Game show game theory

On Tuesday, a friend showed me a wildly entertaining video of a man “winning" a game show by utilising an astonishing understanding of game theory. Without spoiling it too much, the show involves a variant of the Prisoner's Dilemma and invites the final two contestants to “split" or “steal" the pot of prize money. Video link below:</p><p>WARNING: SPOILERS AND BORING ANALYSIS BELOW

The final result, as the audience gasps suggested, was a surprise to many. However, there is more theoretical meat here than what might first appear. Unsatisfied with simply enjoying the show, I sought to explain the strategy and the result by extending the traditional model of the Prisoner's Dilemma, which may present the situation as thus:

According to this payoff matrix, [Split, Split] is never a Nash equilibrium and the (weakly) dominant strategy is to Steal for both players. However, a frequent criticism of the traditional model is that wealth does not equal utility, so to present the payoff matrix in terms of wealth (i.e. the payout of prize money) may miss some key variables in explaining the situation.

One significant variable may be an aversion to inequality, which we denote by A in reference to Atkinson. If A is non-zero and positive, players suffer disutility if the game ends in an unequal outcome. If this is too magnanimous a concept to assume, consider A to represent “an aversion to being screwed over", i.e. players dislike the outcome in which they alone are left with nothing, after having Split whilst their opponent has Stolen. We can now represent the situation as thus [1]:

The inclusion of A results in [Steal, Steal] being the sole remaining Nash equilibrium because Steal is now the strictly dominant strategy for both players. However, this only compounds our problem since this result is even further from what had actually happened on camera. For that, we need to consider the strategies played.

From the start, Nick (Red) played the unconventional strategy of insisting that he is going to Steal. Assuming that Nick's claim is credible, this removed one uncertain half of the game, leaving Ibrahim (Blue) certain that Nick was going to Steal. Given the removal of uncertainty, we can now represent the new game as follows:

This suggests that Ibrahim will Steal simply to avoid getting screwed over by Nick, who would otherwise gain the entire prize pot at the expense of Ibrahim. However, Nick does not stop there. He makes the additional claim that if Ibrahim Splits and Nick wins the entire prize pot, Nick will offer half of it to Ibrahim after the show. Ibrahim is understandably suspicious of this claim, as this is much less credible than Nick's previous claim of definitely Stealing. Even the host hastens to add that Nick's second claim is not legally binding, anxiously promoting the [Steal, Steal] outcome.

However suspicious Ibrahim is, there may be some probability that Nick is indeed honourable and would split the prize pot after the show. Given Ibrahim is the only player making a decision in this version of the game, it is his perception of Nick's honour that matters. Let us denote this probability by H, where H% of the time, Nick honours his vow and splits the prize pot equally after the show, and (1-H)% of the time, Nick runs away with the entire pot. We can now represent the game as thus:

This time, there is no clear dominant strategy for Ibrahim and everything depends on his values of H and A, Ibrahim's perception of Nick's honour and Ibrahim's aversion to inequality, respectively. In order for Ibrahim to choose Split, H(1)+(1-H)(0-A) must be greater than 0. This can be simplified to the inequality H-A+HA>0.

This inequality suggests that the more honourable Ibrahim perceives Nick, the more likely Ibrahim will choose Split. And the more Ibrahim is averse to inequality, the less likely Ibrahim will choose Split. This is in tune with our intuition. The corner solutions for the inequality are as thus:

• If H = 1, 1 must be greater than 0 (which it is!) so if Ibrahim doubtlessly believes that Nick will be honourable, Ibrahim will always Split.
• If H = 0, -A must be greater than 0 so if Ibrahim doesn't think Nick can be honourable at all, Ibrahim will not Split unless he actually prefers being screwed over.
• If A = 0, 2H must be greater than 0 so if Ibrahim doesn't care about being screwed over, as long as he thinks there is some chance of Nick being honourable, Ibrahim will Split.
• If A = 1, 2H must be greater than 1 so if Ibrahim significantly cares about being screwed over, Ibrahim must strongly believe (with >50% probability) that Nick will be honourable in order for Ibrahim to Split.

As can be seen, whether Ibrahim will Split or Steal depends upon his perception of Nick's honour and his aversion to inequality. As the audience, we have the fortunate hindsight of knowing that Ibrahim chose to Split, from which we can infer that he either perceived Nick to be honourable enough or he wasn't too averse to inequality. However, H and A are both hidden to Nick so let us now analyse his possible moves in turn.

Despite insisting that he will definitely Steal, Nick still has the choice to Split or Steal. He knows for certain that he will be honourable (H=1) so that if he Steals whilst Ibrahim Splits, he will share out the prize money equally after the show. However, in the event that Ibrahim Steals after all, if Nick Steals then both players go home with nothing, whereas if Nick Splits then there is a tiny possibility that Ibrahim will be generous and gives him something small (S) in return [2]. We can therefore represent the payoffs Nick faces below:

The above matrix suggests that as long as there is some probability of Ibrahim being generous and giving something to Nick (G and S both being non-zero and positive, even if small), then Splitting is the weakly dominant strategy for Nick. In fact, one might even consider adding an extra positive satisfaction value to Nick if the [Split, Split] outcome is achieved, out of pride that he has “solved" the game, which would result in Splitting being the strictly dominant strategy for Nick.

The final outcome suggests that both the H-A+HA>0 and the G(S)+(1-G)(0)>0 inequalities have been fulfilled, which is why the result was [Split, Split]. Some final takeaway points include:

• This game has deviated substantially from the classic variant of the Prisoner's Dilemma, so one cannot say that Nick has “beaten" the Prisoner's Dilemma. The two contestants were almost playing a different game, with open communication and incomplete information (about H, A, G, S, etc.) This is why the outcome has also deviated so much from the traditional expectation.
• Commitment is key. Not only in increasing H (others' perception of your honourableness), but also in how credible you are in insisting you are definitely going to Steal, a point which we did not explore here. Ironically enough, if Nick was not credible in committing to Steal, then Ibrahim may have been more likely to Steal.
• Aversion to inequality is also important. Studies have shown that most people value relative wealth as opposed to absolute wealth, which suggests that if they were definitely going home with nothing, people would prefer their opponents leave with nothing too, rather than leaving with more money. This spite can almost be described as a certain sadism (increasing utility in others' decreasing wealth), and the outcome may have been drastically different if either player was a strong sadist!

There is still much to be explored in this theoretical goldmine, so please share your comments in the comments section below, and I hope you've enjoyed reading this!

Many thanks to:
Eugene Hwang for sharing the video with me.
Professor Schankerman for teaching me game theory and spurring me to write this.

[1] This payoff matrix only shows A for the losing/Splitting player hence it is more akin to the “aversion to being screwed over". As for a true Atkinsonian aversion to inequality, it is plausible to consider the existence of A for the winning/Stealing player too, as an admission to guilt, but for brevity this is not represented here.

[2] This is a less farfetched assumption than Nick being altruistic. We are also omitting Nick's aversion to inequality here for the sake of brevity, but it does factor into his decision.