My parents are addicted to reality TV, so I often have to endure the sight of Big Brother when I’m in the living room. Normally I just zone out and think of shiny, happy things. But a couple of days ago, when I heard the words “Prisoner’s Dilemma”, my brain sprang to life!
To those unfamiliar with it, the Prisoner’s Dilemma is probably the most famous game in Game Theory – the study of strategic interactions (how to make the optimal choice, given the fact that other people’s decisions will affect your outcome) – which is such a central concept in economics these days.
There are many subtle variations of the story. The details aren’t that important anyway, but I’ll give you my take:
There are 2 prisoners in different cells that committed a crime together. Let’s call them Steve and Joe for kicks. Steve and Joe are individually asked to confess to the crime. Each of them has two options: tell the truth and confess to both of them committing the crime, or lie and deny the charges.
Solving the game
Easy so far?
Now what happens to Steve depends on what Joe says and vice versa. If one of them confesses and the other denies, then the confessor gets a harsh punishment, whereas the person who denies gets off the hook completely. Lucky mug.
If they both confess then they both get a reduced sentence because of their honesty.
Finally, if they both deny it, then they both get a heavier punishment than if they both confess. But because nobody confessed, the prosecutors don’t have enough evidence, so the punishment can’t be as harsh as if one person ratted the other one out.
The key thing is how ‘good’ each situation is for a prisoner. To make this easier to see, we use something called a ‘payoff matrix’. The numbers in there are weights – the higher the number, the better off the person in that outcome. As you’ll see, this makes it much easier to analyse the game than trying to ingest all that wordy stuff:
It’s much more efficient isn’t it? You can see each person’s payoffs (rewards) in their corresponding colours. What we want to do now is, as each player in turn, find our best responses to the other person’s actions. I’ll do one for you and you can reason the rest.
Imagine you’re Steve. Suppose you think Joe is fickle and will confess. What do you do? Well look at the left column, because that’s what happens when Joe confesses. Now look at your choices. If you confess, you’ll get 2, but if you deny then you actually get 3. So your best response is to deny.
If you work out the whole game like this, you’ll see, very interestingly, that the best option for either person is to deny the charges (and this is known in economics terms as the Nash Equilibrium). This is because no matter what the other person does, you’re always better off when you deny, as you can clearly see from the matrix.
Back to Big Brother…
The Big Brother version was not quite as interesting, since the two people were in the same room, and could talk to each other. The game was almost the same, but with different options and slightly different payoff weights in relation to each other – as you can see from the second matrix. This meant that they could form an agreement whereby each person agreed to share the money. This worked in practice, because (apparently) the two women were friends, and so they trusted each other enough to not deviate from the agreement.
This game is a little more complicated, because if the other person takes, then it doesn’t matter what you do. However, in practice, it still makes sense to take rather than share (in this game – “take” is practically the same as “deny” in the classic game) because you can never be worse off than if you shared.
There is no rational reason for anyone to say share (unless you were 100% sure the other would take). The thing is, because this game is only played once, there is absolutely no reason to stick to an agreement. You will never benefit from it, and you can never get punished for breaking your agreement to share. As you can see from the matrix, if someone says “share”, then it’s always best for you to “take” and snatch all the money from under the other person’s nose. What’s more, if you do this and act sneakily, the other person can do nothing other than stare haplessly into space!
So if I was Sara or Lisa, I would happily have chosen take. The other person may choose not to talk to me for the rest of the show, but I’d be £50,000 better off! (I would probably have made the show more interesting by doing that as well…)