Subgame Perfect Nash Equilibrium

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.

[areyes]

I tried this after getting robbed many times and now I'm dead. Thumbs down

[nickm.]

Thank you very much for this. I tried reading Wikipedia pages in 3 different languages and didn't get it at all before I watched your video.

[arabhumanity]

Rubber: Give me a 50$ dollar.
Me: can you wait I need to calculate SPE
Rubber: You need any help?
Me: I did it. look. What do you think about my calculation?
Robber: I think you made a mistake in my payoff. It should be 50 for me and -50 for you.
Me: You are right. Thank you.
Robber: You owe me tutoring fees.
Me: How much you want?
Robber: 50$
Me: Great I will give you 50$
Few years later the robber becomes an expert in Game Theory and Professor.

[simon]

wow, very good video. 3 mins better than a 2 hour lecture. thankyou. i will not be wasting my time with lectures anymore.

[RanRanGaming]

best explanation ever heard! super clear

[royvivat]

I'm only 15 so if this is a dumb question, ignore it, but why doesn't the robber get a payoff of -10, 000 for killing you if you DO give over the money? In that case, he would never kill you and you shouldn't ever give up the money.

[EvilFreeman]

Shouldn't it just be the case that if you give the money and he kills you, the payoff is (-1mil + 10, -99, 990)? Since obviously handing over the money doesn't make dying more awesome or make life sentence shorter. From there it's just a dominant strategy equilibrium.

The robber will never choose "Kill" logically, since the payoff is always worse, so the robber's dominant strategy is always to "Not Kill". Knowing that, you would always choose to not give money.

[bagetblue]

The robber then looks at the game theorist with a weird look, readies his pistol and hides the body. Roll a new character

[JoeTestouri]

Haha, pretty sure 99% of robbers aren't game theorist, hence you may just catch a bullet

[awesomelifeofdands]

But if he kills you then he still kan take your money...

[literallyjustmyname]

The numbers on the upper left should be -1 million, -9990, since in that scenario you give the robber money and he still kills you

[TheDennisgrass]

"Next time you get mugged by a GAME THEORIST, you know what to do."

[haretztj]

in other words, perfect nash equilibrium is a win-win situation in which both payoffs are neither bad

[JaymezF]

If I was able to watch youtube videos like this last year, I would not have failed game theory :'(

[Ahmed-jluh]

I hope you could make more videos about Game Theory.

[bionicbeaver]

I loved the explanation. Im really new to the Nash Equilibrium, so bear with me if it makes no sense at al what I am about to state here.

Given that the money isn't handed over, and the decision is at the robbers table, where you assumed a zero-payoff is better than a minus payoff given the life sentence - which makes perfect sense in an economic theory perspective.

But how does it affect the game and the equilibrium given the chance of getting shot, lets say if the robber is an drug-addict, thus not thinking straight?

Would we have to go back to the first branch of the game and state a whole new set of rules and payoff-matrix?

[mohammedalmahdidewan]

I do not know, I found in matrix formation, there is only one Nash not two.

[NiteeshSood]

is it necessary that a subgame nash equilibrium has to be an overall nash equilibrium?

What I mean is, is it possible that in a subgame there exists a nash equilibrium but when you look at the entire game that particular possibility is not a nash equilibrium?

[changzhenxie]

Actually one needs to do more when robbed to protect oneself. One cannot ensure the robber be arrested but can send signals like that police is near, to increase the probability of crime sentence. If one faces this situation in a region of poor security or cannot memorize the criminal and scene, one has better hand over the money.

[olufisayoadeleke]

for the explanation though...i just hope i get mugged by a rational game theorist