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. Fig. 9.15 When the payoffs for defection are slightly greater, the pattern becomes much more dynamic, with communities of both types constantly expanding and overwhelming one another (a). The grey scale here is the same as in Fig. 9.14. While the community structures change, the average proportion of cooperators and defectors remains more or less the same (b). (Image: Martin Nowak, Oxford University.) But for a value of d greater than 1.8, something interesting happens. Then, the payoff is big enough for a two-by-two cluster of D squares to grow in a 'sea' of C, accumulating more D's around its periphery (particularly at the corners) on each round. This sounds like bad news for the C's, except that, so long as d remains below 2, the same applies to C's: a two-by-two cluster of C can support itself well enough to grow within a sea of D. So we are faced with the interesting situation where a 'critical cluster' of D can invade a C community and vice versa. The patterns in this case become much more dynamic, with blobs of C and D continually expanding, colliding and breaking up (Fig. 9.15a. Under these conditions, there is always a lower proportion of C's than D's in the dynamic steady state: specifically, the landscape contains about 32% of C (Fig. 9.15b). For values of d between 1.8 and 2, the most startling results are obtained when one starts with a sea of C and places a single D invader at its centre. The invader can expand because it exploits all the C's around it; but within this range of d, the C's retain the capacity to fight back. The result is the symmetrical, intricate battle depicted in Plate 24, in which the deployment of troops is constantly changing. Nowak and May claimed that this conflict will eventually generate 'every lace doily, rose window or Persian carpet you can imagine'. The patterns are, in fact, fractalfeatures appear on all possible size scales between the limits of the grid size and the board size. Life is just a game For all the infinite variety of patterns here, one can pick out a menagerie of characteristic forms that tend to recur again and againrather like the coherent structures that occasionally arise out of turbulence. These forms seem to have a life of their ownthey possess certain properties, and carry out specific roles within the community. For example, one grouping of cells appears to glide across the landscape (Fig. 9.16a).the cells themselves don't really move, of course, but the shape of these gliders is faithfully transmitted from place to place. Regions of D are often invaded by Fig. 9.16 Characteristic cooperator structures that survive an propagate in a D community include: (a) Gliders, (b) Rotators and (c) Growers. The last of these expands into a set of 'jaws' that eats its way into the surrounding defectors. Page 241 configurations of C that expand from 'growers' to eat up the D's like a set of jaws (Fig. 9.16c). Nowak and May had seen such 'virtual creatures' before. Their game of cooperators and defectors is yet another cellular automaton, since the behaviour of each cell depends on that of its neighbours. Cellular automata, as we saw in Chapter 3, were the brainchild of John von Neumann and Stanislaw Ulam in the 1930s. Von Neumann was interested in the idea of automatarobotic entitiesthat could interact according to simple rules. His dream was to create automata that could reproduce, and which could give birth to other automata that were more complex and sophisticated than themselves. In this way, he speculated, automata might evolve into thinking machines. Ulam helped von Neumann to develop this idea into a simple, tractable model. Instead of actually trying to build mechanical devices, they envisaged a periodic array of cells that could hold information by existing in one of several different states. In its simplest form, each cell holds a binary bita 1 or a 0. The state of each cell, however, is determined by the states of those around it, according to simple, deterministic rules. In this way, information can be transmitted from place to place, as each cell readjusts its state to reflect those of its neighbours. Von Neumann hoped that it might be possible to write into these cellular automata a pattern of information that would be capable of duplicating itself elsewhere on the checkerboard lattice. There are innumerable ways in which each cell can influence its neighbours, and the spatial Prisoner's Dilemma model of May and Nowak represents just one of the possibilities. Robust, propagating cell clusters with distinct shapes and behavioural characteristics, like those in their scheme, are also a feature of one of the most famous of all cellular automata games, called the Game of Life. This was devised in the late 1960s by Cambridge mathematician John Horton Conway. It is a grand name to call a game, of course: at that time no one was used to thinking of these checkerboard experiments as metaphors for living systems, so to call the game 'Life' introduced a provocative new perspective, even though arguably much more biologically realistic cellular automata have since been proposed. Conway's game resembles that of the cooperators and defectors insofar as it considers two types of cell which 'compete' for dominance of the landscape. (This might seem the most obvious first choice for a cellular automaton, but von Neumann initially considered 29 cell states!) The two states are considered to represent cells that are either living or dead. The state of each cell in each round is determined by that of its eight neighbours in the previous round, according to the following rules: 1. A 'living' cell will stay alive if it has two or three living neighbours. If there are fewer or more than this, it dies. 2. A 'dead' cell will stay dead unless it has exactly three live neighbours, in which case it too comes alive. We can justify these rules in biological terms, although the precise numbers are somewhat arbitrary. Living cells surrounded by too many other living cells die of overcrowdingthey starve. Living cells surrounded by too few others, meanwhile, die of 'exposure'you could say that they don't encounter enough others to reproduce. But groups of a certain size that surround a 'dead' cell can colonize it (make it come alive) by reproducing. OK, it takes a lot for granted about the way life works; but in Conway's Game of Life simplicity is a virtue, because it makes it relatively easy to explore the possible range of behaviour. Fig. 9.17 Denizens of the Game of Life. (a) Honey comb, (b) Long Snake, (c) Aircraft Carrier, (d) Sinking Ship. And that range is extraordinary. From these very simple rules spring forms and patterns that you'd never be able to predict from an analysis of the rules. The only way to appreciate the Game of Life is to play it. As an increasing number of enthusiasts did so in the 1970s, they discovered a diverse zoo of robust cellular groupings, with colourful names such as the Snake, Ship, Beehive Loaf and Pulsar (Fig. 9.17). There were also Gliders, like those found in the game of cooperators and defectors. The ways in which these denizens of the twodimensional checkerboard world interact are suggestive of the encounters between different speciessome ignore each other, some prey on each other, others [...]... simulate the growth of the city of Cardiff (Fig 9. 23) They conducted a simulation of DBM growth constrained by the local geography: by the presence of the coastline to the southeast and the rivers Taff (to the west of the city centre) and Rhymney (to the east) The cluster was seeded from a point between these rivers Its probability of growth became zero (sensibly enough) beyond the coastline; and the rivers... might suggest to planners what the underlying rules are that determine a city's form Fractal geometry provides a means to characterize both the structure of a city and the way that this changes over time In the early 199 0s, Batty and others used the methods of fractal analysis to deduce the fractal dimensions of cities from maps like that in Fig 9. 19a They found that these span a range of values, typically... Batty & Longley 199 4, after Morris 197 9.) Page 246 Fig 9. 21 The Paris metro is a branched network with a fractal form (Image: M Daoud, CEN Saclay.) closer the fractal dimension is to 2, the more dense the city isthe more it resembles a blob that spreads without gaps over the landscape In general, this dimension increases slowly over time, reflecting the fact that more and more of the 'free' space between... that reproduces the observations Now, Batty and Longley realized that the mean fractal dimension of the cities that they and others had analysedabout 1.7is rather close to the fractal dimension of DLA clusters, 1.71 (see p 115) So as a 'baseline' model for urban growth, they decided to use the DLA model developed in the 198 0s In DLA, particles execute random walks until they strike the perimeter of... Alternatively we might compare the city's shape to that of a bubble formed as air is injected under pressure into a liquid-saturated porous rock (Fig 9. 19c) Is this resemblance superficial, or are there really any similarities between these growth processes? That question was addressed in the early 199 0s by Michael Batty from the State University of New York, Page 243 Fig 9. 19 The shape of the city of London,... we saw in the last chapter, Page 251 log-log plots of these scaling relationships give straight lines with slopes equal to the value of n The difference in slope seems rather subtle; but when the researchers plotted the real data for N and S for three different urban environmentsBerlin as it was in 192 0, Berlin in 194 5 and London in 198 1they found that the data seemed to fall on or close to the less... concrete to test their models against But for decades, urban theorists have been stumped by the known scaling laws describing the shapes and growth processes of cities They could measure them, but they couldn't then figure out how these particular laws arose from the underlying economic and demographic processes that determine the evolution of an urban area A physical model that captures the growth and... 9. 24b, c Fig 9. 25 The performance of the correlated percolation model can be tested by looking at the scaling law that it predicts for the number of towns N around the city that have a size S The real data from Berlin in 192 0 and 194 5, and from London in 198 1, all show a power law with an exponent of about 2.06 (which shows up on the log-log plot here as a straight line with a slope of 2.06) The strongly... a range of values, typically between about 1.4 and 1 .9: for example, for London in 196 2 the fractal dimension was 1.77, for Berlin in 194 5 it was 1. 69 and for Pittsburgh in 199 0 it was 1.78 The Page 245 Fig 9. 20 Geometric cities: (a) the grid-iron plan is a common feature of North American cities, such as Washington DC; (b) the radial design of the Renaissance city of Palma Nuova in Italy (Images:... corresponding to the strongly correlated model (Fig 9. 25) In other words, these cities do indeed seem to show correlated growth As a further test, the researchers showed that their model generated a pretty good picture of how city shapes evolve over time: Fig 9. 26a shows how Berlin and its environs have developed from 1875 to 194 5, while Fig 9. 26b shows the kind of growth predicted by the strongly correlated . similarities between these growth processes? That question was addressed in the early 199 0s by Michael Batty from the State University of New York, Page 243 Fig. 9. 19 The shape of the city of London,. time. In the early 199 0s, Batty and others used the methods of fractal analysis to deduce the fractal dimensions of cities from maps like that in Fig. 9. 19a. They found that these span a range. about 1.4 and 1 .9: for example, for London in 196 2 the fractal dimension was 1.77, for Berlin in 194 5 it was 1. 69 and for Pittsburgh in 199 0 it was 1.78. The Page 245 Fig. 9. 20 Geometric