Subgame Perfect Nash Equilibrium

economicsdetective.com/
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.

[@areyes0133]

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

[@nickm.777]

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.

[@simon5543]

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

[@cravarc]

This came in SO handy. Thank you!

[@Evil0Freeman]

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.

[@GhostGlitch.]

EvilFreeman yeah, that bothered me too. Really the only Nash equilibrium should be no money no kill, but of course in reality people aren't always super logical. Also the slightly broken logic there allowed for the rest of the explanation, so it's just a bit of a flawed analogy sorta.

[@user-tx5vr2lu6e]

Old comment but in case anyone comes back: This confused me too, but I think that the gun doesn't represent killing, it represents being serious about the threat. He wouldn't kill you if you gave the money.

[@jommydavi2197]

@@user-tx5vr2lu6e I mean he said the game ends when you give the robber the money, so I suppose you dont get killed.

[@maganashaker167]

I would argue that the killing act itself basically guarantees the robber your money, so handing your money over and the robber killing you anyway is the same option.

[@bagetblue2043]

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

[@ethanp2141]

How do you find the perfect-subgame equilibrium you found by using game tree in your strategic form?

[@Ahmed-jl7uh]

I hope you could make more videos about Game Theory.

[@awesomelifeofdands]

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

[@ericchenli66]

Ok, so -10,000 + 10

[@MrMinevision1]

LOL

[@MrMinevision1]

I'm so late why am i still here commenting

[@wenweiwu3791]

Great Video, Thank u!

[@FFFQQA]

ohhhh i love your videos! thanks for your help (: !!

[@JoeTestouri]

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

[@kiwii7up803]

great solution

[@ceyhunyakupozkardes5231]

Great Video!

[@MohammedAli-fd3rl]

thank you man sincerly a student

[@milxasap7412]

Nice video brother.

[@literallyjustmyname2353]

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

[@davidconsumerofmath]

"gives"

[@MorriganSlayde]

this example was funny and helped me learn through laughing thank you

[@haretztj4682]

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

[@blind1595]

Thank you!

[@mohammedalmahdidewan8188]

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

[@bionicbeaver5188]

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?

[@GarrettPetersen]

Yes, if you wanted to account for the probability that the robber is crazy, you'd have to set it up as a Bayesian game. That's a game where "nature" plays first, secretly and randomly determining the state of the world. In this case, nature could assign a 10% probability to a game where the robber has different payoffs, and the person being robbed doesn't know which payoffs the robber has.

[@bionicbeaver5188]

+The Economics Detective Interesting! Thank you so much for the swift reply! Im gonna start digging! Take Care!

[@TheDennisgrass]

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

[@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?

[@GarrettPetersen]

Yes, an SPNE is always a Nash equilibrium. The biggest problem with Nash equilibrium is that there are often multiple of them (usually an odd number, actually) and Nash gives us no way to say which of these equilibria is the most plausible. SPNE is a way of eliminating some of the less plausible Nash equilibria.

[@NiteeshSood]

Thank you!

[@jerry_zhao]

is this a perfect information game?

[@camilleroesiamae8067]

very helpfuli thank youuu

[@JaymezF]

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

[@Valentin-oc5nh]

u failed game theory? :D

[@sdiptanil]

Hello The Economics Detective, I just have a couple of quick questions : In the table structure we have the perfect structure of both options etc. However, in the tree where is that option where " Even if I give money, the robber can kill me ? " The robber can kill me option pops up under the tree when I am not giving money. But as per the tabular structure there is also this option " Even if I give money, the robber can kill me "..which is 1st row, 1st column. Am I correct ? Secondly, in any situation is it that multiple nash equlibria can't exist and thus we need SPNE or is it that out of multiple NE, SPNE helps to find the best one ?

[@GarrettPetersen]

The "kill" option in the table is killing conditional on no money being given. If the money is given, that node is never reached.

[@sdiptanil]

So, is my understanding correct ? In the tree structure, if money is given, killing CAN'T happen. However, in the table structure, even if money is given, killing can happen. So, in the tree structure, we have 1 option less. Then, why are we going for SPNE when it is not covering all cases ? Is it just to solve the situation where we land up with multiple nash equlibria cases ?

[@GarrettPetersen]

+Diptanil Sengupta You can't kill unless he refuses to give the money. The difference between the tree and the one-stage game is that in the tree, the robber can't pre-commit to the kill strategy.

[@sdiptanil]

+The Economics Detective, thanks a lot for the explanation !

[@royvivat113]

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.

[@Triiiiangle]

Because in the game presented, if you give over the money he will never kill you.

[@kennyyip6625]

why there's no subgame on the left side?

[@ThePian0Man88]

Because the robber only has the gun for the purpose of getting the money. If the individual forks it over without arguing, the robber has the money he sought. So he won't use the gun. So the outcome will only be (-10,10) because the robber won't need to decide whether to use the gun or not. His desirable outcome has been achieved

[@omarkhan5223]

But the robber doesn't know how much money you had so if he suspects you're holding some back even if you're not, you may still get shot because he desparately needs the money, or he might shoot you because you might go to the police otherwise, this way, there's no aleby

[@dinglemingle2546]

guess Imma try that next time.

[@zes7215]

no such thing as so or storix or know or not, cpux, think do , can thnk do any nmw

[@olufisayoadeleke3290]

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

[@KevinBoykin]

Yes me too, but if every mugger was rational, they'd realize there was no point in even mugging! xD

[@esoesminombre7056]

- Hand over your money, or else...! - Before I process your request, may I ask you your stance on game theory?

[@rebeccam8631]

Post more videos!

[@odda132]

Thank you so much!

[@Borzacchinni]

hard to justify this example as he is indifferent between killing and not killing if given money, and not willing to kill if not given money

[@usmanmasood6256]

If you follow his advice, follow it completely: he says, "...the next time you get mugged by a 'game theorist', you know what to do...".

[@InTenMinutes1]

I don't mind your example but it has some subjective qualities such as '-1million' for losing a life and '-10 thousand' for wasting a life in a prison.

[@GarrettPetersen]

B R All that matters is the order of the numbers. I could have made them -1 and -2, or 7 and 5, as long as I maintained the order.

[@herrmanselcher1632]

Why does the robber not get a life sentence for killing you if you give him the money?

[@FOBWebProductions]

Why isn't giving up the money/not getting killed not a nash equilibrium, when it has the same exact pay off as giving up the money/getting killed?

[@xemiliiex]

We assume that the robber is true to his word in that his choice only applies if you don't hand over the money, so the (giving up the money, kills you) option means instead that he WOULD have chosen to kill you if you didn't hand over the money, but given that you did his choice is irrelevant.

[@changzhenxie2904]

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.

[@niveyoga3242]

Instructions not clear, I ended up being killed.

[@SilentAdventurer]

Replace his with her and you are good

[@esoesminombre7056]

But since the human brain is poor at making long run-oriented, rational decisions, the relevance of game theory in predicting decisions in the real world is limited. For instance, our robber could be primarily interested in proving her/his dominance, in which case refusing to give the money would be the worst choice.

[@final0915]

bad example

[@FalseF4CZ]

Good example if both are playing a rational game. But in reality, people are sometimes irrational. He stated if you were to get robbed by a game theorist, this is what you should do.

[@mohdip]

But hey, that's just a theory, A GAME THEORY! Thanks for watching.

[@hadassahsharon3830]

Batman's mom died though, but thanks

[@aaronsun4746]

Dislike, didn't give money and was shot and killed :( haha jk, the video was super helpful!!! tysm. None of the definitions on google made much sense.

[@Horinius]

Theory is theory. If the robber is so GOOD in this logic thing, there's no reason he's going to rob you a few bucks in the street. Either he has found a stable job or planned a bigger heist!

[@StRaWbeRrYsasha]

stubbed by a game theorist lol ...not a robber!