# 1-10 Pareto Optimality

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This video is gonna 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 gonna 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
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, cuz 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.
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. 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 a sk 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. 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. In the battle of the sexes
game, these two 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 gonna 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.