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So this video is going to describe an important property that some games have,

which is called dominant strategies. To begin with, I'm going to start using

this word strategy, that we haven't defined yet, and indeed you're not

going to get a definition for a little while.

So, to begin with, when I use the word strategy, I want you to understand this

just to mean choosing some action. This name in the end is going to be what

we call a pure strategy and it's going to turn out there's another kind of strategy

that I haven't told you about yet. And everything in this lecture is also

going to apply to that kind of strategy, but it's not going to matter for you

right now. So let's just understand strategy to mean

choice of action. So, let's let SI and SI prime be 2

different strategies that player I can take.

And lets let S minus I be the set of all of the other things, that everybody else

could do. I'm going to define two different

definitions of what it means to say the SI strip,

dominates S prime I. So first we have the notion of strict

dominance, and here I'm going to say that SI strictly dominates S prime I, if it's

the case that for every other strategy profile of the other agents.

In other words, for every other thing that they could do, for every other joint

set of actions that they could take, the utility the player I gets where he plays

SI is more than the utility that I gets when he plays S prime I.

By. So, in other words, it might matter to

player i what everybody else does. That might effect his utility,

but it will always be the case that he's happier when he plays s i than he is when

he plays s prime i. And in fact, he's strictly happier

because we have a strict inequality here. So he's going to get strictly more

utility By playing SI, then by playing S prime I.

That means that SI strictly dominates S prime I.

Now, we have another notion of dominance, which I call very weak dominance.

It's almost the same definition as you would have noticed the only difference

here is that I have an weaken equality instead of a strict inequality and so

what this is saying is no matter what everybody else does I'm always at least

as happy playing as I, as I am playing as primate and when that's true I say.

The SI very weakly dominates S prime I. Now, you might wonder why I have this

name very weak, that's because this condition even allows for equality.

So even if it's the case that SI and S prime I are always exactly the same as

each other, I'm still allowed to say that SI dominates SI prime.

And that sounds like a strong thing to say about equalities so we soften it by

saying it's very weak dominance. Now in fact there are also some other

kinds of dominance that kind of live in between these two, that are not quite as

strong as strict dominance and not quite as weak as very weak dominance, but

they're not important for us right now so I won't mention them.

Well, what, what is so important about dominance? Intuitively, when a strategy,

d-, when one strategy dominates another strategy, then I don't really have to

think about what the other agents are going to do in order to decide that i

prefer to play SI than to play SI prime, because I know that my utility is never

worse by playing SI so, regardless of the kind of dominance.

It's sort of a good idea for me just to play SI.

Now, this can get even stronger if one strategy dominates all of the other

strategies. in that case, then this one strategy, si,

is kind of better than everything else. And in that case I can say not just that

it dominates something but I can say that it's dominant.

That it's just kind of the best thing to do, and if I have a dominant stradegy

then basically I don't have to worry about what the other agents are doing in

the game at all, I can just play my dominant strategy and that's going to be

the best thing for me to do. Now, formalizing that notion that this is

just the best thing to do, I can notice, I can claim to you, and it's not hard to

see that it's true, that a strategy profile in which everybody is playing a

dominant strategy has to be in Nash equilibrium.

So, if everyone is playing a dominant strategy, then we've just got a Nash

equilibrium, because none of us wants to change what we're doing.

We already know from the fact that the strategy is dominant that there's nothing

better for me to do. Furthermore, if we all have a strictly

dominant strategies then this equilibrium has got to be unique, because There,

there can't be two equilibriums strictly dominant strategies because that would

mean we prefer these strategies to each other strictly and that, that just can't

happen. So lastly I want to think about the

prisoners dilemma game, and I want to argue to you that the players have a

dominant strategy in this game, so I want to claim to you that player 1 has

the dominant strategy of playing D, and I'm going to do this by a case

analysis. So let's begin by consdiering the case

where player 2 plays C If player 2 plays c, then player 1 is really thinking about

this column of the matrix, he knows he's in this column,

and that means he faces a choice between getting a payoff of minus 1, and getting

a payoff of 0. And 0 is bigger than minus 1, and so

player 1 would prefer to get 0, which means that his best response to c is to

play d. On the other hand, let's consider the

case, where player 2 is playing D. In this case player 1 finds himself in

this green column, and that's kind of too bad for him because now he faces the

choice between the pay off of minus 4, and a pay off of minus 3.

And both of these numbers are smaller than the numbers that he had a choice

about before, so he likes the blue column better than he likes the green column.

But if he is in the green column, he still likes to get minus 3 than to get

minus 4, and that means in this case, he again prefers to play D.

So we can see that, regardless of what player 2 does, player 1 best responds by

playing D, and in both cases, his preference was strict, and that means he

has a strictly dominant strategy in this case.

And so, D is a dominant strategy here. If I argue that player two has a dominant

strategy of playing D, and I do a case analysis about what player one can do,

but the game is symmetric, so the same argument goes through there, as well.