More on Nash equilibrium

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 it's a game theoretical concept and you know 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 this 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 prisoner's dilemma we had two participants in which no 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 they are 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 if we are sitting in state 1 can any of the participants change could change their strategy can gain by changing the 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 Al can have 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 with the other way zooming bill was constant in the denial or you could go the other way if we're sitting here in state four you're sitting here in state four and we assume Alice 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 Nash equilibrium but both of those are true so this is definitely not a Nash equilibrium so not not a Nash equilibrium I gave two examples in which a participant can can gain by a change of strategy as long as the other participant remains unchanged this move was one example and this this was a move by Al with bills denial constant this was a move by Bill with Al's denial constant not a Nash equilibrium now let's think about now let's think about state two if we are sitting in state two if we are sitting in state two can assuming bill is constant can al change to improve his outcome so can L change to improve his outcome in state two al is 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 cannot change his strategy and get again here so so far it's looking good but let's think of it from Bill's point of view so if we are in state two if we are in state two right over here and we assume a Liz 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 3 years in prison so I've given an example of a participant who can gain by a change of strategy as long as all 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 strategy in the confession then this is not a not 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 this is Bill confessing and Al denying so let's first think about Al's point of view if we 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 three two confessing so he could move 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 its 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 this is not a Nash equilibrium now let's go to scenario four and you know where this is going because you watch the last video but now it's a little I'm going through it a little bit in more detail in state four 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 gonna have to stay in this column we're going to say that assume that bill is confessing from Al's point of view if we are in state one 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 ten years so al al can't al cannot gain by a change of strategy as long as all their 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 Alice constant that al is in the confession confession mode so bill right now in state four is confessing his only option is to deny but by doing that he'll go from three years in prison to ten years in prison so he's not going to gain so he too cannot gain so we've just found a state in state four in 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 be aware you are not in this payoff matrix allowing a diagonal move and so no participant can gain need neither Al nor bill holding the other one constant this is a Nash Nash equilibrium equilibrium this one right here and this is a stable state