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More on Nash equilibrium

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In this video we expand our analysis of the prisoners' dilemma to better understand the concept and definition of a Nash Equilibrium. Created by Sal Khan.

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  • primosaur ultimate style avatar for user Jerry
    Can someone give me examples of the Nash Equilibrium in other games, such as Monopoly, Risk, Stratego, etc?
    (8 votes)
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    • mr pants teal style avatar for user Wrath Of Academy
      The Nash Equilibria in Monopoly, Risk, Chess and Go are all fairly boring - it would just consist of whatever the optimal strategy is at each turn in the game.

      Poker and Stratego however are much more complicated, because you don't know what your opponent has done. (In poker their cards are hidden from you, and in Stratego their board of pieces is turned around so that you know locations but not abilities.) In those games determining Nash Equilibria would be incredibly complicated. The NE would involve randomness - ie, in poker you should bluff some but not always, and you need to do so randomly to keep your opponent from predicting you.
      (14 votes)
  • male robot hal style avatar for user Fletch
    A Nash equilibrium is dependent on knowing that others will not change their positions. What kind of assumption is that? I don't see that as even remotely possible in the real world. If it is just a theoretical, or academic exercise that is fine, but it means a Nash equilibrium has no real world application. If that is the case what value is it?
    (4 votes)
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    • blobby green style avatar for user gutzmer23
      Fletch, in your original post you said "A Nash equilibrium is dependent on knowing that others will not change their positions", and in your response you said "It would be a reasonable assumption that every player would be constantly be trying to improve their position." These are both true, and you have basically answered your own question. Keep in mind we are working with the assumptions that

      1. both players are rational
      2. both players know all available moves and the corresponding payoffs

      Given those assumptions, it is inevitable that rational players will end up in a NE. Once there, it doesn't matter whether the players "know" they are in a NE or not, player A simply knows that given B's choice, there is no advantage to be gained from switching positions, and vice versa. (Unless of course they collude and decide to switch together, which we already mentioned)

      I agree with your sentiment that NE may not seem all that realistic in the real world, but I think it is because both assumptions are unlikely to always be true, not because it is dependent on knowing others won't change their positions.
      (11 votes)
  • blobby green style avatar for user Martin Lin
    What is the difference between Nash equilibrium and dominant strategy equilibrium??
    (3 votes)
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  • primosaur tree style avatar for user Mary
    Could there be a situation in which there is NO Nash equilibrium?
    (4 votes)
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    • blobby green style avatar for user Donna Syed
      According to Nash's mathematical proof in his famous thesis entitled, "Non-Cooperative Games" (Princeton, 1950), the answer is no. In it he proved that, ". . . a finite cooperative game always has at least one equilibrium point." The equation proof is pretty hairy but not impossible to follow.
      (2 votes)
  • blobby green style avatar for user Fabian Au
    Is there a real life situation where this can apply to?
    (2 votes)
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    • male robot donald style avatar for user bell506soccer
      Yes, When dealing with chess it is used it programs that try to predict moves of the other player, also there is more complex theory behind this and can be used in any kind of negotiations between people. when it is important for one side to try and be like the saying goes one step ahead of the other person.
      (2 votes)
  • blobby green style avatar for user daniel.corazza.89
    would the situation (confess, confess) be a nash equilibrium so long as the nr. of years are less than the 10 received in the situation (confession, denial) ?
    Meaning, would the situation (confess, confess) be a nash equilibrium even if the duration of incarceration was 9.9 years for both prisoners?
    Just testing out the extreme cases so as to understand better the logic behind it.
    (2 votes)
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    • ohnoes default style avatar for user Tejas
      Yes, if the duration of incarceration for both confessing were 9.9 years each, both confessing would still be a Nash equilibrium. It would remain so as long as it is less than the 10 years.
      (2 votes)
  • blobby green style avatar for user davis.ve
    How would you go about testing a possible Nash Equilibrium for more than one possibility or no possibility?
    (2 votes)
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  • leaf green style avatar for user cyclone975
    In normal games, players are thinking two or more moves ahead. How would this affect the idea of the equilibrium? I.e., if Bill chooses to confess, he knows Al will confess too, therefore resulting in an overall increase of punishment. I realize the example in the video demonstrates a game that is not necessarily turn based, but I was curious if allowing each player to think "two moves" ahead would change anything.
    (1 vote)
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  • leaf orange style avatar for user Nina
    Is there a way to mathematically calculate or prove that confessing is the Nash equilibrium?
    (1 vote)
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    • blobby green style avatar for user Murphy
      It's more logic based than math based.

      Look at it from person A's perspective. He can either confess or not confess. If person B does not confess, it is best for person A to confess. If person B confesses, it is still better for person A to confess. So regardless of what person B does, it is best for him to confess, so therefore he will confess.

      It works the same the opposite way. No matter what person A does, it is best for person B to confess, so therefore he will confess.

      If they could both coordinate and make sure the other does not confess, neither would confess. But that's not the situation. Each knows that no matter what the other does, it is best for them to confess, so they do. They aren't looking for the best overall situation, they are looking to make their situation the best.
      (2 votes)
  • marcimus pink style avatar for user fairview5
    If i am writing a math essay, could i use Nash Equilibrium....is it a mathematical concept, or a microenconomical concept? I think it is both, but im not sure
    (1 vote)
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    • leafers ultimate style avatar for user Caitlin Fischer
      Game theory is an area of mathematics, and Nash equilibrium is a concept of game theory, so we could call it a subset of maths. The thing is that the idea of Nash equilibrium applies very well to microeconomics. We're just applying a mathematical theory, which could be viewed in either a mathematical or economic context.
      (2 votes)

Video transcript

The idea of a Nash equilibrium is important enough that I think it deserves its own video. And you may or may not know, it's named for John Nash, who was played by Russell Crowe in the movie "A Beautiful Mind." And it's a game theoretical concept. And game theory sounds very fancy, but it really is just the theory of games. And this prisoner's dilemma that we talked about in the previous video really is a game. The different players have different strategies, and based on their interacting strategies, you end up in different states. You end up with different outcomes. And here's a definition of Nash equilibrium from Princeton. And that's a good place to get the definition, because that's where John Nash spent a good bit of his career. And it is defined, or this definition says, it's a stable state of a system that involves several interacting participants-- in our prisoners' dilemma, we had two participants-- in which no participant can gain by a change of strategy as long as all the other participants remain unchanged. So let's think about the different states of this system right over here and think of whether any of them meet this criteria. So let's say, let me number this one-- let's say that is state 1. This is state 2. This is state 3, and this is state 4 right over here. So if we are sitting, if we are sitting in state 1, can any of the participants change their strategy-- can gain by changing their strategy, assuming the other participant is constant? So if we are sitting here in state one right over here, so Al is denying and Bill is denying-- well, Al can improve his situation by changing his strategy. He can go from denying, which is the scenario here, to confessing. So Al can change his strategy and gain. Or you could go the other. Assuming Bill was constant in the denial-- or you could go the other way. If we're sitting here in state 4 and we assume Al is constant, Bill can improve his situation by going from a denial to a confession. He can go from two year to one year. So for both of those reasons-- if either of those were true, this would not be a Nash equilibrium. But both of those are true. So this is definitely not a Nash equilibrium. I gave two examples in which a participant can gain by a change of strategy as long as the other participant remains unchanged. This move was one example, and this was a move by Al, with Bill's denial constant. This was a move by Bill, with Al's denial constant. Not a Nash equilibrium. Now let's think about-- let's think about state 2. If we are sitting in state 2, assuming Bill is constant, can Al change to improve his outcome? So can Al change to improve his outcome? In state 2, Al was only getting one year. If Al goes from a confession to a denial he's going to get two years. So Al cannot change his strategy and get a gain here. So far it's looking good. But let's think of it from Bill's point of view. So if we are in state 2, if we are in state 2 right over here, and we assume Al is constant, can Bill do something that changes things? Well, sure, Bill can go from denying to confessing. If he goes from denying to confessing he goes from 10 years in prison to three years in prison. So I've given an example of a participant who can gain by a change of strategy as long as all of the other participants remain unchanged. Both of them don't have to be able to do this. You just need to have one of them for it to not be a Nash equilibrium. Because Bill can have a gain by a change of strategy holding Al's strategy constant, so holding Al's strategy in the confession, then this is not a Nash equilibrium. So this is not Nash, because you have this movement can occur to a more favorable state for Bill holding Al constant. Now, let's go to state 3. Let's think about this. So if we're in state 3, so this is Bill confessing and Al denying-- so let's first think about Al's point of view. If we assume Bill is constant in his confession, can Al improve his scenario? Well, sure. He can go from denying, which is what would have to be in state 3, to confessing. So he could move in this direction right over here. And that by itself is enough evidence that this is not a Nash equilibrium. We don't even have to think about Bill. And it's symmetric. There's actually nothing that Bill could do in this scenario, holding Al constant, that could improve things. Bill would not want to go from here to here. But just by the fact that Al could go from here to here, holding Bill constant, tells you that this is not a Nash equilibrium. Now let's go to scenario 4. And you know where this is going, because you watched the last video. But it's a little-- I'm going through it in a little bit in more detail. In state 4, they are both confessing. Now let's look at it from Al's point of view. And we're going to hold Bill constant. We're going to hold Bill unchanged. So we're going to have to stay in this column. We're going to say that, assume that Bill's confessing. From Al's point of view, if we are in state 1, can he change his strategy to get a better outcome? Well, the only thing he could do is go from a confession to a denial. But that's not going to do good. He's going to go from three years to 10 years. So Al cannot gain by a change of strategy, as long as all the participants remain unchanged. Now let's think about it from Bill's point of view. We're in this state right over here. We're going to assume that Al is constant, that Al is in the confession mode. So Bill, right now, in state 4, is confessing. His only option is to deny. But by doing that, he'll go from three years in prison to 10 years in prison. So he's not going to gain. So he, too, cannot gain. So we've just found a state in state 4 which no participant can gain by a change of strategy as long as all other participants remain unchanged. And this part is important. Because we're not saying that both can change simultaneously. You are not, in this payoff matrix, allowing a diagonal move. And so no participant can gain, neither Al nor Bill, holding the other one constant. This is a Nash equilibrium. This one right here. And this is a stable state.