1-7 Best Response and Nash Equilibrium

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 gonna 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 quote AI, the action sequence of everybody except you, player i, So it's player one and so on until player i - one, i +one and all the rest. So the entire action vector, we call it the action profile, A is really made up of all those a - i and your own action. With that notation, here is the definition. We will say that your response, which we'll call ai-star, it may not be unique, but it's got to be part of the set, as we'll call BR, the best response assuming that all the other agent play A minus I. If it's the case that for any, 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 ai1 star, and the others are playing. a i, a minus i, for whatever it is, has gotta be at least as great as anything else you might choose, a sub i. If that's the case we'll say that a i star is a best response, a very intuitive notion, and that will bring us all, 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 will 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 the best response to the others. Specifically, we'll look at an actual profile here, a, a1 through a n, 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 agent is the best response over t he rest.