This is a continuation of my thoughts on the game I was talking about here.

The other route I can see players taking is a cooperative strategy.

Over the long term, players can’t expect to win more than 1/*n* in each round since there are *n *players splitting a maximum prize of 1 unit per round. To achieve this outcome, players start out by naming 0 or 1 randomly. They do this until they reach a round where they receive the maximum prize (ie. they are the sole player to name 1). From that round forward they name all 0’s except for every *n*th round. So if I win in round 3, I will name 1 only in rounds 3+*n*, 3+2*n*, 3+3*n*, … . Given enough time players will almost certainly fall into a pattern of taking turns naming 1, each earning the prize every *n* rounds.

The problem with this cooperative strategy is that people have incentive to defect. Once the group gets settled into taking turns, one player could just start naming 1 every round and collect 1 prize on his turn and 1/2 of the prize on every other turn. Normally players would try to defend against defectors by switching to a mutually punishment strategy. In this game they could switch to a strategy of picking uniformly from the first million numbers. The defector’s best response in this case would be to name anything from 1-1,000,000*n* and expect a payoff of 1/1,000,000*n* which is definitely less than the 1/*n* he would make from cooperating.

In this game it would be difficult for players to enforce this plan to punish the defector since it’s hard for them to tell whether he is defecting or just randomly picking a lot of 1’s and unable to find the turn where nobody else picks 1. They need to decide upon a statistical test to detect defectors. With any statistical test there will be a chance of a false positive, but but they can choose a test so that the probability of a false positive is arbitrarily low. Also, defectors will be able to defect infrequently enough to still gain an advantage while going undetected.

One simple test would be to choose a large number *M. *Players would play the cooperative strategy until round *M*. By choosing *M* to be large enough we can be arbitrarily certain that all the players will have settled into a turn taking pattern. After round *M* players are ‘on alert’ for defectors. If a player sees anyone else play on a 1 on his turn (ie. he only gets 1/2 a prize on his turn), he will respond by playing 1 in the next*n *rounds to alert the other players and then switch to the punishment strategy. This test has the advantage of catching defectors with certainty. Overall, players should expect to make almost 1/*n* per round over the very long term which is as good as we can hope for. There might be other ways to do this better.

Now the important question is whether players should go for this cooperative strategy or the one-upsmanship game from my last post. It’s tricky to quantify what a specific individual will expect to make in the one-upsmanship game since it depends on how often he can outsmart the other players. Averaged over all the players, they cannot do better than expecting 1/*n* per round although they will likely do much worse. If all the players are equally smart, they should probably cooperate. If (*n*-1) of the players are equally smart they will want to cooperate and the other player would be forced to cooperate. There is some minimum number of cooperators needed for it to get going, but at the moment I can’t think of what this number is. I’ll leave it as an exercise for the reader. lol.

**Listening to:** “One Love” – Nas