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Game Theory with Golden Balls!?!?

Golden Balls is a British TV game show in which contestants compete/work together for a pool of money.  At the end of the game, there are only two contestants left and they have to decide whether to split the pot of money or steal it.  If both players decide to split, then the pool of money is split between the players.  But if one player decides to split and the other decides to steal, then the player who decides to steal takes the whole pool of money and the other player is left with nothing.  If both players decide to steal, then they both leave the game with no money.  It is a game to convince your opponent to share and to determine whether or not to trust/believe that your opponent will do what she tells you she will do.  You have an incentive to share, but a bigger incentive to convince your opponent to share, and then steal the pot of money from him.  It is similar to the prisoner’s dilemma in which both players can benefit if they cooperate, but risk very low payoffs if one player agrees to cooperate and the other one does not.  It is different to the prisoners’ dilemma because the two players are able to communicate to each other their strategy and there is no dominant strategy.  There are a lot of variables in deciding the best strategy such as persuading, coercion, psychological warfare and social expectations, but in the end, this game can be represented by game theory.

Using game theory analysis, the players are the contestants of the game show, say Player A and Player B.  The strategies are both splitting, both stealing, A splits and B steals, and A steals and B splits.  The payoff matrix looks like this:

 

Player B
Player A   Split Steal
Split (1/2,1/2) (0,1)
Steal (1,0) (0,0)

The numbers indicate the fraction of the pool of money each Player received with their strategy, (A,B).

The best way to determine whether or not there is a pure Nash Equilibrium is to determine each player’s best response to each strategy.  Let’s take Player A.  If Player B:

  1. Splits  — then Player A should Steal because it has the highest payoff
  2. Steals —  it doesn’t matter because Player A gets nothing either way and there is no better strategy

These are also the same responses for Player B.  Therefore, there is no pure Nash Equilibrium.  There is also no mixed Nash Equilibrium because when the payoff for Player A choosing split equals the payoff for Player A choosing steal, q, the probability that Player B chooses split, equals 0, but there are no pure strategies.  Despite the fact that you can communicate with your opponent, you still have to convince her to go with your plan, which optimally for the both of you, is to share, but you still have to deduce what your opponent is going to do regardless of what she says.

However, in the video and podcast, there is a special case with players Nick and Ibrahim.  Player A – Nick tells Player B – Ibrahim that he is going to steal.  Nick said that Ibrahim should take split and then once Nick collects the money he will split.  With Nick’s steadfast decision to steal, Ibrahim has two options: split or steal. Given Nick’s ultimatum to steal, Ibrahim’s payoff in both cases is zero.  There is still the possibility that Nick will keep his word and split the money with Ibrahim afterwards.  With little hope of convincing Nick to split, Ibrahim chooses to split since there was a possibility of Nick keeping his word and splitting the money with him after the show.  Hope was better than zero since he was convinced Nick would steal.  But surprisingly Nick also chose split!  The theme is to convince / figure out what the other person will do, so you can act accordingly.  Nick did this by forcing Ibrahim that his best choice was split in the hopes of splitting the money after the show.  Anything else, (since Ibrahim was convinced Nick was going to steal) would have resulted in zero.

Afterwards, NPR interviewed with Ibrahim to see whether or not he still would have split the money if Nick did not tell him that he would steal.  And guess what?  Ibrahim said he would have stolen.  In the end with a little help from psychology, Nick used game theory to persuade Ibrahim to pick split.  This is a wonderful use of prisoner’s dilemma in game shows.

Sources:

http://www.radiolab.org/story/golden-rule/

http://www.npr.org/blogs/money/2012/04/25/151378032/game-theory-explained-with-golden-balls

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