1 - 3 - 1-3 defining games (10-36).txt

8/10/2019 1 - 3 - 1-3 Defining Games (10-36).txt

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Hi folks, welcome back. So this is MattJackson, and we are talking now aboutdefining games and we work to some basicdefinitions of the key ingredients in, ingames.So let's take a look at some of those. Soobviously one of the most obvious ones isthe players in a game. So who is makingthe decisions, are they people?Are we talking about governmentsnegotiating over trade agreements?Are we talking about companies, choosingastrologies for developing the products?do we, do we want to get down to the, thepoint of modeling people within a firm,as opposed to the company as a whole? sothis whole, there's a whole series ofquestions about how we're going to choosethe players, but they're, they're goingto be the central decision makers in whatwe're doing.next we have to decided how we're goingto model the actions.So what can players, what actions can

players actually take? So, when we're,later on in the course we'll be lookingat auctions, they'll have bids, so theycan enter a number of bids.when we're talking about bargaining, itmight be deciding whether or not tostrike.when we're thinking about investing, itcould be that an investor is deciding howmuch of a stock to buy or sell, when tobuy or sell it.how they should react to other people inthe market, how they should be

conditioning their decisions on, onprices.when we think about voters, how do theyvote.So, there's going to be a whole series ofactions, and we'll want to be careful inmaking sure that we have the essentialactions modeled.finally, payoffs.So, what's motivating the players? Dothey care simply about some sort ofprofit.Do they care about other players? So how

are they receiving utility as a functionof what, what the actions lead to in thecontext of the game? So there's basically2 standard representations of games.one is, is what's known as the normalform, and that's what we'll be startingwith in the course,and what it does is it, it's a a verysimple and, and stark representation of agame.


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So it lists what payoffs players get as afunction of their actions.normally, it's, it's thought of as, as,as if players were moving simultaneously,but strategies, and we'll talk about thisin more detail, can, can encode manythings.So, the other alternative representationis what's known as the Extensive Form,and that includes more explicit timing inthe game.So who moves at what, at what point intime.So that's going to be represented oftenas a tree.So, for instance in chess one playermoves first.the white player generally moves first,and the, the black player can see the,the move by the other player, react tothat.And so far.So that's going to be better representedas a tree than than in normal form.

So keeps track of also what players knowwhen they move.So in poker, somebody moves first. Theymay give a bet, but the other player onlysees the bet and not necessarily the cardthat other player sees.So in some cases we'll have sequentialgames, where players will have differentinformation at different place and time.We'll want to talk about modeling thatexplicitly too.So we're going to start out with thenormal form, and then we'll move later in

the course to the extensive form, andwe'll talk about the relationship betweenthese two in more details.OK, so normal form games.What are the key ingredients? again,players.So we're going to have generally we'regoing to think of finite sets of players.So 1 through n, little n will representthe set of players.Generally, we'll index these things by ani so we'll use a little i to representthe, a generic player.

The action set for, for players.we'll represent by a sub i. Okay, sowe'll let that represent the actions ofplayer i, and then we'll talk aboutprofiles of actions which will just be alist of what every player is doing.So for instance are they the, the,deciding to, cooperate or not tocooperate with other players, forinstance.


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In the, in the Prisoner's Dilemma, thatwe'll take about.the utility function is then a payofffunction, which indicates as a functionof all the actions that are played What'sthe payoff for the different players?So for each player i, we end up with afunction which tells us how they evaluateoutcomes of the game.And again, how they evaluate these thingscould could encapsulate many things,and it's going to be very important tomake sure that we were getting the rightrepresentation of what really motivatespeople.Okay.So, often, when we, when we representnormal form games, a very simple ways ofdoing that is just matrix representation.So, let's just look at, at, the, the moststandard representation of very simplegames.writing at two player game as a matrix.So we'll have one player 1, will be the

role player.Player 2 will have, be a column player.So they're going to choose actionsthat'll be represented in a column of thematrix.And the cells, inside the cells will thenrepresent the payoffs.So for instance, the TCP Backoff gamethat was talked about in the earliervideo, can be written as a matrix asfollows.So, the roleplayer, player 1 can bewritten as either C or D.

So this is player 1's choice, generallyknown as the row player.This is player 2's, the column player,and they represent the, the choices thatthey have, and in inside the cells arethe payoffs to the different players.So if player 1 cooperates and player 2cooperates,then these are the payoffs to the 2players.The first payoff, player 1, secondplayoff, player 2. So this is going tothe column player, this one is going to

the row player.Okay? Then we end up, you know, forinstance, if the row player chooses D,and the column player chooses C, then weend up with a payoff here of 0 to the rowplayer, and minus 4 to the column player.So the matrix is a very simple way ofrepresenting all of the, basic elementsof the normal form game visually, so thatwe can actually keep track of exactly


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what the strategic interaction is, and,and what players would like to do as afunction of the game Okay.let's talk about another game that wewon't be able to write down in such asimple form.so let's think of a large collectiveaction, game. So, for instance, whetheror not a population wants to revoltagainst its government So here, we havemany more players.So let's imagine that we have apopulation of 10 million players.So we're not, obviously not going to beable to write that down as a, as a matrixon our screen, so we can do that moreabstractly.But we'll have 10 million players,whether they, whether their actions here,let's keep it very simple.So they have a choice here of eitherrevolting or not.So the action set is binary, two choices.then the payoffs are going to be critical

thing in this game.what happens? Well, let's say that inorder for revolt to be successful, youneed at least 2 million people toparticipate.So in this particular stylized example,what do we end up with then? We, we canrepresent the successful revolt as theplayer getting a path of 1 so "Ui" of theaction profile "A" is equal to 1.If the number of people here, the numberof, of players, j, such that they pickedto revolt, the number of this is at least

2 million.So, if we end up with at least 2 millionpeople, revolting, then player i, gets 1,and note here that this is trueregardless of whether I as one of therevolt's participants, so this is a gamewhere you care about the end outcome, notnecessarily getting utility out of theparticipation.We could change this andhave people get enjoyment out of theparticipation or have cost of theparticipation directly as well.Okay, so what's, what happens if if

things fail.here, if we end up with less 2,000,000,then it depends on whether you were aparticipant in the revolt or not.So, if you, if player i was a participantin the revolt and it fails,then they get a payoff of negative 1.So, this could be in a situation wherethey're punished by the government, orface some other kinds of sanctions,


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and they get a path of 0 if they wereboth not successful and they didn'tparticipate so they weren't one of thepeople that was actually revolting.Now obviously this is very stylized, butwhat it does capture is that players haveto strategically analyze and predict whatother players are going to do and theirpay-offs depend not only on what they'redoing, right so here we have a situationwhere player i's payoff depends onwhether they revolt or not but it alsodepends on what other players are doingand it can depend in fairly complicatedways on what all the players in the gameare doing.Okay, so just in summary, in defininggames we have two different forms, thenormal form and the extensive form.For now, we're starting with the, normalform critical ingredients, players,actions and pay-offs.Later, when we get to the extensive form,that's going to bring in timing,

information, and so forth.So, extra things, that will account formore detailed, representations of, ofthe, strategic interaction by players.

1 - 3 - 1-3 defining games (10-36).txt

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