rational beliefs in extensive games - Economics

GIACOMO BONANNO

RATIONAL BELIEFS IN EXTENSIVE GAMES

ABSTRACT. Given an extensive game, with every node x and every player i a subsct K,(x) of the set of terminal nodes is associated, and is given the interpretation of player I's knowledge (or information) at node x. A belief of player i is a function that assoeiates with every node x an element of the set K,(x). A belief system is an n-tuple of beliefs, onc for cach player. A belief system is rational if it satisfies some natural consistency properties. The main result of the paper is that the notion of rational belief system gives rise to a refinement of the notion of subgame-perfect equilibrium. Keywords : Information, beliefs, subgame-perfect equilibrium.

1.INTRODUCTION ln an extensive game, information sets capture the notion of what a player knows when it is her turn to move: if two decisionnodesx and y belong to the same information set of player i, then player I does not know whether she is making a choice at x or at y. Information sets, however, do not tell us what a player knows at a node that belongs to another player. This raises the question of what information players obtain during the play of a game. For example, in general it cannot be the case that every player always observes which player is moving (even though he may not observe what move was made), as the game of Figure 1 shows. When it is player 3's turn to move, he will either have observed that, before him, only player 1 moved or that both players I and 2 moved. In the first case he will be able to deduce that he is at node x,, while in the second case he will know that he is at node x., . But this contradictsthe fact that, accordingto his information set, player 3 cannot distinguishbetween nodesx2 and x., . The example of Figure 1 also shows that, in general, we cannot think of the play of a game as having a well-definedtemporal structure of the following type: a moveis t h e g a m e s t a r t sa t d a t et : 1 a n d a t e v e r yd a t e t : 2 , 3 , . . . made until a terminal node is reached. In the above example, if player 3 is asked to move at date t:2 he will know that he is at node x. , Theory and Decision 33: 153 116, 1992. 1992 Kluwer Acad.emic Publishers. Printed in the Netherlantls.

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z5 1 1 1

zr

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2 2 2

o 0 o

4 4 o

24

0 0 l

F i g .t .

while if he is asked to move at date /:3 he will know that he is at node x.,. Thus the play of an extensive game requires the presence of an outside agent - we shall call her the umpire - who provides players with different amounts of information as the play of the game unfolds (for example, in the game of Figure 1, if player l chooses a, then player 2 must be informed, while player 3 cannot be informed). In Section 3 we suggest one way of formalizing the information conveyed to each player along every possibleplay of an extensivegame (Section 2 contains some preliminary definitions). With every node -r and every player i we associate a subset K,(x) of the set of terminal nodes, representing what player I knows when node x is reached. The interpretation is that if -when node x is reached-player i's knowle d g ei s g i v e nb y , s a y ,t h e s e t { 2 , , 2 2 1 2 5 , z r } , t h e n p l a y e ri i s i n f o r m e d that the play of the game so far has been such that only terminal nodes z 1 , 2 2 , z s a n d z s c a nb e r e a c h e d . If x is a decision node that belongs to information set lr of player i, we define player I's knowledge at .r as the set of all the terminal nodes that can be reached from nodes in ft. Thus our definition of knowledge gives an accurate representation of a player's information sets. It


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seems that, as long as a player's knowledge at her decision nodes is a faithful representation of her information sets, there is a lot of freedom concerning the specification of her knowledge at decision nodes of other players. Thus the definition put forward in this paper is only one of many possibilities.'An appealing fearure of the definition proposed in this paper is that it provides an intuitive characterization of such notions as perfect recall, perfect information and simultaneity. In Section 4 we show that our approach offers a new way of thinking about the solutions of a game. We first introduce the notion of belief system. We denote by P,@) player i 's belief at node x and we require that B,(x) be an element of the set K,(x). The interpretation is as follows. Supposethat at node -r player I's knowledge is representedby the set K,(t): {zr, zr, zs, z8}. Then, as explainedabove, player i knows (is informed) that only terminal nodes z, , 2,, zs and z, can be reached. If B,@): z, then player i believesthat the final outcome will actually be zr. (Note that z, need not be reachable from x, that is, player i's belief may reflect i's ignorance of the actual play of the game.) Thus a belief of player i is a function that associateswith every node x an element of the set K,(x). A belief system is an n-tuple of beliefs, one for each player. There are some natural consistency properties that one can impose on beliefs. Four simple properties are used in Section 4 to characterizethe notion of rational belief system. It is then shown that from a (not necessarilyrational) belief systemB one can extract a pure strategyprofile o: t(B) in a natural way. The main resuft of this paper is that if B is a rational belief systemthen o : t(B) is a subgame-perfectequilibrium.' While in games of perfect information there is a one-to-one correspondencebetween (pure-strategy) subgame-perfect equilibria and rational belief systems, in games of imperfect information the notion of rational belief system refines that of subgame-perfectequilibrium. We show this by means of an example. Another example is used to show that a rational belief system n e e d n o t b e a s e q u e n t i a le q u i l i b r i u m . ' 2 . P R E L I M I N A R YD E F I N I T I O N S In this section we define some functions that will be used extensively throughout the paper.

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Fix a (finite) extensive game.4Let X be the set of decision nodes and Z the set of terminal nodes. Let T: XU Z. (ln general, we shall denote a decision node by x or y, a terminal node by z and a generic node - decision or terminal - by t.) For every , e f, let

0(t)c z be the set of terminal nodes that can be reached from r. For every ze Z, we set by definition 0(z): {z}. For example' in the game of Figure l, 0(xr): {tr, zo, Zr) . If h is an information set, define

e.(h):

Pot@)

that is, g*(/r) is the set of terminal nodes that can be reached from nodes in h. For example, in the game of Figure l, h: {xr'x.} is p l a y e r 3 ' s i n f o r m a t i o ns e t a n d w e h a v e t h a t e * ( h ) : { z , z r , z t . Z t } ' R e c a l l t h a t a c h o i c ec a t i n f o r m a t i o ns e t h : { x r , . . . , x , , } i s a s e t o f e d g e sc : { ( " , , ! r ) , ( x . , l . ) , . . . , ( x , , , y ^ ) } w h e r e n o d e y * i s a n m). Define i m m e d i a t es u c c e s s oor f n o d e x o ( k : 7 , . . . ,

r . r ( c: )0 ( y ' )u 0 ( Y r u) ' ' ' u 0 ( Y^ ) , that is, pc(c) is the set of terminal nodes that can be reached from nodes in ft following the edges that constitute choice c. For example, in the gameof Figure l. p(R ) : \zz. zr\. 3 . T H E D E F I N I T I O NO F K N O W L E D G E Fix an extensive game. For every player I and for every (decision or terminal) node / we define a subset K,(t) of the set Z of terminal 'player i's knowledge at /'. One nodes, which will be interpreted as way of thinking about the proposed definition is as follows. At the root of the tree, denoted by x0, all players have the same knowledge, namely Z. As the play of the game unfolds and new nodes are reached, an umpire gives (separately) to each player the maximum amount of information that is compatible with that player's informa-

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tion sets, according to the following rules. If z is a terminal node, then every player is informed that the game ended at z.lf node x belongsto information set ft of player i, then player i is told that her information set /r has been reached, but is not told which node in fr was reached. If node x does not belong to player i and all the information sets of player i (if any) that are crossedby paths starting at x consistentirely of nodes that are successorsof x, then player i is informed that node r has been reached (the justification for this rule is that later on, ar any of her information sets, player i will be able to deduce that the play of the game must have gone through node x; hence player i might as well be told at the time when x is reached). When the above condition is not satisfied, player i's knowledge at x either doesn't change (that is, player i is not told anything new) or at most reflects the choice made by player i at the immediate predecessorof x, if that node belonged to player i. The formal definitionis as follows.5 (1) Let x,, be the root of the tree. For every player i set K,(x,,): Z . (2) For every z e Z and for every player l, set K,(z): {z} . (3) If r is a decision node that belongsto information set ft of player r, set K,(x): 0*(h)' (4) For every decision node x and every player i,let H,(x) be the set of information sets of player I defined as follows: h € H,(x) if and only if h is an information set of player i and there is a node y € fr that is a successorof x. Now, if x is not a decisionnode of player I and either H,(*):0 (where 0 denotes the empty set) or, for every heH,(x),0.(h)C 0 ( r ) ( t h a t i s , e v e r y n o d e i n y ' ri s a s u c c e s s oor f -r) then set

K,(x) : 0(x) .


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(5) If node.r is not a decision node of player i and the condition given under (4) is not satisfied (that is, there exists an he H,(x) and a node y c lr such that y is not a successor of x) and x is an immediate successor of decision node / of player i and c is the choice of player I that leads from / to x, then set K,(x):

6 r ( c ).

(6) Finally, if x is not a decision node of player I and conditions (4) and (5) are not satisfied,set K,(x):

K,(n,)

where zr, denotes the immediate predecessor of .r. For example, in the game of Figure 2 we have:

Fig. 2

R A T I O N A L B E L I E F SI N E X T E N S I V EG A M E S By (1):

K , ( r n ): Z : { z . z r , Z t , z 4 , z s , z 6 , z . t } for alli:1.2.3 .

By (2):

X , ( 2 1 ) : { zj }

By (3):

K r ( * r ) : 0 . ( { x , } ) : O ( x r ) : { z t , z 4 , z 5 t z r , ,z t ) .

By (a):

K r ( x r ) : 0 ( xt ) : { z r , Z q , Z s . 2 6 ,z . t } .

By (6):

K.(r,) : Kr(x):

7 .

By (a):

Kr(*r):

0(xr): {2,

By (3):

K.(*r):0*({xr,xr}):

By (4):

K, (r.) : K.(xr) : 0(x) : {zz, 24,zs, z6}

By (3):

& ( x . ) : O * ( { x z , x . } ) : { z r , z r , 2 3 ,Z q , z s , z 6 }.

By (3):

Kr(r):

0 * ( { x o ,x r } ) : { 2 . , z o , z s , z 6 }

By (a):

Kr(r):

Kr(xo):0(x):

By (3):

K , ( x r ) : 0 * ( { x o , x r } ) : { 2 . , z + , z s , 2 6 }.

By (a):

Kr(tr):

for all i:7,2,3

Kr(xr):

159

and for all7:1,...,7

.

zr} .

{ 2 , z r , 2 3 , 2 4 , 2 5 ,z 6 }.

{ z t , z o }.

K : ( x s ) : 0 ( x s ): { z s , z o } .

Remark 1. It is easy to show (seeBonanno, 1991) that for every node / and for every player i, 0(t) is a subset of K,(r). (Recall that 0(r) denotes the set of terminal nodes that can be reached from node r.) Remark 2. It is an immediate consequenceof point (3) of the above definition that if ft is an information set of player l, and x and y are two nodes in ft, then f,(r): K,(y). Thus it makes senseto write K,(ft) for player i's knowledge at her information set ft. An appealing feature of the definition of knowledge given above is that

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it provides an intuitive characterization of such notions as perfect recall, perfect information, simultaneity, etc. For a proof of the f o l l o w i n g p r o p e r t i e ss e e B o n a n n o ( 1 9 9 1 ) . PROPERTY 1. An extensive game with perfect recall satisfies the following property: if node y is a successorof node -r, then' for every player t, K,(y) c K,(x) That is, at every node each player knows at least as much as she knew before that node was reached. The game shown in Figure 3 satisfies the property that if y is a successorof x then K,(y)gK,(x), for every player i, but is not a game with perfect recall. Thus the converse of property 1 is not true. PROPERTY 2. An extensive game with perfect recall has perfect information if and only if at every node all players have the same knowledge, that is, if and only if for every node / and for any two p l a y e r si a n d 7 . K , ( r ) : K , ( t ) .

zg Fig. 3

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Define an extensive game to be simultaneousif every play crossesall the information sets. PROPERTY 3. An extensive game is simultaneousif and only if it satisfiesthe following property: if r is a decision node of player l, then K,(x): Z. That is, when a player has to move she knows as much as she did at the root of the tree.

4.BELIEFS Fix an extensivegame. As before, let x be the set of decisionnodes. Z the set of terminal nodes, and Z the union of X and Z. DEFINITION.

A belief of player j is a function

B,: T--> Z s a t i s f y i n gt h e f o l l o w i n g p r o p e r r i e s :

0) B,G)e KJt), vt e r, (2) lf -r and y belong to the same information set of prayer i (so that

K,(x): K,(y)) rhenB,(x): F,0).

The interpretation is as follows. consider again the game of Figure 2. A t n o d ex , w e h a v et h a t K , ( x , ) : { r . , 2 4 , Z s , 2 6 , z 1 } . T h i s m e a n st h a t player 1 knows that only terminal nodes different from z, and z2can b e r e a c h e d . I f B r Q , ) : z , t h e n p l a y e r I b e l i e v e st h a t t h e p l a y o f t h e game will actually end at node z. (this obviously implies that player 1 believes that player 2 will take action c and player 3 will take action F a n d h e h i m s e l f p l a n s t o c h o o s eG ) . condition (1) in the above definition saysthat what a player believes must be consistentwith what he knows, and condition (2) says that a player cannot have different beliefs at two nodes that belong to one of his information sets, since his knowledge is the same at both nodes. Thus it makes senseto write Bi(h) for player i 's belief at his information sct ft.

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DEFINITION. A belief system is an n-tuple B:(8t,..., w h e r e , f o r e a c hp l a y e ri : I , . . . , f t , F i i s a b e l i e fo f p l a y e rl .

F,),

Let B be a belief system. we say that B is rational if it DEFINITION. satisfies the following properties (which will be discussedimmediately below): (l)

Contraction Consistency. If x and y are two nodes such that K,(x) I K,(y) and F,(t) € K,(y), then B'(y) : F'G).

(2) Tree Consistency. Let h be an information set of player l. Let x e h be the predecessor of B,(h) and let L(x) be the set of immediate successorsof x. Then

F,0) e 0(y)

VYe l(x) .

(3) Individual Rationality. Let h be an information set of player l. Let x e h be the predecessor of B,(h) and let I(x) be the set of immediate successorsof x. Then

U,(8,(h))>U,(F'(y))

VY€ I,(x),

where IJ,: Z-'>ilt is player i's payoff function. (lt denotes the set of real numbers.) (4) Choice Consistency. Let x belong to information set ft of player I' and let c be the choice at h that precedes Br(h). Then, for every player j, if B,@) comes after choice d at h, it must be d: c. Property (1) says that, as the knowledge of a player evolves and becomes more refined, the player will not change his belief unless he has to, that is, unless his previous belief is inconsistentwith the new information. This is a contraction consistency property which is implied, for example, by Bayesian updating. The purpose of property (2) is to rule out situations like the one i l l u s t r a t e di n F i g u r e4 . T h e r e w e h a v e t h a t K r ( h ) : { z r , 2 2 , Z z , 2 4 , 2 5 , zo) where h: {x.,,xr} is the first information set of player 2, and K r ( d : { 2 , z , 2 4 , z s } w h e r e I : { x z , x o } i s t h e s e c o n di n f o r m a t i o n set of player 2.


163

R t t o

o

,6 1 1

Fig. 4.

SupposeBr(h): 26 and PrG): zr. This beliefof player2 is inconsistent becausebelievingin zu at h meansbelievingthat node J2 was reached.Given this belief, if player 2 takesaction/, so that the play of the gameproceedsto informationset g, then node xo must be reached, and from xo terminal node z, cannot be reached. In this example property (2) requires that if Br(h): zu then either Bt( g): zo or Ft(il: zt. The motivation for property (3) is as follows. If terminal node z representswhat player i believesat his information set ft (that is, if z: P,(h)), then it meansthat playeri believesthat he is at that nodex in lr which lies on the play to z. Supposethat U,(z) j is the probability with which player 2 chooses/. Now considera sequence{ot, o',. . .} of completelymixedstrategies whose mth elementis given by o - : ( ( 1- e m - b ^ , e ^ , b ^ ; l * c m , c m l; - d ^ , d ^ ) ; ( l - e ^ , e ^ ) ) T M B L R b t l r with a- ,b^,c^,d^e(O, 1) and convergingto zero as rn tendsto infinity, while e- € (0,1) converges to 1- p. Let ft be the information set of player 2. Now, Prob{/rI o^} : b^ + a^(1 - c^d^) and P r o b { x .I o ^ } : a ^ c ^ ( I - d ^ ) . T h u s , g i v e no - , c^(l - d^)

P r o b { x r , r t: (

*.

r - c^d^

As m goes to infinity, the numerator tends to zero while the denominator either tends to infinity or convergesto a number greater than or equalto 1. Hencev(x):0, that is, player2 must attachzero probabilityto node x, if his informationset is reached.But z(xr) : 0 requires player 2 to play / with probability zero, a contradiction. I

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5.CONCLUSION For every node / and for every player i we defined a subset K,(t) of the set of terminal nodes and we interpreted it as the information given to player i when node / is reached.We then defined a belief of a player as a function that associateswith every node / an element of Kr(r) and a belief system as an n-tuple of beliefs, one for each player. Natural consistency properties were used to define the notion of rational belief system and it was shown that every rational belief system gives rise to a subgame-perfect equilibrium. Two examples were used to show that the notion of rational belief system refines that of subgame-perfect equilibrium and differs from the notion of sequential equilibrium.

MA M E , A P P E N D I XA : D E F I N I T I O NO F E X T E N S I V E . F O R G A finite game in extensive form without chance moves is a sextuple G:

(V, N, P, H, C, U)

where the constituents are as follows. The Game Tree where T is a finite set of nodes and The game tree V is a pair (7,-) '--+' is a binary relation on T. The interpretation of x-+y is'node x immediately precedes node y'. ASSUMPTION 1. Asymmetry.lf x, y € Iand x+1,

t h e n n o t l - - - >x .

An edge is an ordered pair e:(x, y) such that x--->!. We say that e is incident out of x and incident into y. We shall represent edges as arrows [from x to y if e: (x, y)). For each x C T. lel

I(r):

{yeT lr-y}:

s u c c e s s oor sf x . setof immediate

of x . x}: set of immediatepredecessors Il(.r) : {y e T I y --->


169

ASSUMPTION 2. Root. There exists a unique xoe T such that .ftrls II("u) : 0. elementxo is calledthe root of the tree. Thus every node r € Z\{xn} has an immediate predecessor.The of predecessors. following assumptionrequiresuniqueness ASSUMPTION 3. Uniquepredecessor. For everyr € A{x,,}, ll(x) is a singletonand its uniqueelementis denotedby n,. L e t e r : ( x , y , ) a n d e r ( x 2 , y r ) b e t w o e d g e s . W e s a y t h a t e , ,,t u d j a c e nt to e r i f .l r : x z . L e t ( e r : ( x r , / r ) , . . . , € ^ : ( x - , y - ) ) b e f i n i t e s e q u e n c eo f e d g e s( z > 2 ) . l f , f o r e v e r y k : 1 , . . . , m - 7 , e r i s a d j a c e n t t o e k + r , t h e n w e c a l l t h e s e q u e n c ea p a t h f r o m x , t o l ^ . A S S U M P T I O N 4 . N o c y c l e s l.f . ( e , : ( x ' , a path from x, to y^, thenxr* y^.

/r), . . ., €^: (x^, y^))

By the finiteness of Z and Assumptions 2, 3 and 4, for every y e T\{x,,} there exists a unique path from x(.)to y. Let Z: {x€ f ll(x):0}. G i v e n t h e f i n i t e n e s so f Z a n d A s s u m p tion 4, the set Z is non-empty. We call Z the set of terminal nodes (or end nodes). Thc set X: T\Z is called the set of. decision nodes. A play is a path from the root of the tree to a terminal node. As noted above, our assumptions imply that for every z € Z there is a unique play to it.

The Set of Players N:

{ 1 , 2 , . . . , n } i s t h e s e t o f p l a y e r s .P l a y e r sw i l l b e i n d e x e db y i .

The Player Partition The player partition P: {Pr,. . . , P^} is a partition of the set of decision nodes X. P, is the set of decision nodes of player l. We assume that P, is non-empty for every i € N.

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Partition

The information partition H : {Hr, . . . , H,} is defined as follows. For every i € N, a subset h of. P, is called eligible (as an information set) if: (i) ft is non-empty; ( i i ) e v e r y p l a y i n t e r s e c t sf t a t m o s t o n c e ; ' 0 (iii) for every .r and y in /r, the number of immediate successorsof x is equal to the number of immediate successorsof y. 11, is a partition of P, into eligible subsets. An element of I/, is called an information set of player i.

The Choice Partition The choice partition C : {Cr, . . . , Cn} is defined as follows. For every player i and for every information set heHi,let E(h) be the set of edges incident out of nodes in lr. We say that a subset c of E(ft) is eligible (as a choice) if it contains exactly one edge incident out of x, for every xe h. C, partitions the set of all edgesincident out of nodes contained in information sets of player I into eligible subsets.If he Hi and c belongs to an element of C, and also to E(/r), then c is called a choice of player i at information set h. The set of all choices at ft € H, is denoted by C,(h).

The Payoff Function The payoff function U associates with every terminal node z e. Z a vector U(z):(Ur(t),...,U^(z)) € 0 1 " , c a l l e d r h e p a y o f f v e c t o ra t z . The component U,(z) is called player i's payoff at z. DEFINITION. An extensive game is said to have perfect recsll if it satisfies the following property: for every player i and any two information sets h and g of player l, if one noder€gcomes after a choice c at h, then every node y € g comes after this choice. DEFINITION.

An extensive game is said to have perfect information


r7l

if every information set is a singleton.A game which does not have perfect information is called a game of imperfectinformation. DEFINITION. An extensivegame is said to be simultaneousif every play of the game crossesall the information sets. .

APPENDIX B

ln this Appendix we prove Proposition 1. We shall begin with a lemma. Given a pure-strategyprofile o, for every node x we denote by

t @ lo ) the terminal node reached from x by following o (if z is a terminal z ) . O b v i o u s l y .( ( x I o ) e 0 ( x ) . node. we set by definitiontQlo\: LEMMA 7. Fix a game with perfect recsll. Let B be a belief system that satisfies the properties of Contraction Consistency and Choice Consistency. Then, for every player i and for every node x, if B,@) € 0(x), t h e n B , ( x ) : t @ l o ) w h e r eo : t ( B ) . Proof. Let B be a belief system that satisfies Contraction Consistency and Choice Consistency and let o: t(B). Fix an arbitrary node x. If x is a terminal node then ((x I o) : x by definition of ((' ) and B,(x) : x for every player l, since K,(*): {x}. Suppose,therefore, that x is a decision node. Suppose there is a player I for whom : P , Q ) : z ' e . 0 ( x ) b u t z ' * z , w h e r ez { ( x l o ) . L e t x . , x 2 , . . . , x ^ b e the sequence of nodes that form the path from x to z (thus xr:X, x ^ : z a n d , f o r e v e r y k : 1 , . . . , m - 1 , r o i s t h e i m m e d i a t ep r e m.Letxo z for allk:1,..., d e c e s s oor f x o * , ) . O b v i o u s l yt,V k l o ) : be the node at which the path from x Io z and the path from x to z' diverge. (That is, z,z'€ 0(xo) and for every immediate successoryof xo it is not true that both z and z' belong to 0(y).) Note that x. could be x itself (in other words, k : 1 is a possibility). Clearly, x* is a decision node. Let lr be the information set to which xo belongs and 7 be the player to whom /r belongs. Two casesare possible:(1) t:7, and (2) i+ j.

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In case (1), let c be the choice at h that precedesz and c' be the choice at h that precedes z'. By our choice of xp, c+c'. We now obtain a contradiction by showing that o,(h): c'. (If o,(h): c' then following o from .rr. one cannot reach z and this contradicts the fact t h a t z : t @ r l o ) . ) B y R e m a r k 1 o f S e c t i o n3 , 0 ( x o )C K , ( x * ) . T h u s z ' e K , ( x r ) , s i n c e z ' e 0 ( x o ) b y t h e w a y x k w a s c h o s e n .B y P r o p e r t y 1 of Section 3, K,(x) E K,(x). By Contraction Consistency,it must be F , @ o ) : z ' . B y d e f i r i i t i o no f o : f ( B ) , i t f o l l o w s t h a t o , ( h ) : c ' . Now consider case (2), where jl i. Again, let c be the choice at /r that precedesz and c' be the choice at hthat precedesz'. By definition o f o : 6 ( B ) , i t m u s t b e o 1 ( h ) : c . T h u s F , @ o ) c o m e sa f t e r c h o i c ec o f player j at h (even though F,@) need not be a successorof -ro) and F,Qo) comes after choice c' of player i at h, contradicting Choice I Consistency.rr 1. Fix a game with perfect recall. Let B be a belief COROLLARY system that satisfies the properties of Contraction Consistency and Choice Consistency. Then, for every two players i and i, B,Q): F,@o) (where x,.,is the root of the tree). Proof. Since, for every player i, K,(x,,): Z:0(x,,), it follows from I L e m m a 1 t h a t B , ( x o ): { @ u l o ) f o r a l l l . Proof of Proposition l. (The argument of the proof will be illustrated at the end with the help of Figure 6.) Fix an extensive game G with perfect recall. Let B be a rational belief system and o: f(B) the corresponding (pure) strategy profile. Suppose o is not a subgameperfect equilibrium. Then there exist a decisionnode x*, a subgameG' of G with root x*, a player I and a pure strategycr',of player I such that (B.1)

U,(r')> U,(z*)

where z* : t(x* | a) is the terminal node reached from x* by o : ( o , , o , ) a n d z ' : ( ( x * l o ' ) i s t h e t e r m i n a l n o d e r e a c h e df r o m x * b y ( w h e r eo , d e n o t e st h e ( n - l ) - t u p l e ( o r , o':(o',,o,) , oi 1, A i + l , . , . r O r ) ) .

S t e pl . W e f i r s ts h o wt h a t F 1 @ * ) : z * , f o r a l l p l a y e r is: 1 , . . . , r . for everyplayerTeitherH,(x*):0 Sincex* is the root of a subgame,


173

(recall that H,(x*) is the set of information setsof playeri that have a of x*) or for everyhe H,(x*), O.(h)c node which is a successor .|hus 0(x*). K , ( x * ) : 0 ( x - ) . B y L e m m al , F i @ * ) : ( ( x * l r v ) : z * f o r every player7. Step2. Let y be the node at which the path from r* to z* and the path from x* to z' diverge.(That is, z*,2'eO(y) and if w is an of y then it is not the casethat both z* and z' immediatesuccessor belongto 0(w).) Note'thaty: x* is a possibility.Then y is a decision node of player i. By Remark 1 (cf. Section3), 0(y) e K,(y). Thus z* e K,(y). By Property1 (cf. Section3), &(y)G K,(x*). Thus, by ContractionConsistency, F,0): B Qs*): z*. of y on the path from y to Step3. Let y, be the immediatesuccessor zte 0(yr) (hence z' andler zt: Frj). Then, by Tree Consistency, z r * z * ) . T h u s ,b y L e m m a7 , z r : t 0 r l o ) . B y I n d i v i d u aRl a t i o n a l i t y , of y, sinceB,(y) : z* andy, is an immediatesuccessor (B 2)

U , ( tr ) < U , ( r * )

our supposition lf. 2,,: z' the proof is complete,since(B.2) contradicts r h a t U , ( z ' ) > U , ( r * ) . I f z r * z * , p r o c e e dt o S t e p4 . Step4. Let y, be the node at which the path from ytto zt and the path from y, to z' diverge(yr.:y, is a possibility). Then y, is a decisionnodeof playeri. By Remark1 (cf.Section3),0(yr)gK,(yr) and, therefore,sincezre 0(yr), it followsthat zre K,(yr).By Contraction Consistency(since,by Property 1 (cf. Section3), K,(yr)c K , ( v , ) ) ,F , ( v r ) : 2 , . of y, on the path from y, Step5. Let y. be the immediatesuccessor zr€ 0(y.). By Lemma to z' and let z. : F,(yr). By Tree Consistency, l , Z t : { 0 r l o ) . B y I n d i v i d u aRl a t i o n a l i t y , (B.3)

U,(tr)< U,(r,)

a n d , u s i n g( B . 2 ) a n d ( B . 3 ) , (B.4)

U,(tr)< U,(r*)

Step 6. lf z.: z' the proof is complete,since (B.4) contradicts

t74

GIACOMO BONANNO

(B.1). If. z.*z',by p r o c e e d i n gt h e s a m e w a y - s i n c e t h e n u m b e r o f nodes (and hence of information sets) is finite - we must reach a node y, such that: (i) the path from x* to z'and the path from x* to z,= 9,(y,) diverge at y, (and, therefore, y, is a decision node of player i; furthermore. zr : t(y,lo)l: (ii) if i is the imniediate successor of y, on the path from y, to z',

t 0 l o ): t 0 l o ' ) (either becausenone of the nodes on the path from i to z' are decisionnodes of player i, or becauseq and oi coincideat the information setsof player i crossedby this path); (111)U,(2,)U.(2.):0, I n d i v i d u a l R a t i o n a l i t yw o u l d b e v i o l a t e d for player 3. The argument for B.(.r,,): 2., is similar. " B y C h o i c e C o n s i s t e n c y , s i n c e B r ( x , ) : z . , i t m u s t b e e i t h e r F , ( r z, ). :o r p r ( x , ) : 2 , . The latter. however, would require, by Contraction Consistency,that B,(x.):zr, c o n t r a d i c t i n g w h a t w e h a v e a l r e a d y e s t a b l i s h e d ,n a m e l y t h a t g , ( x r ) : 2 . . ' O u r d e f i n i t i o n i s a l o n g t h e l i n e s o f S e l t e n ' s( 1 9 7 5 ) . "' T h a t i s , i f ( e , : ( x , , y , ) , . . . . e , , - ( * , , , , y , , , ) ) i s a p l a y ( t h u s x , - x n a n d y , , ,€ Z ) t h e n the following conditions are satisfied: (a) if r^ E ft for some k : l. . m . y , y ' h a n d .f o r a l l I - 1 . , rn, then, for all j: I xtt ni ( b ) i f y * e h f o r s o m ek : 1 , . . . , m - 1 , t h e n ,f o r a l l I : 1 , . . , m , x , y ' h , a n d , f o r a l l

j+k,y,y'h.

"

It is easy to show that if either Contraction Consistency or Choice Consistency is not satisfied then Lemma I is not true.

REFERENCES Bonanno, G.: 1990, 'Knowledge and beliefs in extensive games', Department of Economics Working Paper No. 361, University of California, Davis. B o n a n n o , G . : 1 9 9 1 , ' P l a y e r s ' i n f o r m a t i o n i n e x t e n s i v eg a m e s ' , t o a p p e a r i n : M a t h e m a t i cal Social Sciences. Greenberg, J.: 1990, The Theory of Social Situations, Cambridge University Prcss, Cambridge. Kreps. D. and Wilson, R.: 1982, 'Sequentialequilibria', Econometrica,50, 863-894. S e l t c n , R . : 1 9 6 5 , ' S p i e l t h e o r e t i s c h eB e h a n d l u n g e i n e s O l i g o p o l m o d e l l s m i t N a c h frAgetragheit', Zeitschrift fiir die gesamteStaatswissenschaft,12,301-324 and 667-6t39. Selten, R.: 1973,'A simple model of imperfect competition where 4 are few and 6 are many', International Journal of Came Theory, 2, 141-201. S e l t c n . R . : 1 9 7 5 , ' R e - e x a m i n a t i o n o f t h e p e r f e c t n e s sc o n c e p t f o r e q u i l i b r i u m p o i n t s i n extcnsive games', International Journal of Game Theory, 4,25,55. Zcrmelo, E.: 1912, Uber eine Anwendung der Mengenlehre auf die Theorie des Schachspicls', Proceedings, Fifth lnternational Congressof Mathematician.s.Vol. 2, pp. 501 50.1.

Department of Economics. University of California. D a v i s . C A 9 5 6 1 6 .U . S . A .