Let us now look at some examples and, of games and Nash equilibria in those games.

So here's the first game, a familiar game, this is of course the prisoner's

dilemma. The if both prisoners cooperate and then

they get a light punishment, and if they do not cooperate, they get a most severe

punishment. If the one cooperates and the others does

not, then the cooperator gets a terrible punishment and the one that does not

cooperate gets off scot, gets off scotfree.

and of course this game has a dominant strategy to defect no matter what the

other agent does, you are better off not cooperating.

And so of course, the only dominant strategy outcome is this one of both

defecting, and indeed, that is the only Nash

equilibrium in this game. So, it's a Nash equilibrium, it's the

best response. If the other person defects, then it's

the best response to defect but in fact, it's much stronger than that, it's best

to defect no matter what the other the other agent does.

So this is an example of one unique Nash equilibrium that happened to be a very

strong one, a dominant strategy, Nash Equilibrium.

So so here's another game. This is a game of pure coordination.

I think of it as walking towards each other on the sidewalk and you both can

decide whether to go to your respective lefts or respective rights.

In both cases, you will do fine and you will not collide,

and of course, if you miscoordinate, if you one goes to the left and the other

to the right, you will collide. So this is a natural game.

And, in fact, you see that you have two Nash equilibria,

the one that I wrote down here. If one one of the players go to the left,

it's the best response to go to the left. And conversely, if the the other player

goes to the right, you're best off going to the right as well.

And the others are not Nash equilibria. So here's an example of a game where

there are two Nash equilibria or two specifically pure strategy Nash

equilibria. Again, we'll discuss why we call these

pure strategy later on. Here's a very different game.

this is often called the game of the battle of the sexes.

Imagine a a couple and they want to go to every two movie and they are considering

two movies. One of them, a a very violent movie

Battle of the Titans, and the other, a very relaxed movie about flower growing,

call this B and F. the wife of course, would prefer to go to

Battle of the Titans, and the the husband would prefer to watch flower growing.

But, more than anything else, they would want to go together and so here are the

paths. If they both go to Battle Of The Titans,

then they're both probably happy the wife more than the husband.

If they go, both go to the flower growing movie, then the husband is slightly

happier than the wife, but if they go to different movies

neither of them was happy. That's that's, that's the that's, that's

the the game. how many how many equilibria we have

here? Well again we have two pure strategy Nash equilibria.