## Thursday, 29 September 2016

### Golden balls and weakly dominant strategies

In ECON100 this week we covered game theory, in which the most famous game is the prisoners' dilemma. One of the tutors then sent me some YouTube links to clips from the game show, Golden Balls. This isn't quite the prisoners' dilemma, but it is close. In this game, the two players have to choose to split or steal. If they both choose split, they share the prize pool. If they both choose to steal, they both go home with nothing. If one chooses to steal and the other chooses to split, the one who steals gets the whole prize pool and the other gets nothing. Take a look:

Why would anyone steal? Stealing is actually a weakly dominant strategy - it's a strategy that is never worse than the other strategy (splitting), but is sometimes better. [*] Most rational players would choose to steal. To see why, consider the game in normal (payoff table) form below.

Consider Sarah's choice first. If Steven chooses to split, Sarah is better off choosing to steal, because £100,150 is better than £50,075. If Steven chooses to steal, it doesn't matter what Sarah does because her winnings will be zero regardless. So, stealing is a weakly dominant strategy for Sarah (it is never worse than splitting, and if Steven chooses to split then stealing is better for Sarah).

Now consider Steven's choice. If Sarah chooses to split, Steven is better off choosing to steal, because £100,150 is better than £50,075. If Sarah chooses to steal, it doesn't matter what Steven does because his winnings will be zero regardless. So, stealing is also a weakly dominant strategy for Steven (it is never worse than splitting, and if Sarah chooses to split then stealing is better for Steven).

So, I would expect the players to choose to steal. Unless they can find some way of changing the game. Which brings me to this clip:

Nick realises that if he chooses to split, he is at risk of going home with nothing if Ibrahim chooses the weakly dominant strategy (steal). So he attempts (successfully) to change the payoffs in the game by convincing Ibrahim that he will choose steal no matter what (and will split the prize pool with Ibrahim afterwards). The game changes to look like this:

Note that neither player has a dominant strategy now. In fact, Nick's strategy choice becomes irrelevant because Nick will get £6,800 if Ibrahim chooses to split, and nothing if Ibrahim chooses to steal, regardless of what he (Nick) chooses. Ibrahim also doesn't have a dominant strategy here. If Nick chooses to split, Ibrahim is better off to steal (because £13,600 is better than £6,800). But if Nick chooses to steal, Ibrahim is better off to split (because £6,800 is better than nothing). Ibrahim does have a minimax strategy to choose split. Minimax is where you look at the worst outcome from each strategy, and choose the strategy that has the worst outcome that is best. In this case, the worst outcome from choosing steal is going home with nothing, and the worst outcome from choosing split is £6,800 (provided Nick agrees to split the prize pool afterwards).

Ultimately though, it comes down to whether Nick's threat to choose steal no matter what (and split the prize pool afterwards) is credible (believable) or not. Once Ibrahim is convinced, then Nick chooses split as well. It's not an equilibrium solution to the game, but by changing the nature of the game Nick managed to make them both better off (than if they had both chosen their weakly dominant strategy).

[HT: My ECON100 tutor, Jae]

*****

[*] Note that this is why this game isn't a classic prisoners' dilemma game. In the prisoners' dilemma, both players have dominant strategies (not weakly dominant strategies), and when both players choose their dominant strategy (which they will do if they are rational) then both players are made worse off.