ON THE EQUIVALENCE BETWEEN SUBGAME PERFECTION AND SEQUENTIALITY * J. Carlos González-Pimienta 1 y Cristian M. Litan 2

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1 Working Paper Economic Serie 16 April 2005 Departamento de Economía Univeridad Carlo III de Madrid Calle Madrid, Getafe (Spain) Fax (34) ON THE EQUIVALENCE BETWEEN SUBGAME PERFECTION AND SEQUENTIALITY * J. Carlo González-Pimienta 1 y Critian M. Litan 2 Abtract We identify the maximal et of finite extenive form for which the et of ubgame perfect and equential equilibrium trategie coincide for any poible aignment of the payoff function. We alo identify the maximal et of finite extenive form for which the outcome induced by the two olution concept coincide. Keyword: extenive form, ubgame perfect equilibrium, equential equilibrium. JEL claification: C72 * Acknowledgement: We would like to thank Lui Corchon, Franceco De Sinopoli, Sjaak Hurken and Francico Marhuenda for helpful comment. Thi reearch ha been partially upported by the grant of the Plan de Formación de Peronal Invetigador de la Comunidad de Madrid, and by the Miniterio de Ciencia y Tecnologia FPI Grant BES The uual diclaimer applie 1 Joé Carlo González Pimienta. Departamento de Economía. Univeridad Carlo III de Madrid, Calle Madrid, 126, Getafe, Madrid. jcgonzal@eco.uc3m.e 2 Critian M. Litan. Departamento de Economía. Univeridad Carlo III de Madrid, Calle Madrid, 126, Getafe, Madrid. clitan@eco.uc3m.e

2 1 Introduction Analyi of backward induction in nite extenive form game provide ueful inight for a wide range of economic problem. The baic idea of backward induction i that each player ue a bet reply to the other player trategie, not only at the initial node of the tree, but alo at any other information et. An attempt to capture thi type of rationality i due to Selten [14], who de ned the ubgame perfect equilibrium concept. While ubgame perfection ha ome important application, it ha the drawback that doe not alway eliminate irrational behavior at information et reached with zero probability. In order to olve thi problem, Selten [15] introduced the more retrictive notion of trembling-hand perfection. The equential equilibrium concept, due to Krep and Wilon [9], require that every player maximize her expected payo at every information et, according to ome conitent belief. They alo howed that trembling-hand perfection implie equentiality, which in turn implie ubgame perfection. Blume and Zame [3] etablihed that for each xed extenive form, equential equilibrium coincide with trembling-hand perfect equilibrium for generic payo. Although a weaker concept than trembling-hand perfection, equential equilibrium eem to be the direct generalization of the idea of backward induction to game of imperfect information. It ati e in uch game all the propertie that characterize ubgame perfection (backward induction) in game of perfect information. 1 However, becaue of di cultie encountered when characterizing the et of equential equilibria, the implication of uing thi more appealing concept a compared to ubgame perfection are eldom dicued in economic model in which agent are uppoed to act according to the backward induction principle. 1 Perfect and proper equilibrium do not atify all thee propertie, ee Kohlberg and Merten [8] for detail. 1

3 In thi paper we nd the maximal et of nite extenive form for which equential and ubgame perfect equilibrium yield the ame equilibrium trategie, for every poible payo function. Thi et can be characterized a follow: in a given extenive form, if for any behavior trategy combination every information et i reached with poitive probability inide it minimal ubform, then ubgame perfection implie equential rationality for any poible payo. Whenever the extenive form doe not have the tructure above, payo can be aigned uch that the et of ubgame perfect equilibria doe not coincide with the et of equential equilibria. However, it may till happen that the et of equilibrium payo of both concept coincide for any poible aignment of the payo function. Thu, we alo identify the maximal et of nite extenive form for which ubgame perfect and equential equilibrium alway yield the ame equilibrium payo. 2 Thi complete the decription of the information tructure where applying equential rationality doe not make a relevant di erence with repect to ubgame perfection. The paper i organized a follow: in Section 2 we brie y introduce the main notation and terminology of extenive form game. Thi cloely follow Van Damme [16]. Section 3 contain de nition. Reult are formally tated and proved in Section 4. In Section 5 we give ome example where our reult can be applied. Section 6 conclude. 2 Notation and Terminology An n player extenive form i a extuple = (T; ; P; U; C; p), where T i the nite et of node and i a partial order on T, repreenting precedence. Furthermore, (T; ) form an arborecence with a unique root. We ue the 2 We actually prove omething tronger. For that et of extenive form, ubgame and equential equilibrium alway induce the ame et of equilibrium outcome. 2

4 notation x < y to ay that node y come after node x. The immediate predeceor of x 6= i P (x) = max fy; y < xg, and the et of immediate ucceor of x i S(x) = fy : x 2 P (y)g. The et of endpoint of the tree i Z = fx : S(x) = ;g and X = T nz i the et of deciion point. We write Z(x) = fy 2 Z : x < yg to denote the et of terminal ucceor of x, and if A i an arbitrary et of node we write Z(A) = fz 2 Z(x) : x 2 Ag. The player partition, P, i a partition of X into et P 0 ; P 1 ; :::; P n, where P i i the et of deciion point of player i and P 0 tand for the et of node where chance move. The information partition U i an n-tuple (U 1 ; :::; U n ), where U i i a partition of P i into information et of player i, uch that (i) if u 2 U i, x; y 2 u and x < z for z 2 X, then we cannot have z < y, and (ii) if u 2 U i, x; y 2 u, then js(x)j = js(y)j. Therefore, if u i an information et and x 2 X, it make ene to write u < x. Alo, if u 2 U i, we often refer to player i a the owner of the information et u. If u 2 U i, the et C u i the et of choice available for i at u. A generic choice c 2 C u i a collection of juj node with one, and only one, element of S(x) for each x 2 u. If player i chooe c 2 C u at the information et u 2 U i when he i actually at x 2 u, then the next node reached by the game i the element of S(x) contained in c. The entire collection C = fc u : u 2 S n i=1 U ig i called the choice partition. The probability aignment p peci e for every x 2 P 0 a completely mixed probability ditribution p x on S(x). We de ne a nite n-peron extenive form game a a pair = (; r), where i an n-player extenive form and r, the payo function, i an n-tuple (r 1 ; :::; r n ), where r i i a real valued function with domain Z. We aume throughout that the extenive form meet perfect recall, i.e. 3

5 for all i 2 f1; :::; ng, u; v 2 U i, c 2 C u and x; y 2 v, we have c < x if and only if c < y. Therefore, we can ay that choice c come before the information et v (to be denoted c < v) and that the information et u come before the information et v (to be denoted u < v). A behavior trategy b i of player i i a equence of function (b u i ) u2u i uch that b u i : C u! R + and P c2c u b u i (c) = 1; 8u. The et B i repreent the et of behavior trategie available to player i. A behavior trategy combination i an element of B = Q n i=1 B i. A common in extenive form game, we retrict attention to behavior trategie. 3 trategie. From now on, we imply refer to them a If b i 2 B i and c 2 C u with u 2 U i, then b i nc denote the trategy b i changed o that c i taken with probability one at u. If b 2 B and b 0 i 2 B i then bnb 0 i i the trategy combination (b 1 ; :::; b i then bnc = bnb 0 i, where b0 i = b inc. 1 ; b 0 i ; b i+1; :::; b n ). If c i a choice of player i A trategy combination b 2 B induce a probability ditribution P b on the et of terminal node Z. If A i an arbitrary et of node, we write P b (A) for P b (Z(A)). If x 2 X, let P b x denote the probability ditribution on Z if the game i tarted at x and the player play according to the trategy combination b. A ytem of belief i a function : X n P 0! [0; 1] uch that P x2u (x) = 1, 8u. Given a ytem of belief, a trategy combination b and an information et u, we de ne the probability ditribution P b; u on Z a P b; u = P x2u (x)pb x. An aement (b; ) i a trategy combination together with a ytem of belief. If player play according to the trategy pro le b, R i (b) = P z2z Pb (z)r i (z) denote player i expected payo at the beginning of the game and R ix (b) = P z2z Pb x(z)r i (z) denote player i expected payo at node x. In a imilar way, R iu (b) = P z2z Pb (z j u)r i (z) = P x2u Pb (x j u)r ix (b) i player i expected 3 We can do thi without lo of generality due to perfect recall and Kuhn theorem, ee Kuhn [10]. 4

6 payo at the information et u with P b (u) > 0. Furthermore, under the ytem of belief, R iu (b) = P z2z Pb; u (z)r i (z) denote player i expected payo at the information et u. Denote by R(b) the n-tuple (R 1 (b); R 2 (b); :::; R n (b)). The trategy b i i aid to be a bet reply againt b if b i 2 arg max b 0 i 2B i R i (bnb 0 i ). Likewie, the trategy b i i a bet reply againt b at the information et u 2 U i if it maximize R iu (bnb 0 i ). The trategy b i i a bet reply againt (b; ) at the information et u 2 U i if b i 2 arg max b 0 i 2B i R iu (bnb0 i ). If b i precribe a bet reply againt (b; ) at every information et u 2 U i, we ay that b i i a equential bet reply againt (b; ). The trategy combination b i a equential bet reply againt (b; ) if it precribe a equential bet reply againt (b; ) for every player. Let ^T T be a ubet of node uch that (i) 9y 2 ^T with y < x; 8x 2 ^T ; x 6= y, (ii) if x 2 ^T then S(x) ^T ; and (iii) if x 2 ^T and x 2 u then u ^T. Then we ay that y = ( ^T ; ^; ^P ; ^U; ^C; ^p) i a ubform of tarting at y, where ( ^; ^P ; ^U; ^C; ^p) are de ned from in ^T by retriction. A ubgame i a pair y = ( y ; ^r), where ^r i the retriction of r to the endpoint of y. We denote by b y the retriction of b 2 B to the ubform y (to the ubgame The retriction of a ytem of belief to the ubform y (to the ubgame y) i denoted by y. y). 3 De nition Let u rt review the concept of Nah equilibrium. De nition 1 (Nah Equilibrium) A trategy combination b 2 B i a Nah equilibrium of if every player i playing a bet reply againt b. We denote by NE( ) the et of Nah equilibria of. If b i a Nah equilibrium, every player i playing a bet reply for the entire game. However, thi doe 5

7 not need to be true for every ubgame. Subgame perfect equilibrium account for thi fact. De nition 2 (Subgame perfect equilibrium) A trategy combination b i a ubgame perfect equilibrium of if, for every ubgame y of, the retriction b y contitute a Nah equilibrium of y. We denote by SP E( ) the et of ubgame perfect equilibria of and by SP EP ( ) = fr(b) : b 2 SP E( )g the et of ubgame perfect equilibrium payo. Sequential rationality i a re nement of ubgame perfection. Every player mut maximize at every information et according to her belief about how the game ha evolved o far. If b i a completely mixed trategy pro le, belief are perfectly de ned by Baye rule. Otherwie, uch belief hould meet a conitency requirement. De nition 3 (Conitent aement) An aement (b; ) i conitent if there exit a equence f(b t ; t )g t, where b t i a completely mixed trategy combination and t (x) = P bt (x j u) if x 2 u, uch that lim (b t; t ) = (b; ). (1) t!1 Therefore, a equential equilibrium i not a trategy pro le but a conitent aement. De nition 4 (Sequential equilibrium) A equential equilibrium of i a conitent aement (b; ) uch that b i a equential bet reply againt (b; ). If i an extenive game, we denote by SQE( ) the et of trategie b uch that (b; ) i a equential equilibrium of, for ome. Moreover, SQEP ( ) = 6

8 fr(b) : b 2 SQE( )g denote the et of equential equilibrium payo. Clearly, SQE( ) SP E( ), for any game. We now introduce ome new de nition that are needed for the reult. De nition 5 (Minimal Subform of an Information Set) Given an information et u, the minimal ubform that contain u i the ubform y that contain u and doe not include any other proper ubform that contain u. We ay that y = ( y ; ^r) i the minimal ubgame that contain u if y i the minimal ubform that contain u. In a given extenive form there are information et that are alway reached with poitive probability. The next two de nition formalize thi idea. De nition 6 (Surely Relevance) An information et u i urely relevant in the extenive form if jc u j > 1 implie that P b (u) > 0; 8b 2 B. A will be een in Propoition 1, the urely relevance property i trictly related with the maximizing behavior of the player who move at that information et. If there i only one choice available at an information et, the player i obviouly maximizing. Hence, we alo conider uch an information et a urely relevant. See Figure 1 for a game form where all information et are urely relevant. De nition 7 (Subform Surely Relevance) An information et u i ubform urely relevant in the extenive form if it i urely relevant in it minimal ubform. See gure 1, 2 and 3 for ome game form where all information et are ubform urely relevant. Converely, ee gure from 4 to 9 for ome game form with information et that are not ubform urely relevant. 7

9 4 Reult The three bet reply concept introduced in Section 2 relate each other, a it i hown in (i) and (ii) in the next lemma. The third aertion of the ame lemma how that the maximizing behavior at an information et i independent of the ubgame of reference. Lemma 1 Fix a game = (; r). The following aertion are true: (i) Given a trategy combination b, if u 2 U i i uch that P b (u) > 0 and b i i a bet reply againt b, then b i i a bet reply againt b at the information et u. (ii) Given a conitent aement (b; ), if u 2 U i i uch that P b (u) > 0 and b i i a bet reply againt b at the information et u, then b i i a bet reply againt (b; ) at the information et u. (iii) If y i the minimal ubgame that contain u and (b y ; y ) i the retriction of ome aement (b; ) to y, then b i i a bet reply againt (b; ) at the information et u in the game if and only if b y;i i a bet reply againt (b y ; y ) at the information et u in the game y. Proof. Part (i) i a known reult. 4 Proof for (ii) and (iii) are trivial. The next propoition con ne the et of extenive form for which the equential equilibrium olution concept actually re ne ubgame perfection. Propoition 1 If i an extenive form uch that every information et i ubform urely relevant, then for any poible payo vector r, the game = (; r) i uch that SP E( ) = SQE( ). If i an extenive form uch that there exit an information et that i not ubform urely relevant, then we can nd a payo vector r uch that for the game = (; r), SP E( ) 6= SQE( ). 4 See Van Damme [16], Theorem

10 Proof. Let u prove the rt part of the propoition. We only have to how that SP E( ) SQE( ). Conider b 2 SP E( ) and build a conitent aement (b; ). 5 Contruct the et U(b; ~ ) = S n o n i=1 u 2 U i : b i =2 arg max ~bi2b i R iu (bn~ b i ) ; we have to prove that it i empty. Suppoe by contradiction that ~ U(b; ) 6= ;, and take u 2 ~ U(b; ), obviouly jc u j > 1. Let y be the minimal ubgame that contain u. By Lemma 1, P b y(u) = 0. However, u mut be ubform urely relevant. Thi provide the contradiction. Let u now prove the econd part of the propoition. Suppoe u 2 U i i an information et that i not ubform urely relevant and let c 2 C u be an arbitrary choice available at u. Aign the following payo : r i (z) = 0 8i if z 2 Z(c) (2) r i (z) = 1 8i elewhere. Clearly any trategy b i = b i nc cannot be part of a equential equilibrium ince playing a di erent choice at u give player i trictly higher expected payo at that information et. We now have to how that there exit a ubgame perfect equilibrium b uch that b i = b i nc. By aumption there exit b 0 uch that P b0 y (u) = 0 in the minimal ubgame y that contain u. Conider the trategy combination b = b 0 nc, it i alo true that P b y(u) = 0. Note that b y i a Nah equilibrium of y ince nobody can obtain a payo larger than one. By the ame argument, b i inducing a Nah equilibrium in every ubgame, hence it i a ubgame perfect equilibrium. Thi complete the proof. 5 A general method to de ne conitent aement (b; ) for any given b 2 B, in an extenive form, i the following: take a equence of completely mixed trategy combination fb tg t! b and for each t; contruct t (x) = P b t(x j u) 2 [0; 1], 8x 2 u, for all information et u. Call k = jx n P 0 j. The et [0; 1] k i compact and ince t 2 [0; 1] k ; 8t, there exit a ubequence of ftg, call it ft j g, uch that t j converge in [0; 1] k. De ne belief a t j = lim j!1 t j. 9

11 Figure 1, 2 and 3 contain ome example of extenive form where the rt part of Propoition 1 applie. The extenive form in gure 4, 5 and 6 illutrate the econd part. Notice that the payo aignment in the previou proof provide a di erence in equilibrium trategie but not in equilibrium payo. The reaon i that thi di erence in equilibrium payo cannot alway be achieved. In gure 7, 8 and 9 there are example of extenive form with information et that are not ubgame urely relevant and, however, the et of equential and ubgame perfect equilibrium payo alway coincide. Propoition 2 provide a u cient and neceary condition for equilibrium payo et to be equal, for any conceivable payo function. Before that, an additional piece of notation i needed. Given a node x 2 T, we call path to x the et of choice P ath(x) = fc 2 S u C u : c < xg. Let y be the minimal ubform that contain the information et u. Contruct the et B(u) = b 2 B : P b y(u) > 0. We ay that player i can avoid the information et u if there exit a trategy combination b 2 B(u), and a choice c 2 C v, with v 2 U i, uch that P bnc y (u) = 0. 6 Therefore, aociated with any information et, we can contruct a lit (poibly empty) of player who can avoid it. The following lemma i ueful for the proof of Propoition 2. Lemma 2 Let be an extenive form uch that every information may only be avoided by it owner. Let (b; ) and (b 0 ; 0 ) be two conitent aement. If b and b 0 are uch that P b y = P b0 y for every ubform y, then = 0. Proof. Let (b; ) and (b 0 ; 0 ) be two conitent aement uch that P b y = P b0 y for every ubform y. Note that b 0 can be obtained from b by changing behavior at information et that are reached with zero probability within it minimal ubform. Hence, without lo of generality, let b and b 0 di er only at one of uch 6 We are retricting to the minimal ubform. Thi i embedded in the notation. For the ake of clari cation: aigning probability zero to an entire ubgame i not the ame a avoiding an information et contained in that ubgame. 10

12 information et, ay u 2 U i, and let y be it minimal ubform. The hift from b to b 0 may caue a change in belief only at information et that come after u and have the ame minimal ubform y. Let v 2 U j be one of thoe information et. If j = i, perfect recall implie that there i no change in belief at the information et v. If j 6= i there are two poible cae, either P b y(v) > 0 or P b y(v) = 0. In the rt cae the belief at v are uniquely de ned, therefore, (x) = 0 (x); 8x 2 v and moreover, (x) = 0 (x) = 0; 8x 2 v uch that u < x. In the econd cae, ince the information et v can only be avoided by player j there exit a choice c 2 C w of player j uch that P bnc y (v) > 0, otherwie player i would be able to avoid the information et u. Let b 00 = bnc and b 000 = b 0 nc, then by the dicuion of the rt cae, 00 (x) = 000 (x); 8x 2 v, furthermore, perfect recall implie 00 (x) = (x) and 000 (x) = 0 (x); 8x 2 v, which in turn implie (x) = 0 (x); 8x 2 v. Remark 1 If in an extenive form every information et may only be avoided by it owner, then belief are alway uniquely de ned (conider b 0 = b in Lemma 2). 7 If we conider extenive form uch that only one choice i available at information et that can be avoided by a di erent player than it owner, then belief are till uniquely de ned whenever they are ueful, that i, whenever an actual choice i faced. However, thi doe not need to be true at information et where only one choice can be taken. Thi enlarged et of extenive form i the one under tudy in the next propoition. Propoition 2 If i an extenive form uch that every information et that i not ubform urely relevant can only be avoided by it owner, then for any poible payo vector r, the game = (; r) i uch that SP EP ( ) = SQEP ( ). 7 A complete characterization i the following: in an extenive form, belief are alway uniquely de ned if and only if juj > 1 implie that u may only be avoided by it owner. 11

13 If i an extenive form with an information et that i not ubform urely relevant and that can be avoided by a di erent player than it owner, then we can nd a payo vector r uch that for the game = (; r), SP EP ( ) 6= SQEP ( ). Proof. Let u prove the rt part of the propoition. We need to prove that 8b 2 SP E( ), R(b) 2 SQEP ( ). Take an arbitrary b 2 SP E( ) and contruct ome conitent belief. If the et ~ U(b; ) = S n i=1 n o u 2 U i : b i =2 arg max ~bi2b i R iu (bn~ b i ) i empty, then b 2 SQE( ) and R(b) 2 SQEP ( ). Otherwie, we need to nd a equential equilibrium (b ; ) uch that R(b ) = R(b). Step 1: Take an information et u 2 ~ U(b; ), obviouly jc u j > 1. Let i be the player that move at thi information et, and let y be the minimal ubgame that contain u: Notice that by Lemma 1, u hould be uch that P b y(u) = 0, hence it i not ubform urely relevant. By aumption, u can only be avoided by player i. Step 2: Let b 0 be the trategy pro le b modi ed o that player i play a bet reply againt (b; ) at the information et u. Contruct a conitent aement (b 0 ; 0 ). Notice that P b0 = P b and, in particular, P b0 y = P b y. By Lemma 2, and 0 aign the ame probability ditribution to information et where more than one choice i available. Step 3: We now prove that b 0 2 SP E( ). For thi we need b 0 y 2 NE( y ). Given the trategy pro le b 0 y in the ubgame y, player i cannot pro tably deviate becaue thi would mean that he wa alo able to pro tably deviate when b y wa played in the ubgame y, which contradict b y 2 NE( y ). Suppoe now that there exit a player j 6= i who ha a pro table deviation b 00 y;j from b0 y;j in the ubgame y. The hypothei on the extenive form implie P bnb00 y;j y = P b0 nb 00 y;j y, which further implie that b 00 y;j hould alo have been a pro table deviation from b y. However, thi i impoible ince b y 2 NE( y ). 12

14 Step 4: By tep 2, U(b ~ 0 ; 0 ) = U(b; ~ ) 1. If U(b ~ 0 ; 0 ) 6= ;, apply the ame type of tranformation to b 0. Suppoe that the cardinality of U(b; ~ ) i q, then in the qth tranformation we will obtain a conitent aement (b (q) ; (q) ) uch that b (q) 2 SP E( ), P b = P b(q), and U(b ~ (q) ; (q) ) = ;. Oberve that, b (q) 2 SP E( ) and U(b ~ (q) ; (q) ) = ; imply b (q) 2 SQE( ), and that P b = P b(q) implie R(b) = R(b (q) ). Therefore (b (q) ; (q) ) i the equential equilibrium (b ; ) we were looking for. Let u now prove the econd part of the propoition. For notational convenience, it i proved for game without proper ubgame, however, the argument extend immediately to the general cae. Suppoe that u i an information et that i not ubform urely relevant. Suppoe alo that it can be avoided by a player, ay player j, di erent from the player moving at it, ay player i. Note that there mut exit an x 2 u and a choice c 2 C v, where v 2 U j, uch that if b = bnp ath(x), then P bnc (u) = 0 i true. Let f 2 C u be an arbitrary choice available to player i at u. Aign the following payo : r j (z) = 0 if z 2 Z(c) r i (z) = r j (z) = 0 if z 2 Z(f) (3) r i (z) = r j (z) = 1 if z 2 Z(u) n Z(f). Let d 2 P ath(x) with d 62 C v, aign payo to the terminal node, whenever allowed by 3, in the following fahion: r k (z) > r k (z 0 ) where z 2 Z(d) and z 0 2 Z(C w n fdg). (4) Player k above i the player who ha choice d available at the information et w. Give zero to every player everywhere ele. 13

15 In word, player j move with poitive probability in the game. She ha two choice, either moving toward the information et u and letting player i decide, or moving away from the information et u. If he move away he get zero for ure. If he let player i decide, player i can either make both get zero by chooing f, or make both get one by chooing omething ele. Due to 4, no player will diturb thi decription of the playing of the game. Thi game ha a Nah equilibrium in which player i move f and player j obtain a payo equal to zero by moving c. However, in every equential equilibrium of thi game, player i doe not chooe f and, a a conequence, player j take the action contained in P ath(x) \ C v. Therefore, in every equential equilibrium, player i and j obtain a payo trictly larger than zero. 8 Thi complete the proof. 5 Example Thee reult can be applied to many game conidered in di erent branche of the economic literature. It allow to identify in a traightforward way, the nite extenive form game of imperfect information for which ubgame perfect equilibria are till conforming with backward induction, when following the minimal interpretation of the principle given by Kohlberg [7]. Beley and Coate [2] propoed an economic model of repreentative democracy. The political proce i a three-tage game. In tage 1, each citizen decide whether or not to become a candidate for public o ce. At the econd tage, voting take place over the lit of candidate. At tage 3 the candidate with the mot vote chooe the policy. Beley and Coate olved thi model uing ubgame perfection and found multiple ubgame perfect equilibria with very di erent outcome in term of number of candidate. Thi may ugget that 8 Equilibrium payo are not necearily equal to one due to eventual move of nature. 14

16 ome re nement might give harper prediction. However, given the tructure of the game that they conidered, it follow immediately from the reult of the previou ection that all ubgame perfect equilibria in their model are alo equential. Thu, no additional inight would be obtained by requiring thi particular re nement. The information tructure of Beley and Coate model i a particular cae of the more general framework o ered by Fudenberg and Levine [5]. They characterized the information tructure of nite-horizon multitage game a almot perfect, ince each period player imultaneouly chooe action, no Nature move are allowed and there i no uncertainty at the end of each tage. A they noticed, equential equilibrium doe not re ne ubgame perfection in thi cla of game. Thi claim can alo be obtained a an implication of Propoition 1 in the preent paper. In their verion of the Diamond-Dybvig [4] model, Adão and Temzelide [1] dicued both the iue of potential banking intability a well a that of the decentralization of the optimal depoit contract. They addreed the rt quetion in a model with a ocial planner bank. The bank o er the e cient contract a a depoit contract in the initial period. In the rt tage agent equentially chooe whether to depoit in the bank or to remain in autarky. In the econd tage, thoe agent who were elected by Nature to be patient, imultaneouly chooe whether to mirepreent their preference and withdraw, or report truthfully and wait. The reduced normal form of the game ha two ymmetric Nah equilibria in pure trategie. The rt one ha all agent chooing depoiting in the bank and reporting faithfully, the econd one ha all agent chooing autarky. The fact that both equilibria are equential i captured by their Propoition 2. A in the other example, becaue of the game form they ued, our Propoition 1 alo cover their claim. 15

17 In the implementation theory framework, Moore and Repullo [11], preent the trength of ubgame perfect implementation. If a choice function i implementable in ubgame perfect equilibria by a given mechanim and the trategy pace i nite, by analyzing the extenive form of the mechanim game, our work perfectly demarcate whether thi very mechanim alo implement in equential equilibria. Namely, if in the extenive form of the mechanim game all information et are ubform urely relevant, then the mechanim alo implement in equential equilibria. If not every information et i ubform urely relevant, then there exit economie for which the mechanim doe not double implement in ubgame perfect and equential equilibria. More example can be found in Game Theory textbook. Good reference are Fudenberg and Tirole [6], Myeron [12] and Oborne and Rubintein [13]. Notice that whenever are preented extenive game where ubgame perfect and equential equilibrium di er, there are information et that are not ubform urely relevant in the game form of uch extenive game. A example conider gure 8.4 and 8.5 in Fudenberg and Tirole, gure from 4.8 to 4.11 in Myeron and gure and in Oborne and Rubintein. 6 Concluion For the cla of extenive form game haring the property that all information et are ubform urely relevant, the et of equential equilibrium trategie i equal to the et of ubgame perfect equilibrium trategie. However, for di erent information tructure the equality of thee et crucially depend on the payo of the player. If we are only concerned with equilibrium payo, we can have information et that are not ubform urely relevant, a long a the player who can avoid it i the ame a it owner. In thi cae, the olution concept of ubgame 16

18 perfection and equential equilibrium yield the ame et of equilibrium payo, for any poible aignment of the payo function. Reference [1] Adão, B., Temzelide, T. "Sequential Equilibrium and Competition in a Diamond-Dybvig Banking Model", Review of Economic Dynamic 1, [2] Beley, T., Coate, S. "An Economic Model of Repreentative Democracy", The Quarterly Journal of Economic 112, [3] Blume, L. E., Zame, W. R. "The Algebraic Geometry of Perfect and Sequential Equilibrium", Econometrica 62, [4] Diamond, D., Dybvig, D. "Bank run, depoit inurance, and liquidity", Journal of Political Economy 91, [5] Fudenberg, D., Levine, D. "Subgame-Perfect Equilibria of Finite- and In nite-horizon Game", Journal of Economic Theory 31, [6] Fudenberg, D., Tirole J. "Game Theory", The MIT Pre [7] Kohlberg, E. "Re nement of Nah Equilibrium: The Main Idea", in Ichiihi T., Neyman A. and Tauman Y. (ed.) Game Theory and Application, Academic Pre, [8] Kohlberg, E., Merten, J.F. "On the Strategic Stability of Equilibria", Econometrica 54, [9] Krep, D.M., Wilon, R. "Sequential Equilibria", Econometrica. 50,

19 [10] Kuhn, H.W. "Extenive Game and the Problem of Information", Annal of Mathematic Studie. 28, [11] Moore, J., Repullo, R. "Subgame Perfect Implementation", Econometrica [12] Myeron, R.B. "Game Theory", Harverd Univerity Pre [13] Oborne, M.J., Rubintein, A. "A Coure in Game Theory", The MIT Pre [14] Selten, R. "Spieltheoretiche behandlung eine oligopolmodell mit nachfragetraheit", Z. Geamte Staatwienchaft 12, [15] Selten, R. "Re-examination of the perfectne concept for equilibrium point in extenive game", International Journal of Game Theory 4, [16] Van Damme, E. "Stability and Perfection of Nah Equilibria", Springer- Verlag

20 A Figure Nature 1/3 2/3 Player 1 Player 1 r r Player 2 r Player 3 Figure 1 19

21 Player 1 Player 2 S SSS Player 1 tu Player 1 Player 2 Figure 2 20

22 Player 1 r Nature r 1/2 1/2 Player 2 q r Player 3 qr Player 2 H HHH H r Nature 2/3 1/3 Player 3 Player 1 Player 1 Player 2 Figure 3 21

23 (1,1) Player 2 S S SSS SSS (1,1) (0,0) (1,1) (0,0) Figure 4 22

24 Player 1 Player 2 p r (1,1,1) r Player 3 r (1,1,1) (1,1,1) (0,0,0) (0,0,0) Figure 5 23

25 Player 1 Player 2 p r (1,1,1) r Player 3 r (1,1,1) (0,0,0) (1,1,1) (0,0,0) Figure 6 24

26 Player 2 S S SSS SSS Player 1 Figure 7 25

27 Player 1 Player 1 Player 2 A A A A A A A A A A A A Figure 8 26

28 Player 1 Player 1 Player 2 Player 1 S S S S SSS SSS SSS SSS Player 1 Player 2 Player 3 S SSS Player 1 Figure 9 27