Subgame Perfect Nash Equilibrium

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In my last video I looked at the concept of a Nash equilibrium. A Nash equilibrium is a set of strategies such that no player has an incentive to change his strategy given every other player's strategy. Now we're going to look at a game where Nash equilibrium doesn't tell the whole story.

Let's say that you are walking home and you meet a robber. The robber tells you that you should give him your money or he'll kill you. Your choices are to hand over your money or not, and his choices are to carry out his threat or not. If you hand over the money you get a payoff of minus ten, and he gets a payoff of ten, since he gets your money. If you don't hand over your money, and he makes good on his threat, you die, and your payoff is minus one million, and he gets a life sentence, so his payoff is minus ten thousand. If you don't hand over your money and he doesn't kill you, both payoffs are zero.

We can find the Nash equilibria in this game the same way we did before. If he won't kill you, you don't give him the money. If he will kill you, you do give him the money. If you give him the money, he is indifferent between carrying out his threat and not carrying it out, since either way he doesn't have to actually kill you. If you don't give him the money, he would rather not kill you and avoid that life sentence.

There are two Nash equilibria in this game, but one of them doesn't make all that much sense. In order to decide which one makes sense and which one doesn't we need another solution concept: the subgame perfect Nash equilibrium. I'll get to defining that later, first let's look at our robber game using a game tree.

This game tree is a different way of writing a game. It carries more information than the simple table, since it tells us in what order the events take place. Each decision is made at a node, like this one and this one, and the game proceeds along a branch for each decision. The payoffs are written at the end of the tree's branches.

First you decide to give the robber your money or not. If you give him the money, the game ends with you losing your money and him gaining your money. If you decide not to give him your money, then the robber decides whether to kill you or not. After he chooses, the game ends.

This game contains two subgames. The first is the entire game, and the second is the robber's choice of whether or not to kill you. A subgame is a set of choices within a game that are also a self-contained game themselves. A subgame always starts from a single node, in this case the robber's choice node.

A subgame perfect Nash equilibrium is an equilibrium in which every subgame is also in Nash equilibrium. We already solved for the Nash equilibria in the entire game, now we need to look for a Nash equilibrium in the other subgame. In this subgame, you have already refused to hand over your money, so the robber has a choice between killing you and going to jail, or not killing you. He's not going to kill you, because a payoff of zero is still better than a life sentence.

So there are two Nash equilibria in this game, but only one is a subgame perfect Nash equilibrium, the one where you don't hand over your money and you don't get killed. So, next time you get mugged by a game theorist, you know what to do.