Games and Decisions. Part I: Basic Theorems. Contents. 1 Introduction. Jane Yuxin Wang. 1 Introduction 1. 2 Two-player Games 2

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1 Games and Decsons Part I: Basc Theorems Jane Yuxn Wang Contents 1 Introducton 1 2 Two-player Games Zero-sum Games Pure Strateges Mxed Strateges Solvng Games Non-zero-sum Games Introducton Game theory s a study of strategc decson makng. Specfcally, t s the study of mathematcal models of conflct and cooperaton between ntellgent ratonal decson-makers. In the comng presentatons, we are gong to see the basc termnologes of Game Theory, and then revew the lterature of Global Games. To understand the presentaton, we 1

2 need some basc Lnear Algebra as well as Analyss. For the Global Game part, some knowledge n Statstcs, or more specfcally, Bayesan updatng would be an advantage (but not a necessty). To start wth, we defne the word game. A game nvolves a number of players N, a set of strateges for each player, and a payoff that quanttatvely descrbes the outcome of each play of the game n terms of the amount that each play wns or loses. A strategy for each player s a plan determned at the start of the game, that descrbes what a player wll do n every possble stuaton. You may conceve a strategy as an agent who does not have ndvdual thnkng ablty and who acts on your behalf: you have to tell t what to do n every possble scenaro to come. Remark. Even though the players mght not move smultaneously, they choose ther strateges smultaneously at the start of the game. Ths s why a strategy need to tell each player what to do at each stage of the game, and why the strateges could be qute complcated. The related lterature usually nvolve cooperatve games and non-cooperatve games. In ths part, we focus on non-cooperatve games. In partcular, we emphass on two-player zerosum games. In part II, we are gong to vst some most recent research result n the feld of global game. 2 Two-player Games In a two-person game, the payoff are usually represented by a payoff matrx (or b-matrx, whch we shall explan n due course). A payoff matrx A = (a ) looks lke ths. We usually call the frst player a row player and the second play a column player, consderng the poston of them n the payoff matrx. In ths matrx, the entres a represent the payoff to player I. In the context of a b-matrx, there would be another matrx B where b represents the payoff to player II. 2

3 Player II Strategy 1 Strategy 2... Strategy m Strategy 1 a 11 a12... a 1m Player I Strategy 2 a 21 a a 2m Strategy n a n1 a n2... a nm Here s an example. Example 1. Two prsoners are suspected to have commtted crmes. They are ntervewed separately, and each has the opton to ether confess or deny. Ther payoffs are shown as follows (as n the negatve of years of mprsonment). Conf ess Deny Conf ess ( 1, 1) ( 3, 0) Deny (0, 3) ( 2, 2) Soluton 1. If you were player II, you would thnk about what prsoner II probably would do. If he confesses, then you would opt for denal; f he denes, then you would also prefer denal. Thus, both partes wll deny and end up n the (2, 2) entry n the matrx. In ths case, both players can fx a strategy. That s, whatever the country party does, you wll always stck to one strategy. As we wll see n the next secton, ths stuaton s called a pure strategy. Remark. Notce that by choosng denal, both players are forgong the a bette opton for both of them,.e., the (1, 1) entry. Ths s because n the (1, 1) entry, both players have ncentves to devate from ther choce. We would later see that (1, 1) entry s not a Nash Equlbrum pont. 3

4 2.1 Zero-sum Games Two-person zero-sum games are represented by a sngle matrx A, as descrbed n the prevous secton. Zero-sum means that the payoff to play II would be exactly the negatve of that to play I. That s, for each outcome, the payoff to play II s a. More generally speakng, constant-sum games can also be reduced to zero-sum games. Constant-sum games are the ones where the payoff to player II s always c a for some constant c. Whle the constant c could be regarded as some sunk costs (n Economcs terms), or smply an entry fee of the game, whch certanly does not affect our analyss of the games. Therefore, our focus les on zero-sum games n ths secton. An mportant purpose of studyng Game Theory s to fnd solutons to the games (or the payoff matrces). Therefore, startng from the next secton, we shall ntroduce a few soluton concepts and see the ustfcaton for them beng solutons to games Pure Strateges A smple defnton for pure strateges are smply that a player fxes one sngle strategy, or wth probablty 1. In contrast to pure strategy s the concept named mxed strategy, whch we shall nvestgate n the next secton. For ths secton, we ntend to see whether the players have a pure strategy. To begn wth, let s consder what you would do f you were player I. You may say, that depends on what player II does. Surely, our dscusson wll start from ths ratonale. Remember, play I s the row player, and he would lke to know what play II would do when he makes the decson. For any gven row, play II wants to choose the mnmum entry n the row,.e., mn a. Then play I wll choose to maxmze ths amount. We denote ths by v := max mn a 4

5 The v here s thus the worst possble stuaton that play I can guarantee. Smlarly, play II would do the same thng and guarantee an amount of v + := mn max a Defnton 1. A matrx game wth matrx A n,m has the lower value v := max the upper value v + := mn v = v(a) = v + = v. max One can easly show that v + v. Proof. Snce v = max mn and be such that v = mn mn a, and a. The game has a value f v = v +, and we wrte t as a and v + = mn a. Then mn max a, let be such that v + = max a, a a max a. Defnton 2. A partcular row and column are called a saddle pont n pure strateges f a a a,. (1) Remark. The and n the precedng defnton may not be unque. Consder a matrx wth all entres beng equal. It s a good tme to ntroduce the concept of Nash Equlbrum (NE) now. In nformal expressons, a Nash Equlbrum s a set of strategy where no player would beneft from unlateral change. That s, nobody would have ncentve to devate from hs or her orgnal choce f everybody else stcks to hs or her choce. Here s a more formal defnton for Nash Equlbrum. 5

6 Defnton 3. Let (S, f) be a game wth n players, where S s the strategy for player, S = S 1 S 2 S n s the set of strategy profles and f = (f 1 (x),, f n (x)) s the payoff functon for x S. Let x be a strategy profle of player and x be a strategy profle for all players except fro player. When each player {1,, n} chooses strategy x resultng n strategy profle x = (x 1,, x n }, then player obtans payoff f (x). A strategy profle X S s a Nash Equlbrum f, x S : f (x, x ) f (x, x ). Lemma 1. A game wll have a saddle pont n pure strategy f and only f v = max mn a = mn max a = v + (2) Proof. : From Defnton 2, we know that v + = mn max a max a a mn But v v + always holds, so we have v = v +. : Suppose v = max mn and be such that v = mn the rght and = on the left, we get a accordng to Defnton 2. a max mn a = v a = mn max a = v +. Let be such that v + = max a, a. Then a v = v + a for any,. Takng = on = v+ = v. Notce that a s a saddle pont Notce that when a saddle pont exsts n pure strateges, Defnton 2 guarantees that no player has ncentve to devate from hs part n the saddle pont. Therefore, n the cases when pure strateges exst, t s optmal to play accordng to the saddle pont for both players. Now let s see an example. 6

7 Example 2. Consder the followng matrx A= Fnd ts lower value and upper value. Also pont out ts saddle pont(s). Soluton 2. For player I, the row mnmums are 3, -3, 2 respectvely. Thus, v = 3. Smlarly, we can fnd that v + = 3. Thus, the game value s 3. The saddle pont s (1, 3). You may check that by adoptng ths strategy, no play would have the ncentve to devate from hs or her orgnal choce. As we can see, the exstence of saddle pont n pure strateges s a very nce property; however, t does not always exst. Example 3. Consder the followng matrx B= Fnd ts lower value and upper value. Does saddle pont of pure strateges exst n ths case? Soluton 3. Check that the lower and upper and lower value do not meet n ths game. Thus, by Lemma 1, saddle ponts do not exst. As shown n the prevous example, saddle pont of pure strateges may not exst n a game. Therefore, we wsh to fnd a soluton concept that always exsts. Ths s the motvaton for our next secton. 7

8 2.1.2 Mxed Strateges From the prevous example, we can see that f a player stcks to a certan strategy, the other party would take advantage of t. Therefore, we would lke to ntroduce the concept of mxed strateges, where players choose a row or column accordng to some probablty process that specfes the chance that each pure strategy wll be played. The probablty vectors are called mxed strateges. Defnton 4. A mxed strategy s a row vector X = (x 1, x 2,..., x n ) for player I and Y = (y 1, y 2,..., y m ) for play II, where n m x 0, x = 1 and y 0, y = 1. =1 =1 Denote the set of mxed strategy wth k components by S k := {(z 1, z 2,, z k ) z 0, = 1, 2,, k, Thus, X S n and Y S m. k z = 1}. =1 Remark. Note that pure strateges can also be represented n terms of mxed strateges. For nstance, supper player I desres the pure strategy of choosng row, then ths s denoted as X = e. Smlar arguments hold for player II. In ths sense, the set of mxed strateges contans the set of pure strateges. Snce probabltes are nvolved mxed strateges, the payoffs can be calculated only n the expected sense. That means the game payoff wll represent what each player can expect to receve and wll actually receve on average f the game s played many tmes. More precsely, we calculate as follows. Defnton 5. Gve X and Y,.e., strateges of player I and II chosen randomly, the expected 8

9 payoff to player I s E(X, Y ) = = = a Prob(I uses and II uses ) a Prob(I uses ) Prob(II uses ) x a y = XAY T As usual, we want to fnd the Nash Equlbra ponts, or the saddle ponts as ntroduced n the precedng secton. Defnton 6. A saddle pont n mxed strateges s a par (X, Y ) of probablty vectors X S n and Y S m that satsfes E(X, Y ) E(X, Y ) E(X, Y ), X S n, Y S m Therefore, (X, Y ) s an equlbrum. As shown n the prevous secton, saddle ponts mght not exst n pure strateges. Thus, we want to fnd out under what condton would saddle ponts, or (mxed) Nash Equlbra exsts n mxed strateges. It turns out ncely that mxed Nash Equlbra always exst. We now proceed to prove ths result. To start wth, we examne a theorem called von Neumann Mnmax Theorem, whch s crucal n establshng our result. The proof of von Neumann Mnmax Theorem utlzes a theorem called Kakutan s Fxed Pont Theorem, whch s qute nvolved n Topology. Thus, for now, we smply cte ths Theorem (and may prove t f tme permts towards the end of ths mn course). Even before that, let s frst ntroduce a few termnologes that wll be used n the followng few theorems. Convex: A set C R n s convex f a, b C and λ [0, 1], λa + (1 λ)b C. 9

10 Closed: C s closed f t contans all lmt ponts of sequences n C. Bounded: C s bounded f t s contaned n a ball of fnte radus. Compact: A closed and bounded subset of Eucldean space s compact. Convex functons: a functon g : C R s convex f C s convex and a, b C and λ [0, 1], g(λa + (1 λ)b) λg(a) + (1 λ)g(b). Now we are ready to present the Kakutan s Fxed Pont Theorem. Theorem 1. Kakutan s Fxed Pont Theorem. Let C be a closed, bounded, and convex subset of R n, and let g be a functon from C to P(C) (the power set of C). Assume that for 10

11 each x C, the set g(x) s nonempty and convex. Also assume that g s upper hemcontnuous 1. Then there s a pont x C satsfyng x g(x ). For now, we take ths Theorem to be true and proceed to von Neumann Mnmax Theorem. Theorem 2. von Neumann Mnmax Theorem. Let f : C D R be a contnuous functon. Let C R n and D R m be convex, closed, and bounded. Suppose that x f(x, y) s concave and y f(x, y) s convex. Then, v = max mn x C y D f(x, y) = mn max y D x C f(x, y) = v+ Proof. Frst, defne the set of ponts where the mn or max s attaned (ths s possble gven that f s contnuous and the domans are compact). For each x C, and for each y D, B x := {y 0 D : f(x, y 0 ) = mn f(x, y)}; y D A y := {x 0 C : f(x 0, y) = max f(x, y)} x C Then these two sets are nonempty, closed and convex. (Nonempty: f s contnuous and the domans are compact; Closed: B x D, A y C; Convex: for any y 0 1, y 0 2 B x and λ [0, 1], f(x, λy 0 1 +(1 λ)y 0 2) λf(x, y 0 1)+(1 λ)f(x, y 0 2) = mn y D f(x, y) f(x, λy0 1 +(1 λ)y 0 2), so they must be equal, and thus λy (1 λ)y 0 2 B. ) 1 A correspondence Γ : A B s sad to be upper hemcontnuous at the pont a f for any open neghborhood V of Γ(a) there exsts a neghborhood U of a such that for all x U, Γ(x) s a subset of V. 11

12 Now defne g(x, y) := A y B x. Kakutan s Theorem says that there s a pont (x, y ) g(x, y ) = A y B x. Thus, we obtan f(x, y ) = max x C f(x, y ) andf(x, y ) = mn y D f(x, y). so that v = max x C mn y D f(x, y) f(x, y ) mn y D max x C f(x, y) = v+ v and f(x, y ) f(x, y ) f(x, y), x C, y D. Ths means that (x, y ) s a saddle pont and v = v + = v = f(x, y ). Now we are ready to examne the exstence of Nash Equlbrum ponts n mxed strateges. Theorem 3. For any n m matrx A, we have v + = mn Y S m max X S n XAY T = max X S n mn XAY T = v. Y S m The common value s denoted as v(a) and s called the value of the game. In addton, there s at least one saddle pont X S n, Y S m so that E(X, Y ) E(X, Y ) = v(a) E(X, Y ), for all X S n, Y S m. Proof. Defne the functon f(x, Y ) := E(X, Y ) = XAY T. For any matrx A, the functon s lnear n both X and Y ; hence, t must be concave n X and convex n Y. Also, f s contnuous 2. The sets S n and S m are certanly convex, closed, and bounded. By von Neumann Mnmax Theorem, the frst part of ths theorem s establshed. The proof for the second half s smlar to the proof for Lemma 1. 2 The contnuty of f s, agan, qute nvolved. For now, we can smply conceve that f s contnuous 12

13 Let X be such that v = mn X AY T and Y be such that v + = max XAY T. Thus, Y X X, Y, XAY T v + = v X AY T. By take X = X on the left hand sde and Y = Y on the rght hand sde, we obtan X AY T v + = v X AY T. Hence, the two sdes must be equal, and X AY T = v(a) (so (X, Y ) s a saddle pont). To get some ntuton, a vsualzed functon wth saddle pont at 1 2, 1 2 looks lke ths. Now that we know mxed strateges must exst, we are ready to see some propertes of such equlbra. Here we lst one of the most mportant (n provng the results to come) propertes. Before that, let s frst see some notatons that we are gong to use. 13

14 E(X, ) means E(X, e ). E(, Y ) means E(e, Y ). A means the th column of A. A means the th row of A. Then comes the mportant property. Lemma 2. Let ω be a real number. If E(, Y ) = AY T ω E(X, ) = X A, = 1, 2,, n, = 1, 2,, m, then ω = v(a) and (X, Y ) s a saddle pont for the game. Proof. If a y ω a x, we have E(X, Y ) = x a y x ω = ω = y x a y = E(X, Y ) Ths gves ω = E(X, Y ). Thus, we have E(, Y ) E(X, Y ) E(X, ) for any,. Thus, E(X, Y ) E(X, Y ) E(X, Y ) for any X, Y, so (X, Y ) s a saddle pont and v(a) = E(X, Y ) = ω. Now that we ve known that mxed strateges always exst n a two-person zero-sum game, the problem remans to fnd out the saddle ponts. The proof of exstence, however, does not provde an algorthm for fndng them. We shall take a look at t n the next secton Solvng Games A convenent way of solvng 2 2 games s by graphs. Let s llustrate the graphcal soluton wth the followng example. Example 4. Suppose we have the matrx 14

15 A= Soluton 4. Frst, we need to check whether t has any pure strateges (f yes, then we cannot use the graphcal method). Snce v = 2 and v + = 3, we know that the optmal strateges must be mxed. We want to fnd the optmal strategy X = (x, 1 x) for player I, where 0 < x < 1. Playng X aganst each column of player II, we get E(X, 1) = XA 1 = 3 2x and E(X, 2) = XA 2 = 2 + 2x. We plot these functons on the same graph. No matter what strategy player II chooses, the expected payoff for player I s always between the two lnes. Recall that player I always plays max mn strateges. Thus, X = ( 1, 3 ), and 4 4 the value of the game s

16 There are more systematcal ways of solvng certan knds of matrces. For nvertble matrces, the soluton s gven by the followng theorem. Theorem 4. Let J n be the 1 n row vector where all entres equal to 1. Assume that 1. A n n has an nverse A J n A 1 J nt v(a) 0. Set X = (x 1, x 2,, x n ) and Y = (y 1, y 2,, y m ), and v 1 J n A 1 J, Y T = A 1 J n T n J n A 1 J, X = J na 1 T n J n A 1 T J n T If x 0 and y 0, we have v = v(a) s the value of the game wth matrx A and (X, Y ) s a saddle pont n mxed strateges. Proof. From Lemma 4, we can see that every optmal strategy for player II must satsfy v(a) E(, Y ) = AY T = v(a). Thus, AY T v(a) = = v(a)jn T. v(a) Note that f v(a) = 0, then Y T = 0, whch s mpossble snce the entres of Y add up to 1. Ths s why we requre v(a) 0. Thus, Y T = v(a)a 1 J T n. Snce y = 1, we obtan J n Y T = 1 = v(a)j n A 1 J T n Therefore, v = 1 and Y T = A 1 J n J n A 1 T J n J n A 1 J. T n 16 T

17 Ths s only a canddate for optmal strategy for player II. By the same method, we obtan If the formula we get for X, Y end up satsfyng the condton that they are strateges, then they are optmal strateges, because for any mxed strategy Y, E(X, Y ) = XAY T = 1 J na 1 J n T J n A 1 AY T = 1 J na 1 J n T J n Y T = 1 J na 1 J n T = v Smlarly, we can check that E(X, Y ) = v, so (X, Y ) s saddle and v s the value of the game. 2.2 Non-zero-sum Games The man dfference between zero-sum and non-zero-sum games s that the later nvolves two matrces, and each player ams at fndng an optmal strategy based on the two strateges. A maor result n ths sort of game s that mxed Nash Equlbrum always exsts n ths sort of games as well. The proof s smlar to that shown n the zero-sum game secton, and we wll not present t agan here. 17