Prelude to Chapter 9
GENERATING ZONES OF COOPERATION IN
THE PRISONER'S DILEMMA GAME
THE famous two-person Prisoner's Dilemma (PD) is widely assumed to raise the problem of cooperation in an arresting way (no pun intended). Two strategies are available to each player: cooperate or defect. In the one-shot game, the dominant strategy is to defect, even though higher payoffs would accrue to each individual were they to cooperate. As when Oedipus solves the riddle of the Sphinx, individual rationality leads to a suboptimal outcome. It is a very elegant puzzle and leads to a core question in social science: How can cooperation evolve in populations whose pairwise interactions have the PD payoff structure?
One gambit has been to introduce repeated play with memory. A “strategy” then specifies a contestant's behavior given some history of interaction. The well-known Tit for Tat (TFT) strategy, for example, is a memory-one strategy: open with cooperation, but then play whatever strategy your opponent played in your last interaction. TFT (not at all uniquely) will sustain cooperation (though it is obviously defeated by Always Defect).1 There is a large literature on cooperation-supporting strategies in the iterated PD with memory. Of course, the higher the memory, the greater the computational load. It strains credulity that humans could ever arrive at “solutions” involving high memory, so as explanations of observed human cooperation, they are not very appealing. Truth be known, the cognitive load of even one-shot play strains my credulity.
My main reason for saying so is this: I have taught the one-shot PD at Princeton. And it requires a lecture to convey, to attentive and analytical students, exactly what the dominant strategy is. Nonetheless, behavior violating it (cooperation), in populations of inattentive and highly nonanalytical folks, is seen as a central anomaly for social science. Why? We're not puzzled if the man on the street can't work out “mate in two moves” chess endings. Why are we puzzled that people don't work out the dominant strategy in the one-shot prisoner's dilemma? Obviously, if people know that the optimal strategy is defect, and they play cooperate anyway, that is anomalous; but they don't know. To me, therefore, the real challenge is to build a model in which cooperation does emerge, but in which people aren't challenged to think much—indeed, at all.
Now, classical evolutionary game theory relieves the cognitive load completely. In the so-called replicator dynamics, a society of thoughtless hardwired agents can evolve precisely the strategy that a perfectly rational player would adopt in the one-shot PD. That is very elegant. But, that strategy is to defect, and we are still left with no explanation for cooperation, which, again, is not very appealing.
So the problem, as I saw it in developing this model, was as follows: posit the most extreme form of bounded rationality, as in the replicator dynamics, but somehow (unlike the replicator dynamics) generate cooperation. The solution is a variation I dubbed the Demographic Game, so-called because it involves spatial, evolutionary, and population dynamics.
Events transpire on a lattice (a topological torus). Agents continually move to random unoccupied sites within their finite vision, playing a fixed inherited strategy of cooperate (C) or defect (D) against neighbors. Unlike the Classes model, agents have no “tags” and are indistinguishable to one another. While payoff orderings are PD, they are negative for mutual defectors and those playing C against D, and positive for mutual cooperators and those playing D against C. Importantly, payoffs accumulate in this model. If these cumulative payoffs exceed some threshold, agents clone offspring of the same strategy onto neighboring sites and continue play. If accumulated payoffs go negative, agents are assumed to die and are removed from circulation. On these assumptions, spatial zones of cooperation emerge. Their stability under various assumptions (e.g., mutation rates) is explored. But here is one way to evolve zones of cooperation in populations of locally interacting boundedly rational agents. Of the points emphasized in the Generative chapter, it is perhaps the role of space that looms largest here.
Given that one evolutionary model (the replicator dynamics) returns the classic result (defect) while another equally evolutionary model (the demographic game) returns cooperate, I am left wondering whether evolutionary theory will end up saving economics or burying it. Time will tell.
Since its original publication, there has been some nice mathematical work formalizing the Demographic PD and proving theorems in agreement with the computational results I report.2
A natural line of future work would be to develop a classification theory for demographic games generally. In a new appendix to the chapter, I present the demographic version of the two-person symmetric coordination game. If we imagine a “rules of the road” interpretation with left-hand and right-hand driving being the only strategic options, traditional game theory offers no way to adjudicate between the equally attractive pure strategy Nash equilibria: drive on the left and drive on the right. The replicator dynamics for the symmetric coordination game predicts convergence to one or the other (i.e., the mixed strategy interior equilibrium is unstable). By contrast to both the traditional and evolutionary game theory pictures, the demographic coordination game eventuates in spatial maps divided into connected regions within each of which a specific norm prevails, with “accidents” on the region boundaries. The agent-based demographic games approach again yields a novel result.
1 Defect wins on play number one, and ties TFT thereafter (since both then defect eternally).
2 Victor Dorofeenko and Jamsheed Shorish, “Dynamical Modeling of the Demographic Prisoner's Dilemma,” Economic Series 124, Institute for Advanced Studies, Vienna. November, 2002.