Hi again. It's Matt, and now we're talking more about strategic reasoning. And in particular let's go through and and analyze the Keynes beauty contest game now and talk about the Nash equilibria of this game. So, remember what the structure of this game was. Each player named an integer between 1 and 100, so you've got a population of players, they're all naming integers. the person who names the integer closest to 2/3 of the average integer named by people wins, other people don't get anything. ties are broken uniformly at random. Okay. So again, what are other players going to do? You have to reason through that and then what should I do in response? So these are the key ingredients of a Nash equilibrium and the Nash equilibrium is everybody's choosing their optimal response, the one that's going to give them the maximum chance of winning in this game to what the other players are doing, that's going to be a Nash equilibrium. Okay. So let's take a look. so, how are we going to reason about this? Suppose that I think that the average play the averaged integer named in this game is going to be some number X. so, I, you know, including my own integer, I think this is going to be the average. Well, what has to be true about my reply to that, my reply should be 2/3 of X, right? I should be naming the integer closest to 2/3 of whatever I believe the average is going to be. So my optimal strategy should be naming an integer closest to 2/3 of X. So here, we're just working through heuristically, we'll, we'll get to formal definitions and analysis in a little bit, but let's just go through the basic reasoning now. Okay, so I should be trying to name 2/3 of what I think the average is going to be. Well, X has to be less than a 100, right? There's no way that the average guess can be more then 100. So the optimal strategy for any player should be no more then 67 right? So if I think that everybody's rational I, so, if I believe that's true, then I think that nobody should be naming an integer bigger than 67. Okay, so what does that mean? Well, that means that I can't think the average is any higher than 67, right? So, if, if the average X is no bigger than 67, then I should be naming no more than 2/3 of 67. Right? Now, you can begin to see where this is going, so that means that if I think everybody else understands the game and understands that nobody should be naming a number bigger than 67 and nobody should be naming numbers bigger than 2/3 of 67. we keep going on this, so nobody should be naming anything more than 2/3 of this, of 2/3 of 67. Now, obviously, when you, if you just keep looking, everybody's going to want to be a little bit lower than everybody else's guess. So wherever the average is you should be lower than that. What's the only number which, everybody can be naming, and consistently choosing the best response they have to what the average guess is. the unique Nash equilibrium of this game is for every player to announce one. Okay? Well that's, yeah, so, so we're driven all the way down to, to announcing one and that's a unique Nash equilibrium, and what happens now, we all announce one we all tie, and somebody wins at random. If, if I try to deviate form that, if I try to announce a higher integer, I'd just be higher than the average guess, so I wouldn't be at 2/3 of the mean. So this is going to be a stable point. Okay? So, let's see what, what actually happens when people play this. So part of this reasoning is you're trying to form expectations of what other players are doing and you need to make sure that those expectations actually match reality. So let's have a peek at some plays of this game. So this, this is a plot here where we're actually giving you the results of the online course of when it was taught last year, we had players play this game, and so these are the results. And here from 2012, we had more than 10,000 people actually participate in this particular game. What do we see? So, down here on this, we have integers going from 0 to 100 and then over here, we have the frequency. So, how many people named the given integer? So the, the 50 right here is the, is the mode, so we get the mode of 50. The most often named integer was 50, 1,600 people named 50. Well, obviously, they hadn't gone through all the reasoning and it takes a while to sort of figure out what the equilibrium of this game is. what's the mean here? So the mean was 34, so actually there's some interesting things. Some people naming 100, a number that could never really win, right? So it's not clear exactly what what, it could, it could end up winning if everybody named 100 then you could end up in a tie there, but then you would be better off naming 67 instead. So so when we, when we end up looking through this, what we end up with is some people naming high numbers, but very few people, then we end up with some interesting spikes a bunch of people just named 50. Not clear exactly what the reasoning is on, on 50. interestingly if you think that a bunch of people are going to do that you might want to name 2/3 of 50. Okay, well, there's a big spike here at 33 where a bunch of people believed that other people were going to name 50. if we keep going, so down here. If we keep going and looking at this, what we see, then we see another spike at 2/3 of 33. So some people said, okay, well, maybe a bunch of people are going to think that the average is going to be 50, they are going to name 33. I'm going to go one better than that. I am going to name something around 22, 23. you know what the winner in this game was? The winner was actually 23. So 2/3 of the average guess here was about 23 because the mean was, was 34 and so one of these people randomly would end up being the winner of this game. Okay? there's actually a spike of people who went all the way to the Nash equilibrium and it's interesting here, because the Nash equilibrium works if you believe that everyone else is going to name the integer one, then that's your best response. But, in situations where a bunch of people don't necessarily understand the game and haven't reasoned through it, then you actually would be better off naming a higher number. So Nash equilibrium is a stable point if everybody figures it out and everybody abides by it, then it's the best thing you can do but it might be that some of the players aren't necessarily figuring out exactly what goes on. Okay. Now suppose you, you start with this game and they're not necessarily playing the Nash equilibrium, but now we have them play it again. Right? So, they get to do this, play it again, and then see what happens. Well, now, these people should realize that they overestimated, right? There's a bunch of people here who are naming numbers too high, they should be moving their announcements to, to lower numbers, right? They should be moving down. And if, if, if I anticipate that everybody's going to adjust and move downwards I should move my announcement downwards as well. So let's have a peek at what happens. So here is, is a subset of players actually from, from one of the classes I, I did on campus, where they got, this is the second play of the game. So after the first play, then we have them play again. Now you can begin to see that things, you know, the, the 50s have disappeared, all the numbers up here have disappeared, people have moved down, and in fact, a lot more people have are moving towards the equilibrium once you get to the second part, the second chance. So if you've played this game, you begin to see the logic of it. You played again and now we get closer to Nash equilibrium. So, Nash equilibrium does is a better predictor here. if from experienced players who have played this game understood it and, and interacting with the same population, you can begin to see things unraveling and moving back towards all announcing one. Okay. So Nash equilibrium, basic ideas, a consistent list of actions, so each player is maximizing his or her payouts given the actions of the other players. Should be self-consistent and stable. the nice parts about this, each players action is maximizing what they can get given the other players. nobody has an incentive to deviate from their action if an equilibrium profile is, is played. someone does have an incentive to deviate from a profile of actions that do not form an equilibrium. So these are the basic ideas and we'll be looking at, at Nash equilibrium in much more detail. So, in terms of of, of making predictions, you know, why, should we expect Nash equilibrium to be played? Well, I, I think there is sort of interesting logic here. in this logic, actually goes back to, to some of the original discussion by Nash. when we want to make a prediction of what's going on a game we want something which if players really understood things, it would be consistent. And the interesting thing is we should expect non-equilibria not to be stable, in the sense that, if players understood it and see what happens in a non-equilibrium, they should move away from that. And we saw exactly that in the, in the, the second round of the, the beauty contest game, then people start moving down toward the Nash equilibrium. So it's not necessarily true that we always expect equilibrium to be played, [COUGH] but we should expect non-equilibrium to vanish over time. And the, there'll be various dynamics and other kinds of settings where there will be strong pushes towards equilibrium over time, but they might have to be learned and they might have to evolve and, and so forth. So, as this course goes on, we'll talk more and more about some of the dynamics and, and things to push towards Nash equilibrium.