Golden Balls: Defeating Prisoner’s Dilemma
This article talks about Golden Balls, a popular British game show. For those who have never seen the show, it works like this. After a series of rounds, the two contestants on the show accumulate some amount of money which adds to their collective jackpot. In the final round, called ‘Split or Steal?’, each of the two contestants is faced with an option of either split or steal the jackpot money. If both of them choose split, then the jackpot amount will be divided equally between them. On the other hand, if one chooses split and other chooses steal, the person who steals gets the entire jackpot money. Finally, if both choose to steal, then both of them will get nothing. This situation can be modelled as a game, whose characteristics are strikingly similar to the Prisoner’s Dilemma. Having said that, there is one caveat here: the contestants are allowed to discuss their strategies before making a decision. In class, we assumed that the choices made by the players are in isolation. However, this opportunity of discussion introduces many behavioural complications in the game, including trust, greed, betrayal, etc.
|
Contestant 2 |
Contestant 1 |
||
| Result | Split | Steal | |
| Split | 50%, 50% | 0%,100% | |
| Steal | 100%, 0% | 0%, 0% | |
Wrong Strategy
Both the contestants usually agree upon to split the money, thereby giving each of them a fair 50% share. However, they also think about back-stabbing the other person and stealing the entire prize money.
The discussion between the contestants usually involves each person trying to convince the other person that he/she will split. However, this is not a good strategy because, let’s say that Contestant 1 promises Contestant 2 that he/she will split the prize money. This changes the payoff matrix drastically since now, Contestant 2 assumes that Contestant 1 will split the money. Contestant 2 will then think about his/her strategy under the new assumption:
|
Contestant 2 |
Contestant 1 |
|
| Result | Split | |
| Split | 50%, 50% | |
| Steal | 100%, 0% | |
As you might have figured it by now, Contestant 2 has an added incentive to steal the money. Therefore, the problem is that if one contestant promises that he/she will split the money, then it implies that the other contestant doesn’t have to worry about the mutual steal option, which leaves the other contestant with a best response of stealing the money.
Defeating Prisoner’s Dilemma
In a famous episode of the show, one contestant, Nick, told the other contestant, Ibrahim, that he will definitely pick the steal option, and he wanted Ibrahim to pick split, so that after the show, he will split the money with him. As peculiar as it may sound, the strategy employed by Nick actually increased their chances of splitting the money. This can be shown mathematically, by building the new payoff matrix under the given assumption:
|
Ibrahim |
Nick |
|
| Result | Steal | |
| Split | 0%, 100% (with a promised split after the show) | |
| Steal | 0%, 0% | |
Now, Ibrahim can also go for steal and force both of them to go home without any money. On the other hand, if he chooses split, he can at least hope that Nick will keep his word. In other words, Nick’s credible threat to steal changed the game in such a way that the best response for Ibrahim was to split rather than steal!
At the end of the episode, it turns out that both of the contestants chose to split, thereby going home with 50% of the money. Therefore, I believe that Nick’s strategy was truly brilliant in the sense that both the contestants went home with decent amount of money, without incurring any damage to their reputation.