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This video is going to tell you about the concept of Pareto Optimality.

Â So far, we've thought about some canonical games from Game Theory and

Â we've thought about how to play them, but we've really been taking the player's

Â perspective. We've been thinking about what, what is

Â the right thing to do in a game? Now I'd like to instead step back and think about

Â the games from the perspective of kind of an outside observer looking in and trying

Â to judge what's happening. And the question that I'd like to ask is,

Â is there a sense in which I can say that some outcomes of a game are better than

Â other outcomes? I, and I'd actually like, I'd like to encourage you to pause the

Â video at this point and just think about this for yourself,

Â see if you can come up with an answer before I, I tell you what my answer is.

Â Well, let me give you a bit of a hint. you can't say that one agent's interests

Â are more important than another agent's interests, because I don't know how

Â important the different agents are, and actually, it turns out, I can't even what

Â scales their utilities are expressed in. There, there isn't necessarilly a common

Â scale for utility between the different agents.

Â And so, in a sense, the, the problem of evaluating an outcome of a game is kind

Â of like trying to find the payoff maximizing outcome when I'm going to be

Â paid an amount in different currencies and I don't know what these currencies

Â are. So, you can kind of think of the outcome

Â of the game as an outside observer just interested in kind of social good of the

Â participants, as kind of being like an outcome where I get player one's pay,

Â player one's pay off in currency one, and I get player two's pay off in currency

Â two, and nobody can tell me what the exchange rate is between currency one and

Â currency two. Now that I've made these a little bit

Â more concrete, let me again invite you to think about whether there is a way that I

Â can Identify outcomes that I would prefer one to another.

Â Well, here is, here is a way we can make this work.

Â we can't do it all the time, but sometimes there's an outcome o that's at

Â least as good for everybody, as some of the outcome o prime.

Â Remember, an outcome is like a cell of the matrix game.

Â So, I got this matrix game and there's some outcome o, which is at least as good

Â for everybody as some other outcome o prime.

Â And furthermore, there's some agent who strictly prefers o to o prime.

Â Well, in that case, I should be able so, so they, let me actually make an example

Â of this. So o might be that player one gets seven

Â units of utility and player two gets eight.

Â And, o prime might be the player one gets seven units of utility and player two get

Â two units of utility. In this case, o is at least as good for

Â everybody and it's, because it's equal for player one and it's strictly better

Â for somebody, strictly better for player two. So, in this case it seems reasonable

Â to say that an outside observer should feel that outcome o is better than

Â outcome o prime. And technically, the way we say this is

Â that outcome o Pareto-dominates o prime. [SOUND] Well, now I can define this

Â concept of Pareto-optimality. An outcome O star is Pareto-optimal If it

Â isn't Pareto-dominated by anything. So that, that's kind of a hard definition

Â because it's defined in negative terms. Let me say it again.

Â An outcome o star is Pareto-optimal if it isn't Pareto-dominated by anything else.

Â So there's nothing else that I can prefer to it.

Â [SOUND] So let's test our understanding of this definition by asking a couple of

Â questions. Is it possible for a game to have more

Â than one Pareto-optimal outcome? As always let me encourage you to think

Â about this for a second before I answer it.

Â of course it is, because it's possible for two outcomes to neither

Â Pareto-dominate each other. If for example, all payoffs in the game

Â are the same, if I have a game where everyone gets a

Â payoff of one no matter what happens, then nothing dominates anything else,

Â because domination requires somebody to strictly prefer something to something

Â else. So this game has more than one

Â Pareto-optimal outcome. Something else I can ask is, does every

Â game have at least one Pareto-optimal outcome or is it possible that just

Â nothing will be Pareto-optimal? Well let's you think about it for a second,

Â but the answer is yes. Every game has to have at least one

Â Pareto-optimal outcome. This is easy to see, because in order for

Â something to not be Pareto-optimal, it has to be dominated by something else.

Â So, in order for, there to be no Pareto-optimal outcomes in a game, we

Â would need to have a cycle in Pareto-dominance.

Â We would need to have it be the case that everything is Pareto-dominated by

Â something different. And it's pretty easy to persuade yourself

Â that we can't have cycles with pareto dominance, the reason we cant have

Â cycles. Is just the way the pareto dominance is

Â defined. That in order for something to be

Â Pareto-dominated, it has to be at least as good for everybody and strictly

Â preferred by somebody. And I'll leave this to you to think

Â about, but, but that definition implies there

Â can't be cycles in the Pareto-dominance relationship.

Â [SOUND] So, finally let's look at our example games that we've thought about

Â and identify Pareto-optimal outcomes. And in each case, I won't say this every

Â time but, I encourage you to pause the video,

Â when I've put up a game, think for yourself about what the Pareto-optimal

Â outcomes are and then I'll identify them for you.

Â So, first of all, we have the coordination game,

Â and here, these two outcomes are both Pareto-optimal.

Â [SOUND] In the battle of the sexes game, these 2 outcomes, again, are

Â Pareto-optimal, the change in payouts here doesn't,

Â doesn't make a difference. In the matching tennis game, this ones a

Â bit trickier, I'll, let you think about it for a

Â minute. Every outcome is Pareto-optimal, because

Â there's no pair of outcomes where everybody likes them equal, likes the two

Â outcomes equally well. There's always kind of a strict trade off

Â that happens because the game is zero-sum.

Â And this is generally true of zero-sum games, that every outcome in a zero-sum

Â game is going to be a Pareto-optimal. Finally, here we have the prisoner's

Â dilemma game, and let me also let you think about this one.

Â Turns out here, all but one outcome is Pareto-optimal.

Â This outcome is not Pareto-optimal because it is Pareto-dominated by this

Â outcome. And now, I'm ready to give you a punch

Â line that we've been building to for a while about the prisoners dilemma game.

Â Here is why the prisoner's dilemma is such a dilemma.

Â The Nash equilibrium of the prisoner's dilemma, is which in fact is a Nash

Â equilibrium in dominant strategy, so it's the strongest kind of Nash

Â equilibrium there is. They're not, the, there's a Nash

Â equilibrium in this game. In fact, everybody should play this

Â equilibrium even without thinking about, even without knowing what the other

Â person is going to do. I can be sure that I should play my

Â strict dominance strategy in this game, get to this outcome, and that is the only

Â non-Pareto optimal outcome in this game. So, almost everything in this game is

Â kind of good from a social perspective and the only other thing in the game is

Â the thing that we strongly predict ought to happen.

Â So, that's why we think the prisoner's dilemma is such a dilemma.

Â