We have nine lectures this week, and in the remaining three lectures, we are going to consider the case where people are not so rational. Okay. So I have been, we have been examining the relationship between players' intelligence and the outcome of, of the game. And the previous lecture showed that if players are hyper rational, okay? If they have unlimited ability of conducting extremely sophisticated reasoning, and if players' rationality are completely understood, then sometimes they can play Nash equilibrium out of rational calculation. Okay, now I focus on low rationality case. Okay? People may not be so rational, and people may not have an ability of conducting sophisticated reasoning, but people are not completely stupid. So this is a case of low rationality. What happens if players are not so, not very smart? That's a question we are going to examine. Okay, well what happens if players are not very smart? Well, outcome may be chaotic. I mean, people can do lots of different things. And maybe game theory doesn't have anything to say about such, such a situation. Or maybe humans tend to make very similar mistakes, okay? Maybe you can see a pattern of irrational behavior when they are confused. Okay. So this is a very important question. And now this important question is systemically addressed by economists and game theorists. So the task here is to find out pattern of low rationality behavior, what kind of common mistakes people commit. By using field data or experimental data and borrowing from, borrowing insights from psychological research. To find out systematic deviation from rational behavior by a human being. So this is the subject of what is called behavioral game theory or behavioral economics. It's a, a, a very active field of research in these days. But in my lecture, I confine my attention to the case where people are rational, or people have, have some kind of rationality. Okay, so even if at the beginning people are confused and their behavior is chaotic and maybe they commit similar mistakes, over time, players may find better ways to play a game. Okay? So over time, player, players may find better ways to play a game, okay? By accumulating experience in the same or a, or a similar game or by observing how others play in the same game or a similar game. Okay? So what you can do is to try various actions and to see which one is better for you. And if you find out a better action, you just switch to better action. Or if other people are adopting a better strategy, you can imitate successful behavior. Okay, so this is what I call trial and error adjustment process, and I argued in one of the previous lectures, this process leads to, sometimes, to a Nash equilibrium. So, let's examine how it works in the location game. Okay. In the last lecture, I showed that extremely sophisticated reasoning by a hyper-rational player in the location game with finitely many locations, eventually they reach, reach the conclusion that they should play Nash equilibrium. And here I'm going to show that low rationality adjustment process, trial and error adjustment process, leads to the same outcome, Nash equilibrium. Surprisingly, by the same basic logic. Okay, so I showed that endpoints 0 or 100 are obviously bad choice. But maybe, you know, Andy, player A, may not be so rational to find out that 0 or 100 are obviously bad locations. So Andy can try various locations. So Andy doesn't have complete rationality. He tries different locations, and eventually, he finds out that choosing endpoints doesn't make any sense. Okay, eventually he finds out endpoints are bad locations. They are strictly dominated strategies. Okay. So sooner or later Andy stopped choosing those endpoints, 0 or 100, okay? So given this behavior of Andy, eventually, now Becky also doesn't know what is the optimal location. Because Becky is not so smart. Okay, so again, she is going to try to find the locations here and there. And given that Andy never chooses the endpoints zero or 100, eventually she finds out that second end points 1 or 99 doesn't make any sense for her. Okay? So eventually she starts avoiding choosing those four endpoints. 0, 1, 99, and 100. And process goes on and on. And then eventually people stop using those endpoints one by one. And eventually players learn to play a Nash equilibrium. Okay, so by means of this trial and error adjustment process, A and B are likely to choose eventually the Nash equilibrium point, the middle point. So let me just summarize. In some games like location games, repeated elimination of obviously bad strategies, repeated elimination of strictly dominated strategies, leads to a Nash equilibrium. Location game with finitely many locations is a perfect example. In such a game, Nash equilibrium might emerge under very different two reasons. So Nash equilibrium may emerge by very careful and sophisticated reasoning. Or Nash equilibrium may emerge out of very low rational trial and error adjustment process. Okay? So if repeated elimination of strictly dominated strategy leads to Nash equilibrium, Nash equilibrium is expected to emerge under a wide range of intellectual capacities of players.