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

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

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

Â going to get a definition for a little while.

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

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

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

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

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

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

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

Â different strategies that player I can take.

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

Â could do. I'm going to define two different

Â definitions of what it means to say the SI strip,

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

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

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

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

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

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

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

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

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

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

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

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

Â That means that SI strictly dominates S prime I.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â dominant strategy has to be in Nash equilibrium.

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

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

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

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

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

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

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

Â happen. So lastly I want to think about the

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

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

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

Â analysis. So let's begin by consdiering the case

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â has a strictly dominant strategy in this case.

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

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

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

Â