Complexity_ A Guided Tour - Melanie Mitchell [105]
Nowak suspected that placing players on a spatial lattice, on which the notion of neighbor is well defined, would have a strong effect on the evolution of cooperation. With his postdoctoral mentor Robert May (whom I mentioned in chapter 2 in the context of the logistic map), Nowak performed computer simulations in which the players were placed in a two-dimensional array, and each player played only with its nearest neighbors. This is illustrated in figure 14.4, which shows a five by five grid with one player at each site (Nowak and May’s arrays were considerably larger). Each player has the simplest of strategies—it has no memory of previous turns; it either always cooperates or always defects.
The model runs in discrete time steps. At each time step, each player plays a single Prisoner’s Dilemma game against each of its eight nearest neighbors (like a cellular automaton, the grid wraps around at the edges) and its eight resulting scores are summed. This is followed by a selection step in which each player is replaced by the highest scoring player in its neighborhood (possibly itself); no mutation is done.
The motivation for this work was biological. As stated by Nowak and May, “We believe that deterministically generated spatial structure within populations may often be crucial for the evolution of cooperation, whether it be among molecules, cells, or organisms.”
Nowak and May experimented by running this model with different initial configurations of cooperate and defect players and by varying the values in the payoff matrix. They found that depending on these settings, the spatial patterns of cooperate and defect players can either oscillate or be “chaotically changing,” in which both cooperators and defectors coexist indefinitely. These results contrast with results from the nonspatial multiplayer Prisoner’s Dilemma, in which, in the absence of meta-norms as discussed above, defectors take over the population. In Nowak and May’s spatial case, cooperators can persist indefinitely without any special additions to the game, such as norms or metanorms.
FIGURE 14.4. Illustration of a spatial Prisoner’s Dilemma game. Each player interacts only with its nearest neighbors—e.g., player P13 plays against, and competes for selection against, only the players in its neighborhood (shaded).
Nowak and May believed that their result illustrated a feature of the real world—i.e., the existence of spatial neighborhoods fosters cooperation. In a commentary on this work, biologist Karl Sigmund put it this way: “That territoriality favours cooperation… is likely to remain valid for real-life communities.”
Prospects of Modeling
Computer simulations of idea models such as the Prisoner’s Dilemma, when done well, can be a powerful addition to experimental science and mathematical theory. Such models are sometimes the only available means of investigating complex systems when actual experiments are not feasible and when the math gets too hard, which is the case for almost all of the systems we are most interested in. The most significant contribution of idea models such as the Prisoner’s Dilemma is to provide a first hand-hold on a phenomenon—such as the evolution of cooperation—for which we don’t yet have precise scientific terminology and well-defined concepts.
The Prisoner’s Dilemma models play all the roles I listed above for idea models in science (and analogous contributions could be listed from many other complex-systems modeling efforts as well):
Show that a proposed mechanism for a phenomenon is plausible or implausible. For example, the various Prisoner’s Dilemma and related models have shown what Thomas Hobbes might not have believed: that it is indeed possible for cooperation—albeit in an idealized form—to come about in leaderless populations of self-interested (but adaptive) individuals.
Explore the effects of variations on a simple model and prime one’s intuitions about a complex phenomenon. The endless list of Prisoner’s Dilemma variations that people have studied has revealed much about