Hi folks, this is Matt again. And, now we are ready to start solving

games, and making some predictions of how people play, in different settings.

And, so we are talking right now about Nash Equilibrium, which is one of

probably the most basic in standard solution concept of all in all the game

theory. Which is named after John Nash, who was a

mathematician in Princeton, and actually some years back won a Nobel Price, for

his work on this subject. And it's a, it's a very basic and

fundamental concept, and in order to sort of motivate it, let's, let's start by

just talking through some a particular game that was described and invented by

another famous person. so this is John, John Maynard Keyne's

Beauty Contest game. So what's the idea here? so let's, let's

think of a basic situation that you might be interested in.

and this was one that Keynes described in some detail.

So the idea was you have a stock, you're holding on to it.

And the stock price is rising, that's great.

you're an investor, you're trying to make profits off of your stock holdings.

And you begin to believe that maybe the stock is too high to be justified by the

value of the company. So, you're thinking that it's possible

this stock is overvalued, maybe there's a bubble in the market and

you're starting to think about selling. Okay, well you'd like to sell it, but

you'd like to wait until the price is at it's peak.

Right, so you'd want to wait until the price is just where it's going to hit its

maximum before you sell it. so you want to get out of the market just

before the other investors do. So this is a game where now you have to predict what

other people think about the stock price and, and what they're going to do and

when they want to get out, so how will they act.

How should you respond to that so, the basic ingredients of Nash equilibrium are

going to be having some prediction of what other players are doing, and then

choosing the optimal strategy in response to that So these are going to be two key

ingredients that we have. so there's a, a very stylized version of

this which is known as the, the Keyne's beauty contest game.

where did it come from? Well, actually Keyne's described the, there was a

newspaper in the in, in England that had a contest where players had to guess

Which picture of, of several women other readers would think was the the, the most

attractive one? So it wasn't to guess what you thought But what you thought

other people were thinking. So, Keyne's likened investing to this,

you, it's not only what you think of, of the stock, but what you think other

people are thinking about the stock that's important in driving your

decisions. Okay,

so, so this now is represented by a very simple game which is played by played by

many people. So what's this game look like? Each

person gets to name an integer between 1 and 100.

Okay? So you get to pick a number between 1 and 100, it has to be an integer.

So 1, 2, 3, etcetera. players are going to move simultaneously.

And the player who names the integer that's closest to 2/3 of the average

integer wins a prize. and the other players get nothing.

So to win this game, you have to guess, you have to guess the average and then

2/3 of it, right? So you'd want, you want to be right at 2/3 of whatever the

average guess is. So a little bit below the average guess.

if there's 2 people who happen to hit the same integer that, that's the right one

then ties are going to be broken uniformly at random.

So we'll just flip a coin or if there's 3 people we'll, we'll roll the dice 3 sided

die, etcetera. Okay, so how would you play this game?

you have to think about what other players are going to do and then forecast

what the best integer is, in a response to that.