The canonical extensive form of a game form Part II ...

Ž .Journal of Mathematical Economics 33 2000 299–338www.elsevier.comrlocaterjmateco

The canonical extensive form of a game formPart II. Representation

Peter Sudholter, Joachim Rosenmuller ), Bezalel Peleg 1¨ ¨Institut fur Mathematische Wirtschaftsforschung, UniÕersitat Bielefeld, D-33615 Bielefeld, Germany¨ ¨

Received 24 April 1998; received in revised form 21 April 1999; accepted 21 April 1999

Abstract

This paper exhibits to any noncooperative game in strategic or normal form a ‘canoni-cal’ game in extensive form that preserves all symmetries of the former one. The operationdefined this way respects the restriction of games to subgames and yields a minimal totalrank of the tree involved. Moreover, by the above requirements the ‘canonical extensivegame form’ is uniquely defined. q 2000 Elsevier Science S.A. All rights reserved.

AMS classification: 90D10; 90D35; 05C05

Keywords: Games; Extensive form; Normal form; Strategic form

1. Introduction

This paper represents the second part of a study devoted to the relationshipŽ .between a strategic game game form and its possible representations by exten-

Ž .sive games game forms . To be explicit, a representation of a strategic game G isan extensive game G the ‘normal’, ‘strategic’ or ‘von Neumann–Morgensternform’ of which is G.

) Corresponding author. Tel.: q49-521-106-4909; fax: q49-521-106-2997; E-mail:[email protected]

1 Also at the Center for Rationality and Interactive Decision Theory, The Hebrew University ofJerusalem, Israel.

0304-4068r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved.Ž .PII: S0304-4068 99 00019-1

( )P. Sudholter et al.rJournal of Mathematical Economics 33 2000 299–338¨300

The aim of this paper is to exhibit the choice of a method to represent strategicgames by extensive games in a way that essentially preserves all symmetries of thestrategic game but in addition satisfies additional axioms. These axioms concernrobustness under restriction and minimality of the tree involved. Eventually, wewant these axioms to characterize the representation and this is the main theorem

Ž .of this paper i.e., Theorem 7.6 .We should emphasize that our starting point is a game in strategic form. The

transition from the extensive form to the strategic form as defined by vonw x ŽNeumann and Morgenstern 5 has already been investigated extensively see

w x .Kohlberg and Mertens 3 for a recent treatment of this topic . The transition in theopposite direction is considered ‘trivial’ and conceptually straightforward. It is thepurpose of our work to show that this is not true: the choice of a representation ofstrategic games by extensive games which respects symmetries of strategic gamesleads to difficult conceptual problems and deep mathematical results.

Ž w x.The previous results obtained see Ref. 6 and the drive of the presentcontinuation can at best be explained by pointing to the ‘Battle of the Sexes’.Some of the problems motivating our treatment are explained at best within thecontext of this example.

A strategic version of this game is represented by the following ‘bimatrix’:

C S C S

C 2 0 C 1 0ž / ž /S 0 1 S 0 2

Here C is interpreted as ‘attending a concert’ and S is ‘mingling with theŽ .crowds at a soccer game’. There are two ‘focal’ equilibria in this game: C,C and

Ž . w xS,S . Myerson discusses these games nicely in Ref. 4 , Section 3.5.There is a standard convention in Game Theory according to which two simple

Žversions of a representation for this game by an extensive game are exhibited see.Fig. 1 .

The above convention which leads to multiple representations has the followingtwo problematic aspects.

Fig. 1. Two representations.

( )P. Sudholter et al.rJournal of Mathematical Economics 33 2000 299–338¨ 301

Ž .1 The transition to the extensive form might influence focality. Consider theextensive game G . It is common knowledge in this game that player 1 moves1

first. Therefore, she has the option to choose C before player two makes hisŽ .choice. Thus, it seems to us that in G the pair C,C of strategies is more likely1

Ž .to be played than S,S . Our feeling is supported by the experimental work ofw x Ž .Rapoport 7 . Clearly, in G the pair S,S may be the dominant focal equilibrium2Ž .rather than C,C .

Ž .2 The transition to the extensive form may destroy symmetry. The game G is‘symmetric’ in the following sense: it has an automorphism which permutes

Žplayers 1 and 2 for a definition of automorphism, i.e., an isomorphism of G tow x .itself, see Harsanyi and Selten 2 , Section 3.4 . This automorphism is given

Ž . w xexplicitly in Example 4.6 1 of Ref. 6 . However, G and G are totally1 2

asymmetric; more precisely, if GsG or GsG , then there is no non-trivial1 2

automorphism of G that respects the temporal ordering of moves in G .The discussion in the last paragraph leads naturally to the following basic

Žquestion: Let G be a game in strategic form. When is G ‘symmetric’? In.particular, is the Battle of Sexes a symmetric game? .

Quite surprisingly there was no satisfactory answer available to this question. Ifwe follow our mathematical intuition and define a strategic game G to besymmetric if all possible joint renamings of players and strategies are automor-

Ž w x .phisms of G see Ref. 2 , p. 71, for the precise definition of renaming , then theclass of symmetric games reduces to the trivial class of all games whose payofffunctions are constant and equal. Also, this definition is incompatible with the

Ž w x .definition of symmetric bimatrix games see Ref. 1 , p. 211 .w xIn Ref. 6 we presented an answer to this basic question. A symmetry of G,

w xaccording to the definition given in Ref. 6 , is a permutation p of the players forŽ . Žwhich there exists an automorphism as p ,w of G here w is a renaming of

. Žstrategies which is compatible with p . Thus, our definition of symmetries of. w xstrategic games is different from that of Harsanyi and Selten 2 , p. 73. The game

G is symmetric if every permutation of the players is a symmetry of G. Thus, inparticular, the Battle of Sexes is symmetric according to our definition. Ourdefinition has the following desirable properties.1. The class of symmetric games is a nontrivial interesting class.

Ž2. It is possible to use similar ideas to define symmetries of extensive games see.Definition 2.14 .

3. It is possible to compare the symmetry groups of a strategic game and itsŽ w x.coalitional form see Theorem 4.11 of Ref. 6 .

As far as we could check, symmetries of games in extensive form whichpreserve the partial ordering on the nodes that is induced by the game tree werenot considered previously. Thus, the treatment of symmetry groups of extensive

w xgames, as offered in Ref. 6 , was entirely new.We now present the solution to the problem of representing the Battle of Sexes

by an extensive game.


Ž .A symmetry of a strategic game G, according to our definition, as presentedw x Ž .in Ref. 6 see Definition 2.13 , is a permutation p of the players for which there

Ž .exists a renaming w of strategies which is compatible with p such that as p ,wis an automorphism of G. The game G is symmetric if every permutation of theplayers is a symmetry of G. Similar definitions have been offered for extensivegames and this way it is seen that Fig. 2 indeed shows a symmetric representationof a symmetric game.

However, it is by no means true that it is the only such symmetric representa-tion.

As a further property it can be seen easily that Fig. 2 indicates a graphŽ .‘minimal’ in the class of all representations of the Battle of Sexes which

preserve the symmetries.It is our aim within this paper to generalize the foregoing construction to all

finite strategic games and to show that there is a ‘canonical’ way of representing astrategic game by a symmetry preserving and minimal extensive game. By variousreasons we speak of games and game forms simultaneously within this context.

Clearly some further difficulties appear at once. Let us list two of them whichappear to be quite diametral.

Ž .1 For example, a 2=3 two-person game has no symmetries. Therefore, a‘canonical’ representation for such a game should have suitable properties of amore general character. If we add the requirement that our representations of 2=3games are consistent with our representations of 2=2 games, then we have an

Ž .additional tool and an additional obstacle to be concerned with.Ž . Ž2 On the other hand, given a ‘square’ n-person strategic game form i.e., all

.players having the same number of strategies , it is not clear how to find for it aminimal and symmetry preserving representation since square game forms inprinciple allow for complete symmetry between the players.

Other than in the example above, when the number of players is greater thantwo, then there is no obvious solution to the problem of representing square game

Ž .forms. For this purpose it is necessary to first define the simplest ‘atomic’representations of square game forms and then build ‘symmetrizations’ of such‘atoms’ in order to obtain symmetry-preserving representations.

Fig. 2. The ‘canonical’ Battle of Sexes.


Based on these procedures we eventually come up with a general canonicalŽ . Žmethod of representing strategic games game forms by extensive games game

. Žforms : we axiomatically justify a certain symmetrization of atoms the one which.in every branch exhibits a clear ‘time structure’ or ‘order of play’ to be the

unique representation which preserves symmetries, respects restrictions and showsa minimal tree.

We now shortly review the contents of the paper. The versions of strategic andextensive game forms and games we are dealing with have been introduced in Ref.w x6 . We will, however, shortly repeat the definitions in order to make this workself contained. The same is true for the basic notions of isomorphisms betweengames and game forms.

w xOne of the major aspects of Ref. 6 is an alternative definition of symmetrydeviating from the one given by previous authors. Actually a symmetry is a

Ž .residual class of automorphisms of game forms games , the elements of whichŽ .differ by an impersonal motion automorphism only. Hence the content of Section

2 is a short review of the necessary definitions.Section 3 is devoted to the definition of faithful representations. An extensive

game from g is a representation of a game form g if for every choice of a vectorŽ . Žu of payoff functions the normal form of the extensive game G g ,u is imperson-

. Ž .ally isomorphic to the strategic game G g,u . A representation g of a game formg is faithful if, for every choice of a vector of payoff functions u, the gamesŽ . Ž .G g ,u and G g,u have the same symmetry group. Theorem 3.5 shows that the

outcome function of a representation does not depend on chance moves.Within Section 4 we set out to construct and characterize faithful representa-

tions of general and square game forms. The main result of this section ispresented in Corollary 4.6. The group of automorphisms of a square game form isthe surjective image of the corresponding group of any extensive game formrepresenting it, if and only if the representation is faithful.

In Section 5 we consider the minimal representations of square and generalgame forms which are called atoms.

Ž .In Section 6 it is first of all shown Theorem 6.4 that symmetrizations of atomsare faithful representations of strategic game forms that are square and general.The main result of this section, Theorem 6.7, proves a converse result: a minimaland faithful representation of a general and square strategic game form must be asymmetrization of an atom.

An atom is time structured if the order of play is the same for all n-tupels ofstrategies. Section 7 is devoted to the proof that symmetrizations of time structuredatoms yield canonical representations of finite strategic games. First, Theorem 7.5shows that minimal and faithful representations of square and general strategic

Žgame forms must be time-structured provided the number of strategies is at least.three .

Ž .A representation in the sense of Theorem 7.6 is a mapping which assigns toŽ .every game form in the domain of the mapping , an extensive game form that


represents it. A representation is canonical if it is faithful, respects isomorphisms,is consistent with restriction, and has a minimality property.

Theorem 7.6, the main result of this work, proves that there exists a uniqueŽ .canonical representation of strategic games which is given by generalized time

structured representations.

2. Prerequisites

The structure presenting the fundamental strata of our considerations is the oneof a game tree. Since it is not unfamiliar in Game Theory we describe it mainly

w xverbally, however, the details are exactly the ones presented in Ref. 6 .Ž .A game tree is a set of data E, $ , P, Q, C, p the elements of which are

described as follows.Ž . Ž .E is a finite set the nodes and $ is a binary relation on E such that E,$ is

Ž Ž .a tree the root is x , the generic element node is j , and the set of endpoints is0.EE . The rank function r defined on the nodes counts the number of steps from the

Ž . � < Ž . 4 Ž .nodes to the root, e.g., r x s0. The set j r j s t \LL E,$ ,t constitutes0Ž . Ž .the leÕel t and R E,$ [Ý r j denotes the total rank of the tree. A pathj g EE

is a sequence of consecutive nodes. A play is any path that connects the root withan endpoint.

P is a partition of EyEE, the player partition. There is a distinguishedŽ .element P gP which may be the empty set representing the chance moÕes. All0

other player sets are assumed to be nonvoid. The elements of P represent sets ofnodes at which a certain player is in command of the next move.

Q represents the information partition. Q is a refinement of P; thus an elementQgQ, Q:P, is an information set of the player who commands the elements ofP. In particular it is required that Q refines P to singletons.0

C is a family of partitions representing choices. To explain this object, letX < X� 4C j [ j j$jŽ .

denote the successors of jgE and define for QgQ

C Q [ C j .Ž . Ž .D

jgQ

Ž . Ž .Now we assume that for any QgQ we are given a partition C Q of C Qsuch that

< <SgC Q ´ SlC j s1 jgQŽ . Ž . Ž .Ž Ž ..holds true. then the system of choices is described by C[ C Q .Q g Q

Ž j . Ž .ps p is a family of probability distributions of chance moves , i.e.,j g P0

p j is a probability on the successors of j for every jgP . We assume that0j Ž X. Xp j is positive for every successor j of j . Intuitively, chance determines the

successor of jgP according to the random mechanism described by p j.0


Fig. 3. An extensive preform.

We are now in the position to discuss preforms. Extensive preforms areobtained by assigning the names of the players to the corresponding elements ofthe player partition of a game tree, more precisely:

Ž .Definition 2.1. An extensive preform see Fig. 3 is a tuple

es N , E,$ ,P,Q,C, p ;ı 2.1Ž . Ž .Ž .where N is a finite set the set of players of at least two elements, the next five

� 4data constitute a game tree, and ı: Py P ™N is a bijective mapping.0

� 4 Ž .Intuitively, ı assigns the nodes of PgPy P to the player ı P gN, who is0

thought to be in charge of choosing a successor when this node occurs during ay1Ž .play. We use the notation ı i sP to refer to ‘the nodes of player i’.i

In order to describe a strategic preform we need much less preliminaryframework.

Ž .Definition 2.2. A strategic preform is a pair es N, S , where N is a finite setŽ < < .the set of players, N G2 and SsŁ S is the product of finite sets S /Øi g N i iŽ . Ž .igN the strategy sets .

Game forms are now obtained by assigning outcomes to the choice of a play orto a strategy profile, respectively. In fact, a strategy profile in the strategic context

Ž .can frequently be identified with an outcome the ‘general’ case . However, it ismore appropriate to choose an abstract set of outcomes and assign its elements toplays or profiles accordingly.

Ž . Ž .Definition 2.3. 1 An extensive game form see Fig. 4 is a tuple

gs e ; A ,h s N , E,$ ,P,Q,C, p ;ı ; A ,h . 2.2Ž . Ž . Ž .Ž .Here, e is an extensive preform and A is a finite set the outcomes while

h:EE™A is a surjective mapping called outcome function, which assignsŽ . Ž .outcomes to endpoints of the graph E,$ . 2 A strategic game form is a


Fig. 4. An extensive game form.

Ž . Ž .quadruple gs e; A,h s N,S; A,h . Here e is a preform, A a finite set andh:S™A is a surjective mapping again called the outcome function. If h is abijection, then g is called general.

ŽThe term ‘general’ is not suitable for extensive game forms as unless in a.special case it is frequently to be expected that different plays influenced by a

chance move result in the same outcome.Ž .We now turn to the notion of extensive and strategic games. In a game we

have payoffs in terms of utilities, i.e., real valued functions. Eventually, thisincludes also the possibility of computing expectations when the chance influenceis taken into account.

Ž .Definition 2.4. 1 An extensive game is a tuple

Gs e ;Õ s N , E,$ ,P,Q,C, p ;ı ;Õ 2.3Ž . Ž . Ž .such that

Õs Õ :EE™R N 2.4Ž . Ž .i igN

represents the system of player’s utility functions depending on endpoints of theŽ . Ž .graph E,$ . 2 A strategic game is a tuple

Gs e ;u s N ,S ;u 2.5Ž . Ž . Ž .Again,

us u :S™R N 2.6Ž . Ž .i igN

constitutes the player’s utility functions defined on strategy profiles.

Remark 2.5. Obviously, games can be generated from game forms by means of aŽ .utility defined on the alphabet. Formally, if g and g are extensive and strategic

game forms and

U : A™R N

Ž .is a ‘utility’ function defined on outcomes, then

Õ[U(h , u[U(h 2.7Ž .


induce games

G[U)g[U e ; A ,h s e ;U(h 2.8Ž . Ž . Ž .and

G[U) g[U e ; A ,h s e ;U(h . 2.9Ž . Ž . Ž .

Next we shortly review the basic ideas of isomorphisms, motions, and symme-w xtries as presented in Ref. 6 .

Isomorphisms for strategic objects are comparably easy to understand, thereforewe start with this notion.

Ž . X Ž X.Given strategic preforms es N,S and e s N,S we consider bijectiveX Ž .mappings p :N™N and w :S™S igN . p renames the players and wi i p Ž i. i

maps strategies of player i into strategies of his double.Ž .The pair p ,w of course induces a simultaneous reshuffling of strategies, i.e., a

mapping

wp :S™SX , wp s [w s igN . 2.10Ž . Ž . Ž . Ž .Ž . Ž .p i i i

Definition 2.6. An isomorphism between strategic preforms e and eX is a familyŽ .p ,w of bijective mappings

p :N™N , w :S™SX igNŽ .i i p Ž i.

such that

p ,w es p ,w N ,S [ p N ,wp S s N ,SX 2.11Ž . Ž . Ž . Ž . Ž . Ž .Ž .holds true.

As to extensive objects, the definition of isomorphisms requires more effort.We start out with game trees. Consider a bijective mapping f which maps thenodes of a game tree onto the nodes of another game tree. It is not difficult toimagine what it means that f respects the structures of the trees, i.e., the

w xsuccessor relation and the partitions — or else consult Ref. 6 . We then quote

Ž .Definition 2.7. A game tree E,$ ,P,Q,C, p is isomorphic to a game treeŽ X X X X X X . X ŽE ,$ ,P ,Q ,C , p if there is a bijective mapping f:E™E an isomorphism

.between the game trees which satisfies the following properties.Ž X. Ž X. Ž X.1. The mapping f respects $ ,$ , P,P , and Q,Q .

Ž . X X f Ž j . Ž Ž X .. j Ž X. X Ž .2. f P sP and p f j sp j holds true for jgP and j gC j .0 0 0Ž Ž . XŽ X.. X X3. For QgQ the mapping f respects C Q , C Q , where Q gQ is theŽ . Xunique information set which satisfies f Q sQ .

Ž . Ž .Note that 2 makes sense in view of 1 , the bijectivity of f, and theunderlying tree structure. Now turn to extensive preforms. The underlying game


trees being isomorphically mapped into each other, we also want to adapt theassignment of partitions to players consistently.

Ž . X Ž X X X X X X X.Definition 2.8. Let es N, E,$ ,P,Q,C, p;ı and e s N, E ,$ ,P ,Q ,C , p ;ıbe extensive preforms. A isomorphism between e and e

X is a pair of mappingsŽ . Xp ,f such that p is a permutation of N,f:E™E with the following properties.

Ž .1. f is an isomorphism between the underlying game trees cf. Definition 2.7 .Ž Ž .. XŽ Ž .. Ž .2. p ı P s ı f P PgP

Ž . Ž .One can imagine that a pair of bijective mappings an isomorphism p ,f actsseparately on all the objects constituting an extensive preform. The exact defini-tions are more or less canonical and will not be explicitly mentioned. To represent

Ž .it in a closed form, the action p ,f on e is described as follows:

p ,f es p ,f N , E,$ ,P,Q,C, p ;ıŽ . Ž . Ž .s p N ,fE,$ f ,fP,fQ,fC, p y1 ;p( ı(fy1Ž .Ž .f

s N , EX ,$X ,PX ,QX ,CX , pX ;ıX 2.12Ž . Ž .Again let us emphasize that isomorphisms respect the ordering of the nodes.

This reflects the notion of ‘focality’ as discussed in Section 1.As to isomorphisms of game forms, our next object of interest is the alphabet of

outcomes. Again we first demonstrate our goal in the range of strategic territory.Consider the following two matrices, each of them representing a strategic gameform.

c aa bFs Gsž / ž /c a b c

The alphabet or outcome set being the same in both cases, we feel that F andG are structurally equal, i.e., isomorphic — nevertheless, this cannot be obtainedby reshuffling the strategies andror the players. In addition to the familiaroperations we should admit a bijection of the alphabet; in the above case thiswould be expressed by a bijection r: A™A, a™c, b™a, c™b. This consider-ation motivates the following definition.

Ž . X Ž X X X .Definition 2.9. Let gs e; A,h and g s e ; A ,h be strategic game forms. AnX Ž . Ž .isomorphism of g and g is a triple p ,w;r such that p ,w is an isomorphism

between e and eX while r:A™AX is a bijection satisfyingy1X ph sr(h( w .Ž .

p Ž .Here, w is the mapping defined in Eq. 2.10 . That is, we havey1 X X Xpp ,w ;r e ; A ,h [ p ,w e ;r(h( w s e ; A ,hŽ . Ž . Ž . Ž . Ž .Ž .

An isomorphism is outcome preserving if AsAX and r is the identity.


Inevitably we have to discuss isomorphisms for the third group of objects underconsideration, i.e., for games. The difference with respect to game forms isconstituted by the presence of utility functions instead of the outcome function.

Ž .Therefore, we have to explain the kind of action a permutation renaming of theplayers induces on n-tuples of utility functions, i.e., on u or Õ, respectively.

Ž . XClearly, if p ,w is an isomorphism between the strategic preforms e and eand u and uX are tuples of utilities defined on S and SX, respectively, then the

Ž .utility of i’s image p i gN should be given by

uXwp s su s igN , 2.13Ž . Ž . Ž . Ž .Ž .p Ž i. i

thus indicating that we rename players and strategies simultaneously.Ž .This defines the action of the pair p ,w on tuples of utility functions via

p ,w u wp s [u s . 2.14Ž . Ž . Ž . Ž .Ž . Ž .Ž .p i i

We have thus explained

p ,w u:SX™R N . 2.15Ž . Ž .

Ž .Analogously, within the extensive set-up, if we have an isomorphism p ,f ofpreforms e and e

X, cf. Definition 2.8, and if Õ:EE™R N is a utility N-tupleŽ . Ž .defined on the endpoints of E,$ , the action of p ,f , i.e.,

p ,f Õ :EEX™R N 2.16Ž . Ž .

is given by

p ,f Õ f j [Õ j jgEE, igN . 2.17Ž . Ž . Ž . Ž . Ž .Ž . Ž .Ž .p i i

Thus we have

Ž . Ž . X Ž X X.Definition 2.10. 1 Let Gs e;u and G s e ;u be strategic games. AnX Ž . Ž .isomorphism between G and G is a pair p ,w such that p ,w is an isomor-

X Ž Ž .. X Ž p Ž ..phism between e and e see Definition 2.6 and Eq. 2.11 and u w s sp Ž i.Ž . Ž .u s igN,sgS . That is, we havei

p ,w Gs p ,w e ;u s p ,w e ; p ,w u s eX ,uX 2.18Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ž .Ž Ž . Ž .. Ž . Ž . X Ž X X.see 2.14 and 2.15 . 2 Let Gs e ,Õ and G s e ,Õ be extensive games.

X Ž . Ž .An isomorphism between G and G is a pair p ,f such that p ,f is anX Ž . X Ž Ž .. Ž .isomorphism between e and e see Definition 2.8 and Õ f j sÕ jp Ž i. i

Ž .jgEE,igN . That is, we write

p ,f Gs p ,f e ;Õ s p ,f e , p ,f Õ s eX ,ÕX 2.19Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ž .

Ž Ž . Ž ..see 2.16 and 2.17 .

We have now finished the discussion of the various concepts of isomorphismsas appropriate for preforms, game forms, and games — in each family withconsideration of the strategic as well as the extensive version. The next and basic


concept is the one of symmetry. This concept applies to games only. As we havew xstressed in Peleg et al. 6 , it does not work to consider just automorphisms of a

Ž .game G, say in strategic form, i.e., isomorphisms p ,w , p :N™N, w :S™Si i p Ž i.Ž .igN , such that the game is preserved, i.e.,

p ,w GsGŽ .w xholds true. The reasons were explained in Ref. 6 ; we refer the reader in particular

to Example 4.6 of this paper. Essentially there are two objections to taking thegroup of automorphisms as to be the group of symmetries: the first is that the onlygame that is preserved under the full group of automorphisms is the constant gamewhereas we definitely feel that, e.g., the ‘Battle of Sexes’, though not constant, issymmetric. The second stems from the opposite observation: a player whosepayoff does not depend on his actions may reshuffle his strategies arbitrarily, thusa rather substantial group of automorphisms is generated — nevertheless theremight be no obvious symmetry in view of another player’s situation.

These considerations lead to the idea of ‘disregarding’ the ‘impersonal’ auto-morphisms which mathematically amounts to factorizing them out or forming the

w xquotient group. More precisely we quote the relevant definitions from Ref. 6 asfollows.

Ž .Definition 2.11. A motion of a strategic game G is an automorphism p ,w of G.Ž . Ž Ž ..A motion p ,w is impersonal if p is the identity and w :S™S igN .i i i

Remark 2.12. Motions form a group. To see this, define the unit motion by

id ,id s id , idŽ . Ž .Ž .N Si igN

Ž . Ž .and the product of motions p ,w and s ,c by

s ,c p ,w s sp ,cmw 2.20Ž . Ž . Ž . Ž .where m is given in a natural manner via

cmw [c (w . 2.21Ž . Ž .i p Ž i. i

Clearly, the impersonal motions constitute a subgroup. This subgroup is normaland therefore ‘disregarding’ or ‘factorizing out’ the impersonal part formallyamounts to constructing the quotient group as to constitute the group of symme-tries of the game.

The precise version is given as follows:

Ž .Definition 2.13. Let G be a strategic game. MMsMM G denotes the group ofŽ .motions. IIsII G : MM denotes the subgroup of G constituted by the imper-

sonal motions. The group of symmetries of G is the quotient group

MM GŽ .SSsSS G sMMrIIs 2.22Ž . Ž .

II GŽ .


A game G is symmetric if its symmetry group is isomorphic to the full group ofŽ .permutations of N called S N , i.e., if

SS G (S N 2.23Ž . Ž . Ž .holds true.

w xThe reader is referred to Example 3.15 of Ref. 6 in order to sharpen hisintuition regarding symmetries and to appreciate or debate our notion of symme-try. We finish this section by adding the completely analogous definitions forextensive games. Clearly, the group of automorphisms of an extensive game is amuch more involved and less easier to view object compared to its strategiccounterpart. Nevertheless, the formal definitions are based on the same type ofarguments.

Ž . Ž .Definition 2.14. Let Gs e ,Õ be an extensive game. A motion p ,f of G is anŽ . Ž .automorphism of G and MMsMM G denotes the group of motions. IIsII G is

Ž .the normal subgroup of impersonal motions, i.e., automorphisms of shape id,f .The group of symmetries is the quotient group

MM GŽ .SSsSS G sMMrIIs 2.24Ž . Ž .

II GŽ .Ž . Ž .and G is symmetric if SS G (S N holds true.

Thus far we have rehearsed the contents of the first part of our venture, i.e.,w xRef. 6 . We now turn to the more involved task of lying the foundations of

representation theory. Eventually we shall construct to every strategic game auniquely defined extensive game ‘representing’ it and admitting for exactly thesame group of symmetries. This is the task described in the following sections.

3. Faithful representations

ŽRoughly speaking, a representation of a strategic object a preform, a game.form, or a game is an extensive object which yields the same von Neumann–

Morgenstern strategic version as the one we started out with. However, ‘the same’is not well defined, as there is no canonical ordering of the strategies when oneconstructs the von Neumann–Morgenstern strategic version. This notion has to bemade precise. We then continue to introduce faithful representations, that is,representations that preserve the group of symmetries.

Let e be an extensive preform. A pure strategy of player i is a mapping thatselects a choice at each information set of player i. If we define for igN

˜ <S [ s s is a pure strategy of player i 3.1� 4 Ž .i i i


Žthen we obtain a strategic preform the von Neumann–Morgenstern strategic.preform

˜ Ñ e s N ,S s N , S 3.2Ž . Ž .Ž . Ž .ž /i igN

N Ñext, if Õ:EE™R is a vector of utility functions, then to any sgS there is acorresponding random variable X s choosing plays in accordance with the distribu-tion induced by s and p. We may define the payoff at each realization of X s tobe the one at the endpoint, thus the expectation

u s sEÕ X s 3.3Ž . Ž . Ž .˜ i i

is well defined. This generates a strategic game

ÑN e ;Õ sNN G [ N ,S ;u s N e ;u e ,Õ 3.4Ž . Ž . Ž . Ž . Ž .Ž .Ž .˜ ˜Ž .which is the ‘von Neumann–Morgenstern’ ‘vNM’ strategic game of G .

Ž .Definition 3.1. N e as defined by 3.1 and 3.2 is called the vNM strategicŽ .preform of e ; NN G as defined by 3.2 and 3.4 is the vNM strategic game of G .

ŽThe term normal form of an extensive game is common in the literature. This weshall not employ in our present context, because the term ‘form’ in our context is

Žused for a particular type of structure strategic and extensive game forms and..preforms .

Our next purpose is to define the relation between isomorphisms of extensiveforms and strategies. To this end, the reader should review Definition 2.7. In this

Žcontext, when we are given the partitions P, Q, C of a game tree a game form, a.game , we would like to refer to the player dependent elements only. Thus we

introduce the following notations. We write

<� 4P [Py P , Q [QyQ where Q [ QgQ Q:P . 3.5� 4 Ž .y0 0 y0 0 0 0

Ž .Now, if p ,f is an isomorphism between our game tree and some other gametree with partitions PX,QX,CX, then we want to consider the induced mappingsgiven by

f P :P ™PX , f Q :Q ™QXy0 y0 y0 y0

where f P P sPX , whenever Psfy1 PX ,Ž . Ž .and f Q Q sQX , whenever Qsfy1 QX . 3.6Ž . Ž . Ž .

Note that both mappings are well defined and bijective, because f is bijectiveand respects the partitions. Also note that

f CŽQ. :C Q ™CX QX 3.7Ž . Ž . Ž .can be defined analogously to f P.


The influence of isomorphism on strategies is then explained as follows.

Ž .Remark 3.2. Note that an isomorphism p ,f between extensive preforms e ande

X induces an isomorphism between the corresponding vNM preforms. Intuitively,a strategy of a player may be thought of as a set of ‘arrows’ indicating actions ofthis player at his information sets. Clearly, reshuffling the nodes and the players inthe extensive preform also rearranges the arrows. More precisely, there exists a

Ž . Ž .mapping J which to any p ,f assigns a p ,w , where w is defined by

Q CŽQ. ˜w s f Q sf s Q QgQ,Q:P ,igN ,sgS . 3.8Ž . Ž . Ž . Ž .Ž .Ž . Ž . Ž .i i i

For the particular case that e equals eX, the mapping J can be seen to respect

compositions of isomorphisms, i.e.,

J s(p ,x(f s s(p ,cmw sJ s ,x J p ,fŽ . Ž . Ž . Ž .Ž . Ž . Ž . Ž . Ž .for automorphisms s ,x , p ,f of e , where J s ,x s s ,c and J p ,f s

Ž .p ,w .

X Ž .Let G and G be games and consider an isomorphism p ,f between them.Ž .Then, of course, p ,f is an isomorphism between the underlying extensiveŽ .preforms, hence J p ,f is an isomorphism between the resulting vNM strategic

preforms. Now we have

Ž . Ž .Lemma 3.3. Let p ,f be an isomorphism between the extensiÕe games Gs e ,ÕX Ž X X. Ž . Ž . Ž X.and G s e ;Õ . Then J p ,f is an isomorphism between NN G and NN G .

Proof: The proof is rather straightforward as all mappings involved are bijective.In view of this fact we restrict ourselves to a mere sketch. The probability at eachchance move of e is fully transported to the corresponding probability at theimage node in e

X, which is also a chance move. Therefore, the expectations ofpayoffs are preserved. q.e.d.

With game forms the situation is more involved. On the other hand here is theclue to the decisive role game forms play in our treatment of symmetries.Therefore, we have to start with the following definition which emphasizes theimportance of the game form with respect to all games it may induce. To thispurpose, we introduce a set of outcomes A and utilities U: A™R N. RecallRemark 2.5 concerning composition of a game form g or g and U to obtain agame U) g or U)g .

Ž . Ž . Ž . ŽDefinition 3.4. 1 Let gs e; A,h and gs e ; A,h be game forms with.identical N and A . Then g is called a representation of g if there is a family of

˜ NŽ . Ž .bijections cs c , c : S™S such that for every U: A™R id,c is ani i g N i i iŽ . Ž . Ž .impersonal isomorphism between U) g and NN U)g . 2 A representation g


of g is said to be faithful if for every U: A™R N the symmetry groups coincide,i.e., if

SS U) g (SS U)g U : A™R N . 3.9Ž . Ž . Ž . Ž .

Our first observation is that representations can only occur if the extensivegame form is of a nature which avoids the introduction of ‘lotteries’ for thecomputation of outcomes resulting from strategies. To this end, for any pure

Ž .strategy s of chance and any strategy profile n-tuple s of the players the0s0 ,s Ž s0 ,s s 0 ,s .resulting play is denoted by X s X , . . . , X . The outcome induced is0 T

Ž s0 ,s .h X . However, it turns out that, given a representation, the outcome does notT

depend on s . More precisely, we have0

˜Theorem 3.5. Let g be a representation of g. Then for all sgS the outcomeŽ s0 ,s .h X does not depend on s .T 0

Ž .Proof: Let id,c be the isomorphism mentioned in Definition 3.4. Let agA. LetŽ . i dsgS be such that the outcome in g is a and let s be the image under c of s,

i.e.,i d y1 � 4sgc h a .Ž .Ž .

Ž s0 ,s .We want to show that h X sa for all pure strategies s of chance. To thisT 0N Ž . Ž . Ž . Žpurpose define U: A™R by U a s1 igN and U b s0 igN,bgA,b/i i

.a . Consider the games GsU)g and GsU) g which is impersonally isomor-Ž .phic to NN U)g . The first game possesses only payoffs 0 and 1 and so does the

Ž .Ž s . Ž . Ž .latter one. Hence it follows that E U (h X s1 igN cf. 3.3 . But thisi

necessarily implies that all plays X s0 ,s yield a payoff 1. q.e.d.

Corollary 3.6. Let g be a representation of g. If g is general, then the impersonalŽ .isomorphism id,c giÕen by Definition 3.4 is uniquely defined.

Ž .Proof: To see this, observe that ‘mixing’ taking expectations can be avoided incomputing the outcomes resulting from strategy profiles in the framework of g .Hence an outcome can be associated to any entry of g. As g is general thisassociation defines a unique mapping. q.e.d.

The same consideration motivates the introduction of a ÕNM strategic gameform of an extensive game form g, even if g does not happen to be general. For,the above mentioned association can be performed in any case, hence thefollowing definition is noncontradictory.

Ž . Ž .Definition 3.7. Let gs e ; A,h be an extensive game form. 1 If, for every˜sgS, the expression

h X s0 ,s \hh s 3.10Ž . Ž .Ž .T


is a constant independently of the pure strategy s of chance, then g is0Ž .nonmixing. 2 If g is nonmixing, then

RRRRR g [ N e ; A ,hhŽ . Ž .Ž .is the vNM strategic game form of g .

Intuitively within the framework of a nonmixing extensive game form takingŽ .lotteries or expectations for that matter is avoided. Every play which chance can

generate yields the outcome determined by the strategy profile of the players.

Remark 3.8. Let g be an extensive game form. Then the following are equivalent.Ž . Ž .1 g is nonmixing. 2 g is a representation of some strategic game form g.

Every representation of a strategic game form is by Theorem 3.5 nonmixing.Conversely, if g is nonmixing, then it is straightforward to verify that it representsŽ .RRRRR g .

Corollary 3.9. Let g be a nonmixing extensiÕe game form and g be a strategic( )game form. 1 g represents g, if and only if there is an impersonal outcome

( ) Ž . Ž . ( )preserÕing IOP isomorphism id,c ,id between g and RRRRR g . 2 If g is generaland g represents g, then the IOP isomorphism mentioned aboÕe is uniquelydetermined.

4. Square general game forms

Our next task is to exhibit faithful representations. This will be done in thecontext of game forms that, in principle, allow for symmetries, i.e., game formswith an equal number of strategies for each player. Since for two persons thestrategic versions of such game forms resemble square matrices, we call suchversions square as well. The following definition provides a formal approach.

Ž . Ž .Definition 4.1. A strategic preform e, game form e; A,h , or game e;u ,Ž . < < Ž .respectively is called square, if we have r e srs S igN .i

Ž .Lemma 4.2. Let gs e ; A,h be a nonmixing extensiÕe game form and letN Ž .U: A™R be a utility profile. Also, let p ,f be an automorphism of e . Then

Ž . Ž . Ž . Ž Ž ..p ,f gMM U)g if and only if J p ,f gMM U) RRRRR g .

The proof is easy and shall be omitted.

Ž .Remark 4.3. The situation is essentially the same if the vNM game form RRRRR g isreplaced by a strategic game form g of which g is a representation; however, we

Ž . Ž .have to observe the IOP isomorphism id,c ,id cf. Corollary 3.9 . Indeed, for


fixed g and g, J induces a mapping QsQ e ,e which carries automorphisms ofe into automorphisms of e via

Q p ,f s p ,cy1 mwmc , where J p ,f s p ,w . 4.11Ž . Ž . Ž . Ž .Ž .

Theorem 4.4. Let g be a faithful representation of the square general strategicgame form g and let U:A™R N be a utility profile. Then

Q : MM U)g ™MM U) g 4.12Ž . Ž . Ž .is surjectiÕe.

Ž .Proof: For rsr e s1 the assertion is obvious, thus we assume rG2.Ž . Ž .1st Step: Let gs e ; A,h and gs N,S; A,h be game forms with the desired

� 4properties. We can assume without loss of generality that S sS s 1, . . . ,ri jŽ � 4.i, jgNs 1, . . . ,n holds true. Indeed, we are going to show that

Q :Aut e ™Aut e 4.13Ž . Ž . Ž .Ž .here Aut denotes the group of automorphisms is surjective. On first sight thismight seem to be a more comprehensive statement, however, in view of oursubsequent proof it will become clear that every automorphism can occur as amotion of a suitable game; hence both claims are in fact equivalent.

2nd Step: First of all consider a utility profile U: A™R N specified as follows.We take

U s s irqs iG3,sgSŽ . Ž .i i

in order to avoid any symmetries between players i, jG3. Furthermore put

U s sU s ,s and U s sU s ,sŽ . Ž . Ž . Ž .1 1 1 2 2 2 1 2

Ž .meaning that U , U depend on the first two coordinates only . In addition U , U1 2 1 2

is specified by

� 40, if s g 1,2 and s G31 2U s ,s sU s ,s sŽ . Ž .1 1 2 2 2 1 ½ s , if s G31 1

and

1 2 1 24.14Ž .1 0 y1 1 y1 0U P,P s U P,P sŽ . Ž .1 2ž / ž /2 y1 0 2 0 y1

Ž . � 4 � 4for P,P g 1,2 = 1,2 .Now we are going to discuss the group of motions corresponding to U) g. To

Ž .this end, let ps 1,2 be the transposition of the first two players. Also let


t i j:S™S be defined by t i j:1™2™1 and let id i j:S™S be the identityi j i j

mapping for i, jgN. Then we have

Ž . Ž .11 22 ny2 12 21 ny2id , id ,id ,id , p , t ,id ,id ,Ž . Ž .Ž . Ž .MM U) g sŽ . Ž . Ž .12 21 ny2 11 22 ny2½ 5p , id ,t ,id , id , t ,t ,idŽ . Ž .Ž . Ž .

c0 , c,s 4.15Ž .½ 53 2c , c

where idny2 is self-explaining.Ž . Ž .As g is faithful, there exists f such that p ,f g MM U)g and J throws

Ž . 3 Ž .p ,f on either c or c . As the group is cyclic, the powers of p ,f are thrownŽ .onto all of MM U) g .

3rd Step: The next utility profile we have to consider is indicated by

1 2 1 24.16Ž .1 0 0 1 0 y1U P,P s U P,P sŽ . Ž .1 2ž / ž /2 y1 y1 2 0 y1

Ž .using the convention established in the 2nd Step . Here the group of motions caneasily be computed as

0 12 21 Žny2. � 0 4MM U) g s c , p , id ,id ,id s d ,d . 4.17Ž . Ž . Ž .� 4Ž .Again using faithfulness, it is at once established that d necessarily has to be

Ž . Ž .the image of some p ,f g MM U)g under Q .4th Step: Now c0, c, c2, c3, and d occurred as motions in a suitable context

but, of course, they are automorphisms of e as well. We may generate similarautomorphisms as images under Q by exchanging any two strategies of players 1and 2 or, for that matter, of any two players. The reader has now to convincehimself that the family of automorphisms created this way generates the full groupof automorphisms of e. q.e.d.

The following example shows that Theorem 4.4 is false if the square gameform is not general. Moreover, it turns out in the next section that ‘square’ cannotbe dropped as a condition.

Ž . Ž .Example 4.5. 1 Let gs e; A,h be the square nongeneral strategic 3-person� 4 � 4 � 4 � 4game form given by S s u,d , S s t,b , S s l,r , and As a . Then g can1 2 3

Ž .be represented by the extensive game form gs e ; A,h as indicated in Fig. 5.Ž . Ž . Ž .Note that for every pgS N there is a unique f such that p ,f gAut e .

Ž . Ž . < Ž . <Indeed, p uniquely determines f j igN and thus f. Hence Aut e s6. Leti

U: A™R be a utility profile. In order to show that g is faithful it is sufficient toŽ . Ž . Ž .distinguish three cases. a If U a igN are pairwise distinct, then thei


Fig. 5. A faithful representation of a nongeneral form.

symmetry groups of U)g and U) g consist of the identity permutation only.However, U) g possesses 8 motions, whereas the identity mapping is the unique

Ž . Ž . Ž . Ž . � 4motion of U)g . b If U a sU a /U a , where i, j,k sN, let us say is1,i j k

js2, and ks3, then

� 4SS U)g (SS U)g ( p ,id ,Ž . Ž .Ž . < Ž . <Here p denotes the transposition of players 1 and 2. whereas MM U)g s16

< Ž . < Ž . Ž . Ž . Ž .and MM Usg s2. c If U a sU a sU a , then1 2 3

SS U) g (S N (SS U)g ,Ž . Ž . Ž .< Ž . < < Ž . <whereas MM U) g s48 and MM U)g s6.

Ž . Ž .This example shows that the mapping Q : MM U)g ™MM U)g is not neces-sarily surjective, if g is not general. In the present case Q is injective.

Ž .2 The situation described in the preceding example cannot occur, if thestrategic game is general. Indeed, if the strategic game form g is ‘the’ correspond-ing general game form represented by the extensive game form g sketched in Fig.6, then a utility profile used in the 3rd step of the preceding proof shows that g is

Žnot faithful. Note that the utility profiles defined in the 2nd step do not yield a.contradiction.

Ž . Ž .For a strategic general game form gs c; A,h every automorphism p ,w of eŽ p y1.induces an automorphism p ,w,h(w (h of g. Analogously, for any nonmix-


Fig. 6. A nonfaithful representation of a general form.

Ž . Ž .ing extensive game form gs e ; A,h every automorphism p ,f of e induces anŽ . y1automorphism p ,f,r of g , where r is essentially given by h(f(h ;Ž . Ž Ž .. y1Ž .meaning that r a sh f j for all jgh a is well defined independently of

Ž .j agA . This fact and the last proof enables us to reformulate Theorem 4.4.

Ž .Corollary 4.6. Let gs e ; A,h be a representation of the general squareŽ .strategic game form gs e ; A,h and let

Q :Aut g )Aut gŽ . Ž .

be defined by

˜ e ,eQ p ,f ,r s Q p ,f ,r p ,f ,r gAut g .Ž . Ž . Ž . Ž .Ž . Ž .˜Then g is a faithful representation of g, if and only if Q is surjectiÕe.

˜ p y1Ž .Indeed, note that Q p ,f,r is an automorphism, because rsh(w (h ,e ,eŽ . Ž .where Q p ,f s p ,w , is satisfied.

The above development suggests to briefly consider automorphisms of exten-sive preforms that leave the corresponding vNM preforms untouched. This kind ofautomorphisms is described by the following definition.


Ž .Definition 4.7. An automorphism p ,f of an extensive preform e is said to be0

chance related if the following holds true:1. ps id

QŽ . Ž . Ž .2. f Q sQ QgQ cf. Definition 2.70 y0CŽQ.Ž . Ž Ž . .3. f C sC CgC Q , QgQ0 y0

Ž . Ž . Ž . Ž .A motion p ,f of a game G is chance related if conditions 1 , 2 , and 3 are0Ž . Ž .satisfied. CC G denotes the subgroup of chance related motions of MM G . Note

Ž .that formula 3.8 of Remark 3.2 implies that J p ,f is the identity, i.e., the0

strategies of the corresponding vNM preform are not disturbed.

Theorem 4.8. The chance related automorphisms of an extensiÕe preform consti-Ž .tute a normal subgroup and, for eÕery extensiÕe game G , the subgroup CC G :

Ž .II G is normal.

Ž .Proof: We have to show that for any automorphism p ,f and any chance relatedŽ . Ž X .automorphism id,f we can find a chance related automorphism id,f such0 0

that

p ,f id ,f s id ,fXp ,fŽ . Ž . Ž . Ž .0 0

holds true. To this end it suffices to show that

fX sf f fy10 0

is chance related. Indeed, we have for igN and Q:Pp Ž i.

fy1 Q :PŽ . i

that is

f fy1 Q sfy1 QŽ . Ž .Ž .0

and hence

f f fy1 Q sQ,Ž Ž .Ž .0

Ž .and analogously for 3 of Definition 4.7 q.e.d.

Theorem 4.9. Let g be a faithful representation of the square general strategicgame form g. Then, for any utility profile U:A™R N it follows that

MM U)g rCC U)g (MM U) g 4.18Ž . Ž . Ž . Ž .holds true.

Proof: Theorem 4.4 implies that Q is a surjective mapping which respectsŽ . Ž .composition Remark 3.2 . It suffices to show that CC U)g is the kernel of this

mapping.Ž . Ž . Ž . Ž .Clearly, if p ,f gCC, then Q p ,f s id,id . On the other hand if Q p ,f

Ž . Ž . Ž .s id,id , then f has to satisfy conditions 2 and 3 of Definition 4.7, for


˜otherwise we can construct a strategy s gS that suffers under the influence of fi iŽ .as defined in 5.5 . q.e.d.

5. Atoms

Our previous results provide us with a possibility to discuss ‘square games’ as afirst attempt to introduce the symmetric canonical extensive version. In order to

Žapproach this program we shall first of all discuss the simplest and ‘nonsymmet-.ric’ version of a representation: an atom.

Definition 5.1.Ž .1 A game tree is atomic, if there are no chance moves and the player partition

Ž .coincides with the information partition. In this case we write E,$ ,P,C . In< Ž . < < Ž X. <particular an atomic game tree is said to be square if C j s C j sr for

X Ž . Ž . Žj ,j gEyEE holds true. 2 A preform es N, E,$ ,P,C;ı is atomic, if E,. Ž . Ž .$ ,P,C is an atomic game tree. 3 An extensive game form as e ; A,h is an

Ž .atom, if e is an atomic preform and h:EE™A is bijective. 4 An atom a and itsŽ .preform and game tree is time structured if every nonvoid level LL E,$ ,t

coincides with one player set PgP. In this case a is called T-atom.

Ž . Ž .Remark 5.2. Let es N, E,$ ,P,C;ı be a preform of an atom and let es N,S˜< < < < Ž .be a strategic preform of a square game form such that S s S igN isi i

˜Ž . Ž . Ž .satisfied, where N e s N,S . Moreover, let id,c be an isomorphism betweenŽ . Že and N e which exists because the corresponding strategy sets have coinciding

.sizes .Ž . Ž .1 If h:S™A is a bijection, then there exists a unique bijective mapping

Ž . Ž . Žh:EE™A such that id,c ,id is an IOP isomorphism between gs e; A,h a. Ž . Ž Ž ..general strategic game form and RRRRR a where as e ; A,h , i.e., a is a

representation of g. An atom which represents g is said to be an atom of g.Ž . Ž .2 If h:EE™A is a bijection, then there exists a unique bijective mapping

Ž . Ž . Ž .h:S™A such that id,c ,id is an isomorphism between gs e; A,h and RRRRR a

Ž Ž ..where as e ; A,h , i.e., the atom a is a representation of the general gameform g.

Example 5.3.Ž .1 For two persons, consider g as indicated by

l r

5.19Ž .t a bž /b c d

There are two atoms as indicated by Fig. 7.


Ž .Fig. 7. Atoms for a 2; 2=2 game.

Ž . < <2 For three persons and rs S s2 for all i, consider the preforms in Fig. 8.i

which may be augmented to game forms representing appropriate strategic forms.The T-atom is ‘time structured’. Assuming that the game is ‘Common Knowl-

edge’, player i is aware that he moves ‘at instant i’. The S-atom seems to exhibitsome symmetry between players 2 and 3.

Ž .3 For four persons and rs2, examples of atoms can be seen in Fig. 9.

Remark 5.4.Ž . Ž .1 Fig. 9 suggests that the underlying tree E,$ of an atom a of a square

Ž . Žgeneral game form gs N,S; A,h is essentially i.e., up to order respecting. Ž .bijective mappings, cf. Section 2 uniquely determined. Clearly, the pair E,$ is

Ž . Ž . Ža tree of a square atom as E,$ ,P,C, ı, A,h of RRRRR a which is general in the. < <sense of Definition 2.3 iff the maximal rank coincides with the number N of

< <players and at every node jgEyEE there are exactly rs S alternatives.iŽ .2 Also, a is an atom of g if and only if it represents g and its total rank is

minimal.Ž .3 In addition note that to any square atom a we can at once construct

isomorphic ones by permuting the players and renaming the outcomes accordingly.For example, there are at once 4! different but isomorphic atoms corresponding toeach one suggested by Fig. 9; all of them being obtained by permuting the playersarbitrarily and renaming the outcomes accordingly.

Ž .Fig. 8. Atomic preforms for a 3; 2=2=2 game.


Ž .Fig. 9. Atomic preforms for a 4; 2=2=2=2 game.

Ž .4 An atom of a general square game form g cannot be a faithful representa-tion of g, because there is no automorphism of the preform which replaces the‘owner of the root’ by any other player.

6. Symmetrizations

In order to obtain faithful representations of square general game forms, wewill now construct ‘symmetrizations’ of square atoms. To this end we shall shortlydescribe a further operation acting on game trees called restriction.

Ž U U U U U U . ŽA game tree E ,$ ,P ,Q ,C , p is a restriction of the game tree E,. Ž Ž U U ..$ ,P,Q,C, p the restriction to E ,$ , if

Ž U U . Ž . Ž U U < U Ž U U .1. E ,$ is a subtree of E,$ i.e., E :E, $ [$ [$l E =E ,EU .EE :EE ,


U � U < 4 U � U < 42. P [ PlE PgP , Q [ QlE QgQ ,U Ž U . � U < Ž .4 Ž .3. C QlE [ SlE SgC Q QgQ ,Ž . U U j j Ž U . Ž4. C j :E and p sp jgP i.e., a chance move together with its0

choices and probability distribution is either completely preserved or disap-. Ž Upears . Note that a possible generalization of this notion the probabilities p

U .should be the conditional probabilities given E is not needed in our presentapproach.Restrictions of extensive game forms and games are defined in an obvious way.

Also, for the strategic versions the definition of a restriction is the straightforwardŽ U . Ž . Uone, e.g., a strategic preform N,S is a restriction of N,S , if S :S holds true.

Ž .Definition 6.1. Let a be a square atom. A game form gs N, E,$ ,P,C, p;ı; A,hŽ .i,i.e., PsQ is a symmetrization of a , if the following conditions are satisfied.1. g is nonmixing.2. The root x of g is the only chance move and p x 0 is a uniform distribution,0

i.e., every edge at x has the same probability.0Ž . j Ž j j j j j3. For every jgC x the restricted game form g [ N, E ,$ ,P ,C ,ı ;0

j .A,h of g obtained by restricting g to the subtree with root j generated byŽ .the edge x ,j is isomorphic to a .0

Ž .4. For every atom b which represents RRRRR a and is isomorphic to a , there existsŽ . ja unique jgC x such that b is IOP isomorphic to g .0

Example 6.2. Consider the case of two persons each of them having twostrategies. Two atoms of the general game form, i.e., of g represented by

l r

6.20Ž .t a bž /b c d

have been indicated in Fig. 7. Clearly, they are isomorphic. A symmetrization isindicated by Fig. 10.

Fig. 10. The symmetrization.


Ž .Thus for the simple 2; 2=2 -case, the symmetrization described in Fig. 10suggests the structure of the ‘canonical’ representation we have in mind.

Already for three persons, this is not so obvious. As Fig. 8 suggests, there areessentially two nonisomorphic atoms: the ‘time structured’ or ‘T-atom’ and the‘S-atom’ which seems to exhibit more symmetry with respect to the players notcalled upon in the first move, i.e., players 2 and 3 in Fig. 8.

Both allow for symmetrizations and at this stage it is not clear which of themwill be a candidate for the canonical version.

Example 6.3. Fig. 11 shows a symmetrization of the T-atom in Fig. 8, that couldbe called TSYM3 . Fig. 12 is the analogous version with respect to the S-atom2=2=2

of Fig. 8. At this state of affairs it may become conceivable that there is a problemarising from the question as to which version of an extensive game represents theŽ .3; 2=2=2 case ‘appropriately’ in view of symmetry considerations.

The next result shows that symmetrizations exist and are faithful.

Ž 3 .Fig. 11. The symmetrization of a T-atom TSYM .2= 2=2


Fig. 12. The symmetrization of an ‘S-atom’.

Theorem 6.4. Let a be a square atom. Then1. a possesses a symmetrization,2. eÕery two symmetrizations of a are IOP isomorphic,3. a symmetrization of a is a faithful representation of eÕery strategic game form

g represented by a .

The proof is decomposed according to the three items claimed.

( ) Ž . Ž .Proof of item 1 : Let as N, E,$ ,P,C;ı; A,h and RRRRR a sg. Furthermore,define

<BBs b b is an atom of g isomorphic to a with tree E,$ .� 4Ž .

Every atom of g which isomorphic to a is isomorphic to some atom of thefinite set BB. Choose a maximal subset AA:BB of atoms which are not IOPisomorphic. Indeed, BB can be partitioned into the equivalence classes of IOPisomorphic atoms. The set AA contains precisely one representative of each

Ž b b b b .equivalence class. Moreover, for every bs N, E,$ ,P ,C ;ı ; A,h gAA take


Ž . Ž .an IOP isomorphism id,c ,id between g and RRRRR b . In view of Corollary 3.9,b

c exists. The extensive game formb

˘ ˘ ˘ ˘gs N , E,$ ,P,C, p ;ı ; A ,h˘ ˘ž /is defined as follows.

˘ � 4 Ž .1. Es 0 j E=AA ,˘ 0 y1Ž . Ž . < < Ž . Ž .2. 0$ x ,b , p x ,b s AA bgAA , where x is the root of E,$ ,0 0 0

˘ X X X X X XŽ . Ž . Ž .3. j ,b $ j ,b , iff bsb and j$j b ,b gAA,j ,j gE .˘ b ˘ � 4 Ž .4. P sD P =b , P s 0 igN ,i b g AA i 0˘ ˘Ž . � Ž Ž . . < 4 Ž .5. C P s D c S ,b S gS igN ,i b g AA b , i i i iŽ . bŽ . Ž .6. h j ,b sh j jgEE,bgAA .˘

By construction g is a symmetrization of a .

( )Proof of item 2 : Let g and d be two symmetrizations of a . By Definition 6.1Ž .3 there is a bijection between the atoms in g and d which maps every atom in g

to an IOP isomorphic atom in d . These IOP isomorphisms together induce an IOPisomorphism between g and d in a straightforward manner.

( )Proof of item 3 : Let g be a symmetrization of a which represents g. Theextensive game form g represents g, because g is nonmixing and every atom in g

Ž . Ž .represents g. Let bs e ; A,h , where es N, E,$ ,P,C, ı , be an atom of gsŽ . Ž .N,S; A,h s e; A,h such that b is isomorphic to a . Assume without loss of

� 4 Ž . Ž .generality that S s 1, . . . ,r igN holds true. Moreover, let p ,w be aniŽ . Ž .automorphism of e and id,c ,id the IOP isomorphism between g and RRRRR b .

y1 Ž .Then r, defined by rsh(w(h , generates an automorphism p ,w,r of g.X Ž X X X X.Define b s N, E,$ ,P ,C ,ı ; A,h and f:E™E as follows.

Ž . Ž . Ž . Ž . Ž Ž ..1 Define f x sx and assume that f j gLL E,$ ,t jgLL E,$ ,t is0 0< < Ž .already defined for some 0-TFns N and 0F t-T. If jgLL E,$ ,T , let us

Ž X. Xsay jgC j and j gP for some igN, then take the unique strategy s gSi i iX Ž . Ž Ž X .. Ž Ž ..such that j gc s and determine zgC f j which satisfies zgc w s ,i i k i

Ž X. Ž .where f j gP . Define f j sz and observe that f is bijective and respectskŽ .$ ,$ .

Ž . X Ž . Ž .2 Put P sf P igN .p Ž i. iŽ . XŽ X . Ž Ž .. Ž .3 Put C P sf C P igN .p Ž i. iŽ . XŽ . Ž y1 .Ž . Ž . Ž .4 Put h j s r(h(f j jgEE , observe that p ,f,r is an isomor-

X X XŽ .phism between b and b , and that b represents g. Indeed, with c s si i� Ž . < X 4 Ž X. Ž . Ž X .c s P lP /Ø lC P igN, s gS the triple id,c ,id is an IOPj i j i i i i

Ž X.isomorphism between g and RRRRR b .j ŽThis procedure applied to every restricted game g where j is a successor of

˜ X. Ž . Žthe root of g yields an automorphism p ,f,r of g note that b is, up to an IOP˜. Ž . Ž . Žisomorphism, a restricted game of g . Clearly, Q p ,f s p ,w cf. Remark 4.3

.for the definition of Q , thus the proof is finished. q.e.d.


Fig. 13. Non-IOP isomorphic atoms.

Remark 6.5. It should be noted that the cardinality of the set AA as defined in theŽ .proof of item 1 varies according to the shape of the atom involved. Most

conspicuously the T-atom requires fewer isomorphic copies than a more complexŽ .version as explained by the following sketches and proved later on .

Fig. 13 shows three copies of non-IOP isomorphic atoms, each of them from aŽ .different equivalence class of BB as mentioned in item 1 . Each of these atoms

admits of 3!s6 further isomorphic atoms which are obtained by permuting theplayers. Of course all of these again stem from different equivalence classes,because permuting players forbids IOP isomorphism. Hence, there are alltogether18 non-IOP isomorphic atoms which, similarly as indicated in Fig. 11, are thenglued together in order to generate the symmetrization.

By contrast, the T-atom for the same general 3=3=3 strategic game form, asrepresented by Fig. 14, generates a symmetrization which is obtained by glueingtogether only the 6 non-IOP isomorphic copies obtained by permuting the players.

Fig. 14. The 3=3=3 T-atom.


Corollary 6.6. Let g be a symmetrization of some square atom which representsg. Then eÕery restriction of g which faithfully represents g coincides with g .

Ž .Proof: To verify this assertion a part of the proof of item 3 has to be repeated.Clearly at least one of the atoms in g , and b , has to occur in the faithful

Ž .restriction otherwise the restriction is not a representation of g . Moreover, forX Ž .another atom b which occurs in g there is an isomorphism p ,w between both

atoms. Applying J and the IOP isomorphisms between the normalizations of theŽ .atoms and g yields an automorphism p ,w,r of g. In view of the proof of

Theorem 4.4 there must be an automorphism of g which is mapped to thisŽ .automorphism of g. Definition 6.1 3 completes the proof. q.e.d.

ŽDifferent atoms of a square general game form possess isomorphic trees e.g.,.the number of nodes coincides . This is no longer true for ‘the’ symmetrizations as

shown in Remark 6.5, i.e., symmetrizations of different atoms of a square generalgame form may have different numbers of plays and, thus, endpoints. Hence, thetotal ranks may differ. Nevertheless, the following result holds true.

Ž .Theorem 6.7. Let gs e; A,h be a general square strategic game form and let g

be a faithful representation of g with minimal total rank. Then g is the sym-metrization of an atom of g. MoreoÕer, the symmetrization of a T-atom of gpossesses minimal total rank.

Proof:< < Ž .1st Step: Let r[ S igN . First of all consider the case that g is thei

Ž < <.symmetrization of a T-atom of g. Clearly g possesses exactly n! where ns NŽ .atoms. Now all plays have the same length i.e., rank of the endpoint which is

nq1. In each of the n! atoms there are r n such paths. Hence the total rank of anyof these atoms is r nn. With respect to g , the corresponding rank originating from

n Ž .each atom is r nq1 , because there is an additional edge joining the atom toŽ .the root of g . There are n! atoms, hence the total rank of the graph E,$ of g is

nŽ . nŽ .r nq1 n!sr nq1 !.2nd Step: Next, we are going to show that a representation which is faithful

nŽ .has total rank which is at least r nq1 !. To this end let g be a faithfulrepresentation of g which has minimal total rank. Without loss of generality it can

Ž . Ž .be assumed that RRRRR g sg holds true. For every sgS and every pgS NŽ . Ž . Ž .choose w p ,s sw and r p ,s sr such that p ,w,r is an automorphism of g

p ˜Ž . Ž .and w s ss is satisfied. Let Q be defined as in Corollary 4.6. Let fsf p ,sŽ .be an automorphism of the game tree of g such that p ,w,r is in the inverse

˜Ž . Ž . Ž .image of p ,w,r , i.e., Q p ,f,r s p ,w,r . The existence of f is guaranteeds0 , s Ž s s .by faithfulness. Fix a pure strategy s of chance and let X s x , x , . . . , x0 0 1 T Ž s.

Ž .be the play generated by s ,s . For every permutation p the outcome0Ž Ž .Ž s .. Ž . Ž .ph f p ,s x coincides with h s , because w p ,s keeps the strategy profileT Ž s.


s. Different strategy profiles lead to different outcomes, because g is assumed tobe general. Counting the number of strategy profiles and the number of permuta-

n Ž .Ž s .tions yields r n! different plays with endpoinds f p ,s x .T Ž s.Ž .The length T s of every play is at least nq1, because every play intersects an

information set of every player and of chance. Indeed, if a player is not involved,Žthen a ‘row’ of g does not depend on the player’s strategy which is impossible,

.because g is general . Moreover, the root g cannot belong to the information setof some player.

Ž . nŽ . Ž3rd Step: The total rank of E,$ is therefore at least n!r nq1 recall that.the length of each play is at least nq1 due to the 2nd step . By minimality of the

total rank and the 1st step it follows that the total rank is exactly equal to thisnumber and the root is the only chance move. Hence the restriction of g to everysubtree generated by a successor of the root is an atom and Theorem 4.4 completesthe proof. q.e.d.

The ‘minimal total rank property’ will be crucial in Section 7. Of course thereare many faithful representations of a general square game form; one is presentedin the following example.

Fig. 15. A faithful representation with multiple chance moves.


Example 6.8. A faithful representation g g of a general square game formŽ .gs N,S; A,h can be constructed recursively on the number n of players. For

Ž .ns2 ‘the’ symmetrization of its atom see Fig. 10 is taken. For nG3 the faithfulrepresentation is constructed as follows. The root x is constructed to be a chance0

move. It possesses n successors j belonging to the different player sets and p x0i

is uniform distribution. There is a bijection from the set of successors k of j toi iŽ .player i’s strategy set S let us say k ™s . The restriction to the subtreei i i

generated by k is the faithful representation g g si of the ny1 person reducedisi Ž � 4 si.strategic game form g s N_ i ,P S ; A,h , defined byj/ i j

hsi s sh s , s .Ž . Ž .Ž . Ž .j i jj/i j/i

Of course every player has only one information set. The straightforward proof offaithfulness is left to the reader. Fig. 15, which sketches the 2=2=2 case, shouldbe compared with Figs. 11 and 12.

7. The canonical representation

Within Section 6, we have characterized the symmetrizations of square atomsas the only representations of square strategic games that respect the symmetriesand satisfy a minimality condition. Apart from the fact that the result holds trueonly in the case that all players have the same number of strategies, the assignmentof an extensive game form to a given strategic game form is not unique. For the

Ž .class of square atoms and their symmetrizations is still remarkably large:compare, e.g., Fig. 9; here we see various nonisomorphic atoms that are capable ofrepresenting a 2=2=2=2-game.

The final task is, therefore, to introduce symmetrizations of time structuredatoms that yield representations in the case of game forms that are not necessarilysquare. In addition we show that this construction admits of an axiomaticallydefined unique mapping, the ‘canonical’ representation.

ŽDefinition 7.1. Let a be a T-atom. An extensive game form gs N, E,$ ,P,.C; p,ı, A,h is a symmetrization of a , if the following conditions are satisfied.

1. g is nonmixing.2. The root x of g is the only chance move and p x 0 is uniform distribution, i.e.,0

every edge at x has the same probability.0Ž . j3. For every jgC x the restricted game form g is a T-atom.0

Ž . Ž .4. For every T-atom b which represents RRRRR a there is a unique jgC x such0

that b is IOP isomorphic to g j.

Ž .Note that the unique up to IOP isomorphisms symmetrization of a T-atom b

has n! branches at the root corresponding to as many T-atoms, which all belong toŽ .the vNM strategic game form RRRRR a .


Generally we will have to accept that a representation can only be defined up tooutcome preserving impersonal isomorphisms. On the other hand the varietyoffered by all atoms is too large. Moreover, we should additionally have faithfulrepresentations of nonsquare game forms.

Clearly the preservation of symmetries as formulated so far cannot help in ageneral nonsquare game, for even in the case of two players there are nosymmetries of a general game at all since there are no bijective mappings of thestrategy sets. However, as our discussion in Section 1 shows, there are symmetriesof restricted versions which should be preserved. Verbally, if two strategiesrac-tions of a player result in the same payoff no matter what his opponents choose todo, then this game is in a well defined sense reducible and the restricted versionmay well have symmetries the preservation of which should be satisfied by a‘canonical’ representation. And if we construct nongeneral game forms with theabove property, then the symmetries obtained this way may indeed be used tofurther reduce the family of representations and hence result in a canonicalrepresentation.

Thus, it will be the interplay of restriction and symmetries that characterizes theŽ .canonical representation of a strategic game form minimality assumed . There-

Ž .fore, we shall add the notion of ‘consistency’ with respect to restriction to ourrequirements concerning representation.

Arbitrary restrictions, however, as defined at the beginning of Section 6 cannotŽ U U U U U U .be admitted. We shall call a restriction E ,$ ,P ,Q ,C , p of a game tree

Ž .E,$ ,P,Q,C, p proper if the root is preserved. Recall that all chance movesŽ .together with their choices and the probabilities are either fully preserved or

completely disappear. The notion is at once extended to game forms and games.There is a further, more formal obstacle to be tackled before we can reach a

rigorous formulation of the ‘canonical’ representation. This is presented by theaim to precisely define a mapping which represents the choice of a canonicalrepresentation. Mappings should be defined on a nice domain — of game forms inour present context. However, if we speak about the ‘set of all game forms’ wemight encounter unpleasant surprises common in elementary set theory, for game

Ž .forms so far are defined with arbitrary finite outcome sets.More than that, if we look closer, we made no restrictions on the underlying

Ž .sets of strategies in a strategic form neither concerning the elements of theŽ .underlying graph in an extensive form . Thus, when speaking about the set of,

e.g., strategic games, at the present state of affairs, we will be forced to speakabout the set of all finite sets several times.

In order to avoid such footangels we should restrict ourselves to a fixed infiniteŽ .set U the alphabet or uniÕerse of letters or outcomes which intuitively first of

Žall is a list of all possible outcomes admitted for game forms strategic and.extensive . I.e., we shall always tacitly assume that for any game form mentioned,

the outcome set satisfies A:U; thus the admissible outcome sets are subsetsof U.


It is no loss of generality to assume in addition that any strategy set Si

mentioned as well as the set of nodes E of a graph involved in our consideration isalso a subset of U. For the present section we set out under this additionalhypothesis.

We feel that this kind of intricacies should be mentioned but not overstressed.Thus, we fix the set of general strategic game forms G and the set of extensiÕegame forms G and define a mapping FF:G™G always assuming that theoutcomes, nodes, strategies . . . involved are given by subsets of U.

Definition 7.2. Let FF:G™G be a mapping.Ž .1. FF is called a representation of strategic game forms if, for any ggG it

Ž . Ž Ž ..follows that FF g is a representation of g cf. Definition 3.4 1 .2. A representation FF is said to be faithful if it preserves symmetries and

respects isomorphisms. More precisely, for any ggG, it should follow thatŽ . Ž Ž ..FF g is a faithful representation cf. Definition 3.4 2 and whenever g and

X Ž . Ž X.g are isomorphic, then so are FF g and FF g .3. A faithful representation FF is said to be consistent if it respects proper

restriction up to impersonal isomorphisms. More precisely, for any ggG andŽ .any restriction g of g it is true that FF g is IOP isomorphic to a proper˜ ˜

Ž .restriction of FF g . That is, ‘FF and the restriction operation commute’.4. A faithful representation FF is said to be minimal, if for every square ggG,

Ž .the total rank of FF g is minimal.5. A faithful, consistent, and minimal representation is said to be canonical.

Remark 7.3. Given our present state of development, we are in the position toconstruct a canonical representation. To this end, assign to every ggG the

Ž .symmetrization of a time structured atom cf. Definition 7.1 . This mapping is notuniquely defined; given ggG, we may apply an impersonal and outcome

Ž .preserving isomorphism to FF g without ‘essentially’ changing the nature of themapping thus defined. In this sense a ‘time structured’ representation is defineduniquely ‘up to impersonal outcome preserving isomorphisms’. It is canonical,because a symmetrization of a T-atom of a restricted game form of a strategicalgame form g is a proper restriction of a symmetrization of a T-atom of g.

Definition 7.4. The time structured canonical representation as described byRemark 7.3 is denoted by TT.

Clearly our next aim is to show that TT is ‘the’ only canonical representation.As it stands now the development in Section 6 and in particular Theorem 6.7 pointto symmetrizations of atoms but not necessarily to the time structured version. Asa first result we shall now prove that the time structure appears necessarily forgeneral square game forms with at least three strategies for each player.


Ž .Theorem 7.5. Let gs N,S; A,h be a square general game form such that< < Ž .S srG3 igN . Let a be an atom of g. Then the symmetrization of a is ai

totally rank minimal faithful representation of g, if and only if a is a T-atom.

� 4 Ž .Proof: Without loss of generality we assume that S s 1, . . . ,r igN holdsiŽ .true. The atom is denoted by as N, E,$ ,P,C;ı; A,h , the symmetrization is g ;

we may assume without loss of generality that a occurs as some g j in the senseŽ .of Definition 6.1 3 .

1st Step: Now we attach labels according to strategies at all nodes of g exceptthe root and its successors. To this end observe first that c identifies elements ofi

˜S and of S as explained in Definition 3.4. Therefore, if player i is in command ati i

node j and chooses s gS , this leads to a well specified successor z of j whichi i

now carries the label s . The process of labeling constitutes a mapping from Si i

into the successors of nodes at which player i is in command. This reflection of astrategy by labels is common in Game Theory and can be viewed in Fig. 5, wherethe labels u, d, t, b, l, r are drawn at the corresponding edges for more clarity.We will not formally define the mapping but frequently refer to it.

2nd Step: The labeling induces an identification of plays in g as well as of allthe atoms in g as follows. First of all any sgS corresponds to a unique play in a

Ž . Ž U .just follow the labels . Next consider the automorphism p ,id of the preform ofg. Here idU is the natural family of ‘identifies’ idU :S™S . To this automor-i i p Ž i.

Ž . Žphism there corresponds a unique automorphism p ,f of the preform of g cf..Corollary 4.6 . f transforms the play in a labeled by s into some other play

Ž .carrying the same label. In particular consider p/ id and ss 1, . . . ,1 orŽ .ss 2, . . . ,2 or etc. Then the second play cannot run through a , because it leads

to the same outcome as the first one — but there is exactly one play carrying anoutcome in each atom. From this we see immediately that f carries a bijectivelyto some other atom in g , say ap, and that, indeed, n! atoms can be identified by

Ž .the permutations id corresponding to a . As the faithful representations of g byg is totally rank minimal, we conclude in view of Theorem 6.7 that g has exactly

p Ž Ž ..the n! atoms a pgS N .Ž3rd Step: We focus the attention on a and recall the definition of r cf.

. Ž . z Ž .Section 2 . For zgE and r z - t let LL t denote the set of nodes on level tthat have z as a common ancestor with respect to the completion of $ . Call a

Ž . Ž .level LL E,$ ,t intact, if it is contained in some P hence equals P andi iŽ .broken otherwise. The nth level is broken! We introduce

<tsmin t LL E,$ ,t is broken .� 4Ž .

It is our final aim to show that t equals n. Assume, on the contrary, that t-nholds true. Let z be such that

z Ž . Ž Ž . Ž .1. LL t ≠ P igN and r z - tiŽ . Ž .2. the rank r z is maximal with respect to 1 .


j Ž .Then every successor of z is the common ancestor of all nodes of LL t . ByjŽ . Ž .maximality of r z the set LL t belongs to one player set. We now claim that

XX j jŽ . Ž . Ž .for different j ,j gC z the corresponding LL t and LL t belong to differ-ent player sets, hence at least r players are in command on level t. This claim willbe confirmed in the 4th Step. Fig. 16 indicates the procedure to be followed duringthe remaining steps of the proof.

Ž . Ž4th Step: To this end let j ,j ,j gC z be different successors of z recall1 2 3j j1 2< < . Ž . Ž .that rs S G3 . Assume without loss of generality that LL t :P and LL ti 1

:P . It suffices to show that LL j 3 lparen t bar rparenis not contained in P . Let2 1j 3Ž .LL t be contained in P . The player who is in command at node z assigns twoi

labels to j and j as described in the first step; let these labels be s2 and s3,2 3

respectively. There is an impersonal automorphism of the preform of g which justtransposes s2 and s3. To this automorphism there exists the corresponding

Ž 23.automorphism of the preform of g given by Corollary 4.6; we call it id,f . Inview of the fact that automorphisms transfer atoms to atoms, we can single out anatom which is the image of a under f 23. By the 2nd Step this atom is of the form

p Ž . pa for some pgS N . We refer to the levels within a by subscript p .

Fig. 16. The preform of a .


Again consider the successor j of z . We have323f Ž j . 23 j 233 3LL t sf LL t :f P sP , 7.1Ž . Ž . Ž .p i i

23 Ž 23.because f respects the ancestor relation and id,f is impersonal. On the23Ž . 2 3 23other hand, f j carries the label s , because j carries the label s and f3 3

has been constructed according to the transposition of s2 and s3. Hence,23f Ž j .3 Ž . Ž .LL t has to be a subset of P . From this and from formula 7.1 wep p Ž2.

Ž .conclude that p 2 s i.Next, perform the same operation for the successor j of z in order to show2

Ž .that p i s2 holds true as well. And, if j is considered, it follows that1Ž .p 1 s1. Clearly this shows that i/1 is true. Hence we have proved the claim

raised at the end of the 3rd Step.5th Step: Without loss of generality we assume is3. Player 3 appears the first

Ž .time on level t in a , as all previous levels are intact. Therefore j is the112Ž .ancestor of some kgP with r k ) t. Let f be generated by the exchange of3

j and j analogously to the construction of f 23 by the exchange of j and j1 2 2 3p 12 Ž p p 23in the 4th Step. The corresponding atom is a as a sa was specified

. 12 Ž . 12 Ž .above . Analogously to the 4th Step we conclude that p 1 s2, p 2 s1, and12 Ž . sp 3 s3. Let sgS be a strategy profile which generates a play X in a

Ž .passing through j and k use the labeling of the 2nd Step . The automorphism1Ž 12 . Ž 12 . Xid,w that induces id,f via Q throws s into some s and, hence, specifies a

sX Ž . X sX

play X in a . On level r k we find exactly one node k on X . The labeling sŽ sX . p 12 12 Ž .corresponds to f X in a . As p 3 s3, the play corresponding to label s

p 12 Ž . Ž X. Xin a will pass through P on level r k , i.e., f k gP . Therefore, k gP3 3 3X j 2Ž Ž .. Xholds as well. Note that k gLL r k due to the construction of s .

13 Ž6th Step: The same procedure argued with the automorphism f constructed. s sanalogously again is now applied to the play X in a . The play X passes

through P first and reaches P at k , Because k is a member of X s and j is an1 3 1SY Ž 13.ancestor of kgP . We conclude that the play X obtained by using w passes3

Ž . Ythe level r k at some node k which is an element of P . By interchanging the1Ž 23 X sX .roles of player 1 and 2 i.e., application of f to k or X , respectively we have

to conclude that kY gP . This is a contradiction which shows that the assumption2

t-n raised in the 3rd Step cannot be true. Hence a is time structured. q.e.d.

The main theorem can now be stated as follows.

(Theorem 7.6. There is a unique i.e., up to impersonal outcome preserÕing) (isomorphisms canonical representation of strategic games oÕer a giÕen uniÕersal

)alphabet and this is the time structured mapping TT.

Proof:ŽAs we have seen in Remark 7.3 the mapping TT has the desired properties of

course TT again is only defined up to impersonal outcome preserving isomor-.phisms . Thus it remains to show uniqueness.


To this end, let FF be a representation enjoying the desired properties. Fix aŽ . U U Ustrategic game gs N,S; A,h gG. Let S sŁ S be such that S =Si g N i i i

< U < < < Ž . U Ž U U U .yields S srGmax S igN for some rG3 and let g s N,S ; A ,hi j g N j

gG be such that g is a restriction of gU. The existence of gU gG is ensured bythe choice of U which renders G to be sufficiently large. By Theorem 7.5 it

Ž U . U Ž U . Ž .follows that FF g \g sTT g holds true up to an IOP isomorphism . LetŽ U . U Ž Ž U ..id,c ,id be the IOP isomorphism between g and RRRRR TT g . This automor-phism in particular carries the subset S of strategies available in g into the

U UŽ Ž ..strategies available for RRRRR TT g , called S . We now define an extensive gameform g with the aid of g

U and cU : All we have to do is to take all plays of g

U

that are images of strategies sgS under cU , i.e., all plays generated by

i d U s p UŽ . Žc S :S . This amounts to taking all plays X in all atoms a of g as.discussed in the proof of Theorem 7.5.

The nodes of gU obtained by pursuing all these plays together with the obvious

binary relation constitute a tree to which all further data of gU may be restricted

in the obvious way. The time structured nature of a symmetrization which ischaracteristically for g

U allows for an easy verification of the fact that therestriction is, indeed, proper. Call the resulting extensive game form g . As TT

respects proper restriction it follows that g is a faithful representation of g.Ž .However, consistency applies as well for FF, hence FF g is IOP isomorphic to

Ž .FF g . q.e.d.

Remark 7.7. Definition 7.2 can be generalized to mappings FF:H™G for anysubset H:G of general strategic game forms without changes; however, H hasto comply with a few additional requirements. This is so because a set H which istoo small may not allow for sufficiently many games, thus the existence require-

Ž .ment of Definition 7.2 3 could be damaged. To avoid this possibility, call Hhereditary, if every restriction of a game form of H belongs to H. For ahereditary H the time structured representation TT restricted to H is clearly

Žcanonical. Uniqueness can be guaranteed repeat the proofs of Theorems 7.5 and.7.6 provided the following condition is satisfied.

For any gs N ,S ; A ,h gH there is a square strategic gameŽ .form gU s N ,SU ; AU ,hU gH such that g is a restriction of gU . 7.2Ž . Ž .

Ž .The remainder of this section is devoted to an example which shows that 7.2cannot be dropped as a prerequisite of uniqueness.

Example 7.8. Let nG3 and H be a hereditary subset of G consisting of game� 4 Ž .forms with strategy sets of cardinality 1 or 2. For Ns 1, . . . ,n and N,S; A,h gH

< < Ž .satisfying S s2, igN define the atom a as follows. The nodes of E arei�Ž . < t4 Ž .given by t,l 0F tFn,1F lF2 . The player sets are specified via t,l gPtq1


Fig. 17. The crossover example.

Ž . Ž . Ž . Ž . Ž . Ž .ts0 or 3F t-n ; 1,l gP ; 2,l gP lF2 and 2,l gP lF3 . Thelq1 3 2

choices are indicated in Fig. 17.Let FF:H™G be the mapping that assigns the symmetrization of a to g and

is arranged consistently otherwise. This mapping is canonical. The clue is foundby an inspection of Fig. 17 and of the proof of Theorem 7.5. As the 3rd strategy ismissing, the overcrossing of player sets P and P cannot be avoided by the2 3

construction supplied in the 5th Step.

Acknowledgements

The authors acknowledge the useful comments of an anonymous referee.

References

w x1 E. van Damme, Stability and Perfection of Nash Equilibria, Springer-Verlag, Berlin, 1987.w x2 J.C. Harsanyı, R. Selten, A General Theory of Equilibrium Selection in Games, MIT Press,

Cambridge, MA, 1988.w x Ž .3 E. Kohlberg, J.-F. Mertens, On the strategic stability of equilibria, Econometrica 54 1986

1003–1037.w x4 R.B. Myerson, Game Theory, Harvard Univ. Press, Cambridge, MA, 1991.w x5 J. von Neumann, O. Morgenstern, Theory of Games and Economic Behavior, Princeton Univ.

Press, Princeton, 1944.w x6 B. Peleg, J. Rosenmuller, P. Sudholter, The Canonical Extensive Form of a Game Form: Part I —¨ ¨

Symmetries. WP 253, Institute of Mathematical Economics, University of Bielefeld, Germany,1996.

w x7 A. Rapoport, Order of Play in Strategically Equivalent Games in Extensive Form, University ofArizona, Tucson, AZ 85721, 1994.

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