Golden Balls: What To Do In the Absence of a Dominant Strategy
A game show in Britain called Golden Balls demonstrates a typical two-player game controlled by game theory. The rules that govern this game show follow, though this previous post on the Networks blog goes into deeper detail on the specific mechanics of the game than I will.
Both players are faced with a choice: split or steal. If they both split, they each walk away with half the final winnings. If one player splits and the other steals, the “stealing” player wins all the money. Finally, if they both steal, they both walk away with nothing. The closest example we covered in class was the hawk and dove example in which neither player had a dominant strategy. In the following payoff matrix I will use H (hawk) to represent the option to “steal” and D (dove) to represent the option to “split.”
| 1\2 | D | H |
| D | 0.5,0.5 | 0,1 |
| H | 1,0 | 0,0 |
For each player, the payoffs are as follows:
If he believes the other player will choose “split,” his best option is to choose “steal” and walk away with all the money. If he believes the other player will choose “steal,” he will walk away with nothing either way.
The three nash equilibria are (D,H), (H,D), and (H,H), but none of these is beneficial to both players. The only strategy in which both players walk away with something is (D,D); however this is unstable, as both players would have increased their payoff by playing the hawk while the other played the dove.
Therefore, the best option for both players is to convince the other that they will play the Dove before switching to the Hawk. In other words, it is to player 1’s advantage to convince player 2 to split the winnings while actually intending to steal. Of course, player 1 runs the risk of player 2 choosing a similar strategy, causing both players would walk away empty-handed.
It becomes clear that even though player 1 would walk away with nothing if player 2 is a hawk, regardless of his choice, he would rather they both walk away with nothing than just him. It is in our nature to search for the largest possible winnings, and it is also in our nature to reject the status of the “sucker.” Players would rather they both walk away with nothing than see the other player win. That is why almost every episode of Golden Balls ended with a split-steal or a steal-steal.
This represents an example in which game theory and strategy can only go so far. As we discussed in class, there is no clear conclusion to this type of game. Both players are pushed to steal, but this tendency leads to a disappointing result for both of them. A player must know more about the other player to determine who will be the hawk and who will be the dove. In Golden Balls episodes, the players take matters into their own hands by constructing elaborate stories in order to convince each other of their evidence. Some players spin sad stories about their financial woes, others play to their innocent looks, and still others whip out confidence-building quotes. All these efforts are meant to drive the other person into the “split” mindset. The game show reveals a disheartening pattern: player after player proclaims their willingness to split before stealing all the winnings for themselves.
This seems like a cycle that will never be broken until a participant named Nick enters into the final round. He realizes that the only way to ensure that his opponent will choose “split” is to drive him into a corner. He does this by assuring his opponent, Ibrahim, that he will steal. Nick guarantees Ibrahim that he will choose “steal” now and will split the money with him after the show. Of course this only works if Ibrahim chooses “split.” The resulting payoff matrix looks like this, letting Nick be player 1 and Ibrahim be player 2.
| 1\2 | D | H |
| D | 0.5,0.5 | 0,1 |
| H | 1,0 | 0,0 |
Nick’s strategy forces Ibrahim into a corner—the bottom left corner to be exact. Faced with the option of a later reward or absolutely nothing, Ibrahim finally relents and chooses “split.”
The interesting fact to notice about Nick’s strategy is that it lays bare game theory—the very force that wraps Golden Balls in frustration and dilemma—and uses it to his advantage. Every other player 1 assured player 2 that he would play the dove. However, if player 2 knows that player 1 will play the dove, his best option is to play the hawk, and vice versa. In contrast, Nick fully exposes the strategies available to both players and, recognizing the unlikelihood that both players will be doves, ensures Ibrahim that he will play hawk. This is more effective than the alternative argument because it is more convincing in narrowing the payoff matrix down to the bottom two boxes. Even though this argument offers an uncertain reward for Ibrahim, the uncertain reward is better than a certain loss.
This is an excellent example in which game theory reaches its limits. Both players can profess their trustworthiness for hours but in the end the mechanics of the game drive both players to steal. It is only when one player dares to crush the strategies that we see a different picture. Interestingly enough, this episode of Golden Balls ended with both players choosing “split” and leaving with half the winnings.
This result demonstrates the brilliance of Nick’s maneuvering. In the linked episode of RadioLab, called “Golden Rule,” both players are brought on for an interview. Ibrahim admits that he never intended to “split”—that he entered the game with the firm belief that he would steal. Nick’s psychological twist rewrote the game and left both players in the rare situation of each playing the dove.
http://www.radiolab.org/story/golden-rule/