The prisoner’s dilemma is one of the great insights in the history of the social sciences. Why would people ever take actions that make everyone worse off? Because we all realize that if everyone took the socially optimal action, we would each be better off individually by cheating and doing something else. Even if we interact many times, that incentive to cheat will remain in our final interaction, hence cooperation will unravel all the way back to the present. In the absence of some ability to commit or contract, then, it is no surprise we see things like oligopolies who sell more than the quantity which maximizes industry profit, or countries who exhaust common fisheries faster than they would if the fishery were wholly within national waters, and so on.
But there is a wrinkle: the dreaded folk theorem. As is well known, if we play frequently enough, and the probability that any given game is the last is low enough, then any feasible outcome which is better than what players can guarantee themselves regardless of other player’s action can be sustained as an equilibrium; this, of course, includes the socially optimal outcome. And the punishment strategies necessary to get to that social optimum are often fairly straightforward. Consider oligopoly: if your firm produces more than half the monopoly output, then I produce the Cournot duopoly quantity in the next period. If you think I will produce Cournot, your best response is also to produce Cournot, and we will do so forever. Therefore, if we are setting prices frequently enough, the benefit to you of cheating today is not enough to overcome the lower profits you will earn in every future period, and hence we are able to collude at the monopoly level of output.
Folk theorems are really robust. What if we only observe some random public signal of what each of us did in the last period? The folk theorem holds. What if we only privately observe some random signal of what the other people did last period? No problem, the folk theorem holds. There are many more generalizations. Any applied theorist has surely run into the folk theorem problem – how do I let players use “reasonable” strategies in a repeated game but disallow crazy strategies which might permit tacit collusion?
This is Sekeris’ problem in the present paper. Consider two nations sharing a common pool of resources like fish. We know from Hotelling how to solve the optimal resource extraction problem if there is only one nation. With more than one nation, each party has an incentive to overfish today because they don’t take sufficient account of the fact that their fishing today lowers the amount of fish left for the opponent tomorrow, but the folk theorem tells us that we can still sustain cooperation if we interact frequently enough. Indeed, Ostrom won the Nobel a few years ago for showing how such punishments operate in many real world situations. But, but! – why then do we see fisheries and other common pool resources overdepleted so often?
There are a few ways to get around the folk theorem. First, it may just be that players do not interact forever, at least probabalistically; some firms may last longer than others, for instance. Second, it may be that firms cannot change their strategies frequently enough, so that you will not be punished so harshly if you deviate from the cooperative optimum. Third, Mallesh Pai and coauthors show in a recent paper that with a large number of players and sufficient differential obfuscation of signals, it becomes too difficult to “catch cheaters” and hence the stage game equilibrium is retained. Sekeris proposes an alternative to all of these: allow players to take actions which change the form of the stage game in the future. In particular, he allows players to fight for control of a bigger share of the common pool if they wish. Fighting requires expending resources from the pool building arms, and the fight itself also diminishes the size of the pool by destroying resources.
As the remaining resource pool gets smaller and smaller, then each player is willing to waste fewer resources arming themselves in a fight over that smaller pool. This means that if conflict does break out, fewer resources will be destroyed in the “low intensity” fight. Because fighting is less costly when the pool is small, as the pool is depleted through cooperative extraction, eventually the players will fight over what remains. Since players will have asymmetric access to the pool following the outcome of the fight, there are fewer ways for the “smaller” player to harm the bigger one after the fight, and hence less ability to use threats of such harm to maintain folk-theorem cooperation before the fight. Therefore, the cooperative equilibrium partially unravels and players do not fully cooperate even at the start of the game when the common pool is big.
That’s a nice methodological trick, but also somewhat reasonable in the context of common resource pool management. If you don’t overfish today, it must be because you fear I will punish you by overfishing myself tomorrow. If you know I will enact such punishment, then you will just invade me tomorrow (perhaps metaphorically via trade agreements or similar) before I can enact such punishment. This possibility limits the type of credible threats that can be made off the equilibrium path.
Final working paper (RePEc IDEAS. Paper published in Fall 2014 RAND.
I don’t understand this bit:
“Any applied theorist has surely run into the folk theorem problem – how do I let players use “reasonable” strategies in a repeated game but disallow crazy strategies which might permit tacit collusion?”
this suggests to me that tacit collusion is permitted by crazy strategies but not by reasonable ones, but I must be reading you wrong because surely there are some reasonable paths to tacit collusion.
Is the example of how collusion can be sustained in oligopoly because of the threat of eternal Cournot competition a reasonable strategy or a crazy one?
Luis – the Cournot strategy seems very reasonable. What I have more in mind is when we condition punishments on complicated history dependence. Even if you restrict to stationary strategies, with a continuous action space there is always room to “encode” any history you want into the state with arbitrarily small harm to your stage game payoff.
Fine summary, thanks!