1 Games in Normal Form (Strategic Form)

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1 1 Games i Normal Form Strategic Form) A Game i Normal strategic) Form cosists of three compoets: 1 A set of players For each player, a set of strategies called actios i textbook) The iterpretatio is that this should be thought of as aythig the player ca do 3 For each player, some prefereces over the set of strategy profiles We will let N deote the set of agets, which is assumed to be a fiite set For each player i N we deote by S i the strategy set To reduce otatioal clutter we let S = i=1s i the Cartesia product), which allows us to write a payoff fuctio for player i as u i : S R Defiitio 1 A ormal form game is a triple G = N, S, u) where N = {1,, } is the set of players, S = i=1s i is the strategy space ad u = u 1,, u ) are the payoff fuctios for player 11 Laguage/Notatio It is coveiet to establish some laguage ad shorthad otatio for future use: 1 We write s i for a geeric elemet i S i ad refer to it as a strategy We write s = s 1,, s ) for a vector cosistig of a strategy for each player We refer to this as a strategy profile 3 We write s i = s 1,, s i 1, s i+1,, s ) for a strategy profile where the ith coordiate has bee removed 4 We will also use S i for j i S j, so that S i cosists of every coceivable s i that the players other tha i may pick 1

2 5 The latter covetio allows us to write s i, s i ) for the strategy profile s 1,, s i 1, s i, s i+1,, s ), which will be coveiet as equilibrium cocepts are defied by havig each player i optimize give s i 1 Domiat ad Domiated Strategies Defiitio s i is strictly domiated by s i if u i s i, s i ) < u i s i, s i ) for every s i S i Defiitio 3 s i is weakly domiated by s i if u i s i, s i ) u i s i, s i ) for every s i S i ad if u i s i, s i ) < u i s i, s i ) for some s i S i Remark 1 This is ostadard i that the typical otio of domiace allows for radomizatios We will retur to this whe talkig about radomized strategies Remark s i is strictly weakly) domiated by s i or s i used iterchageably strictly weakly) domiates s i are Defiitio 4 s i is a strictly domiat strategy if u i s i, s i ) > u i s i, s i ) for every s i S i ad every s i S i \ {s i} Defiitio 5 s i is a weakly domiat strategy if u i s i, s i ) u i s i, s i ) for every s i S i ad every s i S i \ {s i} ad if for every s i S i \ {s i} u i s i, s i ) > u i s i, s i ) for some s i S i 11 The Prisoers Dilemma Suppose that N = {1, }, S 1 = S = {Defect, Cooperate} ad that u 1 Defect, Cooperate) > u 1 Cooperate, Cooperate) > u 1 Defect, Defect) > u 1 Cooperate, Defect) u Cooperate, Defect) > u Cooperate, Cooperate) > u Defect, Defect) > u Defect, Cooperate)

3 Let u 1 Defect, Cooperate) = u Cooperate, Defect) = a u 1 Cooperate, Cooperate) = u Cooperate, Cooperate) = b u 1 Defect, Defect) = u Defect, Defect) = c u 1 Cooperate, Defect) = u Defect, Cooperate) = d, where a > b > c > d i order to capture the prisoer s dilemma rakig It is coveiet to collect this iformatio i a payoff bi-matrix Player Player 1 Defect Cooperate Defect c, c a, d Cooperate d, a b, b ad sice c > d ad a > b it is a strictly domiat to play defect Hece, Defect, Defect) is a equilibrium i domiat strategies 1 A Alterative Story Suppose Alice ad Bob have a law movig operatio They ca either work hard) or shirk, ad the total profit is 10 if both work hard 6 if oe of them works hard ad if both shirks Also assume that the cost of effort is e ad that they split the profit equally This gives the followig payoff matrix Bob Alice Shirk Work hard Shirk 1, 1 3, 3 e Work hard 3 e, 3 5 e, 5 e 3

4 Hece, with cost of effort e = 3 we get Alice Bob Shirk Work hard Shirk 1, 1 3, 0 Work hard 0, 3,, which is idetical a Prisoer s dilemma 13 A Secod Price Auctio with Bidders Vickrey Auctio) Let N = {1, }, S 1 = S = R + ad u 1 b 1, b ) = v 1 b if b 1 > b v 1 b if b 1 = b 0 if b 1 < b u b 1, b ) = v b 1 if b 1 < b v b 1 if b 1 = b 0 if b 1 > b Clearly, there is o strictly domiat strategy as, for example, for every b 1, b 1) such that b 1 > b 1 > b u 1 b 1, b ) = u 1 b 1, b ) = v 1 b Propositio 1 The uique weakly domiat strategy is to bid b i = v i Proof Without loss, cosider i = 1 Suppose that v 1 > b the u 1 v 1, b ) = v 1 b > 0 v 1 b = u 1 v 1, b ) if b 1 > b u 1 b 1, b ) = v 1 b = u 1v 1,b ) < u 1 v 1, b ) if b 1 = b 0 < u 1 v 1, b ) if b 1 < b 4

5 Suppose that v 1 = b the Suppose that v 1 < b the u 1 v 1, b ) = v 1 b = 0 v 1 b = 0 if b 1 > v 1 = b u 1 b 1, b ) = 0 if b 1 < v 1 = b u 1 v 1, b ) = v 1 b = 0 v 1 b < 0 if b 1 > b u 1 b 1, b ) = v 1 b < 0 if b 1 = b 0 if b 1 < b Hece, i each case biddig the valuatio is optimal, so it is weakly domiat Next, geeralize to the case of N = {1,, } v i max j i b j if b i > max j i b j v u i b 1,, b ) = 1 max j i b j if b arg max j N b j i = max j i b j 0 if b i < max j i b j The proof is idetical oce we replace b with max j i b j, so agai it is a weakly domiat strategy to bid the valuatio 14 The Groves Mechaism Suppose that N = {1,, }, that x X is some social decisio with cost fuctio C x) ad that prefereces are give by u i x, v i ) t where v i is a preferece parameter that is also a elemet of some set V i To geerate a ormal form assume that each player i may choose a v i V i ad that give strategy profile 5

6 v = v 1,, v ) a social decisio x v) arg max x X u i x, v i ) C x) i=1 t i v) = C x v)) j i u j x v), v j ) Hece, we have a ormal form game N, V, π) where the payoff π i is give by π i v) = u i x v), v i ) t i v) = u i x v), v i ) + j i u j x v), v j ) C x v)) By defiitio, x v i, v i ) arg max x X [ u i x, v i ) + j i u j x, v j ) C x) ], so v i = v i is a weakly domiat strategy Moreover, pick ay fuctio h i : j i V J R ad ote that if the t i v) = C x v)) j i u j x v), v j ) + h i v i ) π i v) = u i x v), v i ) + u j x v), v j ) C x v)) h i v i ), }{{} j i costat so v i = v i is still a weakly domiat strategy 15 Complete versus Icomplete Iformatio Notice that the ormal form that we have specified assumes that players have complete iformatio about the payoff fuctios Usually, the way we thik about the Groves mechaism is as a way to truthfully elicit prefereces To make sese of this i a complete iformatio setup we ca imagie that a plaer is uiformed, while the agets kow everythig about each other However, whe lookig at domiace i this particular class of eviromet it turs out that the aalysis carries over to the case with icomplete iformatio We may discuss this whe we study games of icomplete iformatio 6

7 16 Why it Works: Iteralizig the Exterality Suppose that i would t exist The the effi ciet choice would be max x X u j x, v j ) C x) S i v i ) j i Hece, the exteral effect o the rest of the ecoomy from beig added to the ecoomy is u j x v), v j ) C x v)) S i v i ), j i so the iterpretatio of the Groves trasfer is that each aget is beig paid the exterality it geerates o all others if positive) or eeds to pay the exterality imposed o all others if egative) 17 A Vickrey Auctio is a Special Case of the Groves Mechaism Let 1 u i x, v i ) = x i v i X = { x R + x i 0 for each i ad i=1 x i = 1 } 3 C x) = 0 for each x X 4 V i = R + The, x v) arg max x x i v i i=1 st x i 0 for each i x i = 1 Obviously ties ca be broke ay way, but oe solutio is that 1 if v x arg max i v) = j N v j i = max j v j 0 if v i < max j v j i=1 7

8 Now t i v) = u j x v), v j ) = j i 0 if v i > max j i v ) j 1 1 max arg max j N v j j i v j if v i = max j i v j max j i v j if v i < max j i v j Hece, the Groves mechaism boils dow to a auctio where the wier does t pay ad the loser is paid the aouced valuatio of the wier However, just add h i v i ) = max j i v j to the trasfer ad we get max j i v j if v i > max j i v j 1 t i v) = arg max j N v j j i v j if v i = max j i v j 0 if v i < max j i v j We ca thus coclude that the secod price auctio is a special case of a Groves mechaism for a eviromet where a sigle idivisible object is to be allocated to a aget 18 Groves Mechaisms for a Biary Public Good Now, assume istead that 1 u i x, v i ) = xv i X = [0, 1] 3 C x) = xc for each x [0, 1] 4 V i = R + Now or igorig the idifferece) x i v) = ) x i v) arg max x v i c x [0,1] i=1 1 if i=1 v i c > 0 0 if i=1 v i c < 0 8

9 so that c j i t i v) = v i which is ofte referred to as a pivot mechaism if i=1 v i c > 0 0 if i=1 v i c < 0, 13 Iterated Elimiatio of Strictly Domiated Strategies The reader should be wared that the stadard approach to domiace as well as iteratios of domiace relies o radomized strategies We will deal with that later However, if a strategy is strictly domiated it is ot a best respose to aythig the oppoets) ca do Ratioality would therefore rule out play of strictly domiated strategies But, assumig that all agets uderstad the structure of the game, the other agets should uderstad that strictly domiated strategies will ot be played, so oe should be able to elimiate these Hece, we ca ask agai whether there are additioal strategies that are strictly domiated give the smaller set of survivig strategies 131 A Simple Example Cosider L C R U 1, 0 1, 0, 1 D 0, 3 0, 1, 0 ad ote: 1 R is strictly domiated by C Hece, player 1 the row player) should make the iferece that R will ot be played ad cosider the reduced game L C U 1, 0 1, D 0, 3 0, 1 9

10 I the reduced game D is strictly domiated by U, so the game that remais is L C U 1, 0 1, Player should the reaso that player 1 will predict that he will ot play the strictly domiated strategy R ad that 1 cosequetly will ot play D 3 Hece, player should pick C ad U,C) is the uique strategy profile that survives iterated elimiatio of strictly domiated strategies 13 A Liear Courot Duopoly Suppose that iverse demad is P Q) = max { Q, 0} ad that two firms produce a homogeous good at a costat uit cost c = 1 Assumig firms maximize profits their payoff fuctios are the π 1 q 1, q ) = max {q 1 1 q 1 q ), q 1 } π 1 q 1, q ) = max {q 1 1 q 1 q ), q 1 } which immediately allows us to coclude that ay q i > 1 is strictly domiated Hece, the problem for i simplifies to max q i q i 1 q i q j ) which has solutio q i = 1 q j, which is strictly decreasig i q j We therefore coclude that ay q i > 1 is strictly domiated as the q i 1 q i q j ) ) q j = q i 1 q i ) ) q i 1 ) q j < q i 1 q i ) ) < 0 where the fial iequality holds because 1 solves the moopoly problem max qq 1 q) But the we may coclude that q i < 1 4 is strictly domiated i the reduced game where quatities 10

11 are picked i [ 0, ] 1 as qi < 1 implies that 4 q i 1 q i q j ) ) 4 4 q j = q i 1 q i 1 ) ) + q i 1 q i 1 ) ) < 0, where the last iequality holds because 1 4 maximizes q i 1 qi 1 ) q i 1 ) ) 1 4 q j }{{}}{{} I geeral, assume that after recursios we are left with [a, b ] The, after + 1 recursios we are left with [a +1, b +1 ] where a +1 = 1 b b +1 = 1 a This ca be see by observig that if q i < 1 b the, q i 1 q i q j ) 1 b 1 1 b ) q j = q i 1 q i b ) 1 b 1 1 b ) b + as 1 b q i 1 q i b ) 1 b 1 1 b ) b < 0 <0 q i 1 b ) b q j ) }{{}}{{} 0 <0 maximizes q i 1 q i b ) Moreover, if q i > 1 a the q i 1 q i q j ) 1 a 1 1 a ) q j = q i 1 q i a ) 1 a 1 1 a ) a + q i 1 a ) a q j ) }{{}}{{} 0 >0 q i 1 q i a ) 1 a as 1 a maximizes q i 1 q i a ) 1 1 a ) a < 0 Hece, the set of strategies that survive iterated elimiatio of strictly domiated strategies is q i [a, b ] for every But 0 a +1 = 1 1 a 1 b +1 = 1 1 a 1 = 1 + a 1 4 = 1 + b

12 so: a +1 > a 1 provided that 1+a 1 > a 4 1 a 1 < 1 ad a 3 +1 < 1 provided that 3 1+a 1 < 1 a < 1 Hece, sice a 3 1 = 0 it follows by iductio that a ) 0, 1 3 for each, implyig that {a } =1 is a icreasig sequece b +1 < b 1 provided that 1+b 1 < b 4 1 b 1 > 1 ad b 3 +1 > 1 provided that 3 1+b 1 > 1 b > 1 Hece, sice b 3 1 = 1 it follows by iductio that a 1, ) 1 3 for each, implyig that {b } =1 is a decreasig sequece {a } =1 ad {b } =1 are takig o values i [0, 1] Mootoe covergece theorem implies existece of limits: Hece a a ad b b where a = 1 + a 4 b = 1 + b 4 or a = b = Iterated Elimiatio of Weakly Domiated Strategies Cosider L R U 1, 0 0, 1 D 0, 0 0, L is weakly strictly too) domiated Oce it is elimiated othig more ca be elimiated Hece, the predictio is U,R) ad D,R) D is weakly domiated Oce it is elimiated L ca be elimiated Hece U,R) is all that remais Hece, the order matters 1

13 Nash Equilibrium Joh Nash suggested that a plausible way to defie equilibria i games is to postulate: 1 Every player does his or her best agaist what other players are doig utility maximizatio), ad; All players have ratioal expectatios This leads to the followig defiitio: Defiitio 6 s S is a pure strategy) Nash equilibrium if u i s i, s i) ui si, s i) for each i ad every s i i S i 1 Justificatios As we will see i examples, sometimes the Nash equilibrium cocept is a bit problematic as it imposes lots of coordiatio However, there are several types of justificatios: If the players agree i advace to play a particular) Nash equilibrium, obody has a icetive to deviate Mass actio/social orms If a strategic situatio occurs frequetly, a Nash equilibrium may emerge as a orm for how to play Predictig o Nash equilibrium play implies that the game theorist uderstads the game better tha the players beig modelled Evolutioary stability Close coectio betwee Nash equilibria ad evolutioary equilibrium cocepts The Best Respose Correspodace I simple games we ca use the defiitio of Nash directly ad check for equilibria by guess ad verify However, whe it gets more complicated we eed to be more systematic 13

14 Defiitio 7 Give a ormal form game G = N, S, u) the best respose correspodece for player i is a mappig β i : S i S i defied by β i s i ) = arg max s i u i s i, s i ) = {s i i S i such that u i s i, s i ) u i s i, s i ) for all s i i S i } Defiitio 8 Give a ormal form game G = N, S, u) the best respose correspodece fis a mappig β : S S where for every s S we have that β s) = β 1 s 1 ),, β s )) Notice that β i s i ) i geeral may be a set as illustrated i the simple example below, Bob where β 1 Left) = {Up, Dow} It is pretty clear that: Alice Left Right Up 0, 0, 1 Dow 0, 4 3, 3, Propositio s is a pure strategy) Nash equilibrium if ad oly if s β s ) for every player i Proof Suppose that s is a pure strategy) Nash equilibrium but that s is ot i β s ) The there at least oe player such that s i / β i s i ) implyig that there exists s i such that u i s i, s i) > ui s ) Hece, there is a cotradictio as this says that s is ot a Nash equilibrium for every player i Next, if s β s ) the s i β i s i ) for each player so that ) s i arg max u i si, s i, s i S i for each player i which meas that s is a Nash equilibrium 14

15 3 Fidig Nash Equilibria By Usig the Best Respose Correspodace 31 The Algoritm Cosider the Game Bob Alice x y z a 1, 1, 0 3, 1 b 1, 1 6, 1 8, 7 c 3, 0 10, 0 6, 5 d 3, 0, 0 5, 1, which does t have ay particular iterpretatio Begi by puttig i the best resposes for Alice by uderliig the correspodig payoff i the payoff bi-matrix as follows Bob x y z Alice a 1, 1, 0 3, 1 b 1, 1 6, 1 7, 7 c 3, 0 10, 0 6, 5 d 3, 0, 0 5, 1, Do the same for Bob Bob Alice x y z a 1, 1, 0 3, 1 b 1, 1 6, 1 7, 7 c 3, 0 10, 0 6, 5 d 3, 0, 0 5, 1, We coclude that c, x), d, x) ad b, x) are Nash equilibria 15

16 3 Courot Duopoly with Liear Demad ad Costat Uit Cost Zero Cosider the Courot duopoly model with iverse demad P q) = q ad C i q i ) = cq i The igorig q 1, q ) such that q 1 + q >, which caot be the case i a equilibrium) π 1 q 1, q ) = q 1 1 q 1 q ) π 1 q 1, q ) = q 1 q 1 q ) We will solve for a Nash equilibrium by usig the best respose Hece, B 1 q ) is foud by solvig I a Nash equilibrium β 1 q ) = arg max q 1 q 1 1 q 1 q ) = 1 q β q 1 ) = arg max q 1 q 1 q ) = 1 q 1 q q 1 = β 1 q ) = 1 q q = β q1) = 1 q 1 Solvig we obtai q 1, q ) = 1 3, 1 3), which is the uique Nash equilibrium of the game We otice that this is also the solutio by itreated elimiatio of strictly domiated strategies This is a geeral fact: Propositio 3 Suppose that s is the oly strategy profile that survives iterated elimiatio is strictly domiated strategies The, s is a Nash equilibrium Proof Let { S k} k=1 be a sequece of subsets of S where Sk is the set of strategies that survive k rouds of elimiatio if the game is fiite, the process will stop after a fiite umber of steps, but this is irrelevat for the argumet) If s survives the process of elimiatio it meas that for each k ad every s i S k i there exists s i S k i such that u i s i, s i ) u i s i, s i ) 16

17 holds Assume that s is ot a Nash equilibrium The, there exists i N ad s i S i such that u i si, s i) > ui s i, s i), which implies that s i, s i) survives the process of elimiatio because s i S i k for every k This cotradicts the assumptio that s is the uique profile survivig iterated elimiatio of strictly domiated strategies Also, Propositio 4 Suppose that s is a Nash equilibrium, the s survives iterated elimiatio of strictly domiated strategies Proof Agai, let { S k} k=1 be a sequece of subsets of S where Sk is the set of strategies that survive k rouds of elimiatio For cotradictio, suppose that s does ot survive iterated elimiatio of strictly domiated strategies The, let k be the first roud where there exist i N such that s i is elimiated, implyig that u i s i, s i ) < u i s i, s i ) holds for every s i S k 1 i But, by costructio, s i S k 1, so u i s i, s i) < ui si, s i), i cotradictig the assumptio that s is a Nash equilibrium 17