To examine the relationship between players' rationality and their behavior in a game, it's very important to define, obviously good strategies and obviously bad strategies, okay? And such things can be formulated by a concept called domination. This is what I'm going to explain in this lecture. So let me start with the familiar example of a game. Prisoner's dilemma game. So let's recall the payoff table or prisoner's dilemma game. So two players, two actions, cooperation and defection. And if they both cooperate. The pay off is minus one. And if they both defect, the payoff is worse minus ten each. And if one player defects, defecting player there is large there, but the defected player gets very large negative payoff minus 15. And the, you exchange those two numbers here, because situation's symmetric here. Okay, so if you examine the nature of better strategy in this game you can see that defection is obviously good strategy, and cooperation is obviously bad strategy. So let's examine how it works. Okay, suppose you are player one. And suppose your opponent is going to cooperate, okay? Which is better, cooperation or defection, for you, as player one? Okay, so, if you switch from cooperation to defection, supposing that the opponent, player two, is cooperating. You're payoff changes form minus one to zero, if you switch from cooperation to defection. So given that the player is going to play cooperation it's better for you to play defection. Okay, so what happens if your opponent is going to defect? So again, if you switch from cooperation to defection, your payoff increases. The payoff increases from minus 15 to minus 10, okay? So defection is always the best strategy for you in Prisoner's dilemma game. Independent of what your opponent is going to do, okay? So if this is true,we say that defection dominates cooperation, okay? So defection is obviously a good strategy. Defection dominates cooperation. And cooperation is a dominated strategy. It's obviously a bad strategy. Okay? So let me just give you the definition of domination by means of a simple picture. so, take any game and horizontal axis measures your payoff. And your payoff partly depends on what other players are going to do. So that the horizontal axis measures other players strategy's, okay? What if you play a certain strategy? Say strategy a. Depending on what other players are going to do, you get certain, you know, payoff. This is a payoff when your strategy is a. And when you switch to another strategy say a strategy b. You have a different, graph. So suppose a graph look like this. Then if you take strategy b, the green strategy. Depending on what other's are going to do, your payoff is described by this blue line uh-huh, this green line. Okay, the blue line is above the green line, so that means no matter what your opponents are going to do, a is strictly better than b. If this is true, we say that strategy a, the blue strategy, strictly dominates the green strategy, strategy b. Is it clear? Okay, sometimes you may get this picture. So payoff to green with blue strategy a, is weakly better than green strategy. And sometimes a is strictly better than b. So sometimes you have tie, and sometimes a is better than b. So if you have this kind of picture we say that a weakly dominates strategy b, okay? So now, let's examine the relationship between rational behavior and domination. Okay, one of your simplification which is going to be repeatedly, you know, adopted, in the following lectures is the following. If you have two strategy, one dominating the other, one strictly dominating the other. So b is strictly dominated by a and the implication of rationality is a rational player never chooses strictly dominated strategy, okay? So maybe you have lots of strategies and maybe you have even better strategies than a. But obviously b is a bad choice, rather than choosing b you could always choose a and to get better payoff no matter what other players are going to get. So rational player never chooses a strictly dominated strategy. Okay, so let me just give you an example of a strictly dominated strategy. So let's remind, recall the location game I explained in the first week. So you have two ice cream stands, a and b. And they are simultaneously choosing their location on a street, okay? But to make my point I'm going to assume that there are finitely many slots. For possible locations of ice cream vendors in b. Okay so I'm going to change the game a little bit and I assume that there are finitely many possible locations. Location one, or slot one, or slot two, slot zero, and so on. Okay, a small remark. So, this is location game with finitely many possible locations. So, the location is between zero and a hundred, so you have 101 possible locations. Okay, I'm going to assume that A and B can occupy exactly the same slot, because the street has two sides. So player A can have the, his stand on one side of the street. And the player B can have its ice cream stand on the other side, okay? So A and B can occupy the same slot, that's a possibility. And again, customers are uniformly distributed over this street and the customer goes to the closest ice cream stand- And if two ice cream stands are located at exactly the same, same slot they split customers equally, okay? And payoff to each ice cream vendor is the number of customers, okay? This is a location game with finitely many possible locations. Well intuitively, the end slot zero and 100 are stupid choice. It's a bad locations of the essay, right? But more precisely you know, you can apply the concept of domination to say that endpoints zero or a 100 are bad locations. So, in terms of game theory, for example, this endpoint zero is strictly dominated by one. This is what I'm going to show, okay? So, suppose you are vendor A. So this means no matter where B locates his ice cream stand. For a one is always better than the end point zero. So this is what strict domination means. Okay, so I'm going to show that location one strictly dominates location zero. Okay, so relative desirability of zero and one depends on where your opponent b is located. So let's consider the situation where B is choosing the end location zero. So if you choose A, if you choose zero as the, as the allocation, it's a tied situation, and half of the customer is coming to you, vendor A. Okay, but if you switch from zero to one, almost all customers are coming to you. And you can increase your payoff. Okay, so given that B is at the end point of zero, one is greater than A. One is better than zero for player A. Okay, another possibility. B is, ice cream stand is somewhere here. And in this case, let's examine the relative you know, profitability of locations zero and one. In this case, if you switch from zero to one, you are moving closer to b. Okay, by switching from zero to one, you are moving to closer to B. And zero for you can steal some customers from B. Thereby increasing your payoff. So switching from zero to one is better for you, given that B is located somewhere here. So again, in this case one is better than zero for player A. So conclusion, no matter where B is located one is always better than zero. So, in terms of game theoretical concept one strictly dominates zero. Or zero is strictly dominated by, by one. And zero is obviously a bad choice. And similarly, 100 is strictly dominated by 99. And those are the really bad locations, okay? The end slots are bad locations. They are strictly dominated. And the conclusion? Rational players never choose those endpoints.