# 1-6 Strategic Reasoning

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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 one and 100, 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 gonna 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 gonna be some number X. so,
I, you know, including my own integer, I think this is gonna 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 naming an
integer closest to two-thirds 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 two-thirds 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
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 than 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 this, of two-thirds 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 two-thirds of the mean. So this is gonna 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 zero to
100 and then over here, we have the frequency. So, how many people nam ed 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 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 down here. If we keep going and looking at this, what we see, then we see another
spike at two-thirds of 33. So some people said, okay, well, maybe a bunch of people
are gonna think that the average is gonna be 50, they are going to name 33. I'm
gonna 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 two-thirds of
the average guess here was about 23 cuz 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 ac tually 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
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 wanna 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, 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.