Split or Steal?

From 2008 to late 2009, there was a game show in Britain called "Golden Balls." At the end of the game, the two contestants are presented with the choices of whether to "Split" or "Steal." The basic concept is explained pretty well by the video below:

A quick glance at the comments show that most viewers were not too pleased with the female player's choice to steal. However, let's take a look at the payoff matrix of this game:

Note that I decided to change the jackpot to 100,000 euros for mathematical simplicity (it's actually 100,150 euros). This situation is slightly different from your typical prisoner's dilemma game, chiefly due to the presence of weak dominance (a case where a the payoff of a strategy is always equal or greater than the payoffs of another strategy as opposed to strict dominance, where the payoff of a strategy is always greater than that of another).

Because the game is symmetrical, I will proceed to analyze the payoffs of only the male player. Note that the payoffs are the exact same for the female player. Given that the female player chooses to split, the best response of the male player is to steal, as a payoff of 100,000 is greater than a payoff of 50,000. If the female player chooses to steal, the male player is indifferent between choosing split or steal. In this game, the male player's strategy of "steal" weakly dominates his strategy of "split."

However, note that this simple analysis is not complete. Because of the existence of weak dominance, two other Nash Equilibria exist: the outcome where the male splits and the female steals, and the case where the male steals and the female splits. If the reason for this is unclear to you, think about the definition for a Nash Equilibrium. The necessary condition for a Nash Equilibrium states that no player can have a profitable deviation. In the outcome where the female splits and the male steals, neither player can increase their payoff by changing his strategy, taking the other player's strategy as given.

The interesting thing about this is that because both "split" AND "steal" are best responses to a particular strategy, the possibility of a mixed strategy Nash Equilibrium arises. A mixed strategy is a scenario where more than one strategy is played at an equilibrium probability (think about a game of rock paper scissors. If you always played one strategy, you would always lose. Therefore, that is not an equilibrium. The only Nash Equilibrium that exists in a game of rock paper scissors is to play rock, paper and scissors with an equal probability of [1/3, 1/3, 1/3] ). To calculate the mixed strategy equilibrium, we must find the probability that the female player chooses to split such that it makes the male player indifferent between choosing split and steal. For a more concise and mathematical explanation, please look at the handout here. In any case, I will provide my calculations below (where p is the probability of the female player choosing split).

Male's payoff of "split" against female player choosing (split, steal) with probability (p, 1-p):
          p (50,000) + (1-p)(0) = 50,000p

Male's payoff of "steal" against female player choosing (split, steal) with probability (p, 1-p):
         p (100,000) + (1-p)(0) = 100,000p

In order for the male to be indifferent between these mixed strategy expected payoffs, his expected payoff from choosing "split" and "steal" must be equal. Hence:

50,000p = 100,000p
=> p = 0

So basically, we learned that both players will choose "split" with a probability of zero in equilibrium. However, this is simply the pure strategy equilibrium of both players choosing "steal." This makes intuitive sense, because the player would only get lower payoffs by choosing "split" were the other player ever irrational and decided to choose "split." This goes back to the domination argument. However, given the case of multiple Nash Equilibria, we must check to make sure that there are no mixed strategy equilibria.

Now you may be thinking that all of this is stupid, because the male player clearly chose "split," a strategy that we said a rational player should never choose. However, the key to this is that we assume the players to be rational. A full analysis of this situation would probably involve a complex discussion that involves behavioral economics. The fact is that a normal person probably doesn't consider the above analysis when making a decision on a game show (but it should perhaps be noted that the outcome is still Nash). However, it does shed some light on the reasons why what the female player did was not, in fact, greedy and corrupt, but simply rational.

Note that this might be slightly different from what you would expect the outcome to be for this game. We are essentially saying that the situation where both players get nothing is rational over the case where both players get half the pot. However, this is no different from our old friend, the prisoner's dilemma. In both of these cases, the outcome is not Pareto efficient (does not maximize total surplus). The potential for higher payoffs ends up reducing equilibrium payoffs to zero.

I'll leave you with another short clip from an episode of "Golden Balls," this time with a slightly different outcome.