JOURNAL

OF ECONOMIC

Notes,

THEORY

52, 406-422 (1990)

Comments,

and

Letters

to the

Editor

Subgame Perfect Equilibrium in Continuous Games of Perfect Information: An Elementary Approach to Existence and Approximation by Discrete Games* MARTIN HELLWIG WWZ, University of Base& Petersgraben 51, Postfach CH-4003 Basel, Switzerland

WOLFGANG LEININGER Lehrstuhl ftir Wirtschaftstheorie, University of Dortmund, Postfach 500 500, 4600 Dortmund 50, West Germany

AND PHILIP J. RENY AND ARTHUR J. ROBSON Department of Economics, University of Western Ontario, London, Ontario, N6A 5C2 Canada Received May 15, 1987; revised January 31, 1990

This paper relates infinite-action (continuous) games of perfect information to finite-action approximations of such games and thereby obtains a new existence proof for subgame-perfect equilibrium (SPE) in the infinite-action case. Accumulation points of SPE paths of approximating finite-action games are shown to be SPE paths of the limiting infinite-action game. However, no such upper hemi-continuity property holds for SPE strategies. The discontinuity in strategies corresponds to a need for forward induction in the SPE construction for the infinite-action game, thus conflicting with the Harsanyi-Selten principle of subgame-consistency. Journal of Economic Literature Classification Number: 026. 0 1990 Academic Press, Inc.

* This paper combines material from two earlier papers of Hellwig and Leininger [9] and Reny and Robson [13]. We thank Tilman Biirgers and Reinhard Selten as well as an anonymous associate editor and referees for stimulating discussions and comments. The research of Hellwig and Leininger was supported by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 303 at the University of Bonn. Robson was partially supported by the Social Sciences and Humanities Research Council of Canada. 406

0022-0531/90 $3.00 Copyright All rights

0 1990 by Academic Press, Inc. of reproduction in my form reserved.

S.P.E. IN CONTINUOUS GAMES

$07

Consider perhaps the prototypical extensive-form game with perfect information. There are T players moving in strict sequence at times I = 1, ...~ T. Player t has perfect information on all previous select from some choice set. A strategy for player f then is a~~~ct~~~ frsm histories before t into this choice set. Each strategy vector yields a unique choice vector and this choice vector can be assumed then to determine all payoffs. HEall choice sets are finite, a subgame perfect equihbriu strategies for such a game is easily constructed by back Player T as the last player may be supposed to follow a strategy such that for any history prior to T, his choice at T just maximizes his payoff over the given choice set. Any other player t < T may be supposed to follow a that, for any history prior to t, his choice at t maximizes his is choice set taking into account to the effect that this own choice has on the subsequent players’ choices through their given st:ategy functions. If player t can only choose among finitely then regardless of the subsequent players’ strategies, problem always has a solution. Proceeding in this way player T- 1, then to player T- 2, up to player 1, one obtains a strategy constellation that is an SPE of the extensive-form ga If choice sets are infinite, however, existence of an E is less obvB~~3. The apparent problem is that the strategies of players t+ 1, ...) T typically lack the continuity properties that automatically guarantee existence of a solution to player t’s maximization problem. Hn spite of this difftcuhy, Harris [7] did prove the existence of an SPE in pure strategies for a large class of infinite-action, extensive form games of perfect ~nformat~~~. In addition, he showed that the mapping from histories into co~ti~uati~~ WE paths is upper hemi-continuous. A somewhat sim gr proof of existence was subsequently given by Hellwig and Leininger papers, choice sets are allowed to depend ~ontinu the game and an infinite horizon is permitted. In this note, we give yet another proof of the existence of ~~re~strateg~ SPE in infinite-action games of perfect information. The merit of this proof is that it is both elementary and simple: Nothing more is involve direct proof of upper hemi-continuity of WE paths as e a~~r~xirnates an ~~~nite~a~t~Q~ game of perfect information by a suita sequence of finite discretizations. The relation between infinite-action continuous games sf perfect i rmation and finite discretizations of such games is of interest in its own right. fn economic applications, the context may suggest the naturas specification of continuous choice variables. However, most extensive-form

408

HELLWIG

ET AL.

game theory has been developed for the case where all strategy sets are finite. Often therefore it is convenient to analyze a finite discretization of the continuous game of interest. But then it is necessary to know how the analysis of the finite discretization bears on the underlying continuous game. For normal-form games with compact metric strategy spaces and continuous payoff functions, the approximation is straightforward. Any sequence of Nash equilibria of successively finer approximating games has a convergent subsequence whose limit must be a Nash equilibrium of the continuous game. (See, for example, Green [6]. Fudenberg and Levine [4] also show that the reverse implication holds for s-equilibria.) For extensive-form games, the problem is more difficult. Here one is not just interested in Nash equilibria, but in subgame-perfect equilibria, that is, strategy constellations that induce Nash equilibria in all subgames of the overall game. Moreover, since strategies are functions from histories into choice sets, the strategy spaces are not generally (sequentially) compact. For the special case of games with perfect information, we resolve these issues by looking at SPE paths rather than strategy constellations. If we use successively closer finite approximations to a continuous game of perfect information, then any corresponding sequence of SPE paths of the discretized games necessarily has a limit path; moreover this limit path is an SPE path of the continuous game. This approximation result is an instance of a more general upper hemicontinuity property of the SPE path correspondence. As such, it can easily be derived from the results of Harris [7], especially from his observation that for any subgame of a given game, the map from histories into continuation SPE paths is upper hemi-continuous. As mentioned above, the merit of the approach presented here is that it provides a direct and elementary access to both the approximation result and existence. Our approximation result cannot be extended to SPE strategy constellations. Even though a sequence of SPE paths of discretized games converges to an SPE path of a continuous game, the corresponding sequence of SPE strategy constellations of the discretized games need not converge to any SPE of the continuous game. This discontinuity in strategies corresponds to a need for a type of forward induction in the construction of SPE for the infinite-action game. This forward induction conflicts with the Harsanyi-Selten principle of subgame consistency, suggesting that perhaps behavioral principles like subgame consistency must be treated rather differently in infinite-action and in finite-action games.

S.P.E. IN CONTINUOUS GAMES

2. Two EXAMEUS The nature of the problem as well as our approach to solving it are best illustrated by means of an example. Consider a game wit f course, player 2 is waiting in the wi .) Players 1 and 3 move in the obvious sequence and have payoffs and CIE[--1,

~,~~,,c,~=~,-c3; U&c,

7 c3)

=

Cl

B]=C,

C3E[4,2]==CS.

. c3;

layer 3’s SPE strategy must be a selection from the est-response correspondence i-u c-4 i PI

MC*)=

21

C,E%--l,0) c* =o Cl&yl: 11~

This has a closed graph, and hence U,(c,, c3) reaches a graph. Indeed this is at the point

This can be supported as an equilibrium

pat

E strategies

Note that the SPE strategy of player 3 breaks ties in favour of player 1. That is, if player 1 chooses c1 = 0 the choice c3 =f3(c1 ) of player 3 is t element of CJJ~(C,)= [ - 1,2] that is preferred by player 1. A tie for player 3 cannot be broken arbitrarily. For example, player 3 chooses e3 - -= 2 ins of c3 = -1 when c1 = 0, then player l’s axim~zat~o~ problem has no For general two-player games with perfect information, it is not bard to show that SPEs are obtained by backward induction given that the second player always breaks ties in favour of the first. A natural ge~erali~ation to N-player games might be to require each player to break a tie in favour of his immediate predecessor. Indeed this is etimes appropriate. suppose our example is modified by the intr ction of the notation overdue player 2. Player 2 moves between players 1 an Uz(c2,

c3)

=

c2.

c,;

CZE

[-1,2-J.

410

HELLWIG

ET AL.

Player 3’s best-reply correspondence is unchanged, so at c1 = 0, he again has a tie. If he breaks this tie in favour of player 2, and player 2 then chooses optimally, the outcome will be c2= cj =2. But as before, this presents player 1 with an insoluble maximization problem. In this example then, player 3 must continue to break ties in favour of player 1 rather than player 2, his immediate predecessor. At the same time, it is unclear how an analysis based on explicit tie-breaking can be carried out in the general case. We can also use the original two-player example to ihustrate our own approach. Replace the choice sets C, and C, by c: = { - 1, - 1 + (2/k), ...) 1) c; = ( - 1, - 1 + (3/k), ...) 2}, where these sets have just k + 1 elements. The resultant finite-action game must have an SPE. (Ties can be broken arbitrarily.) If 0 E C: and player 3 breaks the resulting tie by choosing c3 = -1, the equilibrium of the original infinite-action game is attained exactly. If O$ C:, or if player 3 breaks the tie in any other way, then player l’s equilibrium choice is, for large enough k, cf = -l/k

or -2/k

so that along the SPE path player 3 chooses c’;= -1EC5. Now as k -+ co, these finite-action game SPE paths converge to the SPE path (0, - 1) of the infinite-action game. This is true even if player 3 breaks ties in the “wrong” way in the finite-action game. However, it is not true that the SPE strategies of the finite-action games must converge to SPE strategies of the infinite-action game. For example, if player 3 breaks the tie “incorrectly” with f:(O) = 2, then of course lim f:(O) = 2 k

which is inconsistent with any SPE of the infinite-action game. Nonetheless the complete equilibrium strategies of the infinite-action game can be derived from consideration of equilibrium paths of liniteaction games and those of subgames. To illustrate this procedure with the current example, set f,=c,=limc’f=O f3(0) = lim f:(c:)

= lim c$ = -1.

S.P.E. IN CONTINUOUS CYAMES

(Note that the arguments of the strategies also depend on k.) T completely defines fi, but f3 must also be defined off the equilibrium path Consider then the subgame defined by an arbitrary c1 # 0. Find a sequerace

and define

completing the definition of f3. As we show below, this method generating SPE strategies is quite general.

of

3. THE GENERAL FRAMEWORK

We consider an extensive-form game T with T players t = 1, .~l, T, where T is finite. Each player t has a compact metric choice set C,, taken to be independent of history, and a continuous payoff function u,:clx

... xC,+R.

Players move one after the other, first player 1, then clayey 2, with player T moving last. When any player t with t > 1 moves, he is fully informed about the history e,_ 1 = (c,, .... c, _ i) of the moves chosen by layers I, ~..>t - 1. Let E,- 1 = C, x . .. x C,+, be the set of possible histories for player t. A strategy for player t ( > 1) is a function

ft:E,-I

-+ c,,

which specifies for each history e,- 1 the choice c, =4;(e,A strategy for player 1 is simply a choice

1) t

For t = 1, ~.., T, the set of strategies of player t is denoted as 3,. Any strategy constellation (fi, .... fT) determines a unique play? i.e., a sequence of choices c1 =J;, c,=f2(c1), ~..,c,=fr(c,, .... cTUIj. For player f, the strategy constellation (fi, ...) fT) t us generates the

Wt(fi, ..'2fTI:= U,(fi,fi(fi),f3(fi,fi(f~)), e constellation

-~~fTifl>f2~fi8> -YIP)

(fI , .... fT) is a Nash equ~~~~r~~i~ ifi

w,u, >..‘, f,> ~.~? fT1 3 W,(fl> ...f “L ...>fT1

for b= I, ..,, TandaHl$,ES,.

412

HELLWIGET

AL.

Similarly, for any t > 1 and any history e,- 1 E E,- 1, the strategy constellation (f,, ....fT) d et ermines a unique play c, =f,(e,- 1), c,+ 1 = in the subgame determined by f,+l(e,-l, c,), .... c~=fAe~--l, cr, .... cTel) et-15 having choice sets C, and payoff functions U,(e,- 1, .) for players z = t, .... T. For r = t, .... T, the corresponding payoff of player r is w:(e,- 1, f,, -., fT) := UJe,_ 1, fi(e,- 1), .f,+ ,(e,- lr L(e,- 1)), -., fAe,-

1, fk-

IL -.)I.

A strategy constellation (fi, .... fr) is a subgame-perfect equilibrium, or SPE (Selten [14]), iff it is a Nash equilibrium and moreover

fort=2 ,..., Tandalle,_,EE,_land~ES,.Thus(f~ ,..., f,)isanSPEof the game r iff it induces a Nash equilibrium in r and in any proper subgame of r. The unique play (cl, .... cT) that an equilibrium (fi, .... jr-) induces in r is called the equilibrium path. Equilibrium paths for subgames are defined analogously. 4. THE APPROXIMATION

RESULT

We consider discrete approximations to the game K For k = 1,2, ... and any t, let C: be a finite subset of the choice set C,, and let Uf be the restriction of U, to CT x ... x Cky. For any k , let Tk be the perfectinformation game in which players t = 1, .... T with payoff functions U: choose c, E C: one after the other. In Tk a strategy for player 1 is a choice f T E Cl; for t > 1, a strategy for player t is a function f :: E& 1+ C: where Nash equilibria, subgame-perfect equilibria and Ef-,= cfx 0.. xc:-,. equilibrium paths for Tk are defined in the same way as for the original game r. We say that the sequence {rk’ / approximates the game r iff for each player t, the Hausdorff distance between the choice set Cf in Tk and the choice set C, in r converges to zero as k becomes large, i.e., iff lim k-m

max min dJc,, cf) = 0 cfeCr

c!sCf

for all t, where d,( .) is the metric on the choice set C,. Our approximation result can then be formulated as THEOREM 1. Suppose that {rk} IS a sequence of discrete games that approximates r. Further supposethat for each k, (f f, .... f $1 is an SPE of

S.P.E.

IN

CONTINUOUS

GAEVIES

423

I-“, and that ($, .... c”,) is the corresponding equilibrium path. If (CL,...>CT) is any limit point of the sequence {(2:, .... ?“,)I, then there exists QIZ SPE (fl, .... f,) of T which has (F,, .... CT) as its equilibrium path.

Proof. We give an explicit construction for the desired strategy constellation (f,, ...?fT). On the equilibrium path, jr, .... jT must obviously satisfy the conditions

fT(CI) ...) FT- 1) = CT. The problem is to specify fr , .... fT ofl the equilibrium path. For this purpose, we use the SPEs of the approx~mat~u~ games Tk. For each t and each subgame e, E E, between players t + 1, ...9 r, we construct a subgame equilibrium by taking limits over a suitable subsequence of subgame equilibrium paths in the approximating games. e proceed step by step, beginning with f = 1 and continuing with f = ~~,T- 1. Each stage I defines the equilibrium path for any subgame in which player t viates from his equilibrium strategy, and moreover he is the last to do so. player t does not deviate, then stage t adds nothing to what has already been defined. As a preliminary step, note that for all t, the Hausdorff convergence of 6: to C, implies that for any C,E C, there is a sequence (Zf(c,)) convergmg to c, such that c”F(cz)E Cf for all k. Also, since (Cr , “.., 2,) is a limit point of the sequence (?‘;, -., c”,), there is an (infinite) set D, of indices k sue that

for ah t. The analysis is restricted to this index set D,. Stage 1. Take any c, E C, and consider the subgame determined by e1 = cr. Define an approximating sequence of subgames (er(cr)} by setting

64252i2.12

414

HELLWIG

In Tk, the subgame determined

ET AL.

by e’;(c,) has the equilibrium

path

Z3cI ; 1) =fi(el;(c,)),

%CI;

1)=fk,(4(cl),

$(c,;

l), .... c$-l(~l;

1)).

By compactness, there is a subset D,(c,) c D, of indices such that the limits C,(c,; l)=

lim

C:(c,; I),

k&f?,)

C,(c,; l)=

lim

CF(cl; 1)

kk,&c,)

exist. Accordingly,

set

f2(Cl) = G(c, ; 1) f3(c,, G(c,; 1)) = Cdc1; 1) fT(Cl2 UC, ; 11,-..,CTel(c,; l))=CT(CI;

1).

These conditions determine the strategies of players 2, .... T on what will turn out to be the equilibrium path of the subgame determined by player l’s selection of c1 E C,. In particular, they fully determine the strategy of player 2. 1 As an induction hypothesis, from previous stages : Stage t, t = 2, .... T-

(i) D,(e,)c

suppose that the following

For 7 = 1, ..,, t - 1 and D, such that if t>2,

any e, E E,,

are available

a sequence of indices

for z = 2, .... t-l, all e,-,sE,_,, and c,EC,. (ii) For z = 1, .... t - 1 and any e, E E,, a sequence of subgames such that

4(c) --f e,,

k EDr(erL

S.P.E.

IN

CONTINUOUS

GAMES

415

where e:(e,)~ Et for all k. Furthermore, if r>2 and ZE 12, &..,t- I> then for all e, _ 1 E E, _ I and c, G C, there exists Ef(e, _ 1, c,) E C/: such that

(iii) For P = 1, .~., t - 1, a complete specihcation of the strategy fr,, i of player z + 1. That is, for any e, E E,,

(iv) For z = t + 1, ...~ T, a specification of the strategy f, of phiyer r only on what will turn out to be the equilibrium path of each subgame corresponding to each history e,- 1 E E,- r This ath is it,(e,- 1; t - 11, which is constructed as the limit of approximating pat z = t + 1, .... T. The indices, k, used in the construction are construction is analogous to stage 1 above. Now consider the contribution of stage e. P’ake any e,- I E E,_ 1 and any C,E C, and consider the subgame determined by e,= (e,- r, c,). efine an approximating sequence of subgames (ef(e, _ I, cr)}? k E D, _ r(e, _ r)? by

where

In r”, the subgame determined

by ef(e,) has t e equi~~b~nrn path

e+ l(@,i t) =f “;+ l(@%,))~

By compactness, there is a subset D,(e,) c D,_ I(er- 1j c D, such that the limits c,+l(e,;

t) = !‘t”, Cf+ l(e,; t) k ED,(Q)

416

HELLWIG

exist. Accordingly,

ET AL.

set

ft+l(e,)=F,+,(e,; t) .fr+Aet, ?I+1(5; t)) = Ft+Aer; t) fAe,, ~,+,(e,; t), -., L,(e,;

t)> = ?,(e,; 0

These conditions are consistent with whatever restrictions on f, + 1, .... fT have been imposed in stages 1, .... t - 1. Thus, if e, is of the form (e,-,,f((e,-r)), then by definition of ?f(e,+,, ct), Cf+l(e,-,,f,(e,-l);

t)=S~+l(e~-l(e,-l),f~(e~-l(e,-l))) =c -f+l(e,-l;

t- 11,

where the second equality follows from the construction described in (iv) of the induction hypothesis. Hence, Ft+ r(e,- r, ft(e,- r); t) = Ct+ ,(c,- I; t- 1). Similarly, it follows that c,(e,-r, ft(e,-,); t) = c,(e,-l; t- 1) for z = t -I- 2, .... T. Now we have determined the strategy f, + r of player t + 1 so that

Moreover, we have determined the strategies of player t + 2, .... T on what will turn out to be the equilibrium paths of the subgames determined by histories e, E E,. Thus, the induction argument is complete. By stage T- 1, we have determined a complete strategy (fr , .... fr). We must show that this strategy constellation constitutes an SPE of I’. Suppose not. Then some player t has an alternativer[ which does better thanf, in r or in one of the subgames of I’. If t > 2, let e,- r E Et- L be the subgame in question so that WXe,-l,Af,,f,+l,

.... f7-)> WKe,-l,f2,L+l,

-.,f~).

From stage (t - 1) of our construction, f,, ....fr are defined so that V(e,-,,f,,f,+,, Similarly,

....fr)=

lim

k-co ksD,-Ice,-1)

We:-

,(e,- 1), f F, -., f”,>.

from stage t of our construction f, + I, .... fr. are defined so that WXe,-l,Ji;,f,+l, = lim k-cc keDt(e,-l&e,-1))

.-,ST) W(et-

,(e,- I), ff, ff, 1, --, f”,),

S.P.E.

IN CONTINUOUS

GAMES

417

where$f is any strategy for player t such thatf:(et_ ,(e,- 1)) = i;~(~,(~,_~ I)I. Since Dr(er-li~~:t(et--l))~Dt-l(er-l), it follows that

for any suffcientiy large kED,(e,p,,ft(e,_,) ut this csntradicts the e assumption that assumption that (f:, .... f “,) is an SPE of r”. for some t b 2, ff is better for t than fi in some subgame e,- I E E,- r must be false. A precisely analogous argument shows that for player 1 there is no strategyfr that is better thanf, = F1. Hence (fi, .... J’r) is an WE of J?

Existence

of an SPE of r

Theorem 1 provides us with a new and elementary appr~acb to proving the existence of an SPE of the infinite-action game 6: Since the choice sets C, are compact and metric, each of them has a cQ~ntab~e dense subset {I$, cf, ...) cf, . ..I If we set C:= {cf, .... cf> for k= 1, 2, .... then, as k --, co, t ce between Ct and C, goes to zero. It follows that if we write Tk for scretization of r in which each player t is restricted to the set C/; r than C,), then the given sequence of discrete approximates & in the sense defined in Se choice sets Cf are finite, for each k, a of rk exists. NQW fclr k = I, 2, .... let (E’;, Ci, . ... ?“,I be an SPE pat k. Since C, x . . . x C, is a compact tric space, the sequence ((St) ...) F”,) )F- 1 has a hmit point (C-1, .~.7CT). Theorem 1, (Cr , .... Zr) is an SPE path of IY Thus: THEOREM 2.

The infinite-action

game F has an SPE.

ifferent proofs of Theorem 2 were previously given is [7], and Hellwig and Leininger [lo]. The proof advantages that it is elementary and that it clarifies the relation between discrete and continuous games of perfect reformation. Harris [7j and Hellwig and Leininger [lo] actualBy prove Theorem 2 for the more general case of an infinite-action, pe~fect~~~formation game r with a countable number of players whose e sets may vary ~ont~~~ously with past histories of the game. To t our argument to this general case, we require little more than a in notation. For suppose that r is a game of perfect informatio players I= I, 2, . ..

418

HELLWIG

ET AL.

moving in sequence and having compact metric underlying choice sets C,, continuous and closed-valued constraint correspondences qr: E,- 1 -+ C,, and continuous payoff functions u,: C, x C, x . . . -+ R. For k = 1,2, .... construct a game Tk with k players t = 1, .... k having choice sets Ct, constraint correspondences qf, and payoff functions u: such that -

C: is a finite subset of C,; for t = 2, .... k - 1, C: is a finite subset of C, such that for all e,- i E Cf n qt(et- i) # 0, and cp: is the correspondence given cfx ... xc:_,, by the formula pF(e,-,)=Cfncp,(e,-,) for e,_,ECfx ... XC’-,; - c;=c,xck+lx “-, and qt is the constraint correspondence that maps histories ek- i E E,- i into sequences (c,, ck+ i, ...) satisfying for T=k, k+l, .... CT E vr(ek1f ck, -, c,-~) - for all t, uf= zf,/ Cl; x ... x Ci. Because CE is compact and, for t < k, the choice sets C: are finite, an SPE of Tk exists for each k. For k = 1,2, .... let (Cf, .... $) be an SPE path of Tk, and observe that (Cf, .... L!,“) is an element of C, x C, x . . . . Since C, x C2 x . . . is a compact metric space, the sequence { ($, .... Zt)}F= 1 has a limit point (Ci, C,, .... ). If the discretizations Cf have been chosen in such a way that for each t and for each e,- i E UF= I Cl; x ... x C’f- i, the Hausdorff distance between C’f and C, as well as that between CF n q,,(e,- i) and C, n qt(et- 1) goes to zero as k--t co, then the construction underlying Theorem 1 can be used without change to show that (21, c2, ...) is an SPE path of r. Upper Hemi-Continuity

of the SPE Path Correspondence

Theorem 1 is essentially a statement about the upper hemi-continuity of the correspondence relating SPE paths to the data of the game. Upper hem&continuity of equilibrium paths seems to be the key robustness property of equilibria in extensive-form games. In contrast, the examples in Section 2 show that the correspondence relating SPE strategy constellations to the data of the game will typically not be upper hemi-continuous. The upper hemi-continuity of the SPE path correspondence has been observed before. In the course of his general existence proof, Harris [7] proves upper hemi-continuity of the correspondence relating SPE paths of subgames to prior histories. From this result, which itself is proved by nonelementary methods, Theorem 1 can actually be derived by a simple argument for which we are indebted to a referee. Introduce a dummy player, player 0, with choice set (l/k 1k 2 1 } and constant payoff. Suppose that if player 0 chooses l/k, the approximating game Tk is played by the remaining players. If player 0 chooses 0, the original game r is played. Then all the assumptions of

S.P.E.

IN CONTINUOUS

GAMES

419

arris [ 7] are satisfied. In particular, each of the original players has a feasible choice correspondence which depends continuously on hist Theorem 1 then is an instance of the upper hemi-continuity of the S path correspondence for the subgame of the extended game which begms with player 1. More generally, the preceding argument-or the argument in our proof of Theorem l--establishes upper hemi-continuity of the SPE path correspondence with respect to any parameter that enters payoff functions and constraint correspondences continuously. A systematic discussion of this upper hemi-continuity property of SPE paths in extensive-form games is presented by Biirgers [I]. (Biirgers [2] used this property to study SPE in infinite-horizon games by taking limits of e-SPE paths of ~nite-b~~izQ~ games.) Lower Hemi-Continuity

of the SPE Path Correspondence?

The proof of Theorem 2 shows that at least one SPE of the inliniteaction game r can be approximated by SPEs of the approxi action games Tk. We now discuss the question whether clll S be approximated. In a trivial way, such an approximation is always possible $ the iag games Tk are chosen appropriately. Namely, let (jr3 .... jr) of r, and suppose that for any sufficiently large k, the choice sets Cf, . ... C”, are such that

and

for t = 2, .... T and all e,_ I E E:_ 1. Then for sulIiciently large k, (&) ..,9ST) E of Tk and has the same equilibrium path in Tk as in i’. owever, for an arbitrary approximating sequence Irk), it is generally true that any SPE path of the i~~nite-action game r can approximated by a subsequence of SPE paths of the approximating ga In other words, the correspondence relating S E paths to the &c-&e sets C,, .,., Cr is not lower hemi-continuous. A simple example establishes the point. Set T= 1 so that the ‘“game” reduces to a simple optimization problem. Let C, = [O, 2] and Ui = (1 - c, j2. Then there are two equilibria, f: = 0 and fT* sequence of approximating games with the property that contain c1 = 0, but not c, = 2. For example:

420

HELLWIG ETAL.

Then f;” = 0 is the unique equilibrium of each approximating there is no sequence of equilibria approximating j’;“* = 2.

game, and

6. CONCLUDING REMARKS: SUBGAME-CONSISTENCY IN INFINITE-ACTION GAMES

We conclude this note with a remark on some behavioural aspects of our analysis. In the proof of Theorem 1 as well as in the examples of Section 2, the construction of an SPE involves a kind of forward induction whereby the equilibrium strategy of any player t is at least partly determined by the requirement that the maximization problems of the preceding players 1, .... t - 1 have well-defined solutions. Thus in our first example, player 3 had to choosef,(O) = -1 in order to ensure that the problem

of player 1 has a solution. If player l’s payoff function had been specified as Ul(Cl, c3) = -(cl -G), then the same consideration would have required player 3 to choose f3(0) = 2. Indeed if there were a prior player 0 with choice c0 E [ - 1, 11, and if player l’s payoff function were replaced by Ul(CO,

Cl,

c,)=c,(c1--3),

then player 3’s SPE strategy would have to satisfy

.f\(coA={;l

;; z”z,” 0 .

Thus player 3 would have to condition his behaviour on the choice c,, of player 0 even though his own payoff does not directly depend on co. More generally, the construction underlying Theorem 2 selects the solution of each subgame according to the requirements of the overall game in which the subgame is embedded. This type of forward induction should be contrasted with the principle of subgame consistency proposed by Harsanyi and Selten [S]. According to this principle, the set of equilibria of a game should depend only on the structure of the game and should be independent of any larger game in which the game might be embedded as a subgame. This seems to be the strongest possible expression of the postulate that identical situations

421

S.P.E. IN CONTINUOUS GAMES

should produce identical outcomes, which is at the heart of all theorizing. (The question is of course whether two s~~~a~~o~s with strategically equivalent subgames can be regarded as i entical if they hav reached by different histories. Against subga -csnsistency, see C 33, Kohlberg and Mertens [I I], and Leininger [12].) then does subgame consistency apply to i~~~~te-action games wit information? Our examples in Section 2 and above show th,, subgame consistency cannot in general be achieved in SW been suggested to us, then, that one should forget the altogether and concentrate on finite approximat~o~s~ pies of Section 5 show that the set of subgame perfect equilibria of a finite a~~roxirnat~~g game, while satisfying subgame consistency, is hi sensitive to the chosen discretization. This sensitivity is problematic if t articular discretization which suggests itself as the most natural. pointed out be a referee, there is analogy here between the of subgame consistency and the p lem of the wea~~dom~n~~ce crrterion in infinite-action games. An in&it tion game may not have an that satisfies the weak-dominance criterion. For instance, Bertrand price game with constant costs has a uni this equilibrium involves the use of weakly do~~~ate~ ever, any finite discret~zat~o~ of game has an equiategies. As one approxi tes the i~~~~te-action game by finer and finer iscretizations, these equilibria in un strategies converge to the unique equilibrium of e co~~i~~o~s game. This observation suggests that the wea~-~orn~~~~ criterion to3 is mofe lied to the discrete approximat~o~s of an infinite action game than to the finite-action game itself. More generally, there to be a s~g~i~ca~t ifference between the use of ~e~av~our~~ r ents of equi~~br~~rn in infinite-action games and the use of the same re~ne~e~ts In ~~~te-action games.

I. T. BBRGERS, Upper hemi-continuity of the correspondence of s&game-perfect eqliiiibrium outcomes, Discussion Paper, University of Basei, July 1988. 2. T. BijRGERS, Perfect equilibrium histories of finite and infinite horizon games, J. IZccurr. Theory 47 (1989j, 218-227. 3. 1. M. CHO AND D. PREPS. SignaIling games and stable equiiibria, @pi. J. .!%a~ 102 (198’7), 179-221. 4. D.

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(1985), 261-279. 5. S. GOLDMAN, Consistent plans, Rev.Ecan. Sfzrdies 47 (1980), 533-537. 5. E. GREEN, Continuum and finite-player noncooperative models of competition. Econometrica 52 (1984), 957-993.

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HARRIS, Existence and characterization of perfect equilibrium in games of perfect information, Econometrica 53 (1985), 613-628. J. C. HARSANYI AND R. SELTEN, “A General Theory of Equilibrium Selection in Games,” MIT Press, Cambridge, MA, 1988. M. HELLWIG AND W. LEININGER, A note on the relation between discrete and continuous games of perfect information, Discussion Paper No. A-92, University of Bonn, 1986. M. HELLWIG AND W. LEININGER, On the existence of subgame-perfect equilibrium in infinite-action games of perfect information, J, Econ. Theory 42 (1987) 55-75. E. KOHLBERG AND J. F. MERTENS, On the strategic stability of equilibria, Econometrica 54 (1986) 1003-1038. W. LEININGER, Strategic equilibrium in games with perfect recall, Discussion Paper, Bell Laboratories, 1986. P. J. RENY AND A. J. ROBSON, A simple proof of the existence of subgame perfect equilibrium in infinite-action games of perfect information, Discussion Paper, University of Western Ontario, 1987. R. SELTEN, Reexamination of the perfectness concept for equilibrium points in extensive games, ht. J. Game Theory 4 (1975), 25-55.

7. C. 8. 9.

10. 11.

12. 13. 14.

HELLWIG ET AL.