Split or Steal: An Analysis Using Game Theory

I chanced upon this Youtube video of a British game show called “Golden Balls” in which one of the participants, Nick, played the game in his favour using concepts of game theory.  In particular, he demonstrated a good understanding of the Prisoner’s Dilemma in formulating an unstable but mutually beneficial strategy.

In this blog post, I will explain Nick’s strategy using game theory, and in doing so, illustrate how his strategy effectively allowed both parties to “win” the game by splitting the jackpot halfway – this is the most mutually beneficial outcome of the game.

The game involves a segment called “Split or Steal” in which two participants make the decision to “split” or “steal” to determine how the final jackpot is divided.

• If both contestants choose to split, the jackpot is split equally between them.
• If one contestant chooses to split and the other chooses to steal, the Stealer gets all the money and the Splitter gets nothing.
• If both contestants choose to steal, they both get nothing.

The participants typically play for money in the range of tens of thousands of British Pounds. They will also have the opportunity to negotiate with each other before making their decisions. In most cases, both participants will try to convince each other to split as it is the most mutually beneficial strategy. However, this split-spit strategy, while maximising welfare, is not a stable strategy. Most games end up in either one or both parties defecting. In the former, one party goes home with all the money (and leaving the other with nothing). Or in the latter, which happens more often, both parties loses all the money and goes home empty-handed. Very rarely will both players end up in a split-split scenario.

To understand this, we can draw up a payoff matrix as such:

This is very similar to the Prisoner’s Dilemma as we have studied in class.  A key difference is that, in the standard Prisoner’s Dilemma, if one player chooses to defect (or steal), the other player is better off defecting instead of cooperating (splitting). However, in Golden Balls, as long as one player chooses to steal, neither player has a better strategy. That is to say, all three strategies that involve stealing are Nash equilibria (as shown by the shaded cells) and (Split, Split) is an unstable strategy.In this matrix, Player 1 refers to Nick, the player on the right, and Player 2 refers to Ibrahim, the player on the left.

However, the game is not as simple as that. As evident in the video, there were many facets to Nick’s strategy that involved deeper psychological implications. To model this, we cannot assume that the payoff is solely based on the money each player gets to win. Let’s instead add the additional factor, S, to represent the disutility derived from losing as a result of being swindled by the other player.

Now both players have a dominant strategy of (Steal, Steal). However, this is not ideal to both parties. To counter this, Nick insisted that he will steal regardless of what Ibrahim does. Assuming Nick’s claim is true, the game is now reduced to Ibrahim deciding between Split or Steal, in which Ibrahim should choose to Steal as to avoid a negative payoff (i.e. to get 0 rather than -S).

To further encourage a (Split, Split) outcome, Nick then assures Ibrahim that if he splits and Nick wins the entire pot, Nick will then give Ibrahim half of the prize money. Ibrahim is naturally suspicious of this claim, as Nick could very well keep the winnings to himself. We can represent this suspicion with a new variable, H, which denotes the probability (between 0 and 1) of Nick being honest, and (1-H) denoting the probability that Nick cheats. The payoff matrix is now as such:

Now, given that Nick is going to steal regardless, Ibrahim may have a dominant strategy if (H*1 + (1-H)*-S)) is greater than 0. Effectively, Ibrahim’s decision to split depends on how high he regards his disutility from losing, S, in relation to how much he trusts that Nick is honest, represented by H. To analyse this, let’s consider the following:

• If Ibrahim does not care about the disutility from losing to Nick, then S = 0, and if so, H must be greater than 0, i.e. Ibrahim must have the slightest bit of trust in Nick, before Ibrahim will split.
• If Ibrahim cares about the disutility, S= 1, then H must be greater than 0.5, i.e. Ibrahim must trust that Nick has more than 50% chance of being honest, before Ibrahim will choose to split.
• Similarly, if Ibrahim has utmost trust in Nick that he is being honest, then H= 1, then regardless of the disutility, S, Ibrahim will always split.

What we observe in the video is probably the case of Ibrahim having a value of H somewhere between 0.5 and 1. We can also assume that the value of S is somewhat high as it is in human nature to be reluctant to be taken advantage of. What we know for sure is that (H*1 + (1-H)*(0-S)) > 0 and Ibrahim was convinced that Nick was going to steal. Nick’s decision to split, despite stealing being the dominant strategy for him, was possibly out of altruism or the confidence that his strategy would work and that Ibrahim would split.

What we can take away from this discussion is that while the Prisoner’s Dilemma makes for a good albeit simplified model for such games, it can also be extended indefinitely beyond the classical model that we have learned  in class. In doing so, we can use the Prisoner’s Dilemma model to account for other complex (e.g. psychological and sociological) factors, making it widely applicable in all areas of our lives.

– Shane