[SOUND] Hello, everyone. Welcome to the second week of my lectures. Last week we have seen the basic solution concept in game theory, Nash equilibrium. Nash equilibrium is a situation where all players are doing their best against others. And the mathematical genius John Nash showed that any social problem has such a point. And in this week, we're going to see more examples of Nash equilibrium and we ask the following crucial question: Why do people play a Nash equilibrium? Okay so let's go down to the first lecture. So, I'm going to review the concept of Nash equilibrium first, and then apply this concept to a famous game called Prisoner's Dilemma. Okay so what's the definition of Nash equilibrium? Let me explain the case of a, in the case of two players. So in Nash equilibrium, player one is taking his equilibrium strategy a1 star. And also two, player two is taking his equilibrium strategy a2 star. And those strategies should be mutually best replied. And let me just give you the formal definition. So, take player one, and if everybody sticks to Nash equilibrium, player one's payoff, payoff is represented by g one. G one of a one star and a two star. Ok, so this is mister one's payoff, if everybody is using equilibrium strategy. So what happens if player one deviates from equilibrium action a one star to something else. Okay so, originally, he was taking- Player 1 was taking a one star, but now suppose he switches to something else. The red strategy, a one. Okay, but a two, player two, is still choosing equilibrium action. Okay, so if, everybody, sticks to equilibrium, this is mister one's payoff. But if he deviates to something else, to the red strategy here, his payoff changes from here to here, okay? Nash equilibrium condition says that player cannot increase his payoff by deviating, so this inequality must be true, okay? And this should be true for all strategy, a1. OK, so this should be true for, for all. Red action A one. Okay. And you have symmetric condition for player two. So if those two conditions are satisfied. A one star equilibrium strategy for player one, and equilibrium strategy for player two, a mutual best reply, a one star is a best reply, to player two's equilibrium strategy, and player two is also taking a best reply to player one's Equilibrium strategy. So Nash Equilibrium has a property of mutually best reply and the Nash Equilibrium has the following property, no single player can increase his payoff by deviating by himself. That's the definition of Nash Equilibrium. Okay, so at this point I have to say that the concept of Nash equilibrium should be applied to a very simple class of games. Games without any timing. Okay, so, games without any timing is sometimes called simultaneous move game. Okay, so a simultaneous move game is a static game without any timing. So therefore players take their actions simultaneously, and then that's the end of the game. Okay, so Nash equilibrium concept is applicable to simultaneous move game. And throughout my lecture in first, second, and third week. I confine my attention to this simple class of games, and possibly in, in the last week we are going to see dynamic game with, with some timing, okay? So game theory says that in any simultaneous move game, player's behavior can be predicted by. Nash equilibrium. Okay? So let's apply the concept of Nash equilibrium to the famous game of Prisoner's Dilemma. Okay, so I'm going to explain what it is. So, let's suppose two individuals, say player one and two committed some crime. they, they broke into a house and stole some money. And then they were arrested and now they are put in different rooms and being interrogated by police force. Okay, so the situation can be summarized by a simple table. So, player one has, two choices, two strategies. Either to cooperate or to defect. And likewise player two can either cooperate or defect. Okay, cooperation means remain silent. So let's suppose they stole money but their original intention was to set fire to the house, okay? They- They didn't have a time to set fire to the house and, if they remain silent, if they cooperate, this is not discovered by the police. Okay. So what happens if both players remain silent, if both players chose cooperation? And then, they are put in prison for one year, Okay, they stole some money, so they are going to be put in prison, for one year. So let’s say their payoff is minus one, and minus one. The number here, the first number here represents Player One's payoff, the second number here represents Player Two's payoff. And this is what we call a payoff table. Okay, so if the game was simple, if there are only two players and if the number of actions is small, like, cooperation defection, only two actions, a game can be summarized by a simple table what is called payoff table. Okay. So let's try to fill in the blanks. So what happens if both players defect to confess and then the police finds out that their original intention was to set fire to the house? Okay. So it's a serious offense and they are imprisoned for longer years. Let's say 10 years. So, therefore mutual defection gives payoffs minus 10 and minus 10. Okay. What happens if player one defects and where player 2 cooperates? Well, the defected player who confessed is set free, he's rewarded for telling the truth. Okay, so therefore player one's payoff is 0. On the other hand, player two, who remained silent, was harshly punished. Okay, he's harshly punished and he is imprisoned for a long time. 15 years. So his payoff is minus 15. Okay so I have one remaining blank here. But situation is symmetric so you can just exchange those two numbers zero and 15. And the- this is the set of payoffs here. Okay this is the payoff table of Prisoner’s Dilemma. So now let's analyze what's going to happen in this situation. So suppose you are player one. And think about what you should do. Well, it may depend on what player two is going to do. So let's suppose player two is going to cooperate. Okay? And what happens if you, player 1, switches from cooperation to defection? Okay, so if you switch from cooperation to defection your payoff changes from minus one to zero. Okay zero is larger. So given that your opponent is going to cooperate, defection is better for you. Is it clear? Okay, now suppose player 2 is going to defect. What happens if you switch from defection to cooperation? Okay. Your payoff changes from minus 15 to minus 10. Okay. Minus 10 is larger, better. So, given that your opponent is going to defect again, defection is better for you. Okay so, to sum up, defection is always the best choice for you, okay? And it's independent of what your opponent is going to do. So,. In a Nash equilibrium, players are taking mutually best replies, but the best reply is always to defect. OKAY. So this game has a unique Nash equilibrium, mutual best reply. Best reply is always defection. So this is the only Nash equilibrium in prisoner’s dilemma game. OKAY, why is it called prisoner's dilemma? Well, so if each individual seeks his own interest, rational behavior of each individual leads to Nash equilibrium, mutual defection. OKAY, on the other hand, mutual cooperation is better for the group of two players as a whole. OKAY so group rationality indicates that mutual cooperation is even better. So prisoner's dilemma game clearly shows that group rationality- what is best for the group what is best for the society, Is not always equal to what is best for each individual. Okay, so this gap or difference between group rationality and the individual rationality is very important. And we are going to come back to this issue in the last week. And in the last week we are going to see how to sustain cooperation in this kind of situation.
