In the third week we are asking the following question. What is the relationship between players rationality and the outcome of a game? Okay, to answer this question, you have to make it clear what it means that a player is rational. Well, rationality at this time is a very vague one. Rationality is a vague concept. And the same term, rationality, can mean different things, okay? So, I'm going to present a clear-cut definition of rationality that is widely adopted by game theorists and economists. And by using this particular definition of rationality, I'm going to examine the relationship between rationality and people's behavior in a game. Okay, so I already showed a similar picture in the first week. But let's examine a person's rationality, Mr. A's rationality in some gambling situation. So question is, what does it mean that Player A is rational when he is playing roulette game? What does it mean that he is rational in a horse race? And what does it mean that he is rational in a, in a poker game, okay? So all of those games look fairly similar, roulette and horse race and poker. But as you go down, you know the situation is getting more and more complex. That's what I'm, I would like to explain now. So what does it mean that Mr. A is rational in roulette game? Well, the behavior of roulette is objectively given. So there are objective probabilities that each outcome arises. Okay? And therefore rational behavior means a very simple thing. Okay? Mr. A maximizes expected utility. As I have explained, utility assigned to each outcome reflects Mr. A's attitude towards risk, and he maximize his expected utility. And that's what I mean by rational behavior in roulette game. So roulette game is very simple in terms of rational behavior. What about horse race game? It's very sim, very similar to roulette, but there is a crucial difference. There is no objective probability assigned to the outcome. There's no objective probability that the horse number one wins, no objective probability that horse two wins. Okay, so in this situation, there's no objective probability, and what does he do? Well what should he do? Well, he should carefully think about what's the relative likelihood of different outcomes. If you know horse one is very likely to win or horse two is very unlikely to win, okay, so he can summarize his assessment by means of subjective probability. Okay? So the first step for the rational decision-making in horse race is to assign subjective probabilities on various outcomes and the rest is similar. Then, Mr. A is going to maximize his expected utility. So this is what game theorists means by A's rationale behavior in horse race game. Okay so the definition, or formal definition of rationality, there could be many different definitions, but this is the definition which is widely adopted by game theorists. Okay. The definition says the following, a player or an agent is rational if he or she is aware of all possible events. So first step is to delineate all the possible events. Okay. And the second step, the player should assign objective or subjective probabilities over those possible events. That's the second step, and the third step is to maximize expected utility. Okay, so this is the formal definition, this is a formal definition of rationality which is widely adopted by game theorists. So given this definition, let's try to see the implications of rational behavior. So let's try to apply this definition to human behavior, a game of poker between human and human. Okay, what does what does it mean that Mr. A is rational? Say Andy and Becky, Andy's rationality means, well Becky's behavior is not given by objective probabilities. So, he should assign subjective probabilities about various behavior by Becky, okay? And then A, based on his belief, subjective probabilities, he maximizes his expected utility. This is Andy's rationality. Okay, but the story doesn't end here. I explained this in the first week. But Becky is also a rational decision maker, so she should be thinking in the same way. Okay, so since Becky is also rational, she is forming, she's assigning some, some subjective probability over Andy's possible behavior. So to better predict best, Becky's behavior, Andy have to think about what Becky thinks about his behavior is going to be. And the story doesn't end here, you can go on and on. So to analyze rational behavior in human interaction, you need to think about Andy's belief, about Becky's belief, about Andy's belief about Becky's belief and for a, and so on and so forth. And the story go, goes on and on. This is the problem of infinite regress. Okay, applying the definition of rationality to human interaction leads to the infinite regress problem. So in the remaining few lectures, we are going to see that the, the hierarchy of that kind of sophisticated reasoning, my belief about your belief about my belief. Okay, this kind of sophisticated reasoning can sometimes lead to a surprising outcome, that's what I am going to show you.