Everybody is reacting to the Golden Balls video that I and others have posted. They are saying that the Split or Steal game has been solved. I am not so sure.
- First of all I would like to point out that this solution was suggested here in the comments the first time I (or anybody else I believe) linked to Golden Balls in April 2009. Florian Herold and Mike Hunter wrote: “Perhaps a better strategy would be to tell your opponent that you are going to pick steal no matter what, and then offer to split the money after the show. Pointing out that your offer constitutes a legally binding oral contract, which has been taped, and viewed by hundreds of thousands of witnesses. That way your opponent can opt to pick split, and half the money with you. Or defect in which case you both get nothing.”
- Also, Greg Taylor has a good analysis in the comments to Friday’s post.
- But the successful application of the idea in the most recent video ironically shows the flaw in their reasoning. Consider the player who receives the proposal and is suggested to play split. This is the player on the left in the video. He should ask himself whether he believes that the proposing player will actually play Steal. Florian, Mike, and the rest of the Internet make the observation that Steal is a dominant strategy and therefore a promise to play Steal is credible. But Steal is a dominant strategy for a player with the standard payoffs and the guy who makes this proposal has revealed that he does not have the standard payoffs.
- Now you may respond by saying that the proposal to play (Split, Steal) and divide the winnings at the end is in fact a selfish proposal as it avoids the inevitable (Steal, Steal) outcome. So, you say that the proposer is in fact confirming that he has the standard payoffs and therefore that Steal is a dominant strategy and his promise is credible.
- But let’s look more closely. If he intends to carry out his proposal then he expects to end up with half of the winnings. Indeed he expects to have the full check given to him and either because of altruism, fairness, or reputational incentives to prefer to hand over half of it to the opponent. As he sits there with the balls in his hand and the expectation of this eventual outcome, he can’t avoid concluding that the cheapest way to bring about that outcome is to instead just play Split right now and allow the producers of the show to enforce the agreement.
- Given this the player who is considering this proposal should not believe it. He should believe that the proposer is too nice to carry out his nice proposal. A selfish player faced with this proposal should play Steal because he should expect the proposer to play Split.
- Having dispensed with this try, my personal favorite solution is the one proposed by Evan and elaborated by Emil in which the two men commit to randomize by picking each others’ balls.
- In any case, this video is an essential companion to the original for any undergraduate game theory course.
- Finally, does this Golden Balls show actually exist? In the present? How long ago did this happen? Or is this just some kind of Truman Show like experiment you are all subjecting me to?
Cheap Talk 
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April 23, 2012 at 12:00 am
afinetheorem
While I like the randomization with the 3/8*P payoff, I don’t see how it could be enacted in an incentive compatible way. The show correctly makes each player look at their balls before they talk. How can I commit to randomize in this case? If you pick the ball which I know is split, then why don’t I grab the other ball at the last second?
April 23, 2012 at 12:03 am
jeff
I am assuming they can refuse to look. But yes if they cannot refuse then this one is out the window too.
April 23, 2012 at 7:11 am
wellplacedadjective
jeff – if they can refuse to look (…and it seems not looking would be common knowledge), why would you need to pick your opponent’s ball?
April 23, 2012 at 4:06 am
michaelwebster
My reaction to this is entirely different – I see a poker player trying to push his opponent into making a move that he has a secret good response to.
While I am attracted to the Herold/Hunter scheme, I don’t believe this is what happened. (Well, assuming that anything happened.) One player bullied the other into having to stick to his principles despite playing against a madman. And, once the poker player was sure of that he had forced the right play, he played “split”. This is not an equilibrium play but may be dynamic level k play.
April 23, 2012 at 4:31 am
Tom
The show had a run of success on the UK’s ITV channel a few years back, but sadly ratings dwindled and it was cancelled in favour of random box-picking shows. Golden Balls had its share of randomness admittedly, but as well as Split or Steal, it also had quite interesting earlier rounds where four contestants were voted down to two on the basis of claims of possessing favourable balls (which they carried through). The theme was a tradeoff between survival and establishing a reputation for honesty or cooperativeness in preparation for the final Split or Steal round. By comparison with competitor shows, there never seemed to be enough money in the prize pools though, so there were very few Split or Steal games involving more than a few thousand pounds. Nonetheless, it produced some refreshingly genuine ill-feeling amongst contestants.
April 23, 2012 at 5:54 am
Marco
According to Wikipedia, the show was canned in December 2009. Apparently, this episode is thus pretty old as well.
April 23, 2012 at 6:55 am
BSEconomist
I (mostly) agree with michaelwebster, but I’m not so sure that it isn’t an equilibrium. The problem comes in after the contract has been agreed to (or equivalently after the credible threat has been made) when steal is (supposedly) weakly dominant. With social preferences, the guy on the left will never deviate for any reason.
This is because (1) as you noted the contract is public in front of thousands of witnesses, (2) so the right guy knows that no matter what the left guy definitely plays split (3) that the left guy knows that the right guy can now play split is of no account because if the left guy tried to take advantage HE would be the one to be socially ostrasized for reneging–this is especially true if the right guy is revealed to the audience as picking split and therefore a victim of left’s mendacity. I’m not sure I’d call that an equilibrium in the standard sense, since at the very least it requires this larger social game to function, but it is self-enforcing.
April 23, 2012 at 7:00 am
fitzyman
If the problem is credible commitment, why couldn’t player 1 show the “steal” ball to his opponent? I know of no rule banning such action. Then the opponent can watch and know whether player 1 is actually following through.
Given this setup the “steal” commitment is credible, and the legally binding promise to split some portion (half-half is likely no longer necessary) should incent the opponent to choose “split”.
April 23, 2012 at 7:41 am
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April 23, 2012 at 8:21 am
michaelwebster
Good point about this form of the two person social dilemma having 3 and not 1 equilibrium points.
But, given the uncertainty of whether the offer was genuine or unenforceable for lack of consideration, I don’t think that the outcome row = split and column = steal has the preferences you claim.
April 23, 2012 at 7:43 am
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April 23, 2012 at 7:46 am
proyectocasandra
I think this prisoner’s dilemma is very particular. I will get 0 if the other chooses steal, regardless of my action. If the other guy chooses steal there is no penalty for me to choose split, unlike in the more standard PD.
Suppose a variation of the game in which they both get 1/4 of the prize if they both choose steal. Now the threat of being the sucker (I am the only choosing split) is even greater, since you could secure 1/4 of the prize by choosing steal.
Although, this variation has the same NE as the original game, i think the outcome in the real world would be very different.
April 23, 2012 at 8:14 am
michaelwebster
@BSE Economist,
I am not sure I am following all of your remarks.
Here is how I see the poker player’s problem – he needs to ensure that his opponent who offers to spit is serious and committed, disciplined enough, if you will, to follow through on his promise.
So, the poker player – who likely has a good idea from his opponent’s face and microexpressions which ball is which – tests his opponent’s resolve with an outrageous scheme. Play to give me all the money and we will share – I promise.
He gets a bewildering reply – which is good for him. His opponent is not a poker player. (One correct strategic response for the opponent is to say: write me a cheque for 1/2 amount, pay me to play the game the way you want it played.)
He forces his opponent to see him as a jerk – someone who is going to either get all the money with stupid “promise’ to pay half of it to the opponent. Or perhaps just crazy enough to credible.
But, the poker player is doing more than calculating or solving an equilibrium problem. He is actively creating psychological forces to try to be one step ahead of his opponent. His opponent may be trying the same thing.
The poker player invites curiosity about his play, curiosity trumping prudence.
But, where they end up may not not be a best response to a best response because one of them, or both, will be wrong about where they are. One or both will be surprised when the fog lifts.
(I could be entirely wrong about this, but those were my impressions from watching the interaction. And I hope that my remarks are clearer, now.)
April 23, 2012 at 2:23 pm
Elliot L
A way for the proposer to restore his credibility, as you discussed in #5, would be for him to promise only 30% of his winnings. This brings new issues into play (e.g. What if the other player repeats the sentiment, inducing a game of chicken?), but it then becomes a coherent, time-consistent strategy. “I want .7x rather than nothing, so I’ll bribe you.”
April 25, 2012 at 1:53 pm
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April 25, 2012 at 3:56 pm
Dan K
To all those saying that an oral contract is legally binding, I wouldn’t be so sure. I’m not a lawyer, but I would think that in the context of a television show focusing on deception, deception is an accepted risk of playing just as in the context of a sporting event, actions that might otherwise be called battery (a tackle in soccer or football, say) are acceptable. After all, nobody thinks that a promise to pick “split” is a legally-binding oral contract, do they?
April 13, 2013 at 8:32 pm
Anonymous
Split – always – unless you willing to risk both of you coming with nothing.
January 20, 2014 at 11:20 pm
Anonymous
I havent read through all the comments but shouldn’t player 1 a) guarantee that he is going to steal 2) offer the other player much less than half for splitting? it seems to me that you force the other player to choose between nothing and something instead of allowing them to gamble.
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