The reason he's stupid is that Goldenballs is a one-off (sadly there's more than one episode with Jasper Carrot, but the players are always new). He can, however, feel a bit better - a lot of people make the same mistake as him.
An economist called Güth has done work on a game called the Ultimatum game; a game with two players where player one has a pot of money and must split it with player two. Player one does not have to offer a 50:50 split, and player two decides whether to accept the split or reject it. If the offer is accepted then the deal is done; if player two decides the deal is 'unfair' then neither player receives anything. Just like Goldenballs, the offer is one-off - we can't negotiate. What does game theory tell us to do?
Suppose we're player two, and there's a pot of £100. Obviously the best thing for us would be for player one to offer us all the money, but that's not going to happen. What would we settle for - £50? £40 maybe? or even £30? Game theory says we should accept anything. Suppose we are offered 1p, and player one takes £99.99. Whilst it's not exactly fair, it's still better than nothing so we should accept. If we reject the offer, we're worse off. It would be cutting our nose off to spite our face.
Suppose we're player one. We should put ourselves in player two's shoes, realise that their best strategy is to accept any offer, and offer them 1p.
What's interesting - and what might make our Goldenballs chap feel better - is that most people don't act like this in the game. Why not is unclear - some suggest that we have a sense of fairness and cannot bring ourselves to offer someone just 1p, some (normally economists) label the players as stupid and irrational! There's also the fact that if we think player two is not going to be rational, we play more cautious and offer a fairer deal. Game theory strategies only work if both players act rationally.
I subscribe to the 'irrational player' hypothesis, but am hesitant to label players as stupid. I think that people forget that the game is a one-off. I think the same is true of our Goldenballs case study, and here I'll show when and why splitting can be the right tactic.
Remember the prisoner's dilemma. Imagine the same scenario, but that it's not one-off. If you snitch, your friend will get payback by sending his friends to beat you up. Let's have a look at the original payoff matrix, and what happens if we can send friends round to break your kneecaps.
In the original, John's best tactic was to confess (5 years in prison is better than 10, 0 years in prison is better than 1). Ron's best tactic was also to confess, since the problem is symmetrical.
But what if being beaten up was as bad as 6 years in prison (think of the most gruesome punishment!). Now the payoff matrix looks different.
Now the payoffs are a bit different. Now, if a player confesses they get the original payoff plus an extra six years. So for example if both players confess, they get five years in prison and get beaten up (worth six years in prison) - a total of 11. What's the best strategy now?
Let's be John. If Ron confesses we have the choice of either confessing (and getting the equivalent of 11 years) or staying quiet (and getting the equivalent of 10 years). So we'd choose the latter and keep quiet.
If Ron stays quiet we choose between confessing (and getting the equivalent of six years, all from being beaten up) and staying quiet (with just one year in prison). So we'd choose to stay quiet in this scenario too.
The game has changed - the dominant strategy is now to stay quiet whereas before we were best off if we snitched. "Repeated games", as they're called in the literature, are massively important.
What about Goldenballs? Well, imagine that the guy's friends are all kind-hearted people. If he stole the pot of money, he'd have played perfectly in economic terms but maybe his friends would disown him because he's nasty. His friends are worth £60 (just to put a value on them). Now what would the payoff matrix look like?
The original is first, then the new one.
Goldenballs is a one-off game. But life isn't - we factor in what happens after the game. The more valuable the player's friends are, the more likely they are to split in the new game.
Repeated games are used in loads of situations. OPEC, the organisation of petroleum exporting countries, punishes members if they play the wrong strategy (produce more oil than they should). Next time, I'll look at a special case where repeated games don't work.
