Cooperation in Prisoner’s Dilemma Situations
In class and on this blog, many real-life examples of Prisoner’s Dilemma situations have come up. These include arms races, performance enhancing drug use, corruption, unstable markets, and female shaving (3). For all of these, it is clear that each player’s dominant strategy is the “defect” choice and, so, the only Nash Equilibrium is “defect-defect”. In the actual Prisoner’s Dilemma, this corresponds to both prisoners confessing.
The option where everyone cooperates is always best, but it is unstable. For example, it would be best if no country possessed weapons of mass destruction. In this situation, though, any given country has a huge incentive to acquire such weapons because it would then be more powerful than any other. Once one country has these weapons, all others become vulnerable and need to arm themselves. So, stability is only achieved when every country has the weapons – the worst option overall.
Let’s now imagine a game of this type that can be played repeatedly and indefinitely – the Iterated Prisoner’s Dilemma. Now, all previous game outcomes are known to both players during each round. The goal for each player is, of course, to maximize the sum of the payoffs from each round. So, is the best strategy as simple as always defecting or accepting? Or is the best strategy more complicated and based on previous information? In the 80s, Robert Axelrod asked this same question and held a competition to find out. Competitors submitted their strategies in the form of computer programs. Then Axelrod ran a round-robin matching of each program with every other. The game that each played had a simple symmetric Prisoner’s Dilemma payoff matrix.
The results (1) clearly showed that a relatively simple algorithm called “Tit for Tat” was the best strategy submitted. The rules for this strategy are as follows: a) on the first round cooperate b) on every subsequent round, do whatever your opponent did on the last round. For an in-depth overview of why Tit for Tat is so successful, see the linked notes (1). There are, however, two quick things we can say about the strategy. First, it promotes cooperation by reciprocating any defect quickly and fairly, as the name (coming from “This for That”) suggests. Secondly, Tit for Tat is “the equivalent of an [Evolutionary Stable Strategy] for prisoner’s dilemma” (2) because it holds up to a wide variety of mutations. There are, however, a small set of strategies that can overtake Tit for Tat (1). What’s important when comparing strategies, though, is how they hold up on average over many different strategies. In this sense, Tit for Tat is the best.
One of the most famous examples of putting Tit for Tat to work was the Mutual Assured Destruction (4) understanding during the Cold War. Every day, both sides (the US and USSR) made the choice to either launch an attack on the other or not. This is an iterated Prisoner’s Dilemma situation because of the payoffs (either superpower would have been better off if the other didn’t exist) and the fact that the game is played over and over again. Now, by treating this as an iterated game and have both sides use Tit for Tat, we can avoid the bleak Nash Equilibrium that the single iteration of the game has. Each nation now knows that a nuclear attack on the other would be quickly and equally reciprocated – Tit for Tat. In this situation, the only rational choice for either nation was (thankfully) to not attack. This shows the power of Tit for Tat in promoting cooperative behavior.
So, while playing a Prisoner’s Dilemma game once has the unfortunate rational outcome of both players defecting, there are actually ways to maintain cooperation when the game is played over and over again. Of these, Tit for Tat is the best known strategy.