 Hi again, it's Matt and now we're talking a little more about strategic reasoning and in particular let's go through and analyze the Keynes beauty contest game now and Talk a little bit about the Nash equilibria of this game So remember what the structure of this game was each player named an integer between one and a hundred So you've got a population of players. They're all naming integers The person who names the integer closest to two-thirds 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 average 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 two-thirds of x right? I should be naming the integer closest to two-thirds of whatever I believe the average is going to be So my optimal strategy should be saying naming an integer closest to two-thirds of x So here we're just working through heuristically. 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 two-thirds of what I think the average is going to be Well x has to be less than a hundred right there's no way that the average guess can be more than a hundred So the optimal strategy of any player should be no more than 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 what does that mean? Well, that means that I can't think that the average is any higher than 67, right? So the if the average x is no bigger than 67 Then I should be naming no more than two-thirds 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 There's 67 and nobody should be naming numbers bigger than two-thirds of 67 We keep going on this so nobody should be naming anything more than two-thirds of Two-thirds of 67 now obviously when 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 of what 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 tried to deviate from that if I tried to announce a higher integer I'd just be higher than the average guess, so I wouldn't be a two-thirds 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 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 these 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 have 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 zero to a hundred and then over here we have the frequency so how many people named a given integer so The 50 right here is the the mode so we get the mode of 50 The most often named integer was 50 1600 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 the equilibrium of this game is What's the mean here so the mean was 34 so actually there's some interesting things you say some people naming a hundred a number that could never really win right so It's not clear exactly what? Well, it could end up winning if everybody named a hundred then you could end up in a tie there, but then you would be better off Naming 67 instead so So 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 50 interestingly If you think that a bunch of people are going to do that you might want to name two-thirds 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 here if we keep going and looking at this what do we see then we see another spike at two-thirds 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're going to name 33. I'm going to go one better than that. I'm going to name something around 22 23 You know what the winner in this game was the winner was actually 23 so two-thirds of the average guess here was about 23 The mean 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 really Believe that everybody 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 that'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 start with this game and They're not necessarily playing the Nash equilibrium, but now we haven't played 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 over Estimated right there's a bunch of people here who are naming numbers too high They should be moving their announcements to lower numbers, right? They should be moving down and 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's is a subset of players actually from from one of the classes. I did on campus where They got this is a 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 50s have disappeared all the numbers up here have disappeared People have moved down and in fact a lot more people 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 play it again And now we're getting 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 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 payoffs given the actions of the other players Should be self consistent and it's 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 who does have an incentive to deviate from a profile of actions to do not form an equilibrium So these are the basic ideas and we'll be looking at Nash equilibrium in much more detail So in terms of of making predictions, you know, why should we expect Nash equilibrium to be played? Well, I think there's sort of an interesting logic here And this logic actually goes back to some of the original discussion by Nash When we want to make a prediction of what's going on in the 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 second round of The beauty contest game then people start moving down towards the Nash equilibrium So it's not necessarily true that we always expect equilibrium to be played but we should expect non equilibria to vanish over time and There 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 As this course goes on we'll talk more and more about some of the dynamics and things to push towards Nash equilibrium