Wednesday, October 21, 2009

Split or Steal?

Jasper Carrot teaches Game Theory.

The TV game show Goldenballs concludes with two contestants facing off in a situation that is a variation of The Prisoners' Dilemma. The main workings of the early part of the game are unimportant, what is of interest here is the final round.

Each contestant chooses a ball, either Split, which means they try and split the jackpot with the other contestant or Steal which means they try and steal the entire jackpot for themselves. There are three outcomes as follows:

  • Both players choose Split: The winnings are split equally between them.
  • One player chooses Steal, the other Split: The player who played 'steal' gets all the money.
  • Both players choose Steal: No-one gets any money.

The conclusion of one such episode is shown in the following clip.

The problem is the same as The Prisoner’s Dilemma except it is not quite as pure. This is a one shot game, but the players are in the same room, in fact, they’re looking right at each other, their friends and family are watching and they are given the opportunity to convince the other person of their intention to either Split or Steal. There is more at stake than some money, their reputation amongst all people for one. On top of all of this they have been playing a game for the past half hour and have had the chance to betray each other already.

The similarities with the Prisoner's Dilemma are:

  1. It is a game of cooperation (share) or defection (steal).
  2. Decisions are made simultaneously.
  3. It is a one shot game

The major differences are:

  1. This is a zero-sum game.
  2. The players can communicate.
  3. Steal (defect) is only a weakly dominant strategy

Here is some analysis of the decisions involved:

The worst outcome in this game is for the players to both choose ‘steal’ as that would mean no one wins the jackpot. All other scenarios mean the full jackpot is given to at least one of the players. At initial inspection it may appear that the jackpot will be given out ¾ times and no jackpot a ¼ of the time. But the interesting thing with this game is that assuming all players behave rationally the outcome will actually always be that no one wins the jackpot (i.e. two steals).

If you put yourself in the position as a player, you can see how this works. There are two possible options that your opponent can choose (‘steal’ or ‘split’).

Take scenario 1 where your opponent chooses ‘split’. Here if you choose ‘split’ you will get half the jackpot, if you choose ‘steal’ you will get the entire jackpot. So obviously any rational person will choose ‘steal’ as this will maximise your winnings.

Take scenario 2 where your opponent chooses ‘steal’, in this scenario it is irrelevant whether you choose ‘steal’ or ‘split’ because either way you will get nothing. So given the scenario 2 decision is irrelevant (as ‘steal’ and ‘split’ both result in 0) your decision should be based purely on scenario 1 where it has already been illustrated that any rational person will choose ‘steal’.

So the optimum strategy for any player is ‘steal’! Of course the problem with this is that your opponent has the same options as you and therefore will pick ‘steal’ which means the game ends in two ‘steals’. So going back to the game show assuming that all participants are rational human beings the first 55 minutes of the show are irrelevant because whatever the jackpot ends up being the result of the game will always end up with no one wining anything.

So what actually happens when people are faced with this choice on the show. The show is currently half way through its sixth series and in the five and a half series to date 253 episodes have been broadcast. Data on the 40 episodes in the first series are available here. This gives us a sample of 80 people who were presented with the Goldenballs Dilemma. The average jackpot competed for in the 40 episodes was £12,975.76.

Even though we have shown that 'steal' is the weakly dominant strategy of the 80 contestants, 42 of them chose 'split', or just over 52%, with the other 38 contestants obviously choosing 'steal'.

There were 12 episodes in which both contestants chose 'split' and the jackpot was divided. The average split jackpot was £9,245.49. That leaves 18 people choosing 'split' who had 'steal' played against them and ended up with nothing. The average stolen jackpot was £17,807.14. In the remaining ten episodes both contestants choose 'steal' and the jackpot was lost. The average lost jackpot was £8,742.25.

So the outcomes were:

Both players choose Split:- 12 episodes (30%) Average jackpot = £9,256.49
One player chooses Steal, the other Split:- 18 episodes (45%) Average jackpot = £17,807.14
Both players choose Steal:- 10 episodes (25%) Average jackpot = £8,742.45

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