 Hi, folks. It's Matt again, and now we're ready to start solving games and making some predictions of how people will play in different settings. And so we're talking right now about Nash Equilibrium, which is probably the most basic and standard solution concept in all of game theory. It's named after John Nash, who was a mathematician at Princeton and actually some years back won the Nobel Prize for his work on this subject. And it's a very basic and fundamental concept, and in order to sort of motivate it, let's start by just talking through a particular game that was described and invented by another famous person. So this is John Maynard Keynes' beauty contest game. So what's the idea here? So let's think of a basic situation that you might be interested in, and this is one that Keynes described in some detail. So the idea was you have a stock, you're holding onto 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 that 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 its peak, right? So you'd want to wait until the price was just where it's going to hit its maximum before you sell. So you wanted 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 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 very stylized version of this, which is known as the Keynes' beauty contest game. Where did it come from? Well, actually Keynes described, there was a newspaper in England that had a contest where players had to guess which picture of several women. Other readers would think was the most attractive one. So it wasn't to guess what you thought, but what you thought other people were thinking. So Keynes likened investing to this. It's not only what you think of the stock, but what you think other people are thinking about the stock that's important in driving your decisions. So this now is represented by a very simple game that is played by many people. So what's this game look like? Each person gets to name an integer between 1 and 100. So you get to pick a number between 1 and 100. It has to be an integer. So 1, 2, 3, etc. Players are going to move simultaneously. And the player who names the integer that's closest to two-thirds of the average integer wins a prize. And the other players get nothing. So to win this game, you have to guess the average and then two-thirds of it. So you want to be right at two-thirds of whatever the average guess is. So a little bit below the average guess. If there's two people that happen to hit the same integer 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 three people, we'll roll a dice, a three-sided dice, etc. Okay, so how would you play this game? You have to think about what other players are going to do and then forecast what you think the best integer is in response to that.