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.
