# 1-8 Nash Equilibrium of Example Games

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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 o ften 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 wanna 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,
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 wanna 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.