The Self-Made Tapestry Phần 9 pptx

Page 238

Pavlov is not a new invention: Anatol Rapaport, Tit-for-Tat's creator, knew of it in 1965, and dismissed

it as a 'simpleton' strategy. This is because, if pitched against strategies that always defect, Pavlov does

rather poorly: it switches to cooperation every other round, and so gets repeatedly exploited. But in a

mixed population, Pavlov is canny. It cooperates when it pays to do so (against the Tit-for-Tat police,

for example), but unlike Tit-for-Tat it does not run the risk of being over-whelmed by nice strategies,

such as Generous Tit-for-Tat, because it has no qualms about exploiting them with constant defection, if

it is clear that this will bring no recrimination. The problem with highly cooperative populations is that,

while they fare well amongst themselves, they are constantly at risk of being attacked and overtaken by

defectors (which can arise by random mutations). Pavlov, however, is an exploiter that can masquerade

as a cooperator when it pays to do so. And Nowak and Sigmund found that, if Pavlov has just a small

element of randomness in its responses, it can even resist attack by habitual defectors.

Do real creatures show these strategies? In Axelrod's tournaments one could submit strategies that were

as complicated as you like (and some were highly complicated); but animals (including us) do not base

their interactions on the calculation of detailed probabilities or on the precise recollection of many past

eventsthey tend to adopt very simple strategies. In this sense, Tit-for-Tat and Pavlov are plausible

candidates for behavioural tendencies, since they base their choices on a simple consultation of what

happened last time.

There is some evidence for Tit-for-Tat strategies amongst birds, bats, fish and monkeys. It is always

important in these studies to distinguish between co-operative and sharing behaviour amongst kin, and

that amongst creatures who are not closely related: as I indicated earlier, there are good reasons for the

former behaviour to be genetically programmed irrespective of whether the 'altruistic' creature itself

benefits from the exchange. Gerald Wilkinson of the University of California at San Diego showed in

1984 that vampire bats may share the blood that they have foraged not only amongst kin but also

amongst non-kin members of the community. Significantly, he found that individual bats that behaved

more selfishly could be identified and excluded from sharing by the othersjust the kind of behaviour

that Tit-for-Tat strategies reserve for defectors. Michael Lombardo of Rutgers University in New

Brunswick saw Tit-for-Tat behaviour amongst tree swallows: he made it appear that some non-breeding

birds that were helping parents to tend their young had killed some of the nestlings. The parents

responded with hostility to the 'framed' birds, but returned to a more cooperative interaction when it

appeared that the framed individuals were willing to continue cooperating at the nest. (If this

experiment seems a trifle unjust to the framed suspects, you might be reassured to know that they were

only stuffed models.) And in a remarkable study by Manfred Milinski of the Ruhr University in

Germany, stickleback fish displayed Tit-for-Tat tendencies as they investigated a predator (a pike).

Using a series of mirrors, Milinski persuaded individual sticklebacks that they were accompanied in

their forays by companions who would either cooperate (stay with them) or defect (swim away). The

sticklebacks tended to cooperate with a cooperative 'virtual' partner, continuing to approach the predator

while their partner did so;but they would defectrefusing to approach closelyif the virtual partner

appeared to do likewise.

The magic carpet

So far I've talked only about well-mixed populations, in which everyone encounters everyone else. But

the world is not like that, of courseand we saw earlier that for simple Lotka-Volterra-style relationships

between predators and prey, spatial variability can give rise to complex patterns. What about

evolutionary Prisoner's Dilemma gamesdo they have characteristic patterns too, when played out over

space? We can already see from the discussion above that there is the potential for regional differences

in populations to arise and be sustained. Cooperative strategies do well together, but do terribly amongst

defecting strategies; amongst the latter, only fellow defectors can survive. So we can see the possibility

of segregation between cooperators and defectors. But these divisions need not be rigid or invariant: a

single defector placed amongst a cooperative colony can undermine it, while Tit-for-Tats can convert a

defecting population to a cooperative one.

A naive expectation, therefore, might be to see some crude segregation of cooperators and defectors in

Prisoner's-Dilemma-Land. But Martin Nowak and Robert May got something of a shock when, in 1992,

they set out to study how, in the simplest of scenarios, these two types of creature dispersed across a

twodimensional checkerboard landscape. What they found were astonishing, kaleidoscopic patterns that

put them in mind of Persian carpets (Plate 24). With only the simplest of rules, the strategic landscape

becomes painted in complex and richly varied ways.

Page 239

Nowak and May abandoned all the strategic nuances of Tit-for-Tat, Pavlov and their cousins, and chose

to work with just two kinds of player: those who always cooperated and those who always defected. No

player had any memory of the previous encounter; they just acted out their cooperations or defections

monotonously. And everything was deterministicthere were no errors, no probabilistic changes of

strategy. The rules were simple. Each square of the checkerboard grid contains a player, and each player

interacts with the eight all around (or fewer for sites on the edges of the board).*

The payoffs from each of these interactions are counted up according to the usual rules for the

Prisoner's Dilemma, and for the next round, the square is inherited by whichever of the nine (the

square's original occupant and its eight neighbours) had the highest score. This simulates the

reproductive advantage of the fittest competitor in that group (Fig. 9.13).

Fig. 9.13

The rules of the evolutionary game staged by Nowak

and May. Each square is occupied by a contestant that

competes by unconditional cooperation or defection

against all its neighbours. The points for each if these

interactions (either 1, 0, or a reward d for defection in the

face of cooperation) are added up, and the square is

colonized by a player of the same type as the one

that scored highest amongst each player and all those

it encountered. In the example shown here the players

at the edges of the board have fewer neighbours and so

fewer interactions. (Note that each square also competes

against itself, to make the computation easier; but I haven't

included this self-interaction here for simplicity.) White

squares are cooperators, and grey squares are defectors.

We can see that defectors have an advantage over cooperators: defectors can hold their own amongst

their own kind, but they also do well (much better, in fact) when on their own amongst cooperators.

Lone cooperators, on the other hand, are immediately snuffed out by defectors. So one possibility is that

defectors will just take over the entire board, presenting the depressing sight of an inexorable spread of

selfishness. This will happen if the reward for defecting against a cooperator, designated d, is large

enough (d = 5 in Fig. 9.9, for example). But if this reward is not too great, cooperators can gain a

foothold, because mutual cooperation is more profitable than mutual defection. A cluster of cooperators

can then support each other, while the defectors at the cluster's edges undermine their attempts to

exploit the cooperators by their frustrated attempts to exploit each other too. Under these conditions,

cooperators do better and better the more they spread, while defectors do worse and worse.

Fig. 9.14

Patterns of cooperative and defecting communities.

Black squares denote cooperators and grey squares

defectors. White squares show those sites that have

changed from cooperator to defector in the last round

that is, sites where boundaries are shifting. This

pattern occurs under payoff rules that favour cooperators.

(Image: Martin Nowak, Oxford University.)

Nowak and May found that their communities could settle into states in which the patterns, while

constantly shifting, would maintain a distinctive appearance. The relative proportions of cooperators

and defectors in these 'dynamic steady states' reach an essentially constant value, which depends on the

size of the reward parameter d, Figure 9.14 shows a pattern that results from relatively low rewards

(values of d between 1.75 and 1.8). Here black squares are cooperators (C), grey squares are defectors

(D), and white squares are those that have switched from C to D in the last round. We see that under

these conditions, defectors don't do so

The players also interact with themselves, since this makes the calculations easier. But much the same

behaviour is seen when this rather artificial self-interaction is excluded.

Page 240

wellthey can just about maintain a tenuous web through the background of cooperators. Also notice that

the pattern is pretty staticonly a few squares change the nature of their occupants on each round.