Since everyone is talking about it, here is my two cents on the above ‘Golden Balls’ video.

Let’s suppose that the guy on the left is choosing column and the guy on the right is choosing row. (The first payoff in each cell in the following game is the right guy’s payoff).

Here is the original game.

It has three (pure strategy) Nash equilibria: (Steal, Steal), (Split, Steal) and (Steal, Split). So it is not a traditional prisoner’s dilemma which has one unique equilibrium (Steal, Steal).

Now the left guy’s strategy is to change the payoffs. Specifically, he does so by offering a contract in one cell where he plays steal and the other plays split. That generates a new game.

Notice that there are now only one pure strategy Nash equilibrium. It is just that one of these cells a split like solution. This change occurs because, I suspect, the contract is enforceable because it was a clear offer and acceptance. However, it will likely have some details associated with it and so I have reduced the left guy’s payoff by c to account for those transaction costs. If c = 0, then the (Split, Steal) option is a Nash equilibrium again.

The point is that (Split, Split) — the thing that was actually played is not a Nash equilibrium. A real strategic innovation would be to ensure that. But what is more, unless you believe c = 0, then (Split, Steal) isn’t a Nash equilibrium either.

Now the left guy had recognised that (a) there were three Nash equilibria and (b) by committing to steal so openly he may have pushed the other person to take the split choice as it would make them look better. The point is that, if this is a real innovation, then this game show is dead. But it isn’t sustainable so it will live on nicely.

[Updated: due to early morning thinking error]

My take — Right guy has a taste for fairness and prefers SPLIT,SPLIT above all. Doesn’t know if left guy is selfish or not. He has two options:

1 – pretend to be of the good type and hope the other is of the good type too.

2 – pretend to be of the selfish type. In this way both opponent types would rather play split.

Strategy one 1 is cheap talk. Any type would try to persuade he will play split. 2 gives always the desired result as long as you don’t piss off your opponent (so that he punishes you by playing steal). Promising to give something back as you can see works pretty well as a deterrent….

I guess the proposing contestant could fix the problem of (Split, Steal) not being an equilibrium by adjusting the proposed contract to

6800-x,6800-c+x

for some c<x<6800 in the (split,steal) cell.