[SOUND] People often say that game theory is a science about rationality. So in the third week I'm going to examine the relationship between people's rationality and the outcome of games. Okay, so I'm going to consider the whole spectrum of intellectual capacity of the players. On the one extreme case, I'm going to consider hyper rationality situation. People are rational, and they have unlimited capacity to conduct very sophisticated reasoning, hyper rationality situation. And on the other hand I'm going to also talk about low rational situations. Pe- people may not be so rational, but people may not be completely stupid. So low rational situation I'm going to examine. And also I'm going to talk about absolutely zero-intelligent case. What happens if players absolutely have no intelligence? Still, game theory can predict individuals' behavior. So before giving you a series of lectures about rationality and people's behavior, let's try to examine the outcome of the games I asked you to play in the first week. So in the first week, I asked you to play a simple card game with your friends. And let's examine what actually you did in those card games. In the first lecture, I'm going to give you equilibrium prediction of the card game. So I formulated this card game as a simple two person game. And I solved for the Nash equilibrium. And this gives you game theoretic predictions. And in the second lecture, I'm going to compare the experimental results, your behavior and the Nash equilibrium prediction. Okay, so this is the nature of your card game. There are two players, red and black. And each player has those four cards. And each player carefully chooses, chooses one card and show it simultaneously to their opponent. And if both player choose K, red wins. And if players choose different number the card like 1 and 3 also red wins. In all other cases, black wins. So this was the nature of the game you played in the first week. Okay, so let me summarize the rules of, of the game. Red player wins if both choose K or players choose cards with different numbers like 1 and 3. In all other cases, black player wins. So black player wins if only one player chooses king or players choose cards with the same number such as 1 and 1. Okay? Okay. And then I posed the following three questions. Since the rule, rules are not symmetric, so maybe one of the players may have an advantage. So first question, who has as advantage, red or black? Second question, in particular, what is the winning rate of each player? What's the probability of winning of red and black? Second question. The third question, can you say anything about the distribution of cards of each player? Those three questions are hard questions. Probably the first question, which player is stronger, you can get some idea by inspecting the nature of the game. But the second and third question, calculating the winning rate of each player and calculating the distribution of cards of each player, it's, it's very hard, hard question. And if I knew no game theory, I would have no idea about how to answer a question 2 and question 3. But you can formulate this card game as a game. You can write down payoff table, and you can calculate mixed strategy Nash equilibrium. And then we get the answers to all those three questions. So let's do that. Okay. So, it's this, in this card game just like in rock, paper, scissors game, it's very important to make yourself unpredictable. So the only equilibrium here is a mixed strategy equilibrium. So let's find a mixed strategy equilibrium where each player mixes those four cards with certain probabilities. Okay. The first step to pin down or calculate the mixed strategy equilibrium is to write down the payoff table. So let's try to remember the rules of the game. Okay, so the rules say red player wins if both player choose K, or different numbers are chosen. So if both player, choose K, red player wins, and his payoff is 1, and black player's payoff is 0. Okay? And also, red player wins if they choose different numbers such as 1 and 2 and 1 and 3. And also red player wins and obtains payoff 1, and black player loses, his payoff is 0. In all other cases, black player wins. So this is the payoff table of the game you played in the first week. So, let's try to find the mixed strategy equilibrium. Okay. So your task is to determine the probability distribution over black player's strategies, K, 1, 2, and 3. So let's suppose black player chooses 1 with probability p, 2 with probability q, 3 with probability r. With the remaining probability, 1 minus p minus q minus r, the black player chooses K. Your task is to determine those three numbers. Okay, but since those three number cards 1, 2, 3 have a very similar role, let's guess that black player chooses each number card with an equal probability of q, of p, I'm sorry. Okay, with the remaining probability of 1 minus 3p, black player chooses K. And your task is to determine this number p. Okay, supposing you are red player, and supposing that your opponent, the black player, is mixing those cards with this probability distribution, what happens to you if you chooses K? Okay? So with probability of 1 minus 3p your opponent also chooses K, and you win. Okay? For all other cases, you lose. So if you chose king, your probability of winning is 1 minus 3p. Okay. Your winning rate, winning probability, is 1 minus 3p, if you choose king. Okay. So you can perform a similar computation. What if you choose 1 as the red player? So you win if your opponent chooses different number cards 2 and 3. In those cases you win, and therefore your probability of winning is p plus p, that's 2p. Okay? And the situation is very similar to card 2 and card 3. Okay? And in those cases, your winning probability is the same, 2p. Okay. And the, in the equilibrium red player should be mixing all those cards, king, 1, 2, 3. This implied that red player's winning rate should be identical in all of those four cases, okay? Otherwise, for example, if this number 1 minus 3p is the largest, player 1 would like to choose K with probability 1, okay. Since in the equilibrium probability red player is mixing all those cards, K, 1, 2, 3, those numbers here should be identical, okay? They are equally good. And therefore, red player is mixing between K, 1, 2, and 3. Okay, so equilibrium says that those four numbers, or two number, 1 minus 3p and 2p should be equal. Okay, so you have one equation for one unknown p. So this says 1 is equal to 5p. So p is equal to 1 over 5. Okay? That's 0.2. Okay, so 0.2 times, I'm sorry, 2 times p, 2 times 0.2 is 0.4. And by the equilibrium condition, 1 minus 3p is also 0.4. So the winning rate of red player is always 0.4. And the probability that black player chooses a number card is 0.2. Okay? So we have found those numbers, 0.2 for black players. Okay, so with the remaining probability, the black player chooses king. So the probability is large here. The probability is 0.4. This is the equilibrium distribution of cards by black player. And we have calculated the winning rate of red player, and that was 0.4. Okay, so you can perform similar exercise to determine red player's probability distribution over K, 1, 2, and 3. Okay. So this is the result of calculation. So let's examine what this result says about the winning rate. Winning rate of red player is 0.4 and, of course, the winning rate of black is the remaining probability, 0.6. So the answers to the first and second questions. Okay. So this game favors black player. Black player is stronger. And his winning rate should be 60%. Very sharp prediction. What about the distribution of cards? Well, king is played most often with probability 0.4. And numbers have equal probability of being played. So each number card up here is with probability 0.2. This is the answer to the last question
