[SOUND]. As I promised, our course focuses on basic, conceptual issues about game theory. In the first week, we discussed why we need a game theory. In the second week, we saw the basic equilibrium concept. In the third week, we examined the relationship between rationality and equilibrium. In the final week we focus on cooperation. And, in the first half, I'm going to explain the basic difficulty associated with sustaining cooperation. And in the second half, I'm going to show various ways to sustain cooperation. Okay? So let me begin by talking about the difference between what is rational for the society, and what is rational for each individual. 'Kay, one of the most important messages of game theory is that group rationality is different. Quite often different from individual rationality. That is, what is good for society is not equal, not always equal to, or quite often not equal to, what is good for each individual, okay? To achieve what is good for society, we need to cooperate. But, each individual may have an incentive to shirk. That's a basic difficulty. Okay, so let me give you an example. So, it's best for society if we keep the park clean, okay? So, we need to cooperate to keep the park clean. But, you have an incentive to leave some garbages behind you, okay? So, this is one example, very familiar example where, what is good for society is not equal to what is good for each individual. So, let me give you a more serious example. Okay? Now the temperature of, of our planet, Earth is increasing. It's called global warming and it's very harmful. So, it's best for our society that all countries cooperate to stop global warming. So, this is what is best for society. But, each country has an incentive to pollute. Okay, again, what is good for society as a whole is not equal to what is good for each individual. [SOUND] Okay, so group rationality is quite often different from rationality of individuals. Okay. So let's recall the, famous game prisoner's dilemma which makes this point, very clear. So let's review the game of prisoner's dilemma. There are two players, player one and two. And their action is either to cooperate, remain silent, or to defect, to tell the truth. Okay, so if they cooperate and do not tell the truth they are imprisoned for one year. On the other hand, if they both defect to tell the truth, well, they are put in prison for longer years, for ten years. 'Kay? So, if you compare those two outcomes, mutual cooperation and mutual defection, obviously, mutual cooperation is better for them, okay? And, but what happens if you defect and- and your opponent is going to cooperate. Okay, so let's suppose you are player one, and your opponent is player two. Okay. So, if you defect, that means if you tell the police while your opponent is remaining silent, silent, you are rewarded and, you are set free. Okay, so you spend zero years in prison. But your opponent is punished harshly. He's put in prison for 15 years. Okay. So this is the payoffs here and, the other part here, is symmetric. So you exchange numbers here and there. And this is the payoff table for the prisoner's dilemma. So although mutual cooperation is best for them, you know, they have an incentive to defect. And actually in this game, it's always best to defect, no matter what action your opponent is going to take, 'kay? So rationality of individual leads to a Nash equilibrium on mutual defection. So this is the result of, rationality of individual. They are going to defect. But, the society, for the society as a whole, mutual cooperation is much better. Okay, so this example clearly shows that, what is good for the society is different from what is good for each individual. [SOUND] Okay, so, let me first define what is good for society. I have already, we have already examined in detail, what is good for each player. But now let's examine how we can define what is good for society. Okay, so let me explain it by means of this simple diagram. So society consists of two players, say player A and B. And the, this grey area here represents possible payoffs in the society, okay? The question is how we can define what is best for the society. [SOUND] Okay, but this, it's a, it's a little bit tricky question and it's easier to define a point That is not best for the society. So let me define, what is not best for society first. So let's look at this, black point here. Obviously, this point is not best for the society. [SOUND] Because you can improve both player A and B's payoff simultaneously. So if you move from here to there, player A's payoff increases, and player B's payoff also increases. Okay? So obviously at this starting point, you know, black point is not socially optimal, because we can make everybody happier. Okay, what about this red point here? Again, this is not socially optimum. The explanation is a little bit convoluted here. Well, you can increase Mr. A's payoff without decreasing B's payoff. 'Kay, if you move from here to here, Mr. B's payoff stays constant so he doesn't mind moving from here to there. But A's payoff is strictly increasing. So by moving from here to here, you can make society better. So again, this red point here is not best for society. Okay. So those two points, red point and black point, are not best for society because it's possible to make society better. On the other hand, the yellow points here on those yellow points, there is no better point for society. [SOUND] Okay, so the conclusion is, what is best for society is not a single point, 'kay? Those yellow points here, which correspond to downward sloping part of the boundary, represents the set of what is best for society. Okay? And, those best points for society are sometimes called efficient points or efficient outcomes. Okay, so those yellow points represent the set of best points for society efficient outcome. And group rationality means that players should select one of those best points for society. That's group rationality. But quite often, Nash equilibrium is inside, okay. It's not best point for society. And rational as- rational individual- the behavior of rational individual leads to Nash equilibrium. So quite often, this is the situation. Group rationality is different from rationality of individuals, okay? So, I use this diagram again and again, so please remember what usually happens in the game. [SOUND] Okay. So, this is a situation which usually happens in many games. So this situation is described by- by inefficiency of Nash equilibrium. So this a terminology frequently used by economists. And in computer science, the same situation is often described by what is called price of anarchy. The price of anarchy is the measure of inefficiency of Nash equilibrium, gap between what is rational for the society and what is rational for each individual. The measure of inefficiency is called the price of anarchy in computer science and formally, it's a ratio of maximum total payoff divided by total payoff at Nash equilibrium. 'Kay? Since what you have here at the denominate- the Numerator. A numerator is maximum total payoff so, by definition, it's larger than the denominator so, this ratio is always greater than or equal to 1. The larger this number, price of anarchy is, the larger the inefficiency. Okay.