Now, we are ready to answer the first question I posed in the first lecture in this week. Okay, question number one was the following. What determines the policies of two parties? The Democrats, and the Republicans, okay. Well this is a very vague and general question. And to answer this question, you have to simplify. You look at the reality, and you capture the essence of reality by means of a simple game. Okay. So you have obviously two players in this social situation. The Democratic party and the Republic, Republican party. Two players. And the question is, what are the sets, of possible policies of those two parties. Well they can take many different policies and campaigns and, it's very hard to formulate everything in a simple mathematical model, okay. So, let's suppose that they just choose policies or policy platforms. And let's suppose that policies can be linearly ordered from very liberal to very conservative. Okay, so this is the space of possible policies. Maybe Democrats can implement very liberal policy. And maybe Republicans can implement very conservative policies. Okay. So this is, this segment represents the set of all possible policies. This is the strategy set of each player, democrat and republican, okay. And they simultaneously choose policy platforms on this line, okay. And now I have determined, players and possible strategies. The third item, the payoff to each player. Well, let’s suppose that payoff is equal to the number of votes each party gets. Okay. So Republicans and the Democrats are going to maximize the votes, the number of votes they are going to receive. Okay, so I have specified players and the possible strategies, and payoffs to ea- to all those players. Okay, now let's assume that both these ideal policies are uniformly distributed over this segment. Okay, so some people have very conservative ideas. So their ideal policies may be located here and some people are very liberal, so their ideal policies are somewhere around here. And for simplicity for the moment, let's assume that voters' ideal policies are uniformly distributed. Okay, and let's suppose your ideal policy is here. And let's suppose Democrat's policy is here. And the Republican policy is here. Okay? So then Republican's policy is closer to your ideal policy. So therefore you vote for Republican. Okay? So this determines how people vote. And this determines the payoff of Democrats and Republicans. Okay, so if Democrat chooses a policy platform here, very liberal policy, and the Republican chooses a very conservative policy here, what happens? Well this is a point in-between. And all of the people whose ideal policies is located here, vote for Democrats. And all conservative people, whose ideal policies are located over here, vote for Republicans. And as you can see here, this is identical to the Hotelling's location game. All right. So, you can apply the Nash equilibrium concept to the policy choice game. And the result is, this is the unique Nash equilibrium. Okay, Democrats, and the Republican choose exactly the same policy. Okay so let's re-examine my basic assumption, saying that voter's ideal policies are uniformly distributed. This may not be true, okay. So let's generalize the model by weakening this restrictive assumption. Okay, so let's consider a general distribution of voters’ ideal policies. It may look like this. Okay, then you can determine what is called median. Okay, so median over distribution is determined by the following way. Okay. So, half of the population is here and the other half of the population is here. And this point here is called, the median of the distribution of voters' ideal policies, okay. So intuitively median means the following, okay. So if you linearly order everybody from very liberal to very conservative, you can find someone just in the middle. Okay? That's the median. Okay, so you linearly order everybody from very liberal to very conservative. And then you can find the person just in the middle. That's the median voter. Okay. So our analysis shows that, policies parties when there are only two parties tend to be very similar and they choose this location, Median location where they get half of the- of the voters. Okay. So the answer to the question number one is the following. The answer is given by median voter's opinion, okay. So you linearly order everybody from very liberal to conservative and you find someone, just in the middle, okay. The opinion of this guy, the median voter, determines two parties policy. This is called median voter theorem, in politics. And this is a very prominent application of game theory in political science. Okay, so let’s reexamine this game theoretic prediction. As you can see the prediction is not so accurate. Well, prediction says that the democrats and the republicans should choose exactly the same policy, and in reality it's not true. Okay, but nonetheless, I'd like to argue that game theory is useful. Okay, the game theoretic prediction is not as accurate as Newton's law, like in this example, the prediction is not so accurate, but The prediction captures very important driving force of democrats and republicans. Okay, they are trying to maximize the number of votes they get. And as they move closer they can steal voters from the opponent. Okay, so they are inclined to choose very similar policies. Okay. And game theoretic analysis captures this important driving force of Republicans and Democrats. Okay. And also game theoretic approach is very useful to answer this question here. What determines the policies of two parties? It's a very vague and general question. And if I am asked this question I don't know where to start. But game theory says well, any social problem can be formulated as a mathematical model of game, and then you can apply the solution concept, the Nash equilibrium, and then you get an answer, okay. So I would like to say that game theory provides a useful reference point. Useful benchmark of the analysis. So, let me try to explain what I said by means of a metaphor. So look at this picture here. Okay, so this line here, indicates the trajectory of a falling leaf. And how a leaf falls. It's a very complicated phenomenon. Okay, and it's very difficult to come up- come up with a very simple theory to explain the trajectory of a falling leaf. But you can examine what would happen if there is no air. Okay, and the law of motion in vacuum is given by Newton's Law, and it's very simple. It doesn't give you perfect prediction about the trajectory of a falling leaf. But it gives you a very useful insight, because it captures one of the driving forces of the falling leaf. Okay, so let's suppose, this falling leaf represents human behavior which you'd like to examine. It's very, very complex, and maybe you cannot get a very simple theory to explain everything perfectly, okay. But game theory, gives you an answer, unified answer, what people might behave in any given social situation. Given that they are perfectly rational and given that everybody is doing their best. So game theoretic prediction, captures one of the very important driving forces, okay. Under this complex phenomenon of human behavior can be fruitfully analyzed by getting some benchmark. Okay, and game theory gives you a very useful. Reference point and the by comparing this basic driving force and reality you can get deeper understanding of real human behavior. And probably this is the most important value of game theoretic analysis. Okay, game theory provides you a unifying solution concept, just like Newton's law, but it's not as accurate as Newton's law. But nonetheless, it's very useful in providing you know, a useful reference point, or benchmark of the analysis, and it captures important driving force of human behavior.