In the last lecture we calculated the mixed strategy Nash equilibrium in the card game we played in the first week. So now we have equilibrium prediction about the winning rates of red and black players and the probability distribution of cards. So let's compare this equilibrium prediction with the actual data. So let's examine what we actually did in the first week. Okay, now I'm ready to explain the results of the card game you played in the first week. So I'd like to thank all of you who participated in this experiment. Okay, so I'm going to show you a dataset and players in this dataset where you and your friends, and actually a large number of people 670 pairs, are, were participated in card game as of 11th of February in 2015. Okay, and each pair played card game for 30 times. Okay now let me start with the winning rates. So this is the equilibrium prediction based on Nash equilibrium. Actually in this game, black player is stronger than red player, and more precisely, the winning rate of black player is 0.6, and the winning rate of led, red player is 0.4, okay? So now I'd like to ask you to think about the accuracy of this game theoretic prediction, so those are the theory number, and on the other hand I have real data winning rate calculated from the play of the game in the first week, and I'm going to give you three possibilities about the difference between the actual data and the equilibrium prediction, number one, number two, and number three. And you should, you should think which one fits to your expectation about the accuracy of this game theoretic prediction, so the first possibility so let's talk about red player. The winning rate of red player should be 0.4 according to game theoretic prediction, okay, so the possibility number one says game theoretic prediction works so-so. So the actual data actual winning da- winning rate in the data is, is in the range of say 0.3 and 0.5, okay, so that's possibility number one. Possibility number two the game theoretic prediction works reasonably well, so the actual winning rate of red player is in the range of 0.35 and 0.45. Okay, the last possibility, number three says that the game theoretic prediction works reas- amazingly well, so the actual winning rate of red player is in the range of 0.39 and 0.41. Okay, number one, number two, number three, so please think which one fits to your expectation about the accuracy of game theoretic prediction. Okay, now I'm going to show you the real data. Okay, so this is the actual data. The winning rate of red player was 0.415 and automatically, the winning rate of black player in the actual dataset, taken from the first week, was 0.585. So the game theoretic prediction worked amazing- amazingly well in terms of the winning rates. So next I'm going to explain the equilibrium prediction of the distribution of the cards. Okay, so the Nash equilibrium gives you the equilibrium distribution of cards chosen by red player and the black player and this is the equilibrium card distribution. In Nash equilibrium, king should be chosen with a large probability, in particular, 0.4 and the number cards 1, 2, and 3 should be chosen with a smaller probability of 0.2. Okay, so those are the equilibrium prediction and let me show you the actual data. Okay, again, game theoretic prediction worked amazingly well but I noticed that in this particular dataset number one was played with a larger probability than the equilibrium prediction, 0.24 compared to point 0.20, which is an equilibrium prediction. And my hunch is the following. In the video game instruction card number one was phrased as ace, and usually, ace is a pretty strong card in many card games, so maybe you are inclined to play this strong card, ace, but otherwise the prediction worked pretty well. And also by using this experiment o- of the card game, I'm going to address the three concerns about game theory or about predicting people's behavior by mathematical formula, okay. So in the first week I explained that there are three valid, or natural concerns about predicting people's behavior by using mathematical formula and I use this game to address those concerns at the end of this lecture. Okay, so this game was first invented by Barry O'Neill, he is a professor at the UCLA now, and he discovered that he ran lab experiments and he discovered that people's behavior in this card game is amazingly close to Nash equilibrium prediction, okay? The result was in was reported in one of the leading science journals Proceedings of the National Academy of Sciences in 1987, and I also have conducted a series of experiments in my game theory class in the University of Tokyo. So I have a large number of datasets, so let's- let me explain what I have found in those experiments, okay? So this is my dataset. I know the result of Barry O'Neill's original experiment in 1987, and in the past ten years, I have been conducting this card game with my undergraduate students in my game theory class in the University of Tokyo, and I also had a few occasions which I, I let high school students play this game, 'kay, so this is my dataset, and let me show you what I have found, okay, about the winning rate. Okay, so equilibrium says that winning rate for red is 0.4, and black's winning rate is 0.6. In the original lab experiment by Barry O'Neill, 'kay, the result is very close to the Nash equilibrium, 'kay, so I was surprised when I first read his paper but I thought that he was just lucky, 'kay? I thought he just got this almost perfect fit by chance, so just to see if this experimental result can be reproduced, I- I conducted a series of follow-up experiments in my game theory classes. Okay so the first card game was conducted in 2004 in my class, and again, the outcome was amazingly close to the Nash equilibrium prediction, and I ran similar experiments again and again and every time, you know, the outcome was amazingly close to Nash equilibrium prediction in terms of winning rates. What about the distribution of cards? Okay, equilibrium says that king should be played with a large probability, probability 0.4, and each number card should be played with probability 0.2. That's the equilibrium prediction. In the original experiment by Barry O'Neill, also the outcome was very close to Nash equilibrium prediction, okay? And in my first experiment also, the outcome was very close, and second, it was also close, and as you can see in all those experiments, the outcome was amazingly close to the equilibrium prediction, okay. Surprisingly, people's behavior is closely predicted by Nash equilibrium, okay, so with those results in mind, let me address three concerns about predicting people's behavior by mathematical formula. Okay, so mathematical formula has been proven to be useful to predict natural phenomenon like a falling ball, but now game theory is trying to apply mathematical formula to predict people's behavior, and there are a few common and valid concerns about predicting people's behavior by a mathematical model, okay? So I explained those concerns in the first week, and there were three concerns. The concern number one, well, people have free will, 'kay? The falling ball doesn't have any, you know, free will so it obeys the mathematical law of motion, okay, Newton's law and every time, it follows Newton's laws but human being have free will and we can do anything, right? So if game theory says that this is, this is your behavior, this is your this is the equilibrium prediction, we can always deviate because we have free will. So concern number one about using mathematics to predict the people's behavior says that we have free will, and free will defeats any attempt to predict human behavior by a mathematical model or a mathematical formula, well and second concern, okay the subject of game theory of humans, okay? And humans take certain behavior because they have some intentions, so ultimately, you can use- you can always ask why did you do that, okay, and then you can find out the reason. So concern number two says that you know, mathematical formulation is useless, or we don't need any mathematical model. We can just collect facts, and we can just conduct interviews to find out what was happening, and indeed, this was the- the way we conducted social science research before the invention of game theory. We just used our intuition to explain how people behave, and all you need is fact-finding, what happened, and all we need is interview, why did you do that, so this was the second concern. We don't need any mathematical model in social sciences, and the third concern says I've never heard that game theory works, 'kay? So I'm going to address all those 3 valid concerns about game theory by means of this card game. Okay, concern number one, free will defeats any attempt to predict human behavior by mathematical formula. Well, it worked pretty well in the card game, 'kay? So even if people have free will, they are attracted to the behavior that is best for them, okay? So in this card game, given that other players are choosing you know, equilibrium strategy, it's best for you to follow this equilibrium, so you are naturally attracted to the behavior that is best for you even if you have a free will, okay. Second concern, you can just ask why do you do that. This is very important part so I must stress my answer here. Okay, so this is a result of a card game, and the data shows that people's behavior is amazingly close to Nash equilibrium prediction but ask yourself the following question, why did you choose your card in a certain way, this way? King was large probability and small probability for one, two, and three. Well, I guess lots of you have hard time in articulating why you did it, okay, so you used your instinct or intuition to play this game and outcome was amazingly close to Nash equilibrium, okay. So game theory can uncover the mech- mechanism operating behind your instinctive behavior, so sometimes you take some action out of intuition or out of some reason, but oftentimes, you are- you have hard time in articulating why you did it, and some other mechanisms, some mechanism may be operating behind your behavior. And game theory eh, eh, is, is a very important tool to find out the mechanism operating behind people's behavior, okay? So just asking people how, why you do this? In this particular card game, this kind of research program doesn't really work, okay, so maybe you can find out that the distribution is uncertain, you know, the distribution of the card is this way but by just conducting interview with each individual you can never find out why people chose those distribution. Okay, the third concern, I've never heard that, you know, mathematical approach in social science works, and in the first week, I explained the following episode mathematician Stan Ulam was teasing Paul Samuelson, long-time ago back in 1930s, okay. Stan Ulam was saying that social science is a fake sciences. Natural sciences are the true sciences, 'kay, and he teased Paul Samuelson, an economist, by posing the following question, 'kay? Following challenge, okay, name me one proposition in all of the social sciences which is both true and non-trivial. Social sciences say that such and such thing should be true but almost all of them are trivial, okay, and there is no proposition in social science which is true and non-trivial. That was true back in 1930s, but now with game theory we have many such examples. This card game is one of those examples.