Evolutionary stability of reciprocity in a viscous lattice

Social Networks North-Holland

11 (1989) 175-212

175

EVOLUTIONARY STABILITY IN A VISCOUS LATTICE * Gregory Northwestern

B. POLLOCK

OF RECIPROCITY

**

Uniuersrty

In randomly mixing populations, reciprocity cannot resist invasion by appropriate concurrent multiple mutants (Boyd and Lorberbaum 1987). Here I show how reciprocity can resist such invasion when the clustering (“viscous”) population structure necessary for the emergence of reciprocity in a world of defection is retained after reciprocity has saturated the population. A mutation heuristic is introduced under which only forgiving reciprocity can resist Boyd/Lorberbaum invasion in viscous populations; this provides a selective basis for forgiveness and extends TIT FOR TAT’s collective stability to evolutionary stability under multiple mutation. The results are generalized to n-person games, where Boyd/Lorberbaum invasion is precluded among insular commons, whether or not reciprocators are forgiving. Non-insular commons are. however. invadable, providing a selection rationale for the maintenance of in/out group distinctions under n-person social ecologies.

The little things are what is eternal, and the rest, all the rest, is brevity, extreme brevity. Antonio Porchia, F’0ice.s

1. Introduction: evolutionary stability and mutation Most evolutionary stability arguments assume a single strategy, pure or mixed, which has saturated a population but faces the continual emergence of (single) rival alternatives as mutations (Maynard Smith and Price 1973; Maynard Smith and Parker 1976; Maynard Smith 1982, 1984; Brown and Vincent 1987; Hines 1987). Such an analysis is * This work was supported by a MacArthur Foundation pre-doctoral fellowship in International Peace and Security Studies granted by the Social Science Research Council and a post-doctoral training award from the Graduate Program in Mathematical Methods in Social Science at Northwestern University. Most importantly, without the personal support of Keith Lewis and Malcolm Dow it could never have been written. * * Department of Anthropology, Northwestern University, Evanston, IL 60208, U.S.A.

037%8733/89/$3.50

0 1989, Elsevier Science Publishers

B.V. (North-Holland)

176

G. B. Pollock / Stability in a viscous lattice

restricted to simple games (sensu Brown and Vincent 1987): multiple mutational types never appear simultaneously, limiting the social environment of any strategy to, at most, two types-itself and an unspecified rival. Any emergent inferior is assumed eliminated by the prevalent strategy before emergence of other mutational types, precluding synergetic effects by multiple mutations in the analysis of evolutionary stability. The inclusion of multiple, simultaneous mutations often invalidates conclusions derived from simple games (Boyd and Lorberbaum 1987; Brown and Vincent 1987) providing a more general analysis, but at a cost. While simple games permit the analysis of all mutational variants within the game’s context, the synergetic analysis of multiple mutations requires one to specify the class of invading sets of strategies. If mutations differ in their probability of occurrence, these sets will themselves emerge with differing probabilities. Indeed, differing mutation sets might simultaneously emerge if invaders do not interact randomly with all other invaders. A saturated strategy may do well with some sets but not others, and may recoup its losses from one set by playing exceptionally well against another. While analyses of simple games require a well-specified universe of invaders (Dawkins 1980) the synergetic analysis of multiple mutations must also add specified rates of mutation to differing types, as well as the assortive structure characteristic of invaders. There is thus a tension between the plausibility of multiple mutants and the auxiliary assumptions such an extended analysis entails. Recent work on evolutionary stability under the iterated Prisoner’s Dilemma (PD; Table 1) reflects this tension. Axelrod (1981, 1984) has shown the cooperative strategy TIT FOR TAT (TFT; defined as begin by cooperating and do whatever your partner did the previous game turn thereafter) to be “collectively stable” under appropriate interaction conditions: given a sufficiently high probability of future interaction with a given PD partner, no strategy does better than TFT in an environment otherwise saturated with TFT. But any strategy which begins iterated play by cooperating and does not defect before one’s partner (i.e. a “nice” strategy; Axelrod 1980b) will do as well as TFT when playing TFT. TFT is thus not evolutionarily stable in the strict sense (Maynard Smith 1982) for simple games (Axelrod 1981: 310; Maynard Smith 1984; Selten and Hammerstein 1984; Williams 1984). A population saturated with TFT can become a mix of nice strategies

G. B. Pollock / Stability in a viscous lattice Table 1 Payoffs in the 2-person Individual

Prisoner’s

B

’

Dilemma Individual

Cooperates Defects

177

A

Cooperates

Defects

R/R S/T

T/S p/p

’ Payoffs given as individual A/individual

B with the ordering T > R > P > S > 0 and 2 R > T + S. Individual A’s iterated payoff against individual B is given as Czp=,X, w’, where X, is A’s payoff in game turn I and w is the probability of repeated play (per game turn) against B (Axelrod 1981).

with no phenotypic change in the population. As Williams (1984: 117) used for notes, “if nearly every player uses TFT, the mechanisms ‘defect’ would degenerate, and TFT would be replaced by [ALL]C [i.e. unconditional cooperation]. This in turn would be invaded by [ALLID So long as a mutation is allowed to [i.e. unconditional defection].” out-compete and completely supplant its previously established rival this is true. But TFT outperforms ALL C when both play ALL D. When ALL D emerges in an environment otherwise saturated with TFT and ALL C, TFT can be evolutionarily superior. While TFT is not evolutionarily stable in simple games, in some multiple mutant contexts it clearly is. But this is a Pandora’s box of a solution. The evolutionary superiority of TFT is contingent on the universe of allowable mutations. If some mutation sets make TFT evolutionary stable, other plausible sets force TFT to extinction. Consider, for example, the mutants STFT (SUSPICIOUS TFT, which begins by defecting and duplicates the previous move of its opponent thereafter) and TF2T (TIT FOR TWO TATS, a more tolerant version of TFT, requiring two defections before punishing). Then letting V(X) Y) denote strategy X’s payoff when encountering strategy Y (Axelrod 1981), Table 1 gives V(TFT 1TFT) = V(TFT 1TF2T) = R/(1

(STFT)

1TF2T)

= V(TF2T

- w)

(as both TFT and TF2T V(TF2T

= l’(TF2T

are nice) and

= S + wR/(l = (s + wT)/(l

- w) > V(TFT - w’)

I STFT)

I TFT)

178

G. B. Pollock / S~uhrlrry m LI viscous lattice

when w > (T - R)/( R - S) < 1. Thus, for sufficiently high probability of future interaction, the mutation set { STFT, TF2T) lets TF2T invade a population saturated with TFT (Boyd and Lorberbaum 1987); and, as TF2T can always be invaded by STFT, at best the population will exhibit a balanced polymorphism of nice and first-to-defect strategies. The example generalizes: under random assortment any collectively stable strategy is invadable by a nice (but not collectively stable) strategy given an appropriate mutation set, with “the nature of the selective forces that shape ongoing, potentially altruistic social interactions depend[ent] on the distribution of rare variants that are created by mutation, environmental variation and other processes that maintain phenotypic variation” (Boyd and Lorberbaum 1987: 59). A theory of cooperation then requires a “physiological” theory of variation. If such a theory permits simple deviations in the strategic properties of niceness and provokability (punishment; Axelrod 1980b and below), niceness is unlikely to saturate any population, with cooperation everywhere advantageous. Here I present an approach to the analysis of evolutionary stability in the iterated PD in non-simple environments where mutants are plausible variants (defined below) of pre-existing strategies, including such Boyd and Lorberbaum mutants as STFT and TF2T, yet a balanced polymorphism including first-to-defect strategies does not exist. Retaining the clustered population structure necessary for the emergence of reciprocity in a world of defection (Axelrod 1981, 1984; Axelrod and Hamilton 1981) permits evolutionary stable reciprocity in non-simple games. Low dispersal inducing spatial clustering can isolate mutational variants across evolutionary time, transforming a population-wide non-simple environment into a mosaic of simple games where strategies compete for local rather than population-wide superiority. Assuming collectively stable reciprocity, while reciprocity can initially be invaded given appropriate synergistic mutants, as a locally successful mutant clusters it is itself vulnerable to invasion in a way reciprocity is not. For, under the set of plausible mutants, of those clusters with non-decreasing perimeters, only reciprocity is immune to single-mutant internal invaders. Here the evolutionary stability of reciprocity consists of invading clusters which, after successful emergence, are destroyed from within faster than they can grow. When reciprocity is forgiving (i.e. returns to cooperation after receiving a cooperative apology from a punished defector; Axelrod 1980b), these invaders of invaders are then

G. B. Pollock / Stability in a viscous lattice

179

eliminated by the greater reciprocity-saturated environment; forgiveness thus becomes essential to the maintenance of reciprocity in non-simple, two-person games. This dynamic suggests the clustered population structure necessary for the emergence of reciprocity is also necessary for its maintenance. I begin by characterizing the selective forces behind evolutionary stable reciprocity in simple games as a function of niceness and provokability (section 2). I then show how these forces permit emergence of reciprocity in a viscous linear integer lattice (section 3) and how forgiving reciprocity is non-invadable by multiple mutants under the lattice (section 4). Lastly, I extend the argument to the square lattice (section 5) and consider the stability of reciprocity under the social ecology of the n-person Dilemma (section 6). Unlike the two-person game, when a population consists of mutually exclusive n-person evolutionary stability against groups collective stability implies Boyd/Lorberbaum invasion. When groups are non-insular, however, the general inapplicability of forgiveness in n-person ecologies precludes stability under low dispersal. 2. The selective basis of nice, provokable strategies Consider a population of dyads, where each dyad represents two individuals engaged in an iterated PD. Let all individuals exhibit a nice strategy, although not necessarily the same strategy. Suppose someone switches to ALL D. By definition, this individual will do better, in relative fitness, than his opponent at first defection. Since the defector continues to defect thereafter, he will continue to do better no matter what his opponent does. When exposed to an unconditional defector, a nice player necessarily places himself at a disadvantage; his initial cooperation is, within the dyad, altruistic (Wilson 1983; Michod and Sanderson 1985). If the nice player is also maximally provokable (i.e. always punishes defection with defection; Axelrod 1980b, 1981) the absolute fitness (accumulated payoff) of the defector may also be reduced, relative to what it would be if both were nice, but this does not change the former’s relative standing in the population. Any trait uniquely assigned to a nice individual exposed to ALL D must decline in relative frequency. Neither niceness nor provokability provides an individual advantage to a strategist.

180

G. B. Pollock

/ Stuhi1it.v rn (1 c’ucous lattice

Niceness and provokability persist when dyads with cooperators sufficiently outproduce those in internal conflict so as to provide a global disadvantage to defection in the population (Wilson 1983; Pollock 1988, 1989). A nice individual punishing a defector is always selected against, yet, without punishment, a defector enjoys a population-wide advantage over cooperators (Axelrod 1981); but puked nice, provokable individuals indirectly receive a benefit from this punishment through the depression of the defector’s fitness. Provokability and niceness persist because an individual sacrifices itself to enhance the relative standing in the population of paired duplicates of its strategy (Pollock 1988). At emergence this entails that a nice, provokable strategy appear in a cluster (Axelrod 1981, 1984; Axelrod and Hamilton 1981) to ensure paired duplicates are sufficiently frequent to make such self-sacrifice worthwhile; as the strategy increases in frequency, random assortment alone creates an adequate number of such dyads (Axelrod 1981; Peck and Feldman 1986). Since, by definition, random assortment entails arbitrary encounter while clustering is eroded by such encounter, the former seems both more robust and parsimonious. Indeed, within simple games, any PD strategy which is non-invadable when clustered (defined as being embedded in a viscous lattice; section 3 below) is also non-invadable under random assortment (Axelrod 1984: 158-168). Yet abandoning clustering as reciprocators increase in frequency alters population-wide dispersal and hence the (usually unidentified) selective forces originally responsible for clustering. Random assortment is not parsimonious when derived from a necessary history of clustering. And, in non-simple environments, clustering will be shown as necessary for the maintenance as for the emergence of two-person reciprocity.

3. Emergence of reciprocity in the viscous linear integer lattice Consider

the linear integer lattice

were X, denotes an individual with strategy X at position i, with neighbors engaged in iterated PDs as in Table 1. Individual fitness either cultural clones or haploid genotypes) is (number of “offspring”,

181

G. B. Pollock / Stability in a viscous lattice

proportional to accumulated payoffs (Axelrod 1981; Brown et al. 1982), with offspring dispersed randomly to nearest neighbor and self locations. The population is viscous (sensu Hamilton 1964) with “siblings” naturally clustered across generations. Let the lattice be saturated with unilateral defectors (D) save for two neighboring nice, maximally provokable reciprocators ( C) :

For the C-cluster to expand the reciprocating strategy must retain its initial position and (symmetrically) invade positions i - 2, i + 1. If an offspring of the fittest individual competing at a given location succeeds in occupying that location (termed “deterministic competition” hereafter), growth entails

Jw-,>

’

W-2)

where V( X,) denotes individual Xi’s accumulated payoffs. Converting to the most successful strategy among one’s neighbors and oneself (randomly choosing among maximal ties) after each completed iterated game (a “generation”) provides such a competitive mechanism (Axelrod 1984: 158168). Then

v(c,J=

v(qc)+

V(CID)>

v(Dp)+

V(DlC)=

V(D,_,)

04 v(qc)+

v(cJD)~2v(oJD)=

V(D;_,),

which reduces to (la) as, by definition, Table 1 (la) may be written as R/(1

- W) + [s + WP/(l - W)] > P/(1

(lb)

V( D ( C) > V(D ( D).

- W) + [T+

From

WP/(l - W)]

or w > 1 - (R - P)/(T-

S).

(4

G.B. Poliock / Stability III a viscous lattice

182

If degree of dispersal is invariant to lattice position, condition invariant to offspring dispersal beyond nearest neighbors since V(D,_.)

= V(D;_,)

(2) is

for n 2 3.

Competition for strategy placement across generations is often less deterministic than this, with probability of successful placement of a strategy at a given location proportional to the relative frequency of its “clones” competing for that location (Maynard Smith 1982). Since successful reciprocity rests on contiguous reciprocators, this competitive regime induces a degree of random placement which can isolate reciprocators, leading to their elimination. With increasing dispersal the relative representation of (globally rare) reciprocator offspring decreases at a given position, enhancing the frequency of isolate reciprocators; in this sense viscosity supports but cannot guarantee the integrity of a cluster, and no population is certain to retain emerging reciprocity. Nonetheless, as the local relative frequency of reciprocator offspring rises, success approximates a deterministic process, and lattices precluding the emergence of reciprocity under deterministic competition do so under probabilistic competition as well. Under probabilistic competition, total reciprocator offspring must at least outnumber total defector offspring at positions of possible cluster expansion. In the present case this entails V(L)

> IQ,-,)

+ I0A

or w> l-

(R - 3P)/(T-

s) > 1 - (R - P)/(T-

S),

(2’)

precluding social ecologies with R < 3P. Here I focus on deterministic competition to identify necessary conditions for evolutionary stability in simple viscous lattices. Under the viscous linear lattice deterministic competition permits a single invader to expand if her relative fitness is greater than l/3 at self and adjoining positions. This condition underestimates the robustness of reciprocity under probabilistic competition, where a reciprocity-morph local relative fitness between l/2 and 2/3 would suggest elimination of the invader.

G.B. Poilock / Stability in a viscous lattice

4. Evolutionary

183

stability in the lattice

Consider a nice, collectively viscous linear integer lattice.

stable strategy A which has saturated a Then, for any single mutant B such that

-A,_~-A,_~-B;-Ai+,-Ai+~I”(B)

2 %$+I),

or 2V(BlA) S + T by definition

of the dilemma

(Table

l),

2V(AIA)2T/(AIB)+V(BIA), so V(B(A)

I V(A(A).

When B is first-to-defect 2R > T + S implies I’( B;) < V( A,+,) and V( B 1A) -c V( A ) A). These conditions will be shown sufficient to ensure that no invading B-cluster can expand without the continual introduction of mutants. I then show that if mutation is equally probable throughout the lattice, a simple mutation heuristic ensures that any invading cluster will be destroyed from within faster than it can grow. 4.1. Growth of an invading cluster under multiple mutation Proposition 1. No B-invader may expand (without mutation) A-reciprocator. (This is a special case of Axelrod (1984: proposition 7). If B is to expand it must be clustered and non-nice; collectively stability on A ensures B is non-increasing. Then of B entails

against an 214-215)‘s otherwise, expansion

V(B;) ’ I?&,) in the lattice -Aj_3-A,_2-A;_1-B;-Bi+I-AjL1_Aj+~-Aj+~-~

*. . -B,-.

*. -B;_l-Bj

04

184

G. B. Pollock / Stability in a wscous lattice

or

which is impossible since nice A implies V( A 1A) > V( B ) B) and collective stability implies V( A ( A) > V( B I A). At best, B persists in its cluster, waiting for appropriate mutation along its boarder (e.g. at positions i - 3, i - 2, i - 1) to induce expansion. But continual expansion requires continual mutation: Proposition 2. Suppose two mutants B and E appear in the lattice so as to synergistically increase the absolute frequency of, say, B. Then E is eliminated, necessitating its continual recurrence through mutation to continue expansion of B. Case 1. B and E are both socially -Az_2-A;_I-B,-A;+l-.

s V(Ai+I)

V( El) I V(A,-,)

isolated,

stable

= ‘(A,-,) = ~(A,+I)

and neither mutant increases in frequency. Expansion of one of the invaders is possible, that invader is clustered. Consider

however,

for j = i + 3 if

If both B and E are non-nice -A ,_~-Ai_~-A,_~-Bi-~-B,-Bi+~results if

i.e.

. . -A,_I_EJ_Aj+I_AJ+2->

with j 2 i + 3. Since A is collectively V(Bi)

and competitively

. . . -Bj-~-Bj_A,+~-Aj+~-A,+3-

G. B. Pollock / Stability

in a viscous lattice

185

assuming the boundary of B is stable on the right; E is, however, eliminated, as V((Ai_2) = V( AI_4) > V( Ei_3) by collective stability. Case 2. B and E are socially isolated

but competitively

connected,

i.e.

-Ai_2-Ai_l-B;-Ai+1-Ei+2-Ai+3-A,+4-. If both mutants V(Bi) ’

‘CAi+I)

WA)

%Y+*)

’

V(Ai+3)

’

are nice, no change occurs.

If B is nice but E not,

‘CEi+2)

(the latter by collective

stability)

yields the next generation

lattice

-Ai_2-Ai_I-Bi-B,+I-Ai+2-A;+3-A~+4-. Continual emergence of a non-nice mutant the boundary of A and nice B can expand If neither mutant is nice J$k,)

’ J,‘(R)

I/tAi+3)

’

one position the latter.

removed

from

V(Ei+*),

yielding -A,_~-A~-Bit~-Ai+~-A1+3[arbitrarily designating V(Bi) > V(Ei+*)] which eliminates B in the next generation. Expansion of non-nice B is possible if B is already clustered, as in -Ai_~-A;-_-B;-~-Bi-A;+~-E,+~-Ai+~-, when V(B(B)+

V(BIA)>

Y(AIB)+

V(AIE)

V(B(B)+P’(BJA)>2V(EIA) V(B,_,),

V(B;)=

P’(BlB)+ = V(Ai-2).

V(BIA)>

P’(AIA)+

V@(B)

G. B. Pohck

186

Collective stability implies elimination

/ Stability in a oiscous lattice

of A, when coupled of E.

Case 3. B and E are competitively

with these conditions,

and socially connected,

again

i.e.

-A r-~-A;_,-Bi-E,+,-A,+2-A,+3-. Retention

of contiguous

B and E with expansion

V(B,) ’ %%+I)

B expands

V(E,+,)

the right,

> I/CA,+,)

of B entails either

to

or V( B;) > V( A,_,)

B expands

V( B,) = V( E,,,)

the left.

to

Viscosity precludes expansion to the left for more than one generation. Continual expansion then entails W,)

(3.1)

’ f+;+,)

W,+,)

(3.2)

’ W,+A

which imply W,)

(4)

’ %A+,).

Unpacking

(4):

V(BIA)+P’(BJE)>2V(A(A) but, as A is collectively V(Bl

E) > V(AIA).

Now unpacking

(3.2):

V(E(B)+F’(E(A)>2V(A[A), which implies %W+6Wt),

stable,

V( B ) A) I V( A ( A) so

(5)

G. B. Poilock / Stability in a viscous lattice

contradicting (5), as the summed exceed mutual cooperation. This clustered form the left, as in

fitness of PD partners reasoning also applies

187

may never when B is

As in case 2, expansion of B entails elimination of E. This exhausts the positional placement of the mutants. At best, B persists in its cluster, waiting for an appropriate mutant to arrive as in cases l-3. A’s evolutionary stability is thus contingent on the nature of mutational variance throughout the lattice. I next argue that while the rules of mutation should be invariant to lattice position, their consequences are not. That is, encountered mutants differ significantly by lattice position, especially between homogeneous clusters. This positional variance in mutation ultimately leads to the elimination of a non-reciprocator invading B. 4.2. A mutation heuristic Mutational forms are a function of the pre-existing population (Darwin 1859, 1871; Dawkins 1980). When a strategy saturates a population, mutational variants should be readily derived from the definitional properties of that strategy. A strategy may be defined extensively, by listing its response to all other admissible strategies, or intensively, by providing rules which exhaustively characterize its response to encountered behavior. Extensive definitions are not well defined without ancillary restrictions on the universe of possible strategies, as no strategy may be extensively defined until all other strategies are known; such ancillary restrictions are implicitly intensive. Saturated strategies cannot then be extensively defined, being (extensively) undefined until admissible mutants are known, which are similarly undefined until the saturated strategy is known. Intensive definition, by characterizing a strategy as a response to finite sequences of behavior, provides a ready heuristic for mutation: a mutant is derived from a strategy when the former’s intensive definition entails some minimal change in the latter’s definition. Consider, for example, ALL D. Characterized as unilaterally defect this strategy admits no obvious mutation, as its range of relevant encountered behavior is null. Yet ALL D is maximally provokable, and

188

G. B. Pollock / Stability in a UISCOUS lattice

nice ALL D becomes MASSIVE RETALIATION, a minimal version of reciprocity: never defect before one’s opponent but unendingly punish once the victim of defection (also called FRIEDMAN in Axelrod (1980a)). Defining ALL D by the rules (1) defect first (2) always punish adequately characterizes ALL D if pre-existing behavior is equivalent to defection upon emergence of PD social ecology. The rules are distinct if some behavioral difference exists between initial defection and punishment. Entering the territory of another, for example, is physically non-aggressive, while chasing an intruder is not. Differentiating punishment from initial defection entails recognition of another’s act, which may be advantageous when differential optimal responses exist, even when all are still forms of defection. By distinguishing initial defection from punishment, selection for contingent forms of defection (e.g. passively invade/chase; attack/defend) produces maximal provokability which, upon mutation of first defection to niceness, produces minimal reciprocity. All reciprocators are nice and maximally provokable. In a population saturated with some form of reciprocity, loss of reciprocity entails selection for a mutant which satisfies one of two rule sets:

fl)b e nice (2) be less than maximally provokable

II (1) defect first (2) be maximally

provokable

Set I includes TF2T; set II, STFT. The heuristic thus produces strategies compatible with Boyd/Lorberbaum invasion of reciprocity in non-simple games under random assortment (section 1). I now show how the heuristic guarantees emergence of a mutant within B clusters which eliminates the cluster but cannot itself persist in an environment of (forgiving) A reciprocators. 4.3. Elimination Consider

of clustered non-reciprocators

first the fate of type II mutant

clusters.

G.B. Pollock / Stabrlity in a viscous lattice

Proposition 3. Type II (first-to-defect) non-reciprocator persist when surrounded by reciprocators. From (Ll), at the boundary of A and B V( B,+l) = 2V( B ( B) = 2P,‘(l

- w) + [T+ wP/(l

= V(A) A) + I/(x4 1B) = R/(1

as both are maximally

clusters

cannot

- W)

V( B,) = V( B 1B) + V( B (A) = P/(1 V(A,_,)

189

provokable.

- w) + [S + wP/(l

- w)] - w)]

So

gives w > l-

(R - P)/(T-

S),

which is a necessary condition of emergence of A in a world of defection (cf. conditions (2), (2’) above). THUS, since V( B,) > V( B,, ,), successful emergence of A implies elimination of Type II clusters. Since Type I mutants are nice, they may persist indefinitely in the absence of mutation. A Type I mutant cluster is, however, readily destroyed from within through a simple variant of itself. Proposition 4. The mutation heuristic guarantees construction of mutant which eliminates Type I clusters from within; this mutant then itself eliminated upon encountering a greater environment forgiving reciprocators. Consider a mutant E at position k in (Ll). The heuristic permits E be of the form

a is of to

(1) defect first, then cooperate unless provoked (2) retain the same degree of provokability as B E is the non-nice version of B. Since B is not maximally provokable E does not punish first defection and cooperates for at least one move after its own defection. Then V( E,) = 2V( E I B) = 2[ T-t wR,‘(l > R/(1

- w) + [S + wR/(l

V( E,) > 2[ P + wR/(l

- w)] - w)] = I/@_,)

- w)] = V( B,_,)

190

G.B. Pollock / Stability in a viscous lattice

and E expands

to

. . . -B,_,-B,_2-Ek_l-Ek-Ek+l-Bk+2-Bk+3-

Further

expansion

of E is ensured

..-

if

V(EJE)+V(EJB)>V(BJB)+V(BjE), which reduces

to

P+T>R+S (and is guaranteed

since T > R > P > S), and

V(EIE)+V(EJB)>2V(BjB), or P+T>2R. While the latter does not hold for all Dilemmas, the former prevents elimination of the k - 1, k, k + 1 E triplet. Recurrent mutation to E in the cluster augments the frequency of E by at least one and often three when E is surrounded by B. Expansion of B along the A/B boundary is, however, limited to a single nearest neighbor per mutation (Proposition 2). Since the number of mutational positions expanding B is fixed while the expected frequency of new E mutants increases with B cluster size, E grows faster than B, eliminating the latter. Once E eliminates the cluster it encounters A: -A ,__-Ar_l-E,-Ei+l-.

. . -E,-A~+,-Aj+2-.

Since A and E differ only in niceness max[V(E(A)-V(A(E)]=T-S+w(S-T). Then V( A,_,)

- V( E,) > 0 if

V(AIA)-V(EJE)+V(AJE)-V(EJA)>O, which is ensured

if

R/(1-w)-[P+wR/(l-w)]-[T-S+w(S-T)]>O,

and degree of provokability,

191

G.B. Pollock / Stability in a viscous lattice

or Wl-(R-P)/(T-S), and follows from the (assumed) prior emergence cannot expand against A. Elimination of E also requires

of A (equation

2); E

or V(AlE)>2P+wR/(l-w)-R,

(6)

which, if A is non-forgiving, is impossible for sufficiently provokable E. Suppose, for example, that E is provoked by two defections. For non-forgiving A V(A 1E) = S + wT+

Tw* + Pw3/(1

- w)

and (6) requires w < (2T+

R + S-

2P)/(2T+

2R + S-

3P),

which is incompatible with arbitrarily high w < 1. If, however, A is maximally forgiving (i.e. accepts ative “apology” after punishing): V(AI

E) = S + wT+

as E is not maximally

the first cooper-

Rw*,‘(l - w) provokable.

Equation

(6) is then ensured

when

w>(2P-S-R)/(T-R),

whichrequires

T-R>2P-S-R

or

For social ecologies meeting this latter condition, maximal forgiveness ensures that the B-eliminator is itself eliminated upon exposure to A, leaving the population saturated with reciprocity. This also implies no double mutation on A under the heuristic (i.e. a first-to-defect, less

192

G.B. Pollock / Stability in a uiscous luttrce

than maximally provokable B) can withstand A, precluding successful invasion by the former. Forgiveness is superfluous in simple games, as any collectively stable version of reciprocity must satisfy conditions for collectively stable, non-forgiving reciprocity (propositions 2 and 4 of Axelrod [1984: 207-2111. Selection for forgiveness entails a non-simple environment where more forgiving reciprocators outproduce their less forgiving counterparts when both encounter apologists (Axelrod 1980b, 1981). But apology seems unlikely in a world without forgiveness, offering no advantage in a maximally provokable population. Viscosity removes this mutual dependence, permitting apology to emerge in response to a (local) loss of maximal provokability. While E-eliminators are not apologetic when playing B, as the latter never punish, they apologize (at least once) to maximally provokable opponents. Selection for forgiveness then proceeds through the elimination of such recurrent clusters of apology. No cluster can expand in a population otherwise saturated with collectively stable reciprocity (Proposition 1) without the continual introduction of first-to-defect mutants near the cluster boundary (Proposition 2). Mutation can, however, occur anywhere in the lattice. First-to-defect mutation within an invading cluster’s interior eliminates the cluster faster than it can grow. The eliminator is then itself eliminated by forgiving reciprocity (Proposition 4). Since a cluster eliminator is simply the non-nice version of the invader, it is as likely, under the heuristic, as those boarder mutants which aid the invader. No invading cluster can then persist in evolutionary time. Viscous dispersal saturates the social environment of an initially successful invader with “clones”. The invader must then be collectively stable to resist internal invasion in turn. If an invader of reciprocity must be nice (Proposition 3) then it must also be less than maximally provokable; otherwise it is itself a form of reciprocity. But no less than maximally provokable strategy is collectively stable (Axelrod’s (1984: 62) proposition 4). In this sense viscosity reduces population-wide evolutionary stability to local collective stability. The foregoing need not generalize beyond the mutation heuristic. Proposition 5. Consider a viscous linear lattice saturated TAT (= A) with a clustered mutant B which: (1) defects on the first move;

with TIT FOR

G.B. Pollock / Stability in a viscous lattice

193

(2) cooperates on the second move if its opponent also defects on the first move and plays MASSIVE RETALIATION (MR) thereafter; (3) unilaterally defects on the second move if its opponent cooperates on the first move. Then B is able to resist A along the cluster boarder and may be collectively stable as well. B is stable at its boundary with A if V(BlB)+V(B(A)>V(AIA)+V(AjB). By definition

of B this becomes

[P + WR/(l - W)] + [r+

WP/(l

- W)] > R/(1

- w) + [S + wP/(l

- w)

or T+P>R+S, which is always true. B is collectively stable at positions V(BIB)+V(BIE)>2V(EIB).

k (in (Ll))

if (7)

If E cooperates on the first move V( B I E) > V( E ( B). Also, by construction of B, V( B I B) > V( E ) B) and (7) is ensured. If E defects on the first move, B and E receive identical payoffs until E defects again. Summing payoffs from this second defection onward gives V(E(B)
when

w>(2T-R-S)/(2T-P-S), which is satisfiable for sufficiently large w < 1. Expansion of the B-cluster as in Case 1 of Proposition 2 ensures an arbitrary increase in cluster size, with no possibility of elimination from within through single mutations. B maximally cooperates with itself given that it is

194

G. B. Poilock / Stability WI a viscous lattice

non-nice, retaining maximal provokability to permit collective stability. It is, essentially, a version of reciprocity which uses its first move to distinguish itself from other reciprocators. While not directly invadable, the strategy is vulnerable to its own recognition technique. Consider a mutant B’ clustered within B which: (1) defects on the first two moves; (2) cooperates on the third move if its opponent also defected first two moves and plays MR thereafter; (3) unilaterally defects on the third move otherwise.

on the

The foregoing argument ensures that the boundary of B’ is stable against B, with mutation (again as in Case 1, Proposition 2) expanding the boarder against B just as B expands against A. Two waves are created, of identical speed, converting A to B to B’. Iterating the logic within each embedded cluster unravels cooperation, creating arbitrarily long sequences of unilateral defection. This technique for distinguishing self from a prevalent other is inherently extensive. B is defined as a non-nice, clustered best reply to TFT which is stable within the lattice, rather than as a plausible deviation from the definitional properties of TFT. Such an unraveling of cooperation in the space of all constructed strategies has no analog in the given intuitive space of strategic properties. If mutants conform in definitional structure to their ‘parents’, self-recognizing forms are excluded under a prior universe of reciprocity.

5. Extension to the square lattice The foregoing analysis readily generalizes to a two-dimensional vironment. Consider the viscous square lattice

I

I

I

l

I

I

l

I

I

l

I

I

-x-x-x-x-x-x-x-x-x-

en-

G. B. Pollock / Stability in a viscous lattice

195

where, as before, individuals engage in iterated PDs with nearest neighbors, after which “offspring” are uniformly dispersed to neighboring and self positions. Collective stability of reciprocity (A) entails

which, given 2R > T + S, again implies V(AIA)2

V(BlA)

with strict inequality if B is non-nice. Deterministic emergence of A among ensured in

I

I

I

unilateral

defecting

D

is

I

-D-D-D-D-

I

I

I

I

I

I

-D-A-A-D-

I

I

-D-D-D-D-

I

I

I

I

when

or w>(T+3P-R-

3S)/(T+

2P - 3s).

(8)

As before, probabilistic competition would make emergence more difficult. I next generalize Propositions 1-4 to this lattice under the mutation heuristic. 5.1. Growth of an invading cluster Assume collectively

stable A throughout.

Proposition I ‘. No B-invader an A-reciprocator.

may expand

(without

mutation)

against

G.B. Pollock

196

Expansion

/ Stahility

rn a viscous lattice

of B would entail

nV(BJB)-t(4_n)V(BIA)>41/(A(A)

O
which is impossible as nice A implies V( B ) B) I V( A 1A) and collective stability implies V( B ) A) I V( A ) A). Proposition 2’. Suppose two mutants B and E appear in the lattice so as to synergistically increase the absolute frequency of, say, B. Then E must continually recur through mutation to continue expansion of B. Generalization hence omitted.

I

I

of Cases 1 and 2 of Proposition Case 3 now has the form

I

I

I

I

I

I

I

I

I

2 is straightforward

and

-X-X-X-X-A-

l

I

I

-X-B,-E,-A-A,-

I

I

I

-X-X-X-X-A-

l

I

I

where X may be either A or B. For brevity I invoke Proposition 3’, which shows that, under the mutation heuristic, B must be nice to enjoy a non-contracting border everywhere. Expansion of B along a wavefront induced by E then entails IQ,)

’ IQ,),

(9)

which implies IQ,)

’ 6%).

(10)

(9) and (10) give I’(K)

> I%%)

or nV(B\A)+(3-n)V(B(B)+V(B(E)>4V(AlA) for 0 I n I 3. But I’( B 1B) = V( B ) A) = V( A ( A) by Proposition V(BIE)>

V(AIA).

3’, so 01)

197

G. B. Pollock / Stability in a viscous lattice

(10) gives

nV(EIB)+(4_n)V(EJA)>4~(AIA) for 1 I n I 3 or nV(E~B)>nV(AlA) by collective

stability,

which contradicts

(11) since

V(BJE)+V(EIB)<2V(AJA) by definition of the dilemma. Thus E cannot saturation of B, necessitating E’s continual tation to expand the cluster. 5.2. Elimination

induce a wave leading to recurrence through mu-

of B-clusters under the mutation heuristic

As before, under the heuristic in section 4.2.

B will be either Type I or II as defined

Proposition 3’. Type II (first-to-defect) non-reciprocator persist when surrounded by reciprocators. If B is to persist, along its boarder

clusters

cannot

I -A-

l I I I -B,- B;- Aj- Al I I I -Al

V(Bi)

or V(B,)

(12)

> V(A,).

As both B and A are maximally max[V(B,),

V(B,)]

= 3[T+

since B, must be connected V(BlB)

-C V(BJA).

provokable

wP/(l

-w)]

+ P/(1

-w)

to at least one other B and

198

G.B. Pollock / Stability in a viscous lattice

At the cluster’s boundary:

and inequality w > (3T+

(12) is excluded

P - 3R - S)/(3T-

when 2P - S),

which in ensured when emergence condition (8) is satisfied. For sufficiently high probability of repeated play, Type II B-clusters are eliminated within the greater A environment. Proposition 4’. The mutation heuristic guarantees construction of a mutant which eliminates Type I clusters from within; this mutant is then itself eliminated upon encountering a greater environment of forgiving reciprocators. Consider a mutant E at position k in

I

I

I

I

I

I

I

-B-B-B,-B-BI I - B-B,-

I

Bj- B,- B-

I I I - Bi- Bj- E,I I I - B- Bi- BjI I I

I

I

B,- Bi-

I

I

B;- B-

I

I

I

I

-B-B-B-B-B-

I

I

I

By Proposition 3’ and provokable nice strategy.

the heuristic, B is a less than maximally Let E be the non-nice version of B. Then

V(Ek)=4V(EJB)=4[T+wR,‘(1-w)] V(Bj)=3V(BIB)+V(BIE)=3R/(1-w)+[S+wR/(l-w)] and

‘tEk) ’

G. B. Poilock / Stability in a viscous lattice

as

4T>3R+S. Similarly, v( E,) > v( B;) = 4R/(l yielding

- W)

the lattice

I I I I I -B-B-B-B-BI I I I I -B-B-E-B-B-

I

I

I

I

I

-B-E-E-E-B-

I

I

I

I

I

-B-B-E-B-B-

I

I

I

I

I

-B-B-B-B-B-

I

I

I

I

I

E continues

to expand

[I’ + wR/(l

- w)] + 3[ T + &Z/(1

> 3R/(l-

if

W) + [s +

- w)]

WR/(l - W)]

or

(which is ensured

or

P+3T>R.

by definition

(Table l)), and

199

200

G. B. Pollock / Stability in a oiscous lattice

As in the linear case, failure of the latter merely requires recurrent mutation to E within the B-cluster, which enhances the frequency of E at a rate greater than B’s expansion against A, since B expands asymmetrically while E surrounded by B expands symmetrically (section 4.3). After eliminating B, E encounters A. Since every possible E boundary must include the configuration I

I

I

-E-X-Al

I

I

I

I

I

I

-E-E-E-A-A-

l

I

I

-E-X-A-

l (where

I

I X is E or A), elimination

3V(AlA)+V(AJE)

of E entails

>nV(E(E)+(4-n)V(EIA)

11n54,

or : ’ 0.

3V(A(A)-nV(EIE)-[(4-n)V(E)A)-V(AIE)]

(13)

Now, as A is maximally provokable while E is both first-to-defect less than maximally provokable, for 1 I n 5 3 max[(4-n)V(E(A)= (4 - n)T-

and

V(A(E)] s + w[(4 - n)s

(13) may then be written

- T] + (3 - n)Rw2/(1

- w).

as

3R/(l-w)-n[P+wR/(l-w)] -{(4-n)T-$s+w[(4-n)S-T]

+(3-n)Rw2/(1-W)}

>o,

or 3R-nP+(3-n)Rw-(4-n)T+S-w[(4-n)S-T]

>O

G. B. Potlock / Stability in a viscous lattice

201

and W> [(4-n)T+nP-3R-S]/[T+(3-n)R-(4-n)S], which is bounded

from above by one since

T+(3-n)R-(4-n)S>(4-n)T+nP-3R-S if (6 - n)R > (3 - n)(T+

S) + nP.

The latter is ensured Dilemma (Table 1). For n=4 3l7(&4)+

R > P by definition

of the

V(AlE)>4V(ElE).

If A is non-forgiving 3R/(l-

as 2R > T + S and

and E provoked

by two defections

W) + [S + wT + w*T + w3P/(1

- w)] > 4[ P + wR/(l

- w)] ,

which requires (3R + S + 2T - 4P)/(4R and is unsatisfiable forgiving,

+ S + 2T - 5P) > w

for sufficiently

high

w < 1. If A is maximally

V(AJE)=S+wT+w*R/(l-w) and elimination

of E entails

w>(4P-3R-S)/(T-R), which is possible

when

a weaker condition on the social ecology linear case, where T + S > 2 P.

of the dilemma

than in the

G. B. Pollock

202

/ Stability in a viscous lattice

Nor may E expand along other border configurations, E-advantageous being

I

I

the most

I

-E-E-Al I I I -E-E-A-Ai-A-

I

I I -E-E-A-

I

I

l

I

I I

when 3V(E(E)+V(EJA)
w)] + [iJ+

ws + w2R/(1 - w)] < 4R/(l-

w)

or

Thus for sufficiently high probability of future interaction the Beliminator E is itself eliminated upon exposure to a greater environment of forgiving reciprocity. As in the one-dimensional case, viscosity preserves evolutionary stability under multiple mutation.

6. Evolutionary stability in the n-person game Maximal provokability is lost, either locally in viscous populations (sections 4.1 and 5.1), or globally in randomly mixing populations (Boyd and Lorberbaum 1987), when less punishing strategies fair better against first-to-defect yet potentially cooperative mutants. Such mutants differentially punish reciprocity for being maximally provokable, providing a relative advantage to those more tolerant of defection. When differential treatment of strategies is precluded, reciprocity is no longer vulnerable to multiple invasion. The social ecology of two-person games ensures differential treatment of encountered strategies: individ-

G.B. Pollock / Stability in a viscous lattice

203

uals channel their behavior to uniquely affect specific opponents. In an n-person context, however, this ability is lost; punishment affects all group-mates. In n-person groups individuals may react either to the presence of defection or the number of defecting opponents on any turn. The former is the simpler generalization of two-person strategies, transforming a strategy into an n-person equivalent without modifying the discriminatory ability of players. Such an n-person social ecology enhances the evolutionary stability of reciprocity, eliminating forgiveness as a necessary property when groups are insular (i.e. mutually exclusive). Without insularity, however, evolutionary stability cannot be ensured in viscous populations; stability within a square lattice does not generalize to the n-person game. 6.1. Evolutionary stability among randomly formed, insular commons Consider a population of many mutually exclusive groups, each internally engaged in an iterated n-person Prisoner’s Dilemma or “commons” (Hardin 1968; Schelling 1973; Hardin 1982; Taylor 1987; Table 2) with individual absolute fitness (i.e. number of “clonal” offspring), a monotonic function of accumulated payoffs (Axelrod 1986). After social interaction, offspring mix randomly throughout the population before establishing another generation of groups (Boyd and Richerson 1988; Pollock 1988). Suppose the population saturated with some form of collectively stable reciprocity (A), save for mutants B and E with B nice but not maximally E first-to-defect.

Table 2 Payoffs in the n-person n - k individuals

Prisoner’s

provokable

Dilemma

’

k individuals Cooperate

Cooperate Defect ’

and

Defect

R/R

Tk Is,

%k/T,-k

p/p

Payoffs given as column individuals/row individuals with the orderings Tk > R > P > Sk > 0; Tk > T, Sk > S,, for k < i; and nR > kTk +(n - k)S,_,. Iterated payoffs per individual are calculated as in the 2-person game.

204

G.B. Pollock / Stability in a viscous lattice

Groups may be characterized as having n - k -j A’s, k B’s, and j E ‘s. Denote X’s payoff in such a group as V( X ) n - k -j, k, j). For rare B and E in large randomly mixing populations, k, j I 1. Since most groups are homogeneous in A, a sufficient condition for elimination of B is V(AIn-1,

0, 1) > V(B)n-2,1,

1).

04)

For n 2 3 maximal provokability ensures A continuously defects once E defects. Since B is both nice and not as provokable as A, (14) must be satisfied. For n = 3, non-forgiving A also gives (14). Since A is collectively stable V(AIn,O,O)>

V(EIn-l,O,

1)

(15)

V(AIn,O,O)>V(BIn-2,1,1). The only remaining

possibility

of invasion

is then

Suppose so. Equation (15) implies B is eliminated in the next generation, so E finds itself in (M - k, 0, k) groups. But collective stability implies V(AJn,

0,O) > V(Aln

- k, 0, k)

for 0 < k < n

and E is eliminated (Pollock 1988). Collective evolutionary stability against Boyd/Lorberbaum lar, randomly formed groups. 6.2. Evolutionary dispersal

stability thus ensures invasion among insu-

stability among insular commons formed through viscous

When offspring are viscous, individuals associate preferentially with parental group-mates, enhancing the frequency of co-resident rare B. Suppose offspring from an (n - k - 1, k, 1) [with k < n - I] group associate viscously. Since B is less provokable than A V(A(n-k-l,

k,l)>V(BIn-k-l,

k,l)

G. B. Pollock

/ Srabihv

in a UISCOUSlartice

205

and B’s within-group fitness declines relative to A ‘s; associating samegroup offspring will then exhibit a greater expected frequency of A than in the original parental group (cf. Pollock 1983). Regardless of the contribution made to this group from the surrounding population, this entails a decrease in the expected average frequency of B in groups preferentially derived from (n - k - 1, k, 1) groups. Under viscosity relative migration from or to a given position is bounded by fitness densities at and near that position; offspring at relatively low density disperse no less than those at higher density. Then, since E is non-nice (n-k-l)V(A(n-k-l,

k,l)+kV(BIn-k-l,

k,l)

+ V(E (n - k - 1, k, 1) < nV(A In - k, k, 0) where nV(Ajn,O,O)=(n-k)V(Aln-k,

k,O)+kV(AIn-k,

k,O)

and (n - k - 1, k, 1) is outproduced per capita by its surrounding environment, implying (n - k - 1, k, 1) groups contribute to the formation of no more groups than do (n - k, k, 0) groups. In the absence of sampling error at group formation, B then necessarily declines in population-wide frequency in (n - k - 1, k, 1) environments. With sampling error, B’s fate is contingent on maintenance of B-homogeneous groups. permitting E to uniquely affect both A and B through (n - k, 0, k) and (0, n - k, k) groups respectively. (n k, k, 0) [0 I k I n] groups are phenotypically identical; any degree of migration among them will distribute A and B morphs more homogeneously across next-generation groups, precluding maintenance of (0, n, 0). Group compositions permitting Boyd/Lorberbaum invasion do not persist across generations, and B is ultimately eliminated. 4.3. Relative instability among non-insular

commons

While forgiveness is necessary for maintenance of reciprocity in the 2-person game, it is unnecessary in n-person games. Indeed, forgiveness is inapplicable in most heterogeneous commons. I now show how this

G. B. Pollock / Stability in a viscous lattice

206

inapplicability precludes evolutionary stable reciprocity in a population of non-insular commons. This suggests forgiveness is a necessary condition for stability when non-insular groups are embedded in a viscous lattice, with reciprocity contingent on insularity in n-person social ecologies. Consider the following variant of the square lattice (Pollock 1989): I

I x-

I x-

IXIXIXI x- xIXIXIXI -xx-

x-

-xl

xI

-x-

I x-

xI

individual participates four distinct games with membership. While cannot uniquely another, they uniquely play game. Knowledge others is rather than specific. As (sections 3 5) after interaction “offspring” disperse to and nearest locations (i.e. self and partner locations). emergence of in a of unilateral tion is within the DDDDDD DDDDDD DDAADD DDAADD DDDDDD DDDDDD (omitting

connections

for simplicity

in presentation)

when

w>l-(R-P)/(2T+P-3S) where S = min[S,, S,], T= max[T,, T2, T3] (Pollock 1989). Since reciprocity must emerge in a cluster, invaders of reciprocity may also be clustered. Let B be a (clustered) one-less than maximally

G. B. Pollock / Stability in a viscous lattice

207

provokable version of A (e.g. TF2T when A is TFT). Within this cluster a single, non-nice version of B, say E, is an admissible mutant. The lattice B

B,

B,

Ek

B

B

B

B

B

B

B

B

B

BBB

then becomes B

B

B

B

B

E

E

B

E, E,

E

E

B B

E B

E B

E

B B B B

B

B

since, in the former

lattice,

V( E,) = 4[ Tr + wR/(l

- w)] > V( B,)

= [S, + wR/(l

- w)] + 3R/(l-

w)

and V( E,) > V( B,) = 4R/(l

- w).

The E-cluster is itself stable against B since the maximum adjacent to E, (in the latter lattice) is max[ B 1E,] = 3R/(l

- w) + [S, + wR/(l

fitness of B

- w)]

and max[ B ) E,] < V( E,) = [T, + wR/(l

- w)] + 2[ 7” + wR/(l

+ [P + wR/(l - w)] as 3R + S, -C Tl + 2T, + P.

- w)]

G. B. Pollock / Stability in a viscous lattice

208

Similarly, max[ B ( IT,] = 2R/(lv( E,) = 2[ P + wR/(l

w) + [S, + wR/(l

- w)]

- w)] + 2[ T2 + wR/(l

+ [S, + wR/(l

- w)]

- w)]

and max[B]E,]

< V(E,).

If reciprocity is to persist at arbitrarily eliminate E. But consider A AEEEA A E

A

A

A

Ek

E

A

AEEEA A A

A

A

A

A

high frequency,

A must then

Along the border max[ V(A)]

= 3R/(l

for sufficiently max[I/(A)]

- w) + S, + wT3 + w2T3 + w3P/(1

high w -C 1. Elimination

- w)

of E is then precluded

when


which again is satisfiable for sufficiently high w < 1 as P < R. Non-insular commons thus exhibit a greater expected density of first-to-defect strategies than their 2-person counterparts, with reciprocity relatively less stable in the former. At its boundary E sacrifices itself to reduce the fitness of A opponents, thereby ensuring its replacement by an offspring produced by its clone E,. Viscosity uncouples the competitive and social environments of a strategy such that advantages procured in competitively neutral environments (i.e. at position k where no A’s compete) are channeled to areas where the social environment induces competitive inferiority. Strategy success is a function of the differential success of competing social environments, and a strategy is non-decreasing in population-wide frequency when its homogeneous social environment

G. B. Pollock / Stability in a viscous lattice

209

cannot be supplanted by an opponent’s (Wynne-Edwards 1986; Pollock 1989). In the non-insular 2-person environment forgiveness limits the selfsacrifice of an opponent by forcing the latter into social situations indistinguishable (when strategies are intensively defined) from those which induce cooperation among clones. To preserve the cooperative, fitness-enhancing character of its homogeneous social environment, a strategy can be forced to cooperate with an opponent. If the latter’s homogeneous environment induces greater fitness than the former’s, forgiveness can (when T + 2R + S > 2 P in the square lattice and T + S > 2 P in the linear lattice; Propositions 4’ and 4, respectively) provide a net selective advantage to whichever strategy cooperates first. Forgiveness is impossible when two maximally provokable strategies are in the same commons; punishment, once expressed, unendingly echoes between them. In consequence, non-nice strategies persist at greater expected frequency among non-insular commons, as fitness differentials between heterogeneous and homogeneous social environments are greater than in the non-insular 2-person game with forgiveness.

7. Conclusion: the competitive structuring of populations When multiple mutants simultaneously invade a population, strategy persistence is contingent on social experience beyond that of any single individual. Optimal play restricted to a specific environment cannot by itself ensure evolutionary stability; in this sense our actions transcend calculations of self-interest. This suggests that personally detrimental behaviors may persist because of the relative advantage they provide to the competitive success of social environments experienced by others. Under viscosity the locus of selection is not the individual, but the social environment itself (cf. Wynne-Edwards 1963, 1965, 1986; Pollock 1989). Strategy persistence is contingent on the character of both present and future rivals. Altering behaviorally mediated population structure (i.e. the allocation of social interaction throughout a population) can alter a strategy’s relative standing in the population (Pollock 1988); altering dispersal and the insularity of social ecologies can alter resistance to plausible future invaders. An individual may influence the

210

G.B. Poilock / Stability in a viscous lattice

long-term persistence of her strategy by shaping dispersal and social ecology throughout the population, expending resources not to enhance her current success but, rather, that of (multi-generational) “offspring”, either her own or those of strategy “clones”. Population structure itself becomes a locus of conflict, with strategies vying to shape the social environments experienced by their descendants. For example, future conflict may be enhanced by forcing individuals into non-insular n-person social ecologies. Others might resist such entry and, if unable to do so, expend resources to ensure insularity, suggesting a selective mechanism for maintenance of in/out group distinctions. Individuals seem embedded in social networks of low local density, capable of monitoring the social environments of self and others (e.g. de Waal 1982, 1987; Trivers 1985; Cheney et al. 1986; Freeman and Romney 1987; Freeman et al. 1987). If strategy dispersal is channeled viscously along our interaction networks, such monitoring provides opportunity for directed change in population structure to influence strategy stability against future invaders. Politics may often be devoted, not to the present moment, but to the maintenance of cooperation among future “generations”.

References Axelrod, R. 1980a “Effective choice in the iterated Prisoner’s Dilemma”. Journal of Conflict Resolution 24: 3-25. Axelrod, R. 1980b “More effective choice in the iterated Prisoner’s Dilemma”. Journal of Conflict Resolution 24: 379-403. Axelrod, R. 1981 “The emergence of cooperation among egoists”. American Political Science Review 75: 306-318. Axelrod, R. 1984 The Evolution of Cooperation. New York: Basic Books. Axelrod, R. 1986 “An evolutionary approach to norms”. American Political Science Review 80: 1095-I 111. Axelrod, R. and W.D. Hamilton 1981 “The evolution of cooperation”. Science 211: 1390-1396. Boyd, R. and J.P. Lorberbaum 1987 “No pure strategy is evolutionarily stable in the repeated Prisoner’s Dilemma game”. Nature 327: 59. Boyd, R. and P.J. Richerson 1988 “The evolution of reciprocity in sizeable groups”. Journal of Theoretical Biology 132: 337-356.

G. B. Pollock / Stability in a viscous Iattice

211

Brown, J.S. and T.L. Vincent 1987 “A theory for the evolutionary game”. Theoretical Population Biology 31: 140-166. Brown, J.S., M.J. Sanderson and R.E. Michod 1982 “Evolution of social behavior by reciprocation”. Journal of Theoretical Biology 99: 319-339. Cheney, D., R. Seyfarth and B.B. Smuts and social cognition in nonhuman primates”. Science 234: 1986 “Social relationships 1361-1366. Darwin, C. 1859 On the Origin of Species. London: John Murray. Darwin, C. 1871 The Descent of Man, and Selection in Relation to Sex. London: John Murray. Dawkins, R. stable strategy?” In Barlow, G.W. and J. Silverberg 1980 “Good strategy or evolutionarily (eds.), Sociobiology: Beyond Nature/Nurture? pp. 331-367. Boulder, CO: Westview Press. Freeman, L.C. and A.K. Romney 1987 “Words, deeds and social structure: A preliminary study of the reliability of informants”. Human Organization 46: 330-334. Freeman, L.C., A.K. Romney and S.C. Freeman 1987 “Cognitive structure and informant accuracy”. American Anthropologist 89: 310-325. Hamilton, W.D. 1964 “The genetical theory of social behavior, II”. Journal of Theoretical Biology 7: 17-52. Hardin, G. 1968 “The tragedy of the commons”. Science 162: 1243-1248. Hardin, R. 1982 Collective Action. Baltimore: Johns Hopkins University Press. Hines, W.G.S. 1987 “Evolutionarily stable strategies: A review of basic theory”. Theoretical Population Biology 31: 195-272. Maynard Smith, J. 1982 Evolution and the Theory of Games. Cambridge, UK: Cambridge University Press. Maynard Smith, J. 1984 “Game theory and the evolution of behaviour”. Behavioral and Brain Sciences 7: 95-125. Maynard Smith, J. and G.R. Price 1973 “The logic of animal conflict”. Nature 246: 15-18. Maynard Smith, J. and G.A. Parker 1976 “The logic of asymmetric contests”. Animal Behaviour 24: 159-175. Michod, R.E. and M.J. Sanderson 1985 “Behavioral structure and the evolution of cooperation”. In Greenwood, P.J. and M. Slatkin (eds.), Evolution: Essays in Honour of John Maynard Smith, pp. 95-104. Cambridge, UK: Cambridge University Press. Peck, J.R. and M.W. Feldman 1986 “The evolution of helping behavior in large, randomly mixed populations”. American Naturalist 127: 209-221. Pollock, G.B. 1983 “Population viscosity and kin selection”. American Naturalist 122: 817-829. Pollock, G.B. 1988 “Population structure, spite, and the iterated Prisoner’s Dilemma”. American Journal of Physical Anthropology 77: 459-469.

212

G.B. Pollock / Stabrhty rn a viscous lattice

Pollock, G.B. 1989 “Suspending disbelief: of Wynne-Edwards and his reception”. Journal of Evolutionary Biology 2: 205-221. Porchia, A. 1988 Voices (translated by W.S. Merwin). New York: Alfred A. Knopf. Schelling. T.C. 1973 “Hockey helmets, concealed weapons, and daylight saving: A study of binary choices with externalities”. Journal of Conflict Resolution 17: 381-428. Selten, R. and P. Hammerstein 1984 “Gaps in Harley’s argument on evolutionary stable learning rules and in the logic of ‘tit for tat”‘. Behavioral and Brain Sciences 7: 115-116. Taylor, M. 1987 The Possibility of Cooperation. Cambridge: Cambridge University Press. Trivers, R.L. 1985 Social Evolution. Menlo Park, CA: Benjamin/Cummings. de Waal, F. 1982 Chrmpanree Politics: Power and Sex Among Apes. New York: Harper & Row. de Waal, F. 1987 “Dynamics of social relationships”. In Smuts, B.B., D.L. Cheney, R.M. Seyfarth, R.W. Wrangham and T.T. Struhsaker (eds.), Prrmate Socretles, pp. 421-429. Chicago: University of Chicago Press. Williams, G.C. 1984 “When does game theory mode1 reality?” Behavioral and Brain Sciences 7: 117. Wilson, D.S. 1983 “The group selection controversy: History and current status”. Annual Review of Ecology and Systematics 14: 159-187. Wynne-Edwards, V.C. selection in the evolution of social systems”. Nature 200: 623-626. 1963 “Intergroup Wynne-Edwards, V.C. systems in populations of animals”. Science 147: 1543-1548. 1965 “Self-regulating Wynne-Edwards, V.C. 1986 Evolution Through Group Selection Oxford: Blackwell Scientific Publications.