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Hi again. It's Matt,

Â and now we're talking more about strategic reasoning.

Â And in particular let's go through and and analyze the Keynes beauty contest

Â game now and talk about the Nash equilibria of this game.

Â So, remember what the structure of this game was.

Â Each player named an integer between 1 and 100, so you've got a population of

Â players, they're all naming integers.

Â the person who names the integer closest to 2/3 of the average integer named by

Â people wins, other people don't get anything.

Â ties are broken uniformly at random. Okay.

Â So again, what are other players going to do? You have to reason through that and

Â then what should I do in response? So these are the key ingredients of a Nash

Â equilibrium and the Nash equilibrium is everybody's choosing their optimal

Â response, the one that's going to give them the maximum chance of winning in

Â this game to what the other players are doing,

Â that's going to be a Nash equilibrium. Okay. So let's take a look.

Â so, how are we going to reason about this? Suppose that I think that the

Â average play the averaged integer named in this game is going to be some number

Â X. so, I, you know, including my own

Â integer, I think this is going to be the average.

Â Well, what has to be true about my reply to that, my reply should be 2/3 of X,

Â right? I should be naming the integer closest to 2/3 of whatever I believe the

Â average is going to be. So my optimal strategy should be naming

Â an integer closest to 2/3 of X. So here, we're just working through

Â heuristically, we'll, we'll get to formal definitions and analysis in a little bit,

Â but let's just go through the basic reasoning now.

Â Okay, so I should be trying to name 2/3 of what I think the average is going to

Â be. Well, X has to be less than a 100, right?

Â There's no way that the average guess can be more then 100.

Â So the optimal strategy for any player should be no more then 67 right? So if I

Â think that everybody's rational I, so, if I believe that's true, then I think that

Â nobody should be naming an integer bigger than 67.

Â Okay, so what does that mean? Well, that means that I can't think the average is

Â any higher than 67, right? So, if, if the average X is no bigger than

Â 67, then I should be naming no more than 2/3 of 67.

Â Right? Now, you can begin to see where this is going, so that means that if I

Â think everybody else understands the game and understands that nobody should be

Â naming a number bigger than 67 and nobody should be naming numbers bigger than 2/3

Â of 67. we keep going on this, so nobody should

Â be naming anything more than 2/3 of this, of 2/3 of 67.

Â Now, obviously, when you, if you just keep looking, everybody's going to want

Â to be a little bit lower than everybody else's guess.

Â So wherever the average is you should be lower than that.

Â What's the only number which, everybody can be naming, and consistently choosing

Â the best response they have to what the average guess is.

Â the unique Nash equilibrium of this game is for every player to announce one.

Â Okay? Well that's, yeah, so, so we're driven all the way down to,

Â to announcing one and that's a unique Nash equilibrium, and what happens now,

Â we all announce one we all tie, and somebody wins at random.

Â If, if I try to deviate form that, if I try to announce a higher integer, I'd

Â just be higher than the average guess, so I wouldn't be at 2/3 of the mean.

Â So this is going to be a stable point. Okay?

Â So, let's see what, what actually happens when people play this.

Â So part of this reasoning is you're trying to form expectations of what other

Â players are doing and you need to make sure that those expectations actually

Â match reality. So let's have a peek at some plays of

Â this game. So this, this is a plot here where we're

Â actually giving you the results of the online course of when it was taught last

Â year, we had players play this game, and so these are the results.

Â And here from 2012, we had more than 10,000 people actually participate in

Â this particular game. What do we see? So, down here on this, we

Â have integers going from 0 to 100 and then over here, we have the frequency.

Â So, how many people named the given integer? So the, the 50 right here is

Â the, is the mode, so we get the mode of 50.

Â The most often named integer was 50, 1,600 people named 50.

Â Well, obviously, they hadn't gone through all the reasoning and it takes a while to

Â sort of figure out what the equilibrium of this game is.

Â what's the mean here? So the mean was 34, so actually there's some interesting

Â things. Some people naming 100, a number that

Â could never really win, right? So it's not clear exactly what what, it

Â could, it could end up winning if everybody named 100 then you could end up

Â in a tie there, but then you would be better off naming 67 instead.

Â So so when we, when we end up looking through this, what we end up with is some

Â people naming high numbers, but very few people,

Â then we end up with some interesting spikes a bunch of people just named 50.

Â Not clear exactly what the reasoning is on, on 50.

Â interestingly if you think that a bunch of people are going to do that you might

Â want to name 2/3 of 50. Okay, well, there's a big spike here at

Â 33 where a bunch of people believed that other people were going to name 50.

Â if we keep going, so down here. If we keep going and looking at this,

Â what we see, then we see another spike at 2/3 of 33.

Â So some people said, okay, well, maybe a bunch of people are going to think that

Â the average is going to be 50, they are going to name 33.

Â I'm going to go one better than that. I am going to name something around 22,

Â 23. you know what the winner in this game

Â was? The winner was actually 23. So 2/3 of the average guess here was

Â about 23 because the mean was, was 34 and so one of these people randomly would end

Â up being the winner of this game. Okay? there's actually a spike of people who

Â went all the way to the Nash equilibrium and it's interesting here, because the

Â Nash equilibrium works if you believe that everyone else is going to name the

Â integer one, then that's your best response.

Â But, in situations where a bunch of people don't necessarily understand the

Â game and haven't reasoned through it, then you actually would be better off

Â naming a higher number. So Nash equilibrium is a stable point if

Â everybody figures it out and everybody abides by it, then it's the best thing

Â you can do but it might be that some of the players aren't necessarily figuring

Â out exactly what goes on. Okay. Now suppose you, you start with

Â this game and they're not necessarily playing the Nash equilibrium, but now we

Â have them play it again. Right? So, they get to do this, play it

Â again, and then see what happens. Well, now, these people should realize

Â that they overestimated, right? There's a bunch of people here who are naming

Â numbers too high, they should be moving their announcements to, to lower numbers,

Â right? They should be moving down. And if, if, if I anticipate that

Â everybody's going to adjust and move downwards I should move my announcement

Â downwards as well. So let's have a peek at what happens.

Â So here is, is a subset of players actually from, from one of the classes I,

Â I did on campus, where they got, this is the second play

Â of the game. So after the first play, then we have

Â them play again. Now you can begin to see that things, you

Â know, the, the 50s have disappeared, all the numbers up here have disappeared,

Â people have moved down, and in fact, a lot more people have are

Â moving towards the equilibrium once you get to the second part, the second

Â chance. So if you've played this game, you begin

Â to see the logic of it. You played again and now we get closer to

Â Nash equilibrium. So, Nash equilibrium does is a better

Â predictor here. if from experienced players who have

Â played this game understood it and, and interacting with the same population, you

Â can begin to see things unraveling and moving back towards

Â all announcing one. Okay. So Nash equilibrium, basic ideas, a

Â consistent list of actions, so each player is maximizing his or her payouts

Â given the actions of the other players. Should be self-consistent and stable.

Â the nice parts about this, each players action is maximizing what they can get

Â given the other players. nobody has an incentive to deviate from

Â their action if an equilibrium profile is, is played.

Â someone does have an incentive to deviate from a profile of actions that do not

Â form an equilibrium. So these are the basic ideas and we'll be

Â looking at, at Nash equilibrium in much more detail.

Â So, in terms of of, of making predictions, you know, why, should we

Â expect Nash equilibrium to be played? Well, I, I think there is sort of

Â interesting logic here. in this logic, actually goes back to, to

Â some of the original discussion by Nash. when we want to make a prediction of

Â what's going on a game we want something which if players really understood

Â things, it would be consistent. And the interesting thing is we should

Â expect non-equilibria not to be stable, in the sense that, if players understood

Â it and see what happens in a non-equilibrium, they should move away

Â from that. And we saw exactly that in the, in the,

Â the second round of the, the beauty contest game, then people start moving

Â down toward the Nash equilibrium. So it's not necessarily true that we

Â always expect equilibrium to be played, [COUGH] but we should expect

Â non-equilibrium to vanish over time. And the, there'll be various dynamics and

Â other kinds of settings where there will be strong pushes towards equilibrium over

Â time, but they might have to be learned and they might have to evolve and, and so

Â forth. So, as this course goes on, we'll talk

Â more and more about some of the dynamics and, and things to push towards Nash

Â equilibrium.

Â