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7:19pm on Thursday, 22nd January, 2009:

Strictly a Game


There's a TV programme, Goldenballs, they show in the early evenings on weekdays here. The end of the show has a showdown, in which the two contestants who have made it all the way through to the final round have to decide what they want to do with the prize money they've accumulated: split or steal?

They make their decisions in secret (by choosing a golden ball with the appropriate word inside), then reveal them simultaneously. If they both choose to split, they get half the money each; if they both choose to steal, neither gets any money; if one splits and the other steals, the stealer gets all the money and the splitter gets nothing.

OK, so on the face of it, this is a zero-sum version of the prisoner's dilemma. It's not identical, though, because in the prisoner's dilemma the cases for both players' (doing the equivalent of) splitting and both players' (doing the equivalent of) stealing are reversed.

Both split — both half-win.
Both steal — both lose.
One steals, one splits — the former wins, the latter loses.

Prisoner's dilemma:
Both split — both nearly-win
Both steal — both half-win.
One steals, one splits — the former wins, the latter loses.

Both of these have the same logic to them: the only reason to split is if you're absolutely certain that your opponent will also split, but, if you are absolutely certain they'll split, why would you not instead steal and so end up with all the cash yourself? Therefore, you steal. However, because your opponent thought the same way, you actually end up with only half the money (for prisoner's dilemma) or nothing (for Goldenballs).

OK, so Goldenballs is more dramatic because you both lose everything if you try to stiff each other. However, the prisoner's dilemma has more frisson, because it's non-zero-sum. Goldenballs is strictly zero-sum.

What if it weren't, though?

Suppose that if you both split in Goldenballs then you both got the full amount instead of half: there would be no reason ever to do anything other than split, because you'd always be assured of winning. What if you got £1 less than the full amount? OK, well then you might want to steal so as to get the full amount instead of £1 less, but given that the full amount is maybe going to be several thousand pounds, risking all of it for a possible extra pound is not going to be worth it. What about £2? Well, probably the same story. But £200? Or £2,000? Ah, now it gets interesting.

Let's say the prize money was £10,000. If you both choose to split, you both get £8,000. If they split and you steal, you get £10,000 and they get nothing. So, you're basically betting £8,000 that they'll split, in order to win £2,000. The way it works at the moment, you'd be betting £5,000 in order to win £5,000, which means you have less at stake for a greater reward. If you're only half sure they'll split, then in regular Goldenballs you'd steal; in this version, you'd maybe split.

This non-zero-sum version has much more about it, in my view, even though the logic is the same (ie. you only split if you're sure your opponent will split, but if you are sure then you'd be better off stealing). It makes greed more of a factor and bluff more effective.

Still, as it stands sometime both contestants on Goldenballs do split, and the audience is entertained whatever happens because their fun comes from guessing what the contestants will do.

If I'd designed it, though, I'd have made it non-zero-sum...

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Copyright © 2009 Richard Bartle (richard@mud.co.uk).