In this video, we're going to look at how to compute the Mixed Nash Equilibria of a

normal form game. And, in particular, we're going to go

through the example of battle of the sexes.

So, what you've seen so far about equilibria, kind of suggests that it's

really easy to come up with an equilibrium,

but in fact, as games get big and general, sometimes it can be pretty

tricky to find the equilibria of a game. And Nash's theorom is kind of a funny

theorom, because it tells us that something exists, but it doesn't tell us

how to find it. It just tells us that it has to be there.

It's a nonconstructive argument. So, what I'm going to tell you today is

sort of a starting point to finding an equilibrium, which is enough that it

works in small games. and, in fact, you can turn this into a

general algorithm, but not necessarily the most efficient or, or insightful way

of finding equilbria. So, what I want to tell you today is that

it's easy to compute a Nash equilibrium if you can guess what the support of the

equilibrium is. So, recall what a support is.

A support is the set of pure strategies that receive positive probability under

the mix strategy of the play, of the players.

So, a an equilibrium support is a set of actions that occur with positive

probability. For example, that might be the support of

an equilibrium. So, for battle of the sexes, let's guess

that the support, whoops, let's guess that the support of the

equilibrium is all of the actions. So let's look and intuitively, that, if

there's going to be a mixed strategy equilibrium of this game, that, that

looks like what it should be. So let's guess that that's the support

and then try to reason about what the equilibrium would have to be given that

support. So let's just introduce some notation to

make this work. Let's let player 2 play B with

probability p, and F with probability 1 - p.

Now, if player 1 is going to best respond to this mixed strategy whatever it is and

be playing a mixed strategy in response, then we can reason that player 2 must

have set p and 1-p in a way that makes player 1 indifferent between his own

actions, B and F. So this is an important point in

reasoning about how mixed strategies work, so I encourage you to stop the

video at this point and just think about why that would be true before I tell you

the answer. So the reason why player 1 needs to be in

oh, I don't have the answer on this slide, I'll just tell you.

the reason why player 1 needs to be indifferent is that he's playing himself

a mixed strategy, which means some of the time he's playing

B and some of the time he's playing F. Right? Because these are both in the

support, they both get played with nonzero probability, and if this is an

equilibrium, then this is the best response that player 1 is playing.

Well, if player 1 can play B some of the time and F some of the time and be

playing a best response, he must be indifferent between playing B and F.

If he's not indifferent, if let's say B is better,

then he could get even more utility by reducing the amount of probability he

puts on F and increasing the amount of probability he puts on b.

And in fact, he could get the most utility by putting absolutely no utility

on F and just all of the utility on B. So the only way that he would actually

want to play a mixed strategy is if it's just the same for him to play B and F.

So that means that we can reason that player 2 has set his probabilities p and

1 - p in such a way that it makes player 1 indifferent.

And the reason why we've bothered to think about this is we can actually write

that down in math. So we can say the utility for player 1 of

playing B, and here, I'm kind of abusing notation, you should really understand

this to mean the utility for player 1 of playing B given that player 1 plays p, 1

- p is equal to the utility of player 1 for playing F, again, given that player 2

plays p, 1 - p. So, then we can, we can simply expand

this out taking into account the actual payoffs of the game and learn something

useful. So we can say, if it's the case given the

same probabilities p and 1 - p, that player 1 is indifferent to playing B and

playing F. Then, that means well when he plays B,

then he gets 2 with probability p, and he gets 0 with probability 1 - p.

So that's what we have written down here. And when he plays F, he gets 0 with

probability p, and 1 with probability 1 - p, and that's what we have written down

here. And now we just have a simple equation

and one variable, so if we rearrange it, we end up concluding that the only way

that player 1 can be indifferent between playing B and, B and F is if p = 1/3.

In the same way, we can reason that if player 2 was randomizing which we had

just assumed that he was, then player 1 must make him indifferent.

And, why is player 1 willing to randomize? Because he's simultaneously

being made indifferent by player 2. So, so now lets say that player 1 plays B

with probability q and place F with probability 1 - q.

So, now we can just do the same thing as before, where, again, you should

understand this to mean q, 1 - q, and likewise here and we can say, we can just

expand it out in the same way. So if player 2 plays B, then he gets 1

with probability q, and he gets 0 with probability 1 - q.

And if he plays F, he gets 0 with probability q, and 2 with probability 1 -

q We now again have an equation in one variable,