1-9 Dominant Strategies

So this video is going to describe an important property that some games have, which is called dominant strategies. To begin with, I'm gonna start using this word strategy, that we haven't defined yet, and indeed you're not gonna 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 two 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 gonna 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 gonna 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 st rict 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 gonna 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 wanna claim to you that player one has the dominant strategy of playing D, and I'm gonna do this by a case analysis. So let's begin by consdiering the case where player two plays C If player two plays c, then player one 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 one, and getting a payoff of zero. And zero is bigger than minus one, and so player one would prefer to get zero, which means that his best response to c is to play d. On the other hand, let's consider the case, where player two is playing D. In this case player one 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 four, and a pay off of minus three. 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 three than to get minus four, and that means in this case, he again prefers to play D. So we can see that, regardless of what player two does, player one 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.