So, let's try to summarize what we have learned in this course. Basically, there are five important points you should remember about this course. Okay, first the value of game theory. Game theory provides a generally applicable principle, a governing principle, which can be applied to any social problem. Okay. So there are a variety of social problems. We have seen them in the first week. Political campaigns, negotiation, market competition, traffic allocation, struggle for existence of animals and a parlor game like poker. Okay. So, in any instance, individuals are trying to do their best against others and there are certain set of rules. So, there might be a general theory which can be applied to all those social problems. The question is to find out this general theory, which can be applied to any social problem. Okay, game theory found out that by the same mechanism, which assure us the existence of a vortex point on coffee surface. If you stir coffee, there's always a vortex point. So by the same mathematical reason that a vortex point exists on coffee surface, game theory discovered that any social problem has a point where players are doing their best. That's called the Nash equilibrium. Nash equilibrium is a point where players are doing their best against others. Okay? So, what does game theory do? Well, given any social problem like negotiation, political campaign, traffic allocation, any social problem can be formulated as a mathematical model of game. Okay, so a mathematical model of game specifies players and the possible strategies and their consequences, payoffs. Okay? So, any social problem can be formulated as a mathematical model of a game, and then, by using of the power of mathematics, game theory computes, or finds out, Nash equilibrium. What is Nash equilibrium? Well, Nash equilibrium is a stable situation in a social problem where each player can no longer improve his payoff. So once society is settled in Nash equilibrium, it's difficult to move out. 'Kay? It's a rest point of strategic interaction of players in any social problem. Okay. So, that's what the game theory does. So basic prediction of people's behavior in social problem by game theory is Nash equilibrium. And Nash equilibrium can emerge by quite different reasons. Okay? So we have seen whole spectrum of possible intellectual capacities of players spanning from very rational case and absolutely low rationality, absolutely no rationality case, okay. So in the, in one extreme end, where people are very rational and they have unlimited ability to conduct very sophisticated reasoning. We saw that, you know, if I know that you know that I know, that I know that you know that everybody's rational if players can carefully follow this kind of sophisticated reasoning, sometimes they can reach Nash equilibrium. This is a hyper rationality situation. But in reality people are not so rational, but people are not completely irrational. You know, people know what is better for them and if they find out better, you know, strategies, they may switch to better strategy. That's a low, more realistic low rationality case. So in low rationality case, you can accumulate your experience in the same game or similar game, and you can adjust your behavior by trial and error. And over time, you are getting more and more rational, and over time, you have more and more accurate predictions about other people's behavior. And this process make lead to stable situation Nash equilibrium. Okay? So, this adjustment of story leading to Nash equilibrium applies to allocation of traffic. Okay? So, if all drivers are rushing into a very, you know, a new and a convenient road, it's over congested. So they, sooner or later find out that trying different route is better for them. So they are changing, you know, other routes. Changing to other routes if that is that is profitable for them. And by this kind of trial and error adjustment people play Nash equilibrium traffic allocation. And you have seen the evidence about Hamamatsu City in Japan. Okay, the zero-intelligence case. Well, if you consider the struggle for existence of animals and the plants, they have absolutely no intelligence. In the case of plants, they don't have brain at all. Well, even if players have no intelligence, their behavior may be described by game theory. That's what we've learned because of biological evolution. So strategy of a plant may be determined by the gene and the genes are playing a game. And successful strategy produces more copies of the gene, and eventually, successful strategy have lots of offspring, and it dominates the society. So the outcome of biological, evolution is very much like, Nash equilibrium of a game played by genes. So even for zero-intelligence case, game theoretic prediction sometimes work. Okay, so that's the second message. The third message is about the prediction power of game theory. So game theory provides you with generally applicable principle. The concept of Nash equilibrium can be applied to any social problem. It's guaranteed. Nash equilibrium always exists. But what about the prediction power of Nash equilibrium? Well, it depends. 'Kay. We have seen many examples. The card game we played in the first week and the penalty kicks in soccer game. Game theoretic prediction worked pretty well. So we had a good fit in those examples. Traffic allocations, we have seen that a fit is reasonably good. Roughly 85% of variation of traffic allocation was explained by Nash equilibrium in the Hamamatsu City example in Japan. And what about the policy choice of Democrats and Republican? Well, the very simple game theory model predicts that Democrats and Republicans should choose exactly the same policy, and in reality, policies are different. Okay? So a fit is not perfect in this policy example. 'Kay. So you should remember that sometimes game theory works amazingly and sometimes it doesn't really work. Okay. Well, if it doesn't give you perfect fit, as it really, is game theory really useful? Well I would argue that even though the fit is not perfect, game theory is useful by the following two reasons. Okay? Prediction power of game theory is not perfect, but game theory is useful because of the following two reasons. Okay. First reason, let's go back to this example of policy choice by Democrats and Republicans. The simple game theory framework shows that they should choose exactly the same policy, but in reality, their policies are different. Is game theory completely useless? I would argue to the contrary. Okay? Game theory is pretty useful in this particular example, because it gives you very useful insight. 'Kay? Prediction doesn't work perfectly but you can gain very important insight from game theory. What is the insight? Well, in this situation, parties are tempted to choose very similar policies to steal voters from the opponent. Okay. So this is what game theoretic, model shows. And game theoretic model actually, you know, make it clear, one of the driving forces of political campaign. Okay? So game theory is useful in gaining insight what is one of the main driving forces of policy choice of two big parties. And also game theory provides useful benchmark. Okay? Remember that basic question here was, the following. What determines the policies of Democrats and Republicans? Well, this is a very general and vague question and without the help of game theory, you don't know where to start. Okay? But, game theory assures you that you can always find the Nash equilibrium and Nash equilibrium tells you that they play, they should choose exactly the same policy. But reality, they- their policies are different. So now you can ask, if parties are not choosing the same policy, what could be the reason? Starting with Nash equilibrium and comparing it with reality and, you try to fill the gap. Okay. So this is much, you know, much fruitful, much more fruitful way of examining this basic question about, what determines the two parties' policy rather starting from scratch without any theoretical framework. Okay? So, that's one of the reasons why game theory is useful if the, fit is not perfect. The second reason why Nash equilibrium is useful. Well, there is no reason to believe that people always play Nash equilibrium. Okay, theoretically and empirically there is no guarantee that people always play Nash equilibrium. But nonetheless, Nash equilibrium is very useful because we have what is called the stylized fact. A stylized fact is a mode of behavior that is repeatedly observed in, in the society. And the important goal of social science is to explain stylized fact. And the stylized fact is most likely to be a Nash equilibrium. Well, why? If mode of behavior is not a Nash equilibrium, somebody can gain by deviating. And sooner or later such profitable deviation would be discovered, and the mode of behavior collapses. So, if a mode of behavior is repeatedly observed, a stylized fact, that is most likely to be a Nash equilibrium. An example is this convention in Tokyo subway escalator. You stay to the left so that anyone in a hurry can run on, on the right side. This mode of behavior is repeatedly observed, it's a stylized fact, and actually it's a Nash equilibrium. Okay, so the last message, one of the most important messages of game theory is that, there is a conflict between group rationality and rationality of individuals. Okay, what is good for society is quite often different from what is good for each individual. This global warming story is a wonderful example. Stopping global warming by cooperation of different countries is obviously good for the society as a whole. But each country has an incentive to pollute, okay. And game theory shows how we can possibly overcome this problem. How to fill the gap between individual rationality and group rationality. Well, that's it, and I hope you have enjoyed my course, and I also hope that you will learn more about game theory in the future. And good luck for the final exam.