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Hi folks, welcome back. So this is Matt Jackson, and we are talking now about

Â defining games and we work to some basic definitions of the key ingredients in, in

Â games. So let's take a look at some of those. So

Â obviously one of the most obvious ones is the players in a game. So who is making

Â the decisions, are they people? Are we talking about governments

Â negotiating over trade agreements? Are we talking about companies, choosing

Â astrologies for developing the products? do we, do we want to get down to the, the

Â point of modeling people within a firm, as opposed to the company as a whole? so

Â this whole, there's a whole series of questions about how we're going to choose

Â the players, but they're, they're going to be the central decision makers in what

Â we're doing. next we have to decided how we're going

Â to model the actions. So what can players, what actions can

Â players actually take? So, when we're, later on in the course we'll be looking

Â at auctions, they'll have bids, so they can enter a number of bids.

Â when we're talking about bargaining, it might be deciding whether or not to

Â strike. when we're thinking about investing, it

Â could be that an investor is deciding how much of a stock to buy or sell, when to

Â buy or sell it. how they should react to other people in

Â the market, how they should be conditioning their decisions on, on

Â prices. when we think about voters, how do they

Â vote. So, there's going to be a whole series of

Â actions, and we'll want to be careful in making sure that we have the essential

Â actions modeled. finally, payoffs.

Â So, what's motivating the players? Do they care simply about some sort of

Â profit. Do they care about other players? So how

Â are they receiving utility as a function of what, what the actions lead to in the

Â context of the game? So there's basically 2 standard representations of games.

Â one is, is what's known as the normal form, and that's what we'll be starting

Â with in the course, and what it does is it, it's a a very

Â simple and, and stark representation of a game.

Â So it lists what payoffs players get as a function of their actions.

Â normally, it's, it's thought of as, as, as if players were moving simultaneously,

Â but strategies, and we'll talk about this in more detail, can, can encode many

Â things. So, the other alternative representation

Â is what's known as the Extensive Form, and that includes more explicit timing in

Â the game. So who moves at what, at what point in

Â time. So that's going to be represented often

Â as a tree. So, for instance in chess one player

Â moves first. the white player generally moves first,

Â and the, the black player can see the, the move by the other player, react to

Â that. And so far.

Â So that's going to be better represented as a tree than than in normal form.

Â So keeps track of also what players know when they move.

Â So in poker, somebody moves first. They may give a bet, but the other player only

Â sees the bet and not necessarily the card that other player sees.

Â So in some cases we'll have sequential games, where players will have different

Â information at different place and time. We'll want to talk about modeling that

Â explicitly too. So we're going to start out with the

Â normal form, and then we'll move later in the course to the extensive form, and

Â we'll talk about the relationship between these two in more details.

Â OK, so normal form games. What are the key ingredients? again,

Â players. So we're going to have generally we're

Â going to think of finite sets of players. So 1 through n, little n will represent

Â the set of players. Generally, we'll index these things by an

Â i so we'll use a little i to represent the, a generic player.

Â The action set for, for players. we'll represent by a sub i. Okay, so

Â we'll let that represent the actions of player i, and then we'll talk about

Â profiles of actions which will just be a list of what every player is doing.

Â So for instance are they the, the, deciding to, cooperate or not to

Â cooperate with other players, for instance.

Â In the, in the Prisoner's Dilemma, that we'll take about.

Â the utility function is then a payoff function, which indicates as a function

Â of all the actions that are played What's the payoff for the different players?

Â So for each player i, we end up with a function which tells us how they evaluate

Â outcomes of the game. And again, how they evaluate these things

Â could could encapsulate many things, and it's going to be very important to

Â make sure that we were getting the right representation of what really motivates

Â people. Okay.

Â So, often, when we, when we represent normal form games, a very simple ways of

Â doing that is just matrix representation. So, let's just look at, at, the, the most

Â standard representation of very simple games.

Â writing at two player game as a matrix. So we'll have one player 1, will be the

Â role player. Player 2 will have, be a column player.

Â So they're going to choose actions that'll be represented in a column of the

Â matrix. And the cells, inside the cells will then

Â represent the payoffs. So for instance, the TCP Backoff game

Â that was talked about in the earlier video, can be written as a matrix as

Â follows. So, the roleplayer, player 1 can be

Â written as either C or D. So this is player 1's choice, generally

Â known as the row player. This is player 2's, the column player,

Â and they represent the, the choices that they have, and in inside the cells are

Â the payoffs to the different players. So if player 1 cooperates and player 2

Â cooperates, then these are the payoffs to the 2

Â players. The first payoff, player 1, second

Â playoff, player 2. So this is going to the column player, this one is going to

Â the row player. Okay? Then we end up, you know, for

Â instance, if the row player chooses D, and the column player chooses C, then we

Â end up with a payoff here of 0 to the row player, and minus 4 to the column player.

Â So the matrix is a very simple way of representing all of the, basic elements

Â of the normal form game visually, so that we can actually keep track of exactly

Â what the strategic interaction is, and, and what players would like to do as a

Â function of the game Okay. let's talk about another game that we

Â won't be able to write down in such a simple form.

Â so let's think of a large collective action, game. So, for instance, whether

Â or not a population wants to revolt against its government So here, we have

Â many more players. So let's imagine that we have a

Â population of 10 million players. So we're not, obviously not going to be

Â able to write that down as a, as a matrix on our screen, so we can do that more

Â abstractly. But we'll have 10 million players,

Â whether they, whether their actions here, let's keep it very simple.

Â So they have a choice here of either revolting or not.

Â So the action set is binary, two choices. then the payoffs are going to be critical

Â thing in this game. what happens? Well, let's say that in

Â order for revolt to be successful, you need at least 2 million people to

Â participate. So in this particular stylized example,

Â what do we end up with then? We, we can represent the successful revolt as the

Â player getting a path of 1 so "Ui" of the action profile "A" is equal to 1.

Â If the number of people here, the number of, of players, j, such that they picked

Â to revolt, the number of this is at least 2 million.

Â So, if we end up with at least 2 million people, revolting, then player i, gets 1,

Â and note here that this is true regardless of whether I as one of the

Â revolt's participants, so this is a game where you care about the end outcome, not

Â necessarily getting utility out of the participation.We could change this and

Â have people get enjoyment out of the participation or have cost of the

Â participation directly as well. Okay, so what's, what happens if if

Â things fail. here, if we end up with less 2,000,000,

Â then it depends on whether you were a participant in the revolt or not.

Â So, if you, if player i was a participant in the revolt and it fails,

Â then they get a payoff of negative 1. So, this could be in a situation where

Â they're punished by the government, or face some other kinds of sanctions,

Â and they get a path of 0 if they were both not successful and they didn't

Â participate so they weren't one of the people that was actually revolting.

Â Now obviously this is very stylized, but what it does capture is that players have

Â to strategically analyze and predict what other players are going to do and their

Â pay-offs depend not only on what they're doing, right so here we have a situation

Â where player i's payoff depends on whether they revolt or not but it also

Â depends on what other players are doing and it can depend in fairly complicated

Â ways on what all the players in the game are doing.

Â Okay, so just in summary, in defining games we have two different forms, the

Â normal form and the extensive form. For now, we're starting with the, normal

Â form critical ingredients, players, actions and pay-offs.

Â Later, when we get to the extensive form, that's going to bring in timing,

Â information, and so forth. So, extra things, that will account for

Â more detailed, representations of, of the, strategic interaction by players.

Â