Archive for the ‘ TV & Film ’ Category

Goldenballs, the prisoner’s dilemma and how Ibrahim could have won it all

This clip of an extraordinary game of Goldenballs has been doing the rounds recently, and as I’ve been brushing up on my micro, I thought I’d do a little guide to how it worked.

The idea of the game Goldenballs is based on the Prisoner’s Dilemma in game theory. In this dilemma, two prisoners are arrested and questioned by police. They have two options, Silence or Betrayal. If they both choose Silence, they both are let free. If one chooses Silence and the other chooses Betrayal, then the Betrayer gets a new life in Monte Carlo, while the Silent one is locked up for ten years. If they both attempt to grass up the other, then they both get 5 years in jail.

Let’s put some numbers on to this. If we have person x and person y, then the payoffs are

<Silence, Silence> <5, 5>

<Silence, Betrayal> <-10, 10>

<Betrayal, Silence> <10, -10>

<Betrayal, Betrayal> <-5, -5>

The idea of the game is that there is an equilibrium at <Betrayal, Betrayal>, even though both players would be better of at <Silence, Silence>. This is because, if you think the other person will remain silent, your payoff is maximised by betrayal (5 < 10) and if you think the other person will betray you, your payoff is maximised by betrayal too (-10 < -5). Therefore betrayal is a dominant strategy for both parties (This is the most basic problem in game theory, so although I haven’t used any grids etc a quick google can put you straight if you’re still confused).

The problem is you have no way of co-ordinating your decision in this one-shot game. Goldenballs introduces a co-ordinative aspect, by giving the participants a chance to discuss their decision. And normally it takes the form “We should co-operate”. “Yeah we should.” “Cool let’s co-operate then” then a question of how honest the people are.

But, as Mr Right (the player’s in this game I’m going to call Mr Right and Mr Left, I’ve decided) has worked out, such an attitude to negotiation doesn’t solve the game. The dominant strategy to betray remains, and the Nash Equilibrium of <betrayal, betrayal> does too; regardless of the good vibes you’ve been getting from your partner through out the game. The incentive is still there to steal.

BUT… this is not the same game as outlined above. The payoffs are:

<Split, Split> <6,800, 6,800>

<Split, Steal> <0, 13,800>

<Steal, Split> <13,800, 0>

<Steal, Steal> <0, 0>

If you think the other person is going to Split, there is a clear incentive to Steal. But if you think they are going to Steal, then you are faced with the payoffs of 0 or 0. They are equivalent. Thus, Steal is only a weakly dominant strategy.

Rather than engage in the fantasy that co-operation was possible if Mr L thought R was going to Split (faced with such a proposition, L would always steal), Mr R sets his stall out: I’m going to steal. This infuriates L – he can’t do anything with that. Whatever he does, he’ll get nothing.

But Mr R offers him a chance of the winnings. Although this isn’t taken very seriously by Mr L, it is (quite literally) better than nothing. His payoff from Splitting if Mr Right steals is now 6,800p, where p is the probability that R keeps his promise. p could be infinitesimally small, but as long as it isn’t zero, it’s better than the alternative.

This is presumably the realisation that L has when he sighs “okay, I’m going to go with you.” He splits, so does Mr R, and co-operation in a one shot game has been achieved.

Is this the only solution to the game? Quite simply, no. Mr R constructed a payoff matrix whereby L had nothing to lose by splitting, but the uncertainty in whether or not Mr R will keep his promise is only uncertain to Mr L. Mr R knows exactly what he’ll do. Let’s call “altruistic” Mr R “Nick”, and evil Mr R “Mick”. If Mr R is Nick, then he may as well play Split too, as p = 1, so his payoff from Stealing is 13,600-6,800 = 6,800, so Split and Steal are equivalent. We could even model a negligible benefit ε of being an internet hit and people knowing how nice you are, so his payoff from splitting is slightly higher. But Mick would do no such thing, and would steal the money and then not share. Then he has to play steal, and once he had, even though on the show we wouldn’t know if he would keep his word, we can assume he won’t, as if he was going to cough up he may as well as split. So for what happened to be a proper solution to the game, we have to make assumptions about the nature of Mr R, and his utility gain from (essentially) being nice (perhaps the size of ε).

But Mr L didn’t play in a totally rational manner either. Because rationality means that one can, essentially, work out everything that I’ve just written above before the game is played. And even though L didn’t know who he was playing, evil Mick or nice Nick, he would have been rational to steal in either case. Mr R’s promise is not credible, not just because once he’s one the jackpot no-one can make him share it (a classic intertemporal problem, which I won’t go into here…) but because, as outlined above, an altruistic Mr R (Nick) would just play Split. At which point… Mr L has an incentive to Steal the whole jackpot! So in fact, even though Nick created a payoff matrix for Ibrahim (aka Mr L) where he only had something to gain by Splitting, had Ibrahim thought through the motivations of what Nick was saying, he would have found it rational to cheat. Perhaps Ibrahim didn’t have enough time to think it through, and Goldenballs’ time limit saved Nick and his altruism.

But perhaps Ibrahim is just as irrational as Nick. As they both had a chance to win it all, and neither took it. In a one-shot game, there can be no future punishment, no way of enforcing agreement. So ultimately, although Nick’s strategy seemed to create new payoffs for Ibrahim, its success rested on trust and a belief in non-rational altruism all the same.

Review: Four Lions

Published on, 23rd May 2010

When Chris Morris makes things, there is a predictable pattern of events which follow. First, he finds a suitably controversial subject matter, for example drugs, paedophilia, or in this case, terrorism. Then he turns it into comedy. Then there is a tabloid backlash, radio phone-ins overheat, Channel 4 comes under fire, politicians decry it, then subsequently admit they’ve not seen it but say they are offended in “principle”. Yet, another predictable element of Morris’ work is that it is always absolutely hilarious…

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