III

INFORMATION PATTERNS AND NASH EQUlLIBRlA IN EXTENSIVE GAMES: I

Mamoru KANEKO

Communicated by F.W. Roush

Received 8 June 1984

Revised 3 July 1984

In this paper we explore the relation between information pattern, and Na\h 1 yttliibri,~ 111

extensive games. By information WC mean what players kno\\ about mo\es made h> other\. ,t~

well as by chance. For the most part we confine ourselves to pure wategie\. But w St‘~ti\w _

behavioral strategies are also esamined. It turns out that they can be modeM a\ pure \fr.ttr’s~c‘\

of an appropriately enlarged game. Our results, applied to the enlarged ~!anw. r’;tn thw k

reinterpreted in terms of the behavioral strategies of the original game.

Extensive game: Nash Equilibriunl; information pattern\; folk

theorem.

1. Introduction

The extensive game model is of fundamental importance and captures the in- terplay between information and decision-making. Yet we find that its definiriun. as set forth by Kuhn (1953), is insufficient from certain points of view. It is unsbk to incorporate games with a continuum of players. Also. it often makes for an un-

naturally complex representation. For instance. a game in which II plapm nww simultaneously can be described in the Kuhn framework. But first we ~uld hw : to order the players artificial& and then have them move in sequence Grh suit&l 4

enlarged information sets. If we try to carry this out when n is not finite TV :I WI- tinuum, the difficulty of the procedure becomes clear. Therrtfore w m-e led w develop a variant mode1 which has the feature that several pls_\-ers c’;m IW\ 2 simultaneously at any position in the game. Games of the type in Kuhn (1953) xr’. of course, included as a special case of our set-up.

In Section 2 we develop our mode1 and illustrate it with an esampk. In the rr’! t

016_C-489&‘84~53.00 Y 1984, El: wrier Science Publishers B.\‘. (North-Holl.twi)

II2 P. Dttky and M. Kaneko / Infortnation pattm

of the paper, we focus on the effect on Nash Equilibria (N .E .) that is caused solely

by changes in the information pattern of an extensive game. In Section 3 we show

that if information is refined, without increasing players’ knowledge about chance

moves, then the N.E. of the coarse game do not disappear. But the converse is not

true: in general there is a rapid proliferation of new N.E. In Section 4, we explore

conditions under which this proliferation is arrested. The notion of “no informa-

tional influence’ is introduced. It says that if a single player unilaterally changes his

strategy, then the resultant new outcome tree does not pass through any other infor- mation set of the remaining players than the old one did. This is a purely set-

t ‘heoretic condition and can hold not only in non-atomic, but also in finite, games T_ see the examples in Section 4. We prove that if it holds then a Nash outcome of the refined game is also that of its coarse form, i.e. is not a ‘new’ N.E. brought

about by the increased strategic (threat) possibilities. When we turn to non-atomic games, no informational influence holds in full force and we get: Nash outcomes

arc invariant of the information pattern (see Section 5). This lead:, to the ‘anti-folk theorem” in Section 6: N.E. in each stage.

the N.E. of a repeated game are precisely those which are

simubaeous move form

2. I. Ektensive games: The definition

.An extensive game r in simultaneous move form is a seven-tuple:

r=(!Vu(c), Xv ~9 (S-‘).~E_~, +, (hi]ieN* (&lie.%)* (1)

Let us explain our symbols. (Unless otherwise stated, all sets are assumed to be non-

empty.1 (i) N is the set of all players, and c denotes chance (c@ N). (ii) X is the set of all positions in the game, one of which, x0, is distinguished and

represents the start of the game. (iii) n maps X to 2% (c) l If n(x) is a non-empty subset of N, then it denotes the

set of players who move simultaneously at lhe position x. If n(x) = {c}, then chance moves at x. (Note that players and chance never move together.) Finally, if n(x) = 0, then x is an ending position of the game. tt will be convenient to partition X into the three sets:

Xv= (xEX: n(x)cN, x(x)+0),

x,= (XEX: n(x)= (c}},

x,= {xeX: n(x)=0}.

(Note that XC and XE may be empty.) (iv) For each XE X, Sx is a set of functisns from n(x) to some set Y”. We assume

that n(s) = 0 e 9 = 0. Given s’ E 9, I E Y’ and k n(s). denote by (9 i. I ) thy ~‘wK-

tion from x(x) to Y’ which assigns I to i, and agrees with s’ t‘lw\\ here. .-VW. let $ stand for s’(i). Our assumption on S’ is:

if s’, r’ E SK, then (s’ ,, r;‘) E 9 for all k n(s). (3

and, by (3).

if n(x) is finite. S-’ = fl s;‘ It ‘I\)

(where I7 denotes Cartesian product). Z$ is the set of moves available to pluyer i at the position A- and S‘ is the wt o?i

move sekcfions at s by the players in n(x) that are feasible in the game. (v) # links positions to moves. Put X* = X \ (xc,). Let F he the collection of alI

finite sequences (s’(l. s\‘l, . . . , AT\‘-) with ~“9 E 94 for k = 0. 1% _.. , w. Then 9 ib ;I one-

to-one mapping,

#:X*-+F,

such that: (a) if ~2% s”~~, then (sQ) E @(X*); (b) if (z?‘, ~“1, . . . , s’-: 1, s’m) E Qi(X*). then (s’!‘, s’ . . . . , s’- ) = @(s,,~ k

(c) lf @(x) = (?@, . . . . ~‘-9 and s’ E 9. then (So’.. . . . . s’-, s’) E @(X 9.

Since @ is one-to-one, we will sometimes identi’r s with @(A-). aaxi sq t h;jt S=ts’“* . . . . sy for SEX*. This should cause no confusion. Thus. if

dqx) = (s-y .I., +), we will write (x, s“) for (+. . . . .+, s’). etc.

To describe the rest of the game we need to develop some more terminology . Ii

(AT, s“) =y for some s\’ E 9, then we will sav that v inmediareir e c _ . hollows A- ami N rite

s < /y. If there exist x1, . . . , _v,,~ in X such that s < Isl < I.. . < r.~,p~ < !_v. then J .fi~l/c)w s (or x precedes y), and we write s < _v. (Then < is a partial order on .\’ \\ it h A-\, :te its unique minimal element. Also note that for any s E _X ‘* the cet of all pr&~~ww~ .

of s form a path, under < , from _I-,_) to s; and Qi(s) lists them sequent iall!-. along

with the moves selected at each position that lead from it w its immediate iollo~~t Continuing with our definition:

(vi) A (possibly finite) sequence (_v~,. . . . , _Q. . . . ] is called a pfu~ it

(a) )i, = A-0;

(b) yk < ]yk t l for all k; (c) y+ X, if rp/ is the last element of the sequense. _ (vii) A union of plays A = Uai 4 pR is said to be an OW~YW~ ww (c)r. nwrc \in~-

ly, an otctcorpie) if

El4 P. Dulsey und M. hhneko / Informdon pattwn

for XE A \ ‘)(E,

(s’ES~:(X,S’)EA) = S-\’ if n(x) = (c), a singleton set otherwise. (5)

The set of all outcomes depends upon the five-tuple f -2 (Iv. X, R, (S’& s, @) and will be denoted A (f _ ). Each hi is simply a real-valued function on A (r_ ) and gives the payoffto player i for any outcome of the game. (Normally chance moves have probabilities attached to them, with the result that A gives rise to a probability distribution on plays in f. Then hi is taken to be the expectation of a payoff defin- ed a priori on plays. See Section 7.)

(viii) I’ is a partition of Xi = (X E X : its K(X)) and is called the infornmrion parti- fiort of plu_ver i. If x and y are two positions in the same set of player i’s partition (A-, _V E II E /,), then this means that i cannot distinguish between s and _r. It is natural to impose +3rne constraints on the ( I.}. , ,ES in view of this interpretation. First:

if x,yeueIj, then S:=S’{.

If this were not so, then i could distinguish intrinsically between x and _v. Given (6) we will, without confusion, talk of the set of moves S:’ which is available to i at (each position in) his information set II. Next we assume:

no play passes through an information set more than once, i.e. for any play p and any information set u we must have lpnui s 1, where 1 l 1 denotes cardinality. (For a discussion of (7) see Remark 1.) (7)

This completes the definition of the game f.

Remark 1. The condition (7) follows fr,om the assumption of perfect recall (Kuhn, 1953). Take y in X, e(y) = (s+ , . . . . s’r*f). Then if its n(_q) for some 0~1~13, $j = 1, and x1 E u E 4, we will say that ‘y folllolcws from 1) via the move t of player i' ,

and denote this by (0, t) < iy. The perfect I ecdl assumption may now be stated:

(8)

It says that at any position each player can fully recall the entire history of his previous information and moves. Technically we need only the weaker condition (7), although we feel that it is more natural to postulate perfect recall. Indeed, when we restrict ourselkes in Section 7 to behavioral - rather than mixed - strategies, then (8) is implicitly assumed. For then, by Kuhn’s (1953) theorem, behavioral strategies suffice for the analysis of N.E.

Consider:

AI= (1,2,3,4); X= (-Y&S ,,..., x26);

n(.\;,)= (1,2). n(q)= (ct. n(q)= (34) for f =2, . . . . 6.

and n&)=0 for t=7,...,26;

S$=(V~,}~~} and S$=(8,,&) for r=2,...,6;

S’*=O for t=7,...,26;

The l ,x of this game is shown in Fig. 1, where {a,, a:). (& #L). {y,, yz), id,. 3:) are the moves of players 1 , 2, 3, 4, and (x0, x-2, x-) , {x0, x1, x3, x1 I. -ii, x1- ) xe in A(f_).

2.3. Nash Equilibria

Fix a game r=(NU(c), X, n, (S-‘),,x, 4+ (hJIES, (I,}:,_y) as in Section Z-1. The s?rateu-set of player i in Tis made up of all possible choices of moves a\ ailable to him in Xi under the proviso that he must make the same mow at positions that

are indistinguishable in his information set. It is the set Zf (rj11 consisting of alI maps q from Xi to U,,.V s;\’ which satisfy

(i) q(x)~St, and (ii) q(x)=q(y) if x, y~u~l,.

Thus, we can also think of q as a map q : Ii -+ U;E , 25;‘; and, without confusion. q will be used in both senses. Given a strategwhoice 0 = {G, I1 E \, where each a, E Z; (r), abbreviate {q(x)};, n(_r, by c(x) for A- E X_\. Then o is called &wsibfe ii

a(x) E S” for aII XE x,. (9)

Let C(T) be the set of all feasible stra;egy-choices in r. We assm~e throughour thar C(T) is not empty (see Remark 2). Define the strategy-to-orrtcor?ze map

116 P. Lhbey and M. Kaneko / lnformativn pattern

< : C(f) + A(r_ ) by:

<(a)={xeX:if

Odrm,

Fig. 1.

@(x)=(t-Q, . . . . rvm), x’EX~, and

then + = a(~,)} _ (10)

(Note that chance always picks all of its moves in <(a).) From (5) it easily follows that 4(a) is indeed an outcome in A(C_ ). But not everything in A(r_ ) need be achieved by strategies in C(f ). It will be useful TO define A(r) = <(Z(f )), the set of outcom4s t/rat are feasible in f. (In the example of Section 2.2, the outcome (x0,x1,x3, x13,x4,x16) is in A(r_ ) but not in r(A).) r

Next, given arc and q e&(r), let (a 1 q) be the same as o but with oi replaced by r;. By (3), (a 1 Tj) is also in Z’\r), and therefore our next definition makes sense. The strategy-choice cr EC(~) is called a Nash Equilibrium (N.E.) of Tif, for all ieN:

hi(<(a 1 ri)) I hi([(a)) for all ri e.Zi(f ). (11)

The outcome t(o) produced by an N.E. o of r will be called a Nash outcome of r.

Remark 2. If N is finite then, by (4), it follows that C(f) is non-empty. But in our general set-up we have made no connection between the information sets and the

feasibility condition (9), so it is not possible to deduce that E(r) is non-empty_ \VC find it more economical to assume non-emptiness here rat her than to beck the ewa conditions that will imply it.

3.

in

Pruservation of Nash outcomes

We will focus on the effect on Nash outcomes that is caused .sok& by changes the information pattern of the extensive game. For this purpose we take a prlir

of games, f, f *, which are identical except for their informut~on patterns i IL \i: -_ , , *

v I i ic ,%. Our sharpest result is in the case when N is finite, though many of its CM- ollaries continue to hold in general. We therefore break up this sect ion into t a o

parts.

3. I. T/14 finite-p@er case

For simplicity, denote Ci (r), C(i), 2’i (P), C(P) by Z;, 2, 2:. .X? For cr E Z. define:

Ri(O)=(s~Xi:X~5(O!Ti) for some SJEZ, andjEN\ {ii),

i.e. Ri(o) is the set of positions that are reachable in r via unilateral deGation\ from cr by players in N\ {I ‘i. Also define the sets Ii( I,*(S) to be thz (unique) sets in Ii, Ii* that contain X. (If X@ Xi, then I,(X) is understood to be the empt! WI.)

Proposition 1. Assume (i) N is finite. (ii) o is an N.E. of r (iii) For all i !i? A’:

(a) X, _V E Ri(a)

1i(x) +Il,(Y) 1 = I,*(s) #I,*(y).

(b) Xv J’ E <(a 1 Ti) nxi

for some Ti EC,

>

* I;*(s) = r,yv).

Ii(X) = Ii(y) Then there is an N. E. o * of P such that <*(a *) = <(a). (Here < *. 5 art’ Ihe .~lrclW!?‘-

to-outcome maps in r *, r . )

Proof. For any i E N, put

Ai*= (Ii* :-YE Ri(O)),

Bi*=IF\ Ai*.

Fix ~J*E AT. Then, by (iii) (a),

1lU P. Dubey and hi. Kaneko / hforrnation pattern

Therefore we can define the map pi : Ai* -+ fi by

W,(lJ*) = f,(s) for any SE tJ*nR,(O).

Now construct do * = ( tr&E ,,/ by:

O,*(U*)= ai(vi(U*)) if USE Ai*, arbitrary if v E Bi’.

BY 0) and (41,

d’E2X (14)

SfeP 1. <*(o*) ==<(o): Since <s(o*) and &o) are outcome trees, neither

iYa*)~fWb nor <(o)c<Ya*) is possible. Therefore, if Sqo*)#<(o), there are plays p* and p. with:

P*C cf*W*) \ {(a) and JK<(CX) \ f*(o*).

Let s bc the first position on p, starting from the root x0, which is not on p*, i.e.

s’**j) has the property:

SET \p* and (.v[,. . . .._ \;N) cpnp*.

Clearly, n(.v’“)#O. If n(F)= (c), then x=(x”‘, s\;“)qQP by (S), a contradic-

tion. If n(.C’)#@), then since S”*E Ri(a) for all k/V, we have, by (13):

a,*(.vm] = q(_v”) for all id I&V”),

so .VE pnp*. again a contradiction. This verifies Step I. The proposition will now fallow from:

9ep 2. For any r,* in z;’ there esists a tJ in gj such that <(a 1 tj) = (*(a * I$).

lkfinr

We shall establish (16) and (It) by induction. Ifs= (~‘0, . . . ,s’;), say that the length

of x from x0 is I+ 1. Put:

X’= (xEX: the length of .Y from .qI is (I),

x:=x,nx’ for all k/V.

P. Dubey and Ad. Karreko / Inforrttation pattem IIQ

Denote by ( 16)‘. -‘*’ the statements (la), (17) but with X,. 7, replaced by A’,‘. Xi, respectively . Obsi tit ?t (15)’ and (17)’ are trivially true. So it suffices to dwu that

(16)’ and (17)‘=3 (l(5)‘+’ and (17)‘+‘. (*)

Let $ denote the restriction of jj to <*(cJ* 1 f)fW”. By (17)’ there is an estension of $ to a strategy ,# in JCi. Then, by (16)‘. (IS) and the definitions of .r, <* and C:

<*(a*) r;3nx’+ I =j(afljnx’+ I. ON

By (IS), it follows that

y(dy pnx::* 1 CR,(a) if idV\ {j]. (19

By ( 19) and ( I-t), we pet

G~*(.Y)=o,(.Y) if SE<*(G*~ r,*)n.V:*’ and in.:\ {.;i.

proving (la)’ ’ I. Next take s, IyE <*(a*[ $)nXi * ’ with !,(A-) = I’@). Bx (IX) awl (iii)(b), I*(x) = P(y). Therefore, by (IS), S,(x) =fj(,r-‘). proving (ITI* ‘. This _ establishes (s), and thereby (16) and (17). From ( 17) we see that .!; can be e\;ten&l to a strategy rj in Sj. By (IS) and (la), a **(b* i q*) = <(o T q ). This wrifies Step 9 -._ -. -

An example (Fig. 1). Here

x2 = (A-Q i.

X’ = {_v*. . . ..A&

r, f* have the solid, broken information pattww,

al, 4x2, yl, y2 are moves of player 1,

&. & are moves of player 2,

<(a) = (xu, xl, .q) = play marked with arrow,

R’(o) = (xl,,. S’, .Q),

R2(d = {-vt, 1,

g(a 1 t2)nx2 = (A-,,) for k = I, 2,

Q(a 1 tl )n Xl = {AT,,, .yl ] or {A-,,, s3. x4).

(a is any strategy-choice t hqt

1

I

tz range over all strategies of

It can now easily be checked that (iii) (a) c(a) is Nash in f. it will also be Nash in

leads to the marked pIa> <Cob; ;pnif zi.

players 1, 2.)

and (iii) (b) hold for F. P. o_ Thus. if

t-“.

Fig. 2.

Remark 3. Observe that, by (7):

no chance moves in the game = (iii) (b) automatically holds. (20)

Thus, (iii) (b) says that there is no informationd guin regarding chance moves in going from f to f + at o. However, this needs to be true for player i only under his own unilateral deviations.

Remark 4. Say f < f * if each Ii* is a refintiwreat of Ii, for all i E N. Then

f < f * * (iii)(a) automatically h-Jlds. (21)

We can think of (iii)(a) as a weakening of l-X r*. It requires that, in the region reached by others’ unilateral deviations, the.re is no informational loss in going from f to f * at 6.

Reti 5. The scope of Proposition 1 will become clear below since many of the prckzsitions that follow will be its simple curollrrries wher;r N is finite. Let us point out one such immediately. For any game r let q(f) denote the set of all its Nash outcomes. They, by (20) and (21) we have:

er (a) no chankw mows * R&‘l”)c Q(f ‘1. (22) ot (b) (iii) (b) holds at

(This, in the case of condition (a), is essentiallv the proposit ion in Qubey and _

Shubik, 1981.)

R~IMU& 6. The preceding remark leads one to inwstigate the possibility of Ptoposi- /

tion 1 for the general N case. The difficulty arises in deducing (14) from (13). One would need to make mote measurability-type assumptions on the structure of the game to overczme this difficulty. For instance, consider:

Here 43 is an algebra of subsets of N which inctudes all singleton sets. Aiso require:

TC n(x) TEC

1

* the combination (s& r&. ) \ T) E 9.

s’, r’ E S”

Finally, enlarge Rj(o) in (iii)(a) to include positions reached by player i’s own deviations. Then all these conditions together enable us to go from ( t 3) to (Id), and yield Proposition 1 for general IV. Possibly (**) can be deduced from more &men- tary assumptions on the tree, though we have not explored this.

3.2. Nestedness of Nash Equilibria undet refinement

We now prove (22) without the assumption that N is finite. First note that if r< P, there is a natural sense in which C,Cz;.*: simply identify q E Z! with CT;* E 2’;* where q*(x) = q(x) for ati _Y E Xi.

Proposition 2.1. Assume (i) f < r? (ii) o is a N.E. of f. (iii) Cwzditiwt (iii)(b) of Proposition 1 holds at 0 EC.

Then a is an N.E. of I?

Proof. set n*= 0 and repeat, mutatis mutandis, the argument in Steps 1 and 2 of

the proof of Proposition 1. 2

122 P. Dubey and M. Kaneko / Information pattern

As a3 immediate corollary we get a global version of Proposition 2.1:

PWDTHDSWR 2.2. Assume

(i) r< P. (ii) Cofrdition (iii)(b) of Proposition 1 holds for every B EZ.

Theta q(r)c q(T*).

(Note: If there are no chance moves, then (iii), (ii) of Propositions 2.1 and 2.2 automat icall y hold .)

Proposition 2.2 shows that, if we refine information and if there are no chance moves (or else (ii) holds), then the N.E. of the coarse game are not lost. But there is no dearth of examples to convince one that, more often than not, there is a rapid proliferation of new N.E. Consider the three games, f,, GZ, &, with the informa- tion patterns given below. The payoffs are given in Figs. 3,4 and 5. The Nash plays in each case are marked by X. Those of r1 are preserved in rI+ I (I = 1, 2) in accor- dance with Proposition 2.2.

4. No irfomnatioaal influence

We are interested in investigating conditions under which this proliferation of Nash plays is arrested. The next proposition makes an advance in that direction, and constitutes a partial converse to Proposition 2.1. For 6~ C* define Dig 4* by:

Q(6)= (I;c(X):X~<*(~)nXi), (23)

(2.2.2)

W,11.11)

(3.393)

i13.13.13)

(4.494)

U4,14,14)

(5.5,5)

(15,15,15)

Fig. 3. The game f,.

P. Duky and W. Kaneh-o Inforttmion parrern I23

I /

.

Fig. 4. The game rz.

Fig. 5. The game l-3.

i.e. Q(d) is the collection of i’s information sets through which the trw t’(e51

passes. We say that i has no informational infhrence on j clt (T * in I- * if

Dj(O*)>Dj(O*l T,? for all f,*EZF. (24

I24 P. D&y and M. Kaneko / Information pattern

PqwsWon 3. Assume

(i) f < fsN (ii) U* is a N.E. off *. (ii@ Each playet has no Gnfotntational in$kence (on

P. (iv) r*(a *) E A(r), i.e. {*(CT *) is feasible in r.

Then there is an N. E. tr q,f r strch that c(o) = {*(a *). khich <(a) = <*(a *), is a IV. E. of r .)

every other pia_ver) at a * in

(Indeed every TV E Z(r), *for

PWVO~. By (iv) there is a o in Z such that r(o) = <*(a *). Then it must be true that for all i E IV:

a,(x) = q*(x) if Ii*(x) E &(a *). (25)

Take any q E 2;-. Since f < r* we can define rj*E Zj* by

r]*(X) = 7j(X) for XE Xje (26)

The Proposition will follow if we can show that: <*(o* 1 Tj? = <(CT 1 rj)m If # holds, then there is some x= G(x) = (s\%, . . . ,+) such that

xE<(a i 7j) and Xe<*(a*I rj*) (27)

and

x0, . . .._ rtnEt(a 17j)nt*@*l

Clearly, by (27) and (5), n(x,)# (c}. By (28) aA (iii), Di(a *) for all Therefore, by (25):

and, by

BY (29)

ai(Xm)=Qi*(Xm)

7j(Xm)= 7j*(Xm) if jE R(Xn,). (30)

and (30):

(a* 1 Ti*)(X,)=(~ I7j)(x,)* (31)

and (3 1) the position (x,, (a * l7,3t rm)) =x and is in (“(0 * 1 rj*>n<(a I rj), BY CW I

contradicting (27). 5

To clarify (24) consider the games in Fig. 6. At x0, (a,, a2} and (& ,&} are the moves of players 1 and 2. The solid and broken lines give the information patterns of f and I’*, respectively. At any o* in f * which gives the play x0 to x as an out- come, no player has any informational influence. But if a* gives the outcome {x0 to J+) then player 2 has informational influence on player 3.

The condition (iii) of Proposition 3 is undoubtedly severe, though it is a natural one in the context of a ‘liarge number of small players’, not necessarily non-atomic. SupposeN=(l,..., IOOO}. Let S=(l)..., 500) and T=(501,..., 1000). Thegamef

P. Dubty and hf. KaneA- / Information partem IZC

Fig. 6.

is as follows. First all players in S move simultaneously, and each i E S selects a real number rj in the closed interval [O,l]. The players in S can ebserve 1: E s r[. But

there is a grid on their scale which does not permit very fine measurements. The! can tell only that CiEs ri lies in one of the intervals

[O,lO), [lOJO), . . . . [4!30,500).

After S has moved, then the players in T move simultaneously, and again each of them can select a real number in [0, 11. Suppose there is a Nash Equilibrium in which r. r,Es ri= 145. (One can easily concoct payoffs to make this so.) See Fig. 7. Then no player will have any informational influence at this N. E. The resulting N . E. ph?

is marked in Fig. 7. If any one player in S changes his strategy. this will change thy play but no one in T can observe it because the new play continues to pass through [ 140,150). If we call the below game r* and let r be its coarsening in N hkh pla~ctr+, in T observe nothing (i.e. have the information set marked by dotted lines in Fis. 7) then all the conditions of Proposition 3 are met.

Remark 7. If there are no chance moves, then Propcsirion 3 is Q coroffor>* of Pm- posirion 1 when N is finite. In this case (iv) of Proposition 3 holds automatically. (The trouble with chance moves is that (iv) may not hold in general.)

Remark 8. A stricter version of (24) is:

Dj(0 *) = Dj(0 * I?,*) for all ?i* E Z;*. (2-a) *

126 P. Dubey and M. Kaneko / Information pattern

Fig. 7. The games r and f *.

Then we will say that i has strict& no informational influence on atomic case (Section 5, Lemma 2) it is in fact (24)* that obtains.

S, Nor-atomic games

5. I. The definition

are no chance For simplicity we will as’sume, throughout Section 5, that there moves. (They will be incorporated in Remark 9.) We need to specialize the set- theoretic structure of r to treat non-atomic games. The player-set N is now equipped with a non-atomic measure. Precisely, we have a measure space {JV, B, JJ). B is an a-field of subsets of N which includes the singleton sets (i}, k N; p is a non-atomic probability measure on (N, B). Each Y-’ (for -YE XV) is also assumed to be a measurable space. We now add the following conditions on the constituents of f, over and above those in Section 2.1, (i)-(viii).

(ix) For any XE&, n(x) is a non-null* set in B,

(x) Foa any XE X,, there is a measurable ,correspondence fV from Z(X) to Y”, and S” consists of all measurable selections from f”, i.e. of all functions g : n(x) --* Y” which satisfy:

(a) g(i) E fx(i) and (b) g is measurable. (xi) For any x, YEX .&, tire set (idV:_v~~(x)} is measurable. These conditions are fairly innocuous. The sine qua non of the non-atomic

assumption is in the next, a.nd final, condition. it says that null sets of players and their moves cannot be observed by any of the others.

(xii) If x= (tie, tit, . . . ,+), y = (~~0, r-“l , .-. , Pm), and iE Iv satisfy (where _Y~=x~): (a) XIEVEfiey/EVEIi,

j. In the non-

’ SE J? k called nulf if p(S) = 0; non-null if it is noi null.

I 3 - . .

(b) if _v~,)?IE UE I,, then s;‘i = r;!;, (c) !I( (j E n(.k.,m x(_v,) : 9; ‘=ri”‘))=Cl(n(-\i))=!I(R(~‘i))

for I=O, I, . . . ,111, then

XEVEI; ~_VEVE I;.

This completes our definition of a non-atomic game. Note that (s) easily inqdies

s;’ =f”(i), (32)

if n(x) is a disjoint union of measurable sets n,(x) and nl(x), and g, : n,(x) -+ Y’, gz : xl(s) --, I’-’ are measurable functions which satisfy g,(i) of’ for k nl (s), g,(i) Ef’(i) for ic n#), then the function g : n(s) --+ Y-\’ obtained by put- ting together g, and g:, will belong to S”. (33)

It can be checked that (ix)-(xii) are consistent with the earlier assumptions in (i)-(viii), i.e. there are models of games that satisfy (i)-(G). See the esample in Sec- tion 6.

5.2. Invwiance of Nash plays on information patterns

We will establish that if (i)-(xii) hold for a game, then the Nash plays are in- variant of the information pattern that the same is endowed with.

We prepare for this with

Lemma 1. Let r satisfv (i)-(xi). Then A(IY ) =- A(F).

(Note that since there are no chance moves. outcome trees reduce simply to play )

Proof. Recall that a play is a sequence of immediate follo\vers, starting with the rcwt x0. Given our identification SE @(.I.) = (+, . . . , s‘- ). let p, = (s’ ‘. S’ . . . . . .F’ . . . . 1 E

~(f _ ). Put U, = UiE z,s, I; (.Y,) and C-‘= Uj L;;. For s E c’!, Iet

y’(x) = (i E n(.\-) n n(q) : s E It (xl)).

By (7), if I#!‘, then y’(.\-)n y”(s) = 0. By (xi), each y’(x) is measurable. Thzreforc. by (xi). so is

128 P. D&y and b4. Kaneko 1 Information p&tern

Since (y’(s):I=O, I,...) are disjoint, this cr is well-defined. It can be checked (in- ductively, starting at x0) that &a) =p. It remains to verify that arc. It is clear that if x, YE u for some u E Ii, then q(x) = q(u). Therefore it is sufficient to show that ads-’ for all XEX N- If XE&IU, then a(x)=6(x) and o(x)&\‘, by assumption. If x E V, for some II 0, then n(x) is the disjoint union of (y’(x) : I= 0, I, . . . ) and a(x). By (6) and (32), f”‘(i) =f”‘(i) for i E y’(x). Also, clearly U, ~'(x)c n(q). But then, by construction, the map a(x) on n(x) (given by q(x) for in X(X)) coincides with sq on y’(x) for all 1~0. Hence, o(x) is a measurable selection fror.1 f” on UI v’(x). On the other hand, a(x) coincides with c3(x) on a(x) and is, a fortiori, a measurable selection from f’ on a(x). Therefore by (33). a(x) E S*. cj

Lemma 2. Suppose r satisfies (i)-(xii), and cr E C(T). Then each player has strictlv no informational influence at 0 in r.

Proof. Let 0 = {a(x)},, *, . Consider ri E Fj(r). Put c(a) = (+, ?I, . . . , ?m, . . . ) and 7j)=(byu, fY', l **rPvm, l mm ), where ya,=xo. It will suffice to show that for any I

and any k N \ (j), if x= (s-Q, . . . ,s-Q) and y = (1~0, . . . , r-‘l) then (a), (b) and (c) of (xii) are satisfied. Make the inductive hypothesis that we have shown this for I = 0, 1 ,..., k and consider the case* I=k+ 1. Now xk+, =(s’o ,..., r’&) and y&r =(p”‘,..., rYk). Then, Iby (xii) .

(d).~~.,lEDEI,o_)tk+IEl!‘E~, for iEN\ {j). Hence:

(e) n&l,\ {j)==Wk+I)\{_i) (=Ak+l)- (f) Spf= #khl for i++,.

From (c) and (f):

4+t= ~iER(x~+#%?(y~+~):s~-‘=~~-‘}.

Hence, since p( { j}) = 0,

(g) ~(AA,+,)=~(n(xk+,))=ru(n(Yk+,)).

This verifies the hypothesis for I= k + I. C

Fix a six-tuple L = {IV, X, n, {SX},Ex, @, {A,) , i iEN) for which all the assumptions in (i)-(viii), as wellus (ix) and (x) hold. Denore by V(L) the set of all games obtained by adding information patterns to L subject to (6) and (7), as well as (xi) and (xii). For any f E V(L), recall that q(T) is the set of all its Nash plays.

Proposition 4.1. q(T) = q(F) Jbr any r, i= in V(L).

proof, Denote by {Ii}ieN, {/;-}i,, the information patterns in I’, r. For each i&V,

’ For I=0 the hypothesis obviously holds.

let I,* be the common refinement of /[ and t. i.e.

Consider the game P obtained by adding I,* to f_. We will she\\- th;tt /‘*E I-‘(/ ). Clearly, J* is a partition of X, . For any .Y. J E .Y,, ii E .V : J E I *t-v 1; _

(idkyn~#)~@-)) = (i~N:~EI,(.~))n(i~~;:~E~(.\-)). Since each of thr: last

two sets is measurable, so is the first. and thus /[* satisfies (xi). We omit the

straightforward check that /,’ satisfies (t). Finally. take A- = (.+. .?. . . . . .+ 1.

_v = (r-y c”, . . . , +), and in N (where s~~=.v~,) such that (a*) .V/E V’E li* * Y/E V*LE It*;

(b*) if .q, _v+ v+&*, then s’? = r;‘l; (c*) condition (cl of (xii) hdlds.

In (a*) let us= vf75 for VEIN, 6~17. Then S/E v* =-Y/E v, and _I*+ r:* S_~E F. Frcrtn this it follows that (a *) implies:

i.e. (a) of (xii) holds for Ii. In the same manner (a) of (xii) holds for <. and (b) ot’ (xii) holds for both Ii and 6. (c) is independent of the information pattern and depends only on x and y. To sum up, (a), (b) and (c) of (sii) are satisfied for A-. _v. and i in both I. and i=. Then by (xii):

(d”) XEvEIi*yEUEIi; (e*) x~d+y~d~&.

Let XE W’E Ii*, W”= wni+ for WEli, 6~4. Then SEW and. from (d*), _YE 1’. Similarly, y E r3. Hence, y E w*. In the same way, .v E W* 3 s E w *. This proi es t hai condition (xii) is also satisfied by P. Consequently, ran K’(L).

By construction, r< r* and r< r*. By Proposition 2.2, rl(f )ct~(K*) :uact q(P)cq(T*). Let O* be any N.E. of r*. In the wake of Lemmas 1 and 2. w WI apply Proposition. 3. This tells us that there are N.E. 0 in r and ~5 in /=. such that c(o) = <*(CT*) = f(d). Since CJ* was arbitrary, q(r*)c q(r) and TIC t&h

therefore q(r) = q(f *) = q(r). Kl

5.3. A variation on the theme

The condition (xii) is fairly stringent. Each player has no informational in tlu~rl<:r\ on others, not even on a null set. On the other hand, absolutely no assunlpt icln u ;I\ made on the payoff functions in proving Proposition 4.1. We nw rela.t\ [\ii) IO (xii)* but at the expense of having to add conditions (xiii) and (xiv) O&MI. Then Pro- position 4.1 can still be retrieved, as Proposition 4.2.

Condition (xiii) says, roughly, that if two positions differ only on account of null sets not containing a particular player i, then i cannot tell them apart.

(xiii) If @(x)=(+, s”‘l,...,s’m) and @(.v)=(F”~, r’l, . . . . r’,-) satisfy. for in n(.v)

(with .vO = xg),

I30 P. D&y and M. Kaneko / Informdon puttern

(f) p({j~ n(.~,)nn(.v,) : sJI’= r~))=~l(~(_~,))=y(a(y,)) for I=O, I, . . . . m;

(g) for all I=O, l,..., nt, ie n(x,) @ iE R(y,);

(h) for all I=O, l,..., ~tl, ieR(x,)*&C’=rp;

then iE n(y) and Sf = S;!‘. The next condition (xiv) is on payoffs. It says that they depend on plays ‘modulo’

null sets. (xiv) If two plays p = (s”u, 91, . . . ) and p’ = (r-‘“,~“~, . . . ) satisfy

(i) fl(( je n(x,)n R(vll : $t =ri”)=fl(R(x,))=p(n(y,)) for all 120; (j) k R(x,) m k R(y,), for all /20; (k) i E n(x,) * #I = $‘, for all 1~0;

then II,@) = hi@‘)- In the Eight of (xiii) and (xiv) we weaken (xii) to:

(xii)* No positive informational influence. Each player i has strictly no informa- tional influence on almost all other players (i.e. all except a null set).

Let P be a six-tuple as bekore, but assume this time that the assumptions (i)-(viii), (ix)-(xi), us rspe!I as (xiii) and (xiv) hold. Define G’ *(L *) esactly as F(L) but with (xii) replaced by the weaker (xii)*.

Propodtisn 4.2. q(r) = q(r) for any r and r in F *(t *).

Proof. It is sufficient to show that for any N.E. cr of f, there is a N.E. d of rsuch that <(a) = &).

Let t(a) = (90, ~‘1, . . . ). Select a 6 in Z(f) such that c(d) = <(a). This is possible by Lemma 1.

Suppose c? is not a N.E. of i=. Then there is an fi E&(F) for some idV such that h, (((8 1 f, )) > h(f((d)). Let c(d 1 ii)= (r-‘ll, r-‘l ,...I (yo=xo). Then condition (xii)* implies:

p((jt3r(xl)n2t(yl):$~= r;."))=p(;~~(x~))=p(n(y~)) for all 120.

Choose an riEZi(f) such that if ~J(x)=(~~o, tY1,...,twm) satisfies (f*) p((jER(y,)nR(w,):r~= tj!+))=.u(R(y,))=p(R(w,)) for all I=O, I,...,m;

(g*) for /=O, II , . . ..m. iE R(y,) 0 k R(w,);

(h*) for /=O, 1 ,..., m, if ien then r;” =rff”‘; then ri(x)=?,(.y,,&. .4ssumption (xiii) ensures that this cho& of r, is possible. Let <(a i r, ) = (#“. #‘., . . . ), with ao=xO. From the constrtI Lntion, it is clear that

(i*) for all 120, ~I({IE R(a,)nR(y,):qF=r-))=p(R(a,))=-~(R(_Y,));

(j*) for all 120, itz n(a,) 0 ie n(y,);

(k’) for all 120, if ien( then qp=r$. Therefore, by (xiv), we have h, (<(o 1 ri)) = h,(& j f,)). That is, /~~(<(a f 7,)) =

h,(&cS j i,J)>hi(f(ti)) = hi(<(a))m This is a contradiction. 1

If (xii) and (xii)* are violated, then Propositions 4.1 and 4.2 break down. Non- trivial sounterexamples can easily be obtained by modifying the ‘dilemma game with rumour’ in Kaneko (1981).

The careful reader must have noticed that we have defined a Nash Equilibrium by requiring that arl-’ - as opposed to ‘almost all’ - players must be optimal in .ilc- cordance with ( I I)- This is because, in our opinion. the very basis of an N.E . is in- dividual optimization, and ignoring even a single player would go against the gain of this notion.

Remark 9. If chance nroyes are always countable, then an analogue of Prop&t ion 4.1 (or 4.2) is possible. Take two non-atomic games. /-and r*. differing only in in- formation. Suppose (iii) (b) of Proposition I holds at all strategies in h&r dirty- tions, i.e. in going from f to P and P to f. Then we can show that q(f) = q(P)_

(Naturally, condition (xiv) has to be strengthened to appls to outcome tret\. rrttkr _ than just plays.) If chance mowes are uncountable then we would need additionA measurability assumptions in the spirit of Remark 6.

Remark IO. An asymptotic version of the non-atomic result has been esamineci in part II of Dubey and Kaneko (1982).

6. The anti-folk theorem’

Let f be a non-atomic game in strategic form, i.e. n(s,,) = 3’ and every S’ E S’

constitutes an ending position. Further assume that condition (xi\) holds. in thi\

context that simply says:

i.e. the payoff to any player depends on his strategy and the measurable iun;tiw

of others strategies module null sets. Consider an infinite repetition r” of I-, in which each player c:an observe at each

stage the entire past history of (a) his own moves and pay of t’s and (b) the measurable functions of others’ moves, modulo null sets. The payoffs to piays in r” are assigned by some rule (e.g. lim inf, discounted sum) - it matters little. Then r” satisfies (xii), (xii)* (and, also, of course (i)-(G), (xiii), (xiv)). Consider the game f? obtained by coarsening r”, as shown in Fig. 8, i.e. each player obserwy nothing at the end of any stage in r,. Clearly both Propositions 4.1 and 4.2 appLy.

This says that the Nash plays of f; are identical with the Nash plays of r” . li N e

denote the strategy set of i in r by Zi, then clearly his strategy set in f; is (1. I”. i.e. a strategy for him is to simply pick an infinite sequence each of whose elemrtnts is in Zi. It is a short step from this to verify that nhe Nash plays of f(T (hence of f “) are typically ‘small’. indeed, if we assign the payoff to a play of r” by the dk-

’ That is why the ‘almost all’ variations of assumptions (xii) and (xii)*. (xiii). (11) sould n&It wiiicc

for our results.

’ For a further discussion of the folk thsorsm and the anti-volh rhcorcm vx h;.uwl.~~ 4 IWZ)

I32 P. Dubey and M. Kaneko / lrtformation pattern

Fig. 8.

counted sum’ of payoffs in each stage, then it is obvious that

(a’,o’, .=.,o”, . . . ) is a N.E. of r; 0 each o1 is

a N.E. ot f for I= 1,2 ,... .

This is in sharp contrast with the ‘folk theorem’ (Rubinstein, 1979, and Kaneko, 1982). These players have enormous informational influence, and a stupendous pro- liferation of Nash plays is obtained in P’.

7. B&8vioral !mategies

Our description of extensive games in Se&on 2 permit w model behavioral strategies in Tas pure strategies of an associxed game f, irr lp;e cIzs4 when Nisfinife. The preceding results then apply to P and can be reinterpreted within f. For ease of exposition we shall make the restrictions:

X is a finite set; (34)

there are no chance moves. (35)

Note that (34) implies that not only IV, but also players’ moves and the length of

’ Assuming this will always exist, e.g. by requiring that the payoffs are uniformly bounded in f.

P. Dubty and Al. Kant-A-o .; Inforrnabott pattern

the game are all finite. However, on& the restriction that I\;’ is fink is substantial. all the others are made for notational convenience.

The idea behind going from f to P is roughly as follows. Consider the game I‘ where {a,, a*) and (&, pZ) are the moves of 1 and 2 (Fig. 9). The behavioral

strategies of 1 and 2 are the sets

Construct the game i: as shown in Fig. 10. At x0 players 1 and 2 choose b, and bz from B, and B2. ,4t the resultant position s, ( picks (a,, & ). (~2. /II,). (q. I;+ (c-x2, &) with probabilities b,’ x bi, 6; x b;, h; x b;, 11; x b;. The pa!,oft‘ tk~ the outcome arising from (b,, b2) in P is

h;(b,, b,) = i i b:b&(q, fl,), k-l I=1

where h, is the payoff function of i in I I Then the pure strategies of I= correspond

exactly to the behavioral strategies of f. We now extend this picture to the general

C~SC (assuming (34 and (MN. A behavioral strategy 6; of player i is a function on A> which assigns to each

XE X, a probatiality distribution bi(wX) on $‘, i.e.

E A&,(x)= 1 and 6,,(x)rO for all r&S& 1eS’

This must also satisfy II,(X) = cii(_P) if X, _V E N E Ii. Denote by Bi the set of all

behavioral strategies of player i. Put B = fl tE Lx Bi w An?; b E B induces a map R ‘h : X + !I?, where Q,(x) is the product of the probabilities on all the arcs, going

from x4, to x, assigned according to 6. If we restrict $, to &, then we get a pro-

bability distribution on X,. The payoff to i in f8 is the expectation:

(Since I’has no chance moves and is of finite length, outcomes can be identified with

points in XE and we may view hi as defined on &.) We now proceed to construct P which will represent r” in the format presented

in Section 2. A cap will consistently be used to distinguish constituents of P from IT (a) The player-sets are identical: N= N = ( 1, . . . , n 3.

(b) There is an onto map 6 : 2 \ 2‘. + X which preserves posit ions in the sense: S($ ) = x, for i E N and S(&) = X,.

(cl Folfiowcrs are preserved under ~5, i.e.

3 > _f = S(.i-) > S(_F) for all .\?, 9 E X,.U &.

(d) ri(.f) = n@(s)) for all .iE &. (e) Chance moves come immediately after players’ mobes, and only then, i.e. (e-i) .f E X,, .,’ > ,.t = ri(j) = (c! , (c-ii) ri(.r’) = (c) *there is an 3~ ..f,\- such that 3 > r-i_. (f) For .S x, the moves in P at .e are precisely probability distributions on the

pure st ralegies Sf(.‘) available to i in f, i.e.

(g) Chance moves in p mimic the moves akked with positive probability in f by the immediattly preceding players. In ,>ther words, suppose 3 = (_f, t’) and ri($) -- (cl. By (e-ii) we have .i’~ ,%?, . We require

There is a one-to-one onto map

his pure strategies in r. This map*

where

b is a NE. of Ts Q w(b) is a NE. of k? r_:-r

Thus, to analyze behavioral strategy N.E. of f, it suffices to zonsidzr pux-~-~~ra~g~

N.E. of i;. From the initial pair of games r and f * we derive I= and I= *. Prqw,siFi~~ 113 1. Z

and 3 can be applied to i= and P *. Using the isoFnorphism pii. they MI t Inm lx

transferred to TB and f *. We shall work this out in detail k-r SCWK GWS. Fir\!. for any behavioral strategy-choice b denote by t’(b) the support of z . . i .c.

c’(b)= (sd”: P,(x)>O) is the set of p&tions reached 16th pasitiw pwt~abiht~

under 6. Consider Proposition 3, for instance. To interpret iF Gth b&w i~xd

strategies, take r< r*. It follows immediately that (i)’ PC P*.

Further suppose (ii)’ d* is a NE. of I? (iii)’ No player has informational intluencc at ti* in f *. (iv)’ &P) E A (i;).

136 P. lhdk-y and hf. hizneko / Information pwern

Then, by Proposition -3, any ~5 &Y(f) with ((6) = &t5*) is a N.E. of I’: Put b = v/ - ‘(6) and !I*= v/ - ‘(6*). From our construction of i;+ one can easily

verify that (iii)’ is tantamount to:

To interpret (iv’) first note that:

‘f(i) = &i*, e rp, l(i) = Ip” I,i*).

6 herefore (iv)’ says:

B,, = pd for some de B.

(38)

(3%

WI

Now, using (37) and (39), we have:

which is Proposition 3 in terms of behavioral strategies. As an example, reconsider the game of Fig. 6, but with behavioral strategies for players 3 and 4 in f * (the refin- ed game), as shown in Fig. 11. (Arrows indicate pure moves.) The conditions (38)

Fig. il.

and (40) are met at these strategies. Therefore if the outcome is Nash in the refined game it will also be Nash in the coarsening.

Next consider Proposition 2_ 1 for f and f? Condition (iii) of Proposition Z _ 1 says (in the context of r’) that for ail in N:

So we have, translating Proposition 2.1 from r and I=* to ArB and T&:

~~~~.of~B}~~~isa~.~.*f~~I.

Thus, in the coarse game in Fig. 12, (41) holds at the strategies indicated. We con- clude that if they are Nash in f, then they remain Nash in r *. !Similarly , Proposit ion 1, applied to P and P*, can be interpreted in f B and F? We leaye this to the reader.

d\ /l-d d\ /l-d

____ ___--- ____ / /

/

,&.

1

Fig. 12.

13% P. Dubey and M. Kaneko / hformation pattern

Aclcnowledgments

This work was supported by an O.N.R. Grant NOOOl4-77-0518 issued under Con- tract Authority NRO47-0106, as well as an N.S.F. Grant SES82-10729. The second author was, in addition, provided a travel grant from the National Institute of Public Finance (Japan). This paper is an extensive revision of Part 1 of Dubey and Kaneko (1982).

List of notations

For the reader’s convenience we append a list of notations which are used frequently.

r=extensive game=(NU(c), X, n, (Sx)wEx, @, (hi)ie,vv (Ii)iEN) N = player-set c = chance

X= set of all positions x0 = root = start of game (x0 E X)

n(x) = set of players who move simultaneously at x, or (c) , or empty X, = set of player i’s positions = (s cf X : i E n(,u))

X, = players’ positions = Uie N Xi X,=positions for chance moves={.u~X:R(X)={c)} X, = ending positions = {x E X : n(x) = 0) Sx= set of move-selections at x by n(x) (Note.= V = 0 e n(x) = 0) Z# = set of moves of i at x (for iE E(X))

9)(x) = (S-Q , . . . ,s-Ym) = path from x0 to x and moves picked along it x < /y = y immediately follows x (i.e. )’ = (x, Sr) for some s’ E S-r) x ( J’ = y follows K (x precedes y)

p=(xo, ..*, Xk, . . . I= play (i.e. x0+*. <,xk +..) A = outcome tree (union of plays on which chance picks all its

moves) f_. = same as r withlout {h,)ieN a:_,d. {h)iEN

A(11 ) = set Iof outcome trees in r_ hi : A(r_ ) + R = payoff function of pL%yer i

Ii = player i’s information partitioa on Xi Zi(r) = strategy-set of player i in r C(T) = strategy-selections feasible in ,I-

6= element of c(r) ai= player i’s strategy in d

4 : C(r) -+ A (r- ) = strategy-to-outcome map /1(r) = C&W )) = set of outcomes feasible in

h(x) = information set of player i x64)

r thin contains x (Ii(x) is empty if

P. D1tbe-v and .%l_ Kaneko ’ Infbrrnutron puttem

q(f) = set of Nash outcomes ;n F f < P= P is an information-refinement of f

T8- the game with behavioral-strategies on I- f= enlargement of r so that F” corresponds to i= (:&MU: in i: we

consider only pure strategies) Bi = set of behavioral strategies of i in I- B= product of Bi over k Iv 6 = element of B bi = player i’s behavioral strategy in h

References

P. Dubey and M. Kaneko, Information about moles in extensive !gaum: I &i II. C‘kN 1; \ 1 ,wI\J.lli,~l:

Discussion Papers Nos. 625 & 629 ( 1982). P. Dubey and M. Shubik. Information conditions. communication and gencrnl cqurlrbrumr.

Mathematics of Operations Research 6 (1981) 186-189. M. Kaneko, The conventionally stable sets in noncooperative games \\ith limited obw\ation\: ~ki~nr-

tions and introductory arguments, Cowtes Foundation Discussion Paper No. 601 I t Ml 1. Xl. Kaneko, Some remarks on the folk theorem in game theory. \lathematEcsl Sc~~ird %ciclncc\ 3 I 89SLb

281-290. Erratum: 5 (1982) 233.

H.W. K,rhn, Estensive games and the problem of information, in: H.\Y. Kuhn and .A.\\‘. Iuchcr. cd\.

Contribution to the Theory of Games, Voi. II. Annals of Mathematics Studie$ 28 (Prifxeton 1 mk crk~ -

ty Press, 1953) pp. 193-216. A. Rubinstein, Equilibrium in supergames with overtaking criterion. Journal of Ec~w~ww 1 htxv? 2 I

(1979) l-9.