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. why is that? Well, if either of them goes to the Battle of the Titans, then the other one wouldn't want to go there to, because if they go to a different one, they would get zero rather than whatever they get here, one or two depending on whether the husband or the wife. And then conversely, on the, on the flower watching movie, flower growing movie and so, in both cases, they, the best response is to go to the movie selected by the other party. So, on the face of it, it looks very similar to the game of pure coordination that we have here, but we do see a slight difference here, and we'll revisit that later on when we speak about not pure strategies, but mixed strategies. Here is here is another example, the last one we'll look at this is a game called matching pennies. Imagine each of us, two players, needing to decide, needing to decide on some side of a of a coin, heads or tail. If we decide on the same sides, heads or tail, but we decide on the same one then, then I win. If we decide on different sides, you heads and me tails or vice versa, then you win. and so we see this here, if we both decide on heads or we both decide on tails, I win, and otherwise, you win. By winning I mean I get one and you get minus one, so this is a zero-sum game. The sum of our payoffs is zero. what is a pure strategy Nash Equilibirum here? Well, let's think about it. Suppose I pick head, what is your best response? Well, your best response then is to pick tails, because you get one rather than minus one. But if you pick tails, then my best response is now to play tail, because I want to coordinate with you, because then, I will get one rather than the minus one that I would be getting here. But, now if I play tails, you'd rather play heads, because you'd get one rather than the minus one you're getting here. But again, if you're playing a tails, I want to if you're playing heads, I want to play heads to match. So we have this cycle where the best responses are leading us in the cycle. And so there is no pure strategy Nash equilibrium in this game of matching pennies.
