So far, we have seen lots of games and Nash equilibria. So we, we have a clear idea of what kind of situation the concept of Nash equilibrium represents. And now it's time to ask the following crucial question, why do people play a Nash equilibrium? In the following few lectures, I’m going to give you some answers. And in part one A, this lecture concerns when people are rational. Okay. So the definition of Nash equilibrium let's try to recall what was the definition. Nash equilibrium has mutual best reply property. So if there are two players, one and two, Player 1 is taking best re- reply against, you know, two, Player 2's action. And Player 2 is also taking a best reply. Okay? Mutual best reply is a property of Nash equilibrium. The question is how do players find out such a point? Okay? This is a question I'm going to ask. So there are several reasons why players might play a Nash equilibrium and I'm going to give you three main reasons. Maybe players are very rational and they, if rational players calculate what they should do. Sometimes, they can reach Nash equilibrium. So this is reason number one. Player are very rational. Reason number two says that, well player communicate before playing the game, about how to play the game. Okay? So players get together and talk, how they should play the game. And by means of this kind of pre-play communication, people are settled in a particular Nash equilibrium. That's the second reasoning. The third reasoning is that players might not be so rational and they may not have an opportunity to talk to each other. So they just play the game and outcome may not be Nash equilibrium. And then they play the same game again or similar game and as they accumulate the experience, eventually they converge to a Nash equilibrium. That's the reasoning given by trial and error adjustment. Okay. And which reasoning applies depends on the nature of game under consideration and the nature of context. Okay. So let's explain the first reasoning. So player plays the game only once and before playing the game, they really think hard what they should do. Rational thinking sometimes leads to a Nash equilibrium. Okay. So this reasoning perfectly applies to the game of prisoner's dilemma. OK so this is the pair-off table of Prisoner's Dilemma. And if suppose you are playing player one and suppose you think very hard what you should do in this game before playing Prisoner's Dilemma. If you- your opponent player two is going to play C, cooperation, and if you switch from C to D. Your payoff increases from minus 1 to 0. So given that your opponent play is cooperation C, your best reply is to play D. Okay? And the same is true for the case where player two is going to defect. So, if you switch from C to D, your payoff increases from minus 15 to minus 10. So D is always best for you no matter which strategy our opponent is going to choose. So clearly, if you are try to maximize your payoff, D is your optimal strategy. So, rational players play Nash equilibrium. So rational reasoning leads to a Nash equilibrium in Prisoner's Dilemma game. Okay? But this reasoning to obtain Nash equilibrium may not always work. Okay. So let's consider a game of dating, which is sometimes called the Battle of the Sexes game. So this game has two players, a man and a woman and they are going for a date. And possible destinations is either a football stadium or a shopping mall. Okay? So, if they are going to different places say, man going to football stadium and woman is going to shopping mall, they are not meeting. They are not going out together, so they are very unhappy. And let's say their payoff's zero in those cases. But if they go to the same place, they are happy. So, if they both go to football game, they are happy. But man is very happy, his payoff is three. And woman is not so happy, but better than going for other places. Okay. If they go for shopping together, women are extremely happy and man is not so happy. Okay. So this game, obviously has two Nash equilibria. Everybody going to football and everybody going to shopping. Okay, let me give you the reasoning. If my partner goes to football, it's my optimal strategy to go to football. Okay? So therefore, everybody goes to football is a Nash equilibrium. Okay? Likewise, if my partner goes for shopping, then I should go for shopping. So there's another equilibrium, shopping equilibrium. Okay? And intuitively, it should be clear that the rational reasoning alone does not tell which equilibrium to play. So, if man and woman don't have any opportunity to talk to each other and if they just think what they should do before going for a date tomorrow, the outcome isn't so clear. So rational reasoning alone usually does not, you know, lead to a Nash equilibrium. Okay. To obtain, this is very important part here, okay? To obtain a Nash equilibrium, you need rationality plus something. Okay? What is plus something here? Well, its correct beliefs about what others are going to do. Okay? Rationality and correct beliefs about other players’ actions lead to a Nash equilibrium. Okay? Okay. And player- players’ reasoning alone is not sufficient to form correct beliefs. And I'm going to give you two more reasons why people play a Nash equilibrium. And one is pre-play communication, the other is trial-and-error adjustment. Those two reasoning provides certain mechanism to form correct beliefs.