 Hi everyone, this is Alice Gao. In this video, I'm going to start talking about the concept of Pareto optimality. Let's first recall the last game that we've discussed. I've copied this slide from the last lecture. We talked about this game where Alice and Bob are choosing between two activities, dancing and running. And if we look at the payoff matrix based on our intuition, this is a coordination game. Alice and Bob want to coordinate on choosing the same activity. They're both happier when they're doing the same thing. So they can go dancing together or they can go running together. Both of these outcomes are pretty good. And this intuition is indeed confirmed by the solution concept Nash equilibrium. If we use Nash equilibrium, then it tells us that both of these outcomes are stable and are Nash equilibria of this game. But our intuition tells us more than this. It also tells us that if we look at the payoffs, if we look at the utilities, then coordinating on dancing seems better than coordinating on running. In fact, if they coordinate on dancing, then both of them receive the highest utility possible. And that it's really good, right? Unfortunately, Nash equilibrium does not say anything about this, about the fact that going dancing seems to be better than going running. In fact, whenever we have a game where there are multiple Nash equilibria, the Nash equilibrium concept doesn't say anything about which equilibrium is more likely to happen in the future. It simply is incapable of giving us that information. So in this case, is there any way to formalize our intuition? Is there any way to compare different Nash equilibria and say maybe one of them is better than the others? Turns out we can use the concept called Pareto dominance and Pareto optimality to compare different outcomes of a game. On this slide, I have some definitions. I'll first define Pareto dominance, then talk about Pareto optimum outcome, and then I'll discuss some nuances in these definitions. So first of all, Pareto dominance, Pareto dominance is a way for us to compare multiple outcomes of a game. So in this definition, we are comparing outcome O with outcome O prime. Now, what do I mean when I refer to an outcome? An outcome of a game is a combination of actions of the players, one action for each player. So in this case, for our two by two game, let me come back here for a quick second, we have four possible outcomes, dancing, dancing, dancing, running, running, dancing and running, running, right? So Pareto dominance gives us a way of comparing two outcomes of the game. So Pareto dominance says that we can say outcome O Pareto dominates another outcome O prime if the following two properties are both satisfied. The first property is that every player weekly prefers O to O prime. Okay. And the second player, the second property is that at least one player strictly prefers O to O prime. If I write out some logical formulas to express these two statements, they would look like this. So for every player, the player's utility in outcome O is greater than or equal to the player's utility in the outcome O prime. And then for the other, for the second requirement is there exists a player such that the player's utility in outcome O is strictly better than the player's utility in outcome O prime. So this property gives a way of comparing two outcomes, but notice that this comparison is a pretty strong relationship, right? It's basically saying everyone thinks O is better and at least one person thinks O is strictly better. Right. So we could encounter situations where if you have two outcomes, it's possible that O does not Pareto dominate O prime and also O prime does not Pareto dominate O. Okay. So we cannot establish this relationship for any pair of outcomes. Next, given that we have defined a way of comparing two outcomes using Pareto dominance, now we're ready to define what does it mean for an outcome to be Pareto optimal. So from the name, you can see that here we're trying to, in some sense, define an outcome that's the best among all the possible outcomes of a game, right? So an outcome O is Pareto optimal if only if no other outcome O prime Pareto dominates O. So there does not exist another outcome that Pareto dominates this outcome O. If this is true, then the outcome O is Pareto optimal. This definition is pretty tricky because in your mind, before I actually gave you the definition, you probably thought that O, if I want to define an outcome being the best, shouldn't I just say the outcome is the best if it Pareto dominates every other outcome, right? Notice that that's different from the definition that I gave. So to point out the subtle difference, I've given you an additional question at the bottom of the slide. So look at the two statements at the bottom and think about what's the difference between the two. So the first statement is the stronger statement that we could have made. So this statement says, an outcome O Pareto dominates every other outcome, right? This is one way we can use to define that O is the best outcome in some sense. And then the statement we actually made was that an outcome O is Pareto optimal if it is not Pareto dominated by any other outcome. For now, you can take some time to think about the difference between these two statements, but as we see more examples, the difference between the two will become much more clear. Let's now practice this concept on the dancing or running game. So given this dancing or running game, how many of the four outcomes are Pareto optimal? Now this question may already sound weird, because well, you might have thought that Pareto optimal is a way to define the best outcome, right? So why are there, why could there be multiple best outcomes? Well, it turns out that's possible a game could have multiple Pareto optimal outcomes. Okay, now if you accept that, then let's look at this example. How many of the four outcomes are Pareto optimal? Think about this yourself, apply the definition, carefully apply the definition, and then keep watching for the answer. This answer is relatively short, so I'm going to include it in this video. The correct answer is B, there's only one Pareto optimal outcome, and that outcome is dancing dancing. This is a simple, relatively simple application of the definition, because in this case, dancing dancing does Pareto dominate every other outcome. So in other words, if we compare dancing dancing and running running, dancing dancing Pareto dominates running running. So therefore running running does not Pareto dominate dancing dancing, right? Similarly, for example, dancing dancing Pareto dominates dancing running. So dancing running does not Pareto dominate dancing dancing, right? So you can do have the same reasoning for all three other outcomes. Therefore dancing dancing is the Pareto optimal outcome. We will see more examples where it's a little trickier to apply this concept, but for this question, it's relatively straightforward. That's everything for this video. Thank you very much for watching. I will see you in the next video. Bye for now.