This video will explain to you the intuition behind a new game theoretic

solution concept called Correlated Equilibrium.

Let's think again about the battle of the sexes game.

And let's think about its nash equilibria.

Remember there's a nash quilibria where both players play B.

There's a nash equilibria where both players play F.

And, there's another nash equilibrium, where both players randomize according to

these probabilities. And that means, that all 4 of the

outcomes can happen. So, both of these outcomes can happen,

but these ones can happen as well. Now, lets think about, intuitively, what

makes the most sense to us, about the battle of the sexes.

It's really that this outcome would happen half the time and this outcome

would happen half the time. Right, if you and your partner really did

have different preferences about what to do and you wanted to be together, you

would work it out where you each went half the time.

And something that might have struck you about Battle of the Sexes is that it

doesn't give us a way to say that that's a stable outcome.

And the reason why we don't say its stable is that, if both of these things

can happen, then both players have to be playing mixed strategies with full

support. And that means the miscoordination

actions have to be possible as well. Well, here's another example that we can

think about that will give us some intuition for why the what's really going

on here and how we can do something about it.

So this is the traffic game. So this is a marvelous situation where 2

cars are coming together at an intersection and they have to decide

whether to wait for the other person to go through first or whether to just go

through. Well, if, if one of them goes, and the

other one waits, then the one that goes gets the most utility, and the one that

waits gets a bit less utility. If they both wait, they're both strictly

less happy, because they both waited, and they still have to decide what to do.

If they both go, the worst of all things happens because

they crash into each other. So, in total, we have 2 pure strategy

equilibria, that are again asymmetric, kind of like in Battle of the Sexes.

And of course it's also possible to have a mixed strategy equilibrium here.

But let's think about what would really happen in the world, when this kind of

situation where we have an intersection and there's the possibility that cars

might collide. What we really do is we have a traffic

light. Now, let's think about what a traffic

lights mean game theoretically. It's a fair randomizing device that

recommends actions to the agents. It tells one of them to go, it tells the

other one to wait, and it's fair in the sense that at different times it makes

different recommendations. Well, the benefits here are that the

negative payoff outcomes are completely avoided.

We can achieve something fair, and in general, although not in this example,

it's possible to end up with a sum of social welfare that exceeds that that can

be achieved in any Nash Equilibrium. Well, we can use the same idea to achieve

the fair outcome in the battle of the sexes game.

So we could have a situation where the husband and wife flip a coin and

depending on how the coin comes up they go together either to the ballet or to

the football. Well this is essentially the idea of a

correlated equilibrium. A correlated equilibrium is a randomized

assignment of action recommendations to the agents.

Such that everybody wants to follow the action recommendations.

So, we have some randomizing device, that tells me b some of the time, and f some

of the time. And, it's a, and A correlated way,

potentially correlated way tells you b some of the time and tells you f some of

the time. So flipping a coin where if it's heads we

both get the recommendation b, if it's tails we both get the recommendation f,

is a randomizing device like this, although it's possible for it to be more

complicated. And in principle, if it comes up heads

and we understand that to mean that we're both getting the recommendation B, this

doesn't compel us both to go to the ballet.

We still get to freely decide how to interpret that recommendation,

but it's a correlated equilibrium if neither of us would want to deviate from

the recommendation. And you can see in this example with

battle of the sexes that indeed neither of us would want to deviate.

Because if the other is following the recommendation, and I deviate, then I'm

just going to get a pay off of 0 instead of getting a positive pay off.

So, a correlated equilibrium is any such randomized assignment of these possibly

correlated action recommendations that leaves nobody wanting to deviate.

It. It's a generalization of the idea of Nash

Equilibrium, because if these action recommendations are not correlated at

all, then we just get back to mixed strategies like we had before.

So we can capture any Nash Equilibrium this way.

But we can also get new things like we've just seen.

So it's this strict Kind of weakening of the concept of Nash equilibrium and it

includes more things and it can get us these kinds of nice fair outcomes.