Radiolab ran a fantastic segment this weekend on the game theory of the “Golden Balls” game show. Listen at http://www.radiolab.org/story/golden-rule/. In the show, contestants must choose whether to “split” or “steal”: if both split, they split the money equally; if one splits and one steals, the “thief” gets everything; and if both steal, neither gets anything. The drama of the show is that, usually, both contestants attempt to convince one another that they plan to share, but then steal anyway. In one famous episode, however, Nick Corrigan took a very different tack when playing with Ibrahim Hussein, telling him:
“Ibrahim, I want you to trust, 100%, I’m going to choose the ‘steal’ ball. I want you to choose ‘split’ and, I promise you, I will share the money after the show … If you do ‘steal’, we’ll both walk away with nothing.”
The Freakonomics blog discussed this same episode back in April 2012, under the title “UK Game Show Golden Balls: A New Solution to the Prisoner’s Dilemma” http://freakonomics.com/2012/04/25/uk-game-show-golden-balls-a-new-solution-to-the-prisoner%E2%80%99s-dilemma/
The most important thing about the Split-or-Steal game, however, is that THIS IS NOT A PRISONERS’ DILEMMA. For this to be a Prisoners’ Dilemma, each player must have a dominant strategy to steal, i.e. stealing must make them strictly better off regardless of what the other player does. In the Split-or-Steal game, however, each player is (financially) indifferent between splitting or stealing when the other person steals. (Whether we both steal or only the other player steals, I get nothing.)
What this means is that whether a player wants to split or steal, when the other player steals, will always depend on considerations outside of the rules of the game. Ibrahim himself explained to Radiolab why he himself would prefer to steal, if he knew that Nick were stealing:
“I was always gonna steal … the reason being, if I split and the other guy steals, I get nothing. I’d rather both of us walk away with nothing, than someone embarrass me [by stealing when I split].”
For Ibrahim, the fear of embarrassment “broke the tie” in favor of stealing, should he believe that the other player was planning to steal. For others, undoubtedly, other emotional considerations such as revenge or envy also might break the tie in favor of stealing. But one could just as well imagine that, if some kind-hearted person were playing the game, he would genuinely prefer for the other person to get the money and hence share if he believed the other player planned to steal.
The beauty of Nick’s idea was to change Ibrahim’s payoff in the outcome “Nick Steals + Ibrahim Splits”, to give Ibrahim at least some hope that he would walk away with some money if he chooses “split” … whereas Ibrahim is SURE to get nothing if he chooses “steal”. It’s important to note here that Ibrahim doesn’t necessarily even need to believe Nick for this tactic to work. As long as there is ANY chance that Nick is telling the truth, and as long as Ibrahim cares more about the prospect of getting the money than the risk of being embarrassed, then Nick has now “broken the tie” for Ibrahim in favor of splitting the money.
This example illustrates the importance, in game-theory analysis, of paying close attention to “payoff indifferences,” situations in which players’ primary motivators (e.g. money in Golden Balls) don’t dictate their incentives. In these situations, secondary considerations (e.g. embarrassment for Ibrahim) will determine player strategy. For those looking to change games for the better, these situations are especially easy picking, as even a small change in the system (e.g. a chance that Nick is telling the truth and will share the money) is enough to dramatically change the natural outcome of the game.