via static payoff dence of conditions 1-3. conditions is pragmatic: because identifying stable points of a Given the widely recognized claim about TFT's evolutionary dynamic system is usually very difficult, it is important to have stability-be it in sense 1, 2, or 3-it was quite surprising when simple static conditions which imply desired dynamic proper- Boyd and Lorberbaum (5) stated in the title of their Nature ties. Ultimately, it is what conditions 1-4 tell us about a paper that "No pure strategy is evolutionarily stable in the strategy's dynamic behavior that will provide the key to repeated Prisoner's Dilemma game." Their result has attracted resolving the conceptual confusions. much attention and has been widely cited. In an important review article in Science, Axelrod and Dion (7) recognized the Dynamics: Assumptions and Notation Boyd-Lorberbaum result as a major theoretical development. Others have even based some radical conclusions on it. Farrell Before beginning our examination of dynamics, we must first and Ware (10), for instance, having generalized Boyd and comment on the set of assumptions underlying our analysis. Lorberbaum's nonexistence result to finitely mixed strategies, Due to the proliferation of results in this field, it is easy to went on to infer that "evolutionary stability is too strong or too become confused about which sets of assumptions lead to demanding as a solution concept" (ref. 10, p. 166).§ which conclusions about evolutionary stability. We therefore Boyd and Lorberbaum's result is best explained by their wish to emphasize that our assumptions are the same as those original example. In this example the ecology consists of 1 - of the works (1-6) involved in the debate. [Clearly, different s TFTs and E mutants-TF2T and suspicious TFT (STFI), assumptions can generate different results (7).] dividing an arbitrarily small s in any proportion. (STFT is The core idea of an evolutionary game is that a strategy's defined as a defect in period 1 with subsequent mimicking of payoff, summed across all pairwise games in an ecology, the partner's previous move. TF2T is defined as cooperate in represents the strategy's fitness in the subsequent reproduction periods 1 and 2, and thereafter defect in period k > 2 if and of behaviors. A strategy with a higher fitness has a higher only if the opponent defected in periods k-2 and k- 1.) TF2T reproduction rate. We will call any replication dynamic that outscores TFT (for sufficiently high w) since it cooperates satisfies this condition an evolutionary process. Under standard indefinitely with both partners, whereas TFT's immediate simplifying assumptions (11) the process involves a sequence provocability creates a vendetta with STFl. In this construc- of nonoverlapping generations, with the population's total size tion TF2T is a neutral mutant of TFIl and STFT is chosen to remaining fixed. inflate TF2T's fitness versus TFT's. With a population of these We further assume that populations are composed of or- ganisms that play a finite number of strategies. In a population which consists of strategies ...,,N played with relative §Apparently a proof that no such pure or mixed strategies exist was jl, obtained independently by Pudaite (13), as noted by Axelrod and frequenciesp1, . .. , PN, strategy k's fitness, V(jk), equals (for Dion (ref. 7, p. 1388). An extension of ref. 10 beyond finitely mixed appropriately normalized payoffs) PlV(ik, Il) + * + PNV(Jk, strategies was recently reported by Lorberbaum (14). jN), where k = 1, ..., N. Downloaded by guest on October 1, 2021 3598 Evolution: Bendor and Swistak Proc. NatL Acad Sci. USA 92 (1995) Dynamic Properties of Evolutionarily Stable Strategies: x y z Two Types of Stability
A fundamental task of evolutionary analysis is to identify x 1 strategies that, when used by everyone in a population, can 1,1 1, 0,0 resist a small invasion of any mutant strategies. These so-called evolutionarily stable strategies constitute the notion of an evolutionary equilibrium. More precisely, i is evolutionarily y 1,1 0,0 1,0 stable if there exists an so > 0 such that for all e < so the population playing the native strategy i can resist any invasion of mutants. This essential idea of an evolutionarily stable strategy becomes a definition after we specify the other z 0,0 0,1 0,0 concepts involved in the formulation. "Resist an invasion" may be understood in two intuitively FIG. 1. Payoff matrix of a one-shot game where the pure strategy acceptable ways: stronger, if the invaders decline in frequency x is the unique strategy that satisfies the Maynard Smith condition 2. under the evolutionary process, and weaker, if they do not Yet given a heterogeneous invasion, x may not be maximally fit, no increase. These correspond to two types of stability which, for matter how small the invasion. Consider an s invasion with£y ofy and the purpose of this paper, we label as strong and weak Ez of z (s = Ey + Ez). Note that V(x) = (1 - Ey - sE)-V(x, x) + syV(x, evolutionary stability. (These two types of stability are related y) +sEZV(x, z) = 1 -Ez; similarly, V(y) = 1 - sy and V(z) = 0. Clearly, to what are called asymptotic stability and stability in standard x's fitness is less than y's whenever sy < sz, no matter how smalle is. dynamic analyses.) But because x fulfills condition 2, from Fact 1, part iii, we know that Strong stability is clearly a stronger condition. If natives play its fitness exceeds the population's average fitness (for sufficiently V = - - + + Ez * a strongly stable strategy, then after an invasion occurs the small E): V(x) > (1 Ey sz)-V(x) y-V(Y) V(z). evolutionary dynamic will restore the population's original by a mutant in a heterogeneous invasion of arbitrarily small stable state-the mutants will die off. Weak stability does not size. What does follow from x's fulfilling Maynard Smith's imply the extinction of mutant strategies. It requires only that condition 2 is that, for sufficiently small s, x's fitness exceeds the native, when invaded, does not decrease in frequency. Thus the average fitness in the population. under weak stability certain mutants may have the same fitness While "beating the average" does- not imply stability under as the natives and hence may drift in the population indefi- all evolutionary processes, it is sufficient to imply stability nitely. under the proportional fitness rule (PFR), a standard speci- Dynamic Consequences of Conditions 2 and 4 fication in biological models (3, 8). This rule says that if strategy i has proportion p and fitness V(i) in generation t, Some of the conceptual confusions surrounding the issue of where the average fitness is V, then i's proportion in the next evolutionary stability might have their roots in the somewhat generation equals p[V(i)/V]. Thus, strategies with above av- convoluted history of the meaning and implications of condi- erage fitness grow, and those with below average fitness tion 2. A brief analytical account of this history, captured in the decline. Clearly, to increase under the PFR a strategy need not following fact, should help clarify matters. be unbeatable. In the example of Fig. 1, the fact thatx exceeds Fact1. The following three conditions (iiii) are equivalent. the population's average fitness suffices forx to increase under (i) (Maynard Smith's definitional condition) Strategy i satis- the PFR even though x will not enjoy maximal fitness as long < fies condition 2 for all strategiesj # i. asEy sz. The dynamic under the PFR can be easily imagined. In all ecologies where i has a sufficiently large frequency; Take the initial population composition as assumed above. (ii) (homogeneous invaders) if all invaders play the same Both x andy will have above-average fitness as long as 6y
one-shot games with multiple stages. Thus, condition 2 can be A B satisfied only by strategies in one-shot games with a single stage. Clearly, given the ubiquity of neutral mutants, the problem A is that once any pure native is invaded by a neutral mutant, the 1,1 1,1 mutant strategy will not die out-it will drift in the population with a fitness equal to the native's. Hence, the best the native strategy can do is to prevent the mutant from spreading. This B is not a mere technicality: that both condition 2 and strong 1,1 2,2 stability in general are unattainable in repeated interactions describes an important phenomenon. Take, for example, a FIG. 2. In this one-shot game, pure strategy A satisfies the collec- population of TFTs. Since everyone in this ecology always tive stability condition 1 yet is fundamentally unstable. SinceA obtains cooperates, TFT's potential for retaliation is not exercised. As V(A,A) = V(j,A), for all strategiesj,A is collectively stable. Take now an unused capacity may decay, some TlFTs may change to a less a population of 1 - s As and s > 0 pure strategy Bs. Strategy A's provocable strategy, such as TF2T. Because this "invasion" expected fitness, V(A) = 1, is smaller then B's, V(B) = (1 - e) + (2s) produces no new behavior, it is unobservable, so that TFT = 1 + s. Thus, V(A) < V(B) for all E * 0. Hence A is fundamentally cannot stop TF2T from drifting into the population. Hence all unstable. we can hope for in populations of strategies in repeated games is weak stability. In this sense weak stability is the only Suppose 1 - s of the population play actionA; the rest play B. A attainable form of evolutionary stability in repeated games. simple calculation (see Fig. 2) shows that while A satisfies < Similarly, Boyd and Lorberbaum's result on the unattain- condition 1, and hence is collectively stable, V(A) V(B) for ability of condition 4 in the IPD also generalizes enormously: all values of s. Thus B will take over the entire population no remaining unbeaten in the face of all heterogeneous invasions matter how few Bs had initially invaded. Therefore strategyA, is infeasible in any repeated game of interest, as our first though it satisfies condition 1, is fundamentally unstable. theorem shows.11 Hence we are left with condition 3. The first point to observe THEOREM 1. For sufficiently high w, no pure strategy is is that condition 3 does not suffer from the ontological flaw of unbeatable (i.e., satisfies condition 4) in any repeated nontrivial conditions 2 and 4: strategies satisfying condition 3 do exist in game. the IPD. TFT, for example, satisfies condition 3, as is implied We define a trivial one-shot game as a symmetric game in by Axelrod and Hamilton's proof. which each player has a strategy that yields the game's maximal The next question is, what kind of stability does condition 3 payoff regardless of the opponent's play. Lacking any strategic imply? The following two theorems provide the answer. interdependence, such games are truly trivial. We have proven THEOREM 2. A strategy is weakly stable under the PFR ifand Theorem 1 elsewhere (18). Thus Boyd and Lorberbaum's result only if it satisfies condition 3. does not depend in any way on the specifics of the IPD or on THEOREM 3. If a strategy does not satisfy condition 3, then it the substantive problem of cooperation; the impossibility of is fundamentally unstable. being unbeatable turns out to be a generic property of repeated Both theorems are proved in ref. 18. Two points are worth games. noting about these results. First, Theorem 2 tells us that Given the above, the issue in the context of repeated games condition 3 is equivalent to weak stability under the PFR, is not what type of stability should be analytically adopted (e.g., which is a much "cleaner" relation than that between condition strong or weak), but what kind of stability is feasible. 2 and its implied forms of stability (Fact 1). Second, condition 3 is the minimal condition that a strategy i must meet for it to Dynamic Implications of Conditions 1 and 3 be stable in any acceptable sense: if i violates this condition it is fundamentally unstable. Hence Theorems 2 and 3 uncover
Consider the following situation. Assume that a 1 - s fraction two serious theoretical reasons for introducing condition 3-a of a population play i and an e > 0 fraction play a mutantj such condition that so far may have seemed to be only an "onto- that for all values of a,j is more fit than i: V(j) > V(i). In this logical correction" to condition 2. case i's frequency not only will decrease right after the Theorem 2 and the fact that TFT satisfies condition 3 invasion, it will continue decreasing across generations until i together imply that TFT is weakly stable under the PFR. It is vanishes entirely, regardless of which evolutionary process this type of stability-weak stability under the PFR-that we governs replication. Certainly, a strategy like i is unstable in a examine more closely in our last section. very extreme sense. We will call such a strategyfundamentally unstable. This notion will prove useful in uncovering the Results on Stability in the IPD problems of collective stability (condition 1) and the virtues of condition 3. In contrast to the nonexistence of strategies satisfying condi- The drawback of the concept of collective stability is that it tions 2 and 4, the next result shows that the difficulty con- does not imply stability in a dynamic sense (5, 19, 20). The fronting condition 3 is just the opposite: the IPD has a one-shot game of Fig. 2 reveals the essence of the problem. profusion of strategies which are weakly stable under the PFR. And while TFT is one of these, if w is sufficiently big, so are less as next "Because heterogeneous invasions are essential in the proofs of Boyd many cooperative strategies, the result indicates. and Lorberbaum and of Theorem 1, a comment on the plausibility of Theorems 4, 5, and 6-proven in ref. 21-are derived under such perturbations is warranted. Observe that if a population of, for the most general assumption of heterogeneous invasions that example, TFTs is invaded by a neutral mutant, the mutant will persist may include either deterministic (pure) or probabilistic indefinitely; thus whenever a new invasion occurs the ecology will (mixed) strategies.tt then be composed of the native TFT and two mutant strategies. Such evolutionary games must be modeled, of course, under the assump- tion of heterogeneous mutations. Hence, in analyzing repeated games ttAs suggested by an anonymous reviewer, all of these results can be the homogeneity assumption is not only empirically unrealistic (8, 9), rewritten by using condition 3* instead of 3, where a strategy i it is theoretically inadequate as well. And, as with the possibility of satisfies 3* if, for allj,j is either a neutral mutant of i or satisfies 2. neutral mutants arising, the possibility of such heterogeneous inva- While 3* has some stronger implications, we have decided to use 3 sions is no mere technicality; it reflects how hard it is for native because of its other theoretically important properties, as given in strategies to adapt to all possible heterogeneous invasions. Theorems 2 and 3. Downloaded by guest on October 1, 2021 3600 Evolution: Bendor and Swistak Proc. Natl. Acad ScL USA 92 (1995)
THEOREM 4: THE PROFUSION OF EVOLUTIONARILY STABLE is maximal. The growth of TFT and TF2T will be offset by the STRATEGIES. For sufficiently high w, any degree of cooperation, decline of STFT. In the new equilibrium, STFT will be extinct; from 0% to 100%lo, can be supported by a strategy that is weakly the only strategies remaining in the ecology will be TFT and stable under the PFR. TF2T. Hence, behaviorally, the preinvasion state of universal This result reveals a problem: being weakly stable under the cooperation will be restored. PFR is not a distinctive property in the IPD. Fortunately, these This observation holds in general. The proof of Theorem 5 strategies differ in a crucial and empirically meaningful way. actually yields a slightly stronger conclusion than that of part What differentiates them is their robustness-the minimal ii. It implies that the nice and retaliatory native will be more relative frequency a native must have to resist all possible fit than average as long as any non-nice strategies remain in the invasions. We call this a strategy's minimal stabilizingfrequency. ecology. With such strategies present, the native will increase (Technically, this delineates a strategy's basin of attraction.) (under the PFR) in the next generation. This dynamic will end The following definitions provide abbreviations in the re- only when all non-nice strategies have died out. The resulting sults below. We call a strategy nasty if it never cooperates first, equilibrium will exhibit universal cooperation. Hence as long retaliatory if it defects in period t + 1 whenever its opponent as the nice and retaliatory native is in the majority, any defected in t, and almost-nice if in an infinite game with a clone perturbation (invasion) will be behaviorally inconsequential: it cooperates in all but a finite number of periods. For brevity, when the "shadow of the future" is sufficiently long, the the theorems refer to strategies as being "evolutionarily sta- preinvasion state of universal cooperation will always be ble" instead of "weakly stable under the PFR." restored. THEOREM 5: THE ROBUSTNESS OF COOPERATION. In games with sufficiently high w; (i) no strategy has a minimal stabilizing We thank John Aldrich, Robert Axelrod, Dave Baron, Randy frequency s {).5; (ii) any nice and retaliatory strategy is a Calvert, Gordon Hines, Hugo Hopenhayn, Susanne Lohmann, Man- maximally robust evolutionarily stable strategy, and its minimal cur Olson, Ed Schwartz, Jeroen Swinkels, and Peyton Young for stabilizingfrequency converges to 0.5 as w converges to 1; (iii) all helpful comments. Suggestions of two anonymous referees improved maximally robust evolutionarily stable strategies must be almost- the paper considerably. P.S. thanks the General Research Board of nice. Graduate Studies and Research at the University of Maryland, whose THEOREM 6: THE FRAGILITY OF DEFECTION. The minimal Summer Research Award enabled him.to work on this project. stabilizing frequency of any nasty strategy converges to 1 as w converges to 1. 1. Maynard Smith, J. (1982) Evolution and the Theory of Games Theorem 5 establishes in part i that to be stable against all (Cambridge Univ. Press, Cambridge, U.K.). 2. Axelrod, R. (1984) The Evolution of Cooperation (Basic Books, possible invasions, any strategy must be in the majority. Part New York). ii states that strategies like TFI, which are nice and retaliatory, 3. Dawkins, R. (1989) The Selfish Gene (Oxford Univ. Press, are maximally robust. Once in the majority (with sufficiently Oxford). large w) such strategies will be weakly stable under the PFR. 4. Axelrod, R. & Hamilton, W. (1981) Science 211, 1390-1398. Unlike other results (3) that establish evolutionary stability 5. Boyd, R. & Lorberbaum, J. (1987) Nature (London) 327, 58-59. only for "s small" invasions, part ii shows that nice and 6. Axelrod, R. (1981) Am. Polit. Sci. Rev. 75, 306-318. retaliatory strategies have a remarkably large basin of attrac- 7. Axelrod, R. & Dion, D. (1988) Science 242, 1385-1390. tion. Thus, such strategies can withstand large invasions. This 8. May, R. (1987) Nature (London) 327, 15-17. feature is significant not only for the stability of cooperation 9. Nowak, M. & Sigmund, K. (1992) Nature (London) 355, 250-253. but also for its emergence. Part iii shows that all maximally 10. Farrell, J. & Ware, R. (1989) Theor. Pop. Biol. 36, 161-166. robust strategies are essentially fully cooperative. Hence these 11. Hines, W. (1987) Theor. Pop. Biol. 31, 195-273. 12. Selten, R. & Hammerstein, P. (1984) Behav. Brain Sci. 7, cooperative strategies are unique. They generate the most 115-116. robust state of the population: the easiest to attain, and the 13. Pudaite, P. (1985) Mimeo (Merriam Laboratory for Analytic hardest to disrupt. Less cooperative strategies require higher Political Research, Univ. of Illinois, Urbana, Champaign). frequencies in order to be weakly stable under the PFR. The 14. Lorberbaum, J. (1994) J. Theor. Biol. 168, 117-130. least cooperative strategies are maximally vulnerable to inva- 15. Maynard Smith, J. & Price, G. (1973) Nature (London) 246, sion (Theorem 6). 15-18. The implications of part ii for the evolution of cooperation 16. Zeeman, E. C. (1979) in Proceedings ofthe Conference on Global is significant; hence, a short elaboration is warranted. Suppose Theory ofDynamical Systems, Evanston: Northwestern, eds. Dold, a majority of the population in generation 1 use a nice and A. & Eckmann, B. (Springer, Berlin), pp. 471-497. retaliatory strategy, such as TFT. Everyone else can play an 17. Thomas, B. (1984) Theor. Pop. Biol. 24, 49-67. arbitrary mix of strategies; to revive Boyd and Lorberbaum's 18. Bendor, J. & Swistak, P. (1992) Research Paper (Grad. School of these are TF2T Business, Stanford Univ., Stanford, CA), No. 1183. example, say and STFT. Theorem 5, part ii, 19. Sugden, R. (1988) in Compromise, Negotation and Group Deci- ensures that the nice and retaliatory native will not decline sion, eds. Munier, B. & Shakun, M. (Reidel, Dordrecht, The across generations. In fact, in Boyd and Lorberbaum's exam- Netherlands). ple, TFT, though not maximally fit, will be more fit than 20. Swistak, P. (1989) Behav. Sci. 34, 151-153. average; hence TFT will grow under the PFR. Naturally, TF2T 21. Bendor, J. & Swistak, P. (1992) Research Paper (Grad. School of will also increase, and more quickly than TFT since its fitness Business, Stanford Univ., Stanford, CA), No. 1182. Downloaded by guest on October 1, 2021