Collusion conundrums...

Nick asked at the end of class the other day about the Nash equilibrium in the collusion game. the idea is that in the collusion game the men are deciding whether to continue colluding to not hire each other's wives, and thus be able to extract monopoly rents from the wife (her husband becomes her only employer, a monopsonist). Each man can, however, cheat on the agreement and hire some of the other wives at the low wage and thus make a temporary profit. if, however, the strategy of the men is to "collude until someone cheats, and then cheat thereafter", once one man cheats, all the others will cheat.

In the collusive game as presented, the payoffs are identical. And, time is not continuous. Time happens in discrete periods, and if you cheat on your mates in one period, and they don't (play is simultaneous) then you get the benefit and they are the suckers. So you evaluate the benefit of cheating compared with continuing to collude. Of course, everyone else is making the same evaluation.

So here is Nick's question. if your discount rate of the future is such that you decide it is worthwhile to cheat, then it must be the case that everyone else has the same calculus, and so everyone will cheat simultaneously. So, Nick suggest, you get no benefit from cheating when you cheat because everyone else will do the same thing. So the condition for colluding is that the discounted stream of profits from colluding be greater than the discounted stream from the competitive equilibrium (which it is by assumption). But... can this really be the Nash equilibrium condition for sustaining collusion? It is an easier condition to have hold that the one we have, where the discounted stream of profits from colluding has to be greater than the one period return from cheating plus the future discounted return of the competitive equilibrium. Suppose that we were in an equilibrium where everyone was colluding, and everyone continued to collude because they though the condition for colluding was this easier condition. but now one person would come along and say:, "Everyone else is going to be a sucker; they think collusion is going to continue, so they will play collude, but I'll cheat and get the one period profits." So that cannot be a Nash equilibrium strategy for he players.

The Folk Theorem says that this repeated game setup of a collusive possibility repeated infinite amount of tie has many different Nash equilibria, involving complex strategies that depend on the history of play. The strategy we looked at is sometimes called the "grim trigger strategy" since when it is played if someone cheats everyone loses forever. A different less grim strategy might be to say that if someone cheats you will cheat for 10 periods after, or until you see them collude, and then you collude thereafter.

One can go a step further and ask, in Nick's question was suggested a problem with the idea of the men being identical- didn't that mean they all had the same thoughts and each knew the other had the same thoughts? But this last part is not quite the idea of game theory, where each actor is independent. While the payoffs may be identical, and the valuation of those payoffs identical also, nobody can exert any kind of "mind control" that "makes" the other person decide the way you want them to. Moreover, in game theory, it is typically assumed that you cannot exercise that mind control over yourself! That is, you cannot commit your future self to behave in a way that your present self would like your future self to behave ("Ill eat the ice cream today because I'll make my future self go to the gym, when tomorrow arrives.")