 We'll now speak about how we predict that people will play in a game, and specifically about the concepts of best response and Nash equilibrium. Now let's for a moment assume that you are one of the players, and the other players are going to play in a way that you know. Then we can speak about what is your best thing to do, and we'll call that the best response. And technically speaking, let's use the following notation. Let's call AI the action sequence of everybody except you, player i. So it's player 1 and so on and so on, and player i minus 1, i plus 1, and all the rest. So the entire action vector, we call it the action profile, A is really made up of all those A minus I and your own action. With that notation, here's the definition. We will say that your best response, which we'll call AI star, it may not be unique, but it's got to be part of the set that we'll call BR, the best response assuming that all the other agents play A minus I. If it's the case that for anything you might be thinking of doing A sub I, for all A sub I, it's got to be the case that your utility for playing that, in other words, when you're playing AI star, and the others are playing AI, A minus I, for whatever it is, has got to be at least as great as anything else you might choose A sub I. If that's the case, we'll say that AI star is A best response. Very intuitive notion, and that would bring us almost all the way there. The problem, of course, is that you don't know what the others will do. But that's okay because we'll use the notion of best response as a building block to define what we call the Nash equilibrium. And the Nash equilibrium briefly is a set of actions, one for each of the agents, such that each is a best response to the others. Specifically, we'll look at an action profile here, A1 through An. And we'll say that it's a Nash equilibrium. And later on, we'll tell you why we call it specifically a pure strategy, Nash equilibrium. If it's the case that for every agent, that action, A sub I associated with agent, is their best response to all the rest.