This video is going to tell you about the concept of Pareto Optimality. So far, we've thought about some canonical games from Game Theory and we've thought about how to play them, but we've really been taking the player's perspective. We've been thinking about what, what is the right thing to do in a game? Now I'd like to instead step back and think about the games from the perspective of kind of an outside observer looking in and trying to judge what's happening. And the question that I'd like to ask is, is there a sense in which I can say that some outcomes of a game are better than other outcomes? I, and I'd actually like, I'd like to encourage you to pause the video at this point and just think about this for yourself, see if you can come up with an answer before I, I tell you what my answer is. Well, let me give you a bit of a hint. you can't say that one agent's interests are more important than another agent's interests, because I don't know how important the different agents are, and actually, it turns out, I can't even what scales their utilities are expressed in. There, there isn't necessarilly a common scale for utility between the different agents. And so, in a sense, the, the problem of evaluating an outcome of a game is kind of like trying to find the payoff maximizing outcome when I'm going to be paid an amount in different currencies and I don't know what these currencies are. So, you can kind of think of the outcome of the game as an outside observer just interested in kind of social good of the participants, as kind of being like an outcome where I get player one's pay, player one's pay off in currency one, and I get player two's pay off in currency two, and nobody can tell me what the exchange rate is between currency one and currency two. Now that I've made these a little bit more concrete, let me again invite you to think about whether there is a way that I can Identify outcomes that I would prefer one to another. Well, here is, here is a way we can make this work. we can't do it all the time, but sometimes there's an outcome o that's at least as good for everybody, as some of the outcome o prime. Remember, an outcome is like a cell of the matrix game. So, I got this matrix game and there's some outcome o, which is at least as good for everybody as some other outcome o prime. And furthermore, there's some agent who strictly prefers o to o prime. Well, in that case, I should be able so, so they, let me actually make an example of this. So o might be that player one gets seven units of utility and player two gets eight. And, o prime might be the player one gets seven units of utility and player two get two units of utility. In this case, o is at least as good for everybody and it's, because it's equal for player one and it's strictly better for somebody, strictly better for player two. So, in this case it seems reasonable to say that an outside observer should feel that outcome o is better than outcome o prime. And technically, the way we say this is that outcome o Pareto-dominates o prime. [SOUND] Well, now I can define this concept of Pareto-optimality. An outcome O star is Pareto-optimal If it isn't Pareto-dominated by anything. So that, that's kind of a hard definition because it's defined in negative terms. Let me say it again. An outcome o star is Pareto-optimal if it isn't Pareto-dominated by anything else. So there's nothing else that I can prefer to it. [SOUND] So let's test our understanding of this definition by asking a couple of questions. Is it possible for a game to have more than one Pareto-optimal outcome? As always let me encourage you to think about this for a second before I answer it. of course it is, because it's possible for two outcomes to neither Pareto-dominate each other. If for example, all payoffs in the game are the same, if I have a game where everyone gets a payoff of one no matter what happens, then nothing dominates anything else, because domination requires somebody to strictly prefer something to something else. So this game has more than one Pareto-optimal outcome. Something else I can ask is, does every game have at least one Pareto-optimal outcome or is it possible that just nothing will be Pareto-optimal? Well let's you think about it for a second, but the answer is yes. Every game has to have at least one Pareto-optimal outcome. This is easy to see, because in order for something to not be Pareto-optimal, it has to be dominated by something else. So, in order for, there to be no Pareto-optimal outcomes in a game, we would need to have a cycle in Pareto-dominance. We would need to have it be the case that everything is Pareto-dominated by something different. And it's pretty easy to persuade yourself that we can't have cycles with pareto dominance, the reason we cant have cycles. Is just the way the pareto dominance is defined. That in order for something to be Pareto-dominated, it has to be at least as good for everybody and strictly preferred by somebody. And I'll leave this to you to think about, but, but that definition implies there can't be cycles in the Pareto-dominance relationship. [SOUND] So, finally let's look at our example games that we've thought about and identify Pareto-optimal outcomes. And in each case, I won't say this every time but, I encourage you to pause the video, when I've put up a game, think for yourself about what the Pareto-optimal outcomes are and then I'll identify them for you. So, first of all, we have the coordination game, and here, these two outcomes are both Pareto-optimal. [SOUND] In the battle of the sexes game, these 2 outcomes, again, are Pareto-optimal, the change in payouts here doesn't, doesn't make a difference. In the matching tennis game, this ones a bit trickier, I'll, let you think about it for a minute. Every outcome is Pareto-optimal, because there's no pair of outcomes where everybody likes them equal, likes the two outcomes equally well. There's always kind of a strict trade off that happens because the game is zero-sum. And this is generally true of zero-sum games, that every outcome in a zero-sum game is going to be a Pareto-optimal. Finally, here we have the prisoner's dilemma game, and let me also let you think about this one. Turns out here, all but one outcome is Pareto-optimal. This outcome is not Pareto-optimal because it is Pareto-dominated by this outcome. And now, I'm ready to give you a punch line that we've been building to for a while about the prisoners dilemma game. Here is why the prisoner's dilemma is such a dilemma. The Nash equilibrium of the prisoner's dilemma, is which in fact is a Nash equilibrium in dominant strategy, so it's the strongest kind of Nash equilibrium there is. They're not, the, there's a Nash equilibrium in this game. In fact, everybody should play this equilibrium even without thinking about, even without knowing what the other person is going to do. I can be sure that I should play my strict dominance strategy in this game, get to this outcome, and that is the only non-Pareto optimal outcome in this game. So, almost everything in this game is kind of good from a social perspective and the only other thing in the game is the thing that we strongly predict ought to happen. So, that's why we think the prisoner's dilemma is such a dilemma.
