 Hi folks, welcome back. So this is Matt Jackson and we're talking now about defining games and we'll work through some basic definitions of the key ingredients in games. So let's take a look at some of those. So obviously one of the most obvious ones is the players in the game. So who's making the decisions? Are they people? Are we talking about governments negotiating over trade agreements? Are we talking about companies choosing their strategies for developing new products? Do we want to get down to the point of modeling people within a firm as opposed to the company as a whole? So there's a whole series of questions about how we're going to choose the players but they're going to be the central decision makers in what we're doing. Next we have to decide how we're going to model the actions. What actions can players actually take? So later on in the course we'll be looking at auctions that'll have bids so they can enter a number of bids. When we're talking about bargaining they might be deciding whether or not to strike. When we're thinking about investing it could be that an investor is deciding how much of a stock to buy or sell, when to buy or sell it, how they should react to other people in the market, how they should be conditioning their decisions on prices. When we're thinking about voters, how do they vote? So there's going to be a whole series of actions and we'll want to be careful in making sure that we have the essential actions modeled. Finally, payoffs. So what's motivating the players? Do they care simply about some sort of profit? Do they care about other players? So how are they receiving utility as a function of what the actions lead to in the context of the game? So there's basically two standard representations of games. One is what's known as the normal form and that's what we'll be starting with in the course. And what it does is it's a very simple and stark representation of a game. So it lists what payoffs players get as a function of their actions. Normally it's thought of as if players are moving simultaneously, but strategies, and we'll talk about this in more detail, can encode many things. So the other alternative representation is what's known as the extensive form and that includes more explicit timing in the game. So who moves at what point in time? So that's going to be represented often as a tree. So for instance in chess, one player moves first, the white player generally moves first and the black player can see the move by the other player, react to that, and so forth. So that's going to be better represented as a tree than in normal form. So it keeps track of also what players know when they move. So in poker, somebody moves first, they make a bet, but the other player only sees the bet and not necessarily the cards that the other player sees. So in some cases we'll have sequential games where players will have different information at different points in time. We'll want to talk about modeling that explicitly too. So we're going to start out with the normal form and then we'll move later in the course to the extensive form and we'll talk about the relationship between these two in more detail. Okay, so normal form games, what are the key ingredients? Again, players. So generally we're going to think of finite sets of players. So one through n, little n will represent a set of players. Generally we'll index these things by an i. So we'll use a little i to represent a generic player. The action set for players will represent by a sub i. So we'll let that represent the actions of player i and then we'll talk about profiles of actions which will just be a list of what every player is doing. So for instance, are they deciding to cooperate or not to cooperate with other players, for instance, in a prisoner's dilemma that we'll talk about. The utility function is then a payoff function which indicates as a function of all the actions that are played, what's the payoff for the different players. So for each player i, we end up with a function which tells us how they evaluate outcomes of the game. And again, how they evaluate these things could encapsulate many things and it's going to be very important to make sure that we're getting the right representation of what really motivates people. So often when we represent normal form games, a very simple way of doing that is just in the matrix representation. So let's just look at the most standard representation of very simple games, writing a two player game as a matrix. So we'll have one player one will be the row player. Player two will have to be a column player. So they're going to choose actions that will be represented in the column of the matrix. And the cells inside the cells will then represent the payoffs. So for instance, the TCP back off game that was talked about in the earlier video can be written as a matrix as follows. So the row player player one can choose either C or D. So this is player one's choice, generally known as the row player. This is player two's, the column player. And they represent the choices that they have and then inside the cells are the payoffs to the different players. So if player one cooperates and player two cooperates, then these are the payoffs to the two players. The first payoff, player one, second playoff, player two. So this is going to the column player. This one is going to the row player. Then we end up, for instance, if the row player chooses D and the column player chooses C, then we end up with a payoff here of zero to the row player and minus four to the column player. So the matrix is a very simple way of representing all of the basic elements of the normal form game visually so that we can actually keep track of exactly what the strategic interaction is and what players would like to do as a function of the game. Okay, let's talk about another game that we won't be able to write down in such a simple form. So let's think of a large collective action game. So for instance, whether or not a population wants to revolt against this government. So here we have many more players. So let's imagine that we have a population of 10 million players. So we're obviously not going to be able to write that down as a matrix on our screen. So we can do that more abstractly. But we'll have 10 million players. What are their actions here? Let's keep it very simple. So they have a choice here of either revolting or not. So their action set is just binary. Two choices. Then the payoffs are going to be the critical thing in this game. What happens? Well, let's say that in order for a revolt to be successful, you need at least two million people to participate. So in this particular stylized example, what do we end up with then? We can represent a successful revolt as the player getting a payoff of one. And the payoff of the action profile A is equal to one. If the number of people here, the number of players J, such that they picked to revolt, the number of this is at least two million. So if we end up with at least two million people revolting, then player I gets one. And note here that this is true, regardless of whether I is one of the revolt participants. So this is a game where you care about the end outcome, not necessarily getting utility out of the participation. We could change this and have people get enjoyment out of the participation, or have costs of the participation directly as well. Okay, so what happens if things fail? Here, if we end up with less than two million, then it depends on whether you were a participant in the revolt or not. So if player I was a participant in the revolt and it fails, then they get a payoff of negative one. So this could be in a situation where they're punished by the government or face some other kinds of sanctions, and they get a payoff of zero if the revolt's not successful and they didn't participate. So they weren't one of the people that was actually revolting. Obviously, this is very stylized, but what it does capture is that players have to strategically analyze and predict what other players are going to do, and their payoffs depend not only on what they're doing, right? So here we have a situation where player I's payoff depends on whether they revolt or not, but it also depends on what other players are doing, and it can depend in fairly complicated ways on what all the players in the game are doing. Okay, so just in summary, in defining games, we have two different forms, a normal form and an extensive form. For now, we're starting with the normal form, critical ingredients, players, actions, and payoffs. Later, when we get to the extensive form, that's going to bring in timing, information, and so forth. These are the things that will account for more detailed representations of the strategic interaction by players.