More on Nash Equilibrium Looking more closely at the definition of Nash Equilibrium
More on Nash Equilibrium
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- The idea of Nash equilibrium is important enough but I think it deserves it's own video.
- And you may never know it's name for John Nash, who's played by Russel Crowe in the movie of Beuatiful Mind.
- And it's a game-theoretical concept and you know game theory sounds very fancy but it really is just theory of
- games. In this Prisonner's dilemma, that we have talked about in the previous video, really is a game.
- Different players have different strategies and based on the intearacting strategies, you can adopt in different states
- and you can put different outcomes. And here's a definition of Nash equillibrium from Princeton.
- That's a good place to get a definition, because that's were John Nash spent a good bit of his career.
- And it's defined or this definition - it's a stable state of a system, but involved several interacting participants in
- our Prisoner's dilemma. We had 2 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 wheter 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're sitting in state 1
- can any of the participants change the strategy, can gain by changing the strategy assuming that other
- participants just constant. So if we're sitting here in state 1 right over here, so Al is denying and Bill is denying.
- But Al can improve his situation by changing his strategy. He can go from denying, which is a scenario here
- to confessing. So Al can change his strategy and gain or you could gain the other assuming Bill was consent into
- deny or you can go the other way. If we're sitting in state 4 and we assume all is consent Bill can improve his
- situation by going from deny to confession. We can go from 2 year to 1 year. So for both of those reasons if I
- do this true but this would not be Nash equilibrium but both of those are true so this is definitely not a Nash equillibrium.
- I give 2 examples in which A participant can gain by change of strategy as long as the other participant
- remains unchanged. This move was one example and this is a move by Al, what Bill's deny to consent.
- This was a move by Bill what Al's deny to consent. Not a Nash equillibrium. Now let's think about state 2.
- If we're sitting in state 2 can assuming Bill is constant can Al change to improve his outcome. In state 2 Al is
- only getting 1 year. If Al goes from confession to deny he's going to get 2 years. So Al cannot change his
- strategy and get again here. So sofar it's looking good. But let's think from a Bill's point of view. So if we're in
- state 2 right over here and we assume Al is constant can Bill do something that changes things. We're sure
- Bill can go from denying to confessing. If he goes from denying to confessing he goes from 10 years
- in prison to 3 years in prison. So I give an example of a participant who can gain by change of strategy
- as long as all other participants remain unchanged. Both of them don't have to be able to do this.
- You just need one of them for to not to be Nash equillibrium. Because Bill can have again
- by change of strategy holding Al's strategy constant. So holding Al's strategy in the confession -
- that is not a Nash equillibrium. Because you have this movement can a curve to a more favourable
- state for Bill holding Al constant. Now let's go to state 3. Let's think about this. So for in state 3, so this is
- Bill's 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. We're sure he can go from denying which is what have to be
- about we're in state 3 to confessing. So he can move in this direction right over here. And that by self is
- not enough evidence this is not a Nash equillibrium. We don't need to think about Bill, that's the metric,
- we actually nothing that Bill can do in this scenario holding Al constant. That could improve things
- Bill would not go from here to here but just by the fact that Al can go from here to here holding Bill
- constant tells you that this is not a Nash equillibrium. Now let's go to scenario 4 and you know where this
- is going as you watch the last video. But now it's going to a little bit in more detail. In state 4 they're both
- confessing. Now let's look at from Al's point of view and we're holding Bill constant. We're holding Bill
- unchanged. So we stay in this corn, we assume that Bill is confessing. From Al's point of view if we're
- in state 1 can he change his strategy to get better outcome? But the only thing he can do is go
- from confession to deny but that is not going to be good. He going to get 3 to 10 years. So Al cannot
- gain by change of strategy as long as all of the participants remain unchanged. Now let's think about
- 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 will go from 3 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 particapants can gain by change of strategy as long
- as all participants remain unchanged. And this part is important because we're not saying that both can
- change simultaneously. You were not in this pay-off matrix allowing a diagonal move. And so no participant
- can gain neither Al nor Bill holding the other one constant. This is a Nash equillibrium. This one right here
- and this is a stable state.
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