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.