Okay. Now, I'm going to examine what happens if players mutually know that they are rational, okay? And their surprising outcome comes out in the location game. This is one of the highlights in the, the third week. Let's have some fun. Okay, so before examining people's mutual knowledge about rationality in location game, let's consider the following questions. Let's consider the following different situations. In situation one, players are rational. But they may not know that other players are rational. Okay? Players are just rational. That's a situation 1. Situation 2, players are rational, and moreover, players know that they are rational. So I know that you are rational, and you know that I am rational. Okay? Obviously, situation 1 and 2 are different, right? You can expect that different outcome or different behavior is observed in situation 1 and 2. But let's consider the third situation, where not only players are rational and player know that they are rational. Players know that they know that they are rational. So I know that you know that I am rational. Okay. So what is the difference between 2 and 3? Intuitively, you know, 2 and 3 doesn't make any difference. That's my intuition. And you can go further and a fourth situation. Players know that they know, that they know that they are rational, okay? Oh the question I would like pose is the following. Are there any difference between situation 2 and 3 or situation 3 and 4 and so on? Well, we're going to see a very surprising answer to this question. Okay. So let's examine this question in terms of location game. Okay. As we have seen in the first week the location game has a unique Nash equilibrium. So A and B chooses the middle location. So this is the only Nash equilibrium in location game. So let's consider a hyper rational situation. Player A and B are rational, and also they have unlimited capacity of conducted sophisticated reasoning. Okay? And also, let's assume that players' rationality is completely known by the, those, by those two players. Okay. I'm going to show that just by using rational calculation or rational reasoning, I'm going to show you that they must play Nash equilibrium. Okay. Okay, so the explanation goes as follows. In the previous lecture we learned that endpoints are obviously bad location. So no rational player chooses 0 or 100. So that what I explained in the last lecture. So if Andy is rational, he never chooses 0 or 100. Avoid end points, that's the implication of rationality. So now let's suppose the other player, Becky, is also rational. But let's suppose she knows that Andy is rational. Okay, so if Andy is rational, he tends to avoid the end points 0, 100. But you should understand the following point really well. The next point 1 and 99 may be a rational choice for Andy. Okay? If the other player Becky is stupid, and if Becky is stupid and chooses 0, this location 1 is the best point for Andy. Because by locating here at point 1, Andy can get almost all customers. So the next endpoints, 1 and 99 may be a good choice for Andy if Becky is stupid. Okay? So now suppose that both players are rational, and now suppose that Becky knows that Andy is rational. Okay, since both players are rational, they avoid choosing end- end slots, 1 or- 0 or 100. And now Becky knows that Andy is rational. So since Becky knows Andy is rational, she knows that Andy never chooses 0 or 100. Okay, so in this situation we can conclude that Becky never chooses say 1 because 1 is dominated by 2 if Andy never chooses 0 or 100. Okay, so Becky knows that Andy is rational and therefore Andy won't choose 0 or 100. And if you exclude the possibility of 0 and 100, in the smaller game location 2, strictly dominates location 1. Okay? What does it mean? Well, it means that no matter where Andy is located between 1 and 99, okay, 1- 2 is always better than 1 for Becky, okay? The reasoning is identical to the reasoning I explained in the last lecture. Okay? So after excluding the possibility of 0 and 100 by Andy because Andy is rational, 2 dominates 1, okay? So the conclusion is, you know, the next end point 1 or 99 is obviously bad choice for Becky given that Andy never chooses end points 0 and 100. Okay. So, if player B knows that A is rational, he can, she can exclude the second end points 1 and 99. Okay. And the rest is very similar. Okay, so but let me go slowly. Okay, so next let me examine what happens if Andy knows that Becky knows that Andy is rational. So the picture looks like this. Okay, now both players are rational, so they won't chose the end point. And both player know that the other player is rational so they won't choose second end points, 1 or 99. But now Andy further knows that Becky knows that Andy is rational. Okay. So the conclusion is well, since Becky knows that I am rational, she won't choose those two end points here and two end points here. So her behavior is distributed between 2 and 98 somewhere, okay? So if that's a possibility, by the same argument the third endpoint are obviously bad choice for Andy. Okay? So location 2 and 98 is bad choice for Andy given that Becky's location is somewhere between 2 and 98. And you can go on and on, by the same kind of reasoning. Okay, so what's the conclusion? Well, if players know that they know that they know that they know that they are rational. 'Kay. If the word know appears many times here, and then you can exclude those bad choices one by one. And eventually the conclusion is, you must choose the middle point. Okay, so let's examine how many times the word know should appear to reach the conclusion that middle point is chosen. Okay, so if a player Andy is rational he never chooses location 0. And if he knows that Becky is rational if this word know appears once, then you can exclude point 1. Okay, and since you have say, from 1 to 49, since you have 49 points to be excluded, this word know should appear 49 times. So if players know that they know that they know that they know, and that this word know appears 49 times, the conclusion is they must play Nash equilibrium. They must choose the central location. Okay, so let me just summarize, our argument. Okay, if this statement is true, if players know that they know that they know that they are rational and this word know appears many times, and then they should play the Nash equilibrium. Players play in Nash, players play Nash equilibrium in the location game. Okay. And a situation where players know that they know that they are know that they are rational. And this word, know, can appear in any number of times. If this is true, we say that the rationality of players are common knowledge. Okay? So this is the situation where players are completely understanding that they are mutually rational. Okay? So they have complete mutual knowledge of rationality. So my lecture shows that in some games like a location game with finitely many locations sometimes in some games, common knowledge of rationality leads to a Nash equilibrium.