The Existence of Subgame-Perfect Equilibrium in Continuous Games with Almost Perfect Information: A Case for Public Randomization Author(s): Christopher Harris, Philip Reny, Arthur Robson Source: Econometrica, Vol. 63, No. 3 (May, 1995), pp. 507-544 Published by: The Econometric Society Stable URL: http://www.jstor.org/stable/2171906 Accessed: 05/12/2008 21:50 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=econosoc. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected].

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Econometrica, Vol. 63, No. 3 (May,1995),507-544

THE EXISTENCE OF SUBGAME-PERFECT EQUILIBRIUM IN CONTINUOUS GAMES WITH ALMOST PERFECT INFORMATION: A CASE FOR PUBLIC RANDOMIZATION' BY CHRISTOPHERHARRIS, PHILIPRENY, AND ARTHUR ROBSON2 The startingpoint of this paperis a simple,regulardynamicgame in which subgameperfectequilibriumfails to exist. Examinationof this exampleshowsthat existencewould be restored if players were allowed to observe the output of a public-randomization device.The main resultof the papershowsthat the introductionof publicrandomization yields existencenot only in the example,but also in a large class of dynamicgames.It is also arguedthat the introductionof publicrandomizationis the minimalrobustextension of subgame-perfectequilibriumin this class of games. KEYWORDS: Existence, subgame-perfect, infinite-action games, imperfect information, stage games, counterexample, correlated equilibrium.

1. INTRODUCTION

of a dynamic game. Firms set out with exogenously specified capacities (which are known to all). In period one they simultaneously choose investment levels (possibly on a random basis), and are then informed of one another's choices. The result is a change in capacities. In period two firms simultaneously choose production levels (within their capacity constraints), and are then informed of one another's choices. The result is a change in inventories. In period three firms simultaneously choose prices. The vector of prices chosen affects the vector of demands for their products, but so do certain exogenous random factors. Firms are informed of one another's chosen prices and of the final realized demands. (The demands bring about a second change in inventories.) The three period cycle of choices then begins afresh. And so on. This game is an example of a game of the following general type. Time is discrete. There is a finite number of active players. There is also a passive CONSIDER THE FOLLOWING EXAMPLE

1

The present paper combines results from two separate papers. Harris (1990) gave the first counterexampleto the existence of subgame-perfectequilibriumin games of almost perfect information,proved the existence and upper semicontinuitytheorems that obtain once public randomizationis introduced, and drew attention to the fact that the introductionof public randomizationis essentialif there is to be robustnesswith respect to errorsin the extensiveform. Harrisalso pointed out the interestof a minimalcounterexample.Reny and Robson (1991),who had been workingindependentlyon the problemof findinga counterexampleto the existenceof subgame-perfectequilibriumin games of almost-perfectinformation,gave the first minimalcounterexample.Section2 of the presentpaperis based on Reny and Robson(1991).The remainderof the paperis based on Harris(1990). 2The authorswould like to thank Philippe Aghion, PatrickBolton, Drew Fudenberg,Martin Hellwig,Daniel Maldoom,Andreu Mas-Colell,Eric Maskin,Jean-FranqoisMertens,Meg Meyer, Leo Simon,Max Stinchcombe,Jean Tirole, John Vickers,Bill Zame, and participantsat seminars andworkshopsat Oxford,StonyBrook,and Harvard.Reny and Robsonacknowledgethe supportof the Social Sciencesand HumanitiesResearchCouncilof Canada. 507

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player,Nature. In any given period: all players(both active and passive)know the outcomes of all previousperiods;the set of actions availableto any active player is compact, and depends continuouslyon the outcomes of the previous periods; the distributionof Nature's action (which is given exogenously)depends continuouslyon the outcomesof the previousperiods;the players(active and passive)choose their actionssimultaneously;and the outcomeof the period is simplythe vector of actions chosen. The outcome of the game as a whole is the (possiblyinfinite)sequence of outcomes of all periods, and players'payoffs are bounded and depend continuouslyon the outcome of the game. We shall refer to a game of this type as a continuousgame of almostperfect information. The class of continuousgames of almost perfect informationis as regulara class of dynamicgames as one could ask for. A counterexampleshows,however, that games in this class need not have a subgame-perfectequilibrium.3'4It is therefore necessary to weaken the equilibriumconcept in such a way that existenceis restored.5 What is the most naturalweakeningof the equilibriumconcept? One clue as to the answerto this questionis providedby the followingconsiderations.First, if we simplifythe class of games under considerationby requiringthat players' action sets are always finite, then subgame-perfectequilibriumalways exists. Hence the nonexistenceproblem appearsto relate specificallyto the fact that we allow a continuumof actions.Secondly,suppose insteadthat we simplifythe class of games under considerationby assumingthat players' action sets are independent of the outcomes of previous periods. Then a natural way of approximatinga game is to considersubsets of players'action sets that consist of large but finite numbersof closely spaced actions.6Moreover,if one takes a sequence of the increasinglyfine approximations,and a subgame-perfectequilibriumof each of the approximations,then it is naturalto expect that any limit of the sequence of equilibriumpaths so obtainedwill be an equilibriumpath of the original game.7One can, however,find examples in which the limit point

3 Van Damme (1987) gives an example of a game with a continuum of actions that does not possess a sequential equilibrium. His game does, however, possess a subgame-perfect equilibrium. Our example is more regular than his-it is a game of almost-perfect information-and it shows nonexistence of a weaker form of equilibrium, namely subgame-perfect equilibrium rather than sequential equilibrium. 4 The problem is reminiscent of that of the nonexistence of Nash equilibrium in pure strategies. But there is an important difference: if agents' action sets are always finite, then subgame-perfect equilibrium does exist. s We argue below that the game should be extended to allow for public randomization in order to restore existence. In related but independent work, Iorio and Manelli (1990) consider van Damme's (1987) example of the nonexistence of sequential equilibrium in a continuous signalling game. They argue that such games should be extended to allow for cheap talk in order to restore existence. 6 Hellwig et al. (1990) were the first to consider such an approximation, in the context of finite-horizon games of perfect information. 7Convergence of equilibrium paths, as used in Harris (1985a), Hellwig et al. (1990), and B6rgers (1989) seems more relevant than the convergence of strategies considered by Fudenberg and Levine (1983) and Harris (1985b).

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involves a special form of extensive-formcorrelation:at the outset of each device.8'9"0 period agents observethe output of a public-randomization A second, related clue is obtainedby consideringthe possibilityof bounded rationality.It would appear that, in games of the class with which we are concerned,subgame-perfects-equilibriumdoes exist." This being the case, it would be naturalto expect that, if one takes a sequence of s-equilibriumpaths with E tendingto zero, then anylimit point of the equilibriumpaths so obtained will be an equilibriumpath of the game. Once again, however,the limit point may involvepublic randomization. These considerationssuggest that, at the very least, the equilibriumconcept should be weakened to allow for public randomization.That such a weakening may also be sufficientis suggestedby the workof Simonand Zame (1990).Their work implies that, if a two-stagegame is extended to allow the observationof device at the outset of the second stage, the output of a public-randomization then equilibriumwill exist in the extendedgame.12"13 8The concept of extensive-form correlation was introduced by Forges. See Forges (1986), for example. 9 Fudenberg and Tirole (1985) consider two such games. They point out that the obvious discretizations of these games have a unique symmetric subgame-perfect equilibrium, and that the limiting equilibria obtained as the discretization becomes arbitrarily fine involve correlation. Their games do not, however, yield counterexamples to the existence of subgame-perfect equilibrium. For both games possess asymmetric equilibria which do not involve correlation. 10Borgers (1991) makes a related point. He considers a family of discrete games indexed by n E-N, and shows that the limiting equilibria obtained as n -X oo involve correlation. His example is more relevant for our purposes, since in his example the limiting game is a game in discrete time. "1This result was first pointed out by Chakrabarti (1988). We believe his result, but cannot vouch for the proof. 12 Strictly speaking, the work of Simon and Zame applies to normal-form games in which there is an upper semicontinuous and convex valued payoff correspondence. They do, however, motivate their framework by reference to two-period games. They point out, in particular, that in two-period games in which only one player moves in the second stage, the continuation-payoff correspondence for stage 1 will be upper semicontinuous and convex valued. 13 We should also mention related work on stochastic games by Mertens and Parthasarathy (1987) and Nowak and Raghavan (1992). Mertens and Parthasarathy consider stochastic games in which: there is a finite number of players; the set of actions available to a player is compact, and depends measurably on the current state; the payoff to a player from the current period depends measurably on the current state and continuously on the profile of actions chosen; and the probability distribution of the next state depends measurably on the current state and norm continuously on the profile of actions chosen. They show that, for such games, subgame-perfect equilibrium exists. Nowak and Raghavan consider the more restrictive class of games in which there is, in addition, a single probability measure over the state space with respect to which the probability distribution of the next state is absolutely continuous for all possible choices of the current state and all possible profiles of actions. They show that, if such a game is extended to allow for public randomization, then there exists a subgame-perfect equilibrium in Markov strategies. (Note that any stochastic game of the class studied by Nowak and Raghavan can be reduced to a stationary game of the same class by including time in the state space. Hence the existence result just stated follows from the existence of subgame-perfect equilibrium in stationary Markov strategies in stationary games. It is this latter, stronger, result that Nowak and Raghavan actually prove.) Comparing the work of Mertens and Parthasarathy and Nowak and Raghavan, it can be seen that public randomization plays a different role in the result of Simon and Zame from the role that it plays in the result of Nowak and Raghavan. In the class of games considered by Simon and Zame, public randomization is used to ensure the existence of subgame-perfect equilibrium. In the class of games considered by Nowak and Raghavan, subgame-perfect equilibrium already exists. Public randomization is used to ensure the existence of subgame-perfect equilibrium in Markov strategies.

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The present paper shows that if a continuous game of almost-perfect information is extended to allow for public randomization, then the extended game will possess a subgame-perfect equilibrium. It shows, furthermore, that the set of equilibrium paths of the extension of a game depends upper semicontinuously on the game.'4 These results appear to be the appropriate generalization, to the present context, of Harris' (1985a) results for games of perfect information. It follows from the upper semicontinuity result that any limit point of equilibrium paths of finite approximations to a game is an equilibrium path of the extended game.15 It can also be shown, although we shall not do so here, that any limit point of e-equilibrium paths of a game is an equilibrium path of the extension of the game.16 So the extension we use suffices to capture, albeit in an idealized form, all the phenomena that can be obtained by the two most classical methods for circumventing nonexistence. It is not the minimal extension with these properties, as can be seen by considering any finite game.17 It is, however, the minimal robust extension of subgame-perfect equilibrium. In other words, suppose that we require, first, that the set of equilibrium paths of a game be contained in the set of equilibrium paths of the extended game; and secondly, in the spirit of Fudenberg, Kreps, and Levine (1988), that every equilibrium path of every nearby game be near an equilibrium path of the game itself.18 Then the introduction of public randomization is the minimal extension for which both these properties hold. Or, to put the same point another way, the graph of the mapping which takes a game into the set of equilibrium paths of the extension of the game is the closure of the graph of the mapping which takes a game into the set of equilibrium paths of the game itself.19 The organization of the paper is as follows. In Section 2 we present our counterexample to the existence of subgame-perfect equilibrium. The example involves a two-period game with two players in each period. One of the players in period 1 has an infinite action set, but the remaining three have only two actions each. The example is therefore a minimal example of a game in which subgame-perfect equilibrium fails to exist.

14 More precisely, it is the set of marginals over the original game of paths of subgame-perfect equilibria of the extension of the game that depends upper semicontinuously on the game. 15Hellwig et al. (1990) and Borgers (1991) both stress the importance of applying the upper semicontinuity of the equilibrium correspondence in this way. 16 The proof is a straightforward combination of the methods of the present paper with those of B6rgers (1989). Cf. Harris (1990). 17 If we require that the extensive form of a finite approximation to a game is contained in the extensive form of that game, then the set of limit points of equilibrium paths of finite approximations to a finite game is precisely the set of equilibrium paths of the game itself. (In Section 4.4 we adopt a slightly wider definition of finite approximation for which this statement would not hold.) And the set of limit points of e-equilibria of the game itself is again precisely the set of equilibrium paths of the game itself. (Here there is no ambiguity in the definition.) 18 It should be stressed that the set of perturbations with respect to which the equilibrium concept is required to be robust includes perturbations not only of payoffs but also of the extensive form itself. 19 The word "mapping" is used loosely. For the space of games and the space of paths are both classes, not sets.

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Section 3 formulates the class of games with which the present paper is concerned, and introduces the concept of the extension of such a game. Section 4 shows that subgame-perfect equilibrium always exists in the extension of a game, and that the set of equilibrium payoff vectors varies upper semicontinuously with the game. The proof, an overview of which is given at the beginning of the section, exploits insights from several earlier papers, including Harris (1985a), Hellwig et al. (1990), Simon and Zame (1990), Borgers (1991), Mertens and Parthasarthy (1987), and Duffie et al. (1994). Section 5 refines the results of Section 4, showing that the set of equilibrium paths varies upper semicontinuously with the game. The proof is closely analogous to that of Section 4. In our view, the ease with which the proof used in Section 4 extends to the more general problem treated in Section 5 is an important argument in its favor. Section 6 attempts to develop a perspective on the main result. It is shown there that there are at least three types of game in which the introduction of public randomization is not necessary to obtain the existence of subgameperfect equilibrium: games of perfect information, finite-action games, and zero-sum games. The proofs, which naturally we only sketch, once again involve only simple variations on the format used in Section 4. In particular, we arrive at yet another proof of the existence of subgame-perfect equilibrium in pure strategies in games of perfect information.20 Section 7 shows that introducing public randomization is the minimal robust extension of subgame-perfect equilibrium in the class of games with which this paper is concerned. And Section 8 concludes.

2. THE COUNTEREXAMPLE

Any counterexample to the existence of subgame-perfect equilibrium in continuous games of almost perfect information requires at least two stages, since otherwise standard existence results for Nash equilibrium apply. There must be at least two players in the first stage, since, with only one, the upper semicontinuity of the second-stage Nash equilibrium correspondence suffices to guarantee existence. There must also be at least two players in the second stage since, with only one, the best reply correspondence for that player is convexvalued and the results of Simon and Zame (1990) ensure existence. Finally, of course, at least one player must possess an infinite choice set.

20 This proof combines elements of those of Harris (1985a), Hellwig et al. (1990), and B6rgers (1989). And it yields measurability, as in Hellwig and Leininger (1987). (Note that it is not necessary to adopt the approach of Hellwig and Leininger (1987) in order to obtain measurability. Indeed, if the assumptions of Harris (1985a) are specialized appropriately (by assuming that the embedding spaces for players' action sets are compact metric instead of compact Hausdorff), then it is simple to show that exactly the construction given there leads to strategies that are measurable. Indeed, in the notation of page 625 of Harris (1985a), all that is necessary is to ensure that the functions gi can be chosen to be measurable; and that this is possible follows from Lemma 13 of the present paper.)

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2.1. The Basic Example

The followinggame is minimalin all of the above senses. In stage 1, players A and B simultaneouslychoose a e [-1,1] and b E {L, R} respectively.In stage 2, playersC and D are informedof the choices made by playersA and B in stage 1. They then simultaneouslychoose c E {L, R} and d E {L, R} respectively, and the game ends. Player D's payoffdepends only on his own action and that of playerA: if he chooses L he gets a payoff of - a, and if he chooses R he gets a payoffof a. Player C likewiseobtains -a if he chooses L and a if he chooses R. PlayerB's payoffdependson his own action and that of player C: if he chooses L then he gets 1 if player C chooses L and -1 if C chooses R; and if he chooses R then he gets 2 if C chooses R and -2 if C chooses L. Finally, there are three contributionsto player A's payoff.First, if B and C make the same choice he gets - Ia , whereas if they make differentchoices he gets al. Secondly,if C and D make the same choice he gets 0, whereas if they make differentchoices he gets - 10. Thirdly,he gets - la 12 A's overall payoff is the sum of these three contributions. This completes our descriptionof the game. The reader will note that each player's action set is compact and independent of the actions of the other players,and that each player'spayofffunctionis continuous. The motivationsof the four players can be understood loosely as follows. Players C and D are agents of player A, and will follow his instructions providedthat they are unambiguous:they will choose L if a is strictlyto the left of zero, and R if a is strictlyto the right.PlayerB wishes to guess the choice of player C. Player A has to trade off three objectives.First, he would like to prevent player B from guessing the action of player C. Moreover both the benefitof doing so and the cost of failingto do so are precisely Ia 1.Secondly,he wants players C and D to coordinate.Thirdly,there is a cost 1a2 associated with any nonzero action on his part. We begin the formalanalysisof the game in stage 2. If a < 0, then C and D both strictly prefer L, and subgame perfection demands that c = d = L. If a > 0 then they both strictly prefer R, and we will have c = d = R. If a = 0 then they will both be indifferent between L and R. They may therefore choose to randomize,and all we can say in general is that their choices will be independent. Let Yb and 8b be the probabilitieswith which C and D respectively choose R when a = 0 and B's action is b. Turningto stage 1, supposethat B chooses R with probability,. Then it may be checked that A's payoff from any a # 0 is (1 - 2/3)a - la2, and that his payoff from a = 0 is at most zero. Hence, if 3> 2 then A's best response is unique, and attachesprobabilityone to a = 1 - 2/3 < 0. This impliesthat C will choose R. But then B can improvehis payoffby setting /8= 1 -a contradiction. In other words,if /3< 2 then B is more likely to choose L. By choosing a > 0, A ensures that C chooses R, and thereby maximizesthe probabilitythat B's guess is wrong.And by choosing a = 1 - 2/3, A achieves the optimal trade-off

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between the gain (1 - 2,13)afrom the bias in B's guess and the cost 'a2 of his actions. Similarly, if 13> 2 then A will choose a = 1 - 2/3 < 0 with probability one, C will choose L, and B will switch to /8 = 0-another contradiction. The only possibility, then, is that 13= 2. In this case A's payoff from any a # 0 is just - 1a2, and his payoff from a = 0 is again at most zero. So the only possible best response for A is to set a = 0 with probability one, and it must be that his payoff from doing so is exactly zero. But A's payoff from a = 0 is zero iff the actions of C and D are perfectly correlated. Since the actions of C and D are also chosen independently it must be that, for each b E {L, R}, either Yb = ab = 1 or Yb = ab = 0. But this in turn implies that B's payoff from b = L is either 1 or - 1, while his payoff from b = R is either 2 or - 2. Hence B cannot be indifferent between L and R, and will deviate from 8 = 2. This third and final contradiction establishes that our game possesses no subgame-perfect equilibrium. The reason for the breakdown in existence in this problem should be clear. A needs to ensure two things. First, the choices of C and D must be perfectly correlated lest he incur a penalty of 10. Secondly, they must be random if B is to be prevented from guessing what they will be. A can achieve both these objectives if he himself randomizes, let us say by choosing a = - E with probability 2 and a = E with probability 2. But in so doing he incurs a cost of 12~~2 1E2. He can make this cost arbitrarily small, but he cannot reduce it to zero. The problem is that, in the limit as E - 0, the signal which A generates for C and D degenerates, and C and D can no longer coordinate their actions in a random way. 2.2. IntroducingPublic Randomization Suppose that we extend the game so that, at the outset of stage 2, C and D observe not only the actions a and b taken by A and B in stage 1, but also the realization of a public signal distributed uniformly on [0, 1]. As long as a : 0, C and D will ignore the public signal, choosing R if a > 0 and L if a < 0. We can therefore argue exactly as above that B will choose R with probability 13= 1, that A will set a = 0 with probability 1, that A's payoff is precisely zero, and that the actions of C and D must be perfectly correlated. When a = 0, however, the actions chosen by C and D may depend on the realization of the public signal. Let Ab be the probability with which C and D choose R when a = 0 and B's action is b. Then B's payoffs from L and R are 1 - 2AL and 4AR - 2 respectively. B will be indifferent between these actions iff 1 - 2AL= 4AR- 2, and we therefore obtain an equilibrium iff AL E [0,1] and AR= (3 -2AL)/4. Restoring existence in this way seems entirely natural. If A gives players C and D unambiguous instructions (i.e. a # 0), then they will carry these out to the letter. So why should they not be allowed to carry out the spirit of A's instructions in the case where these are ambiguous (i.e. a = 0)? Why, for example, should they be forbidden from tossing a coin and both choosing R if

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the coin comes up heads and both choosing L if the coin comes up tails (i.e. AL=AR= 2)? 2.3. Taking Finite Approximations Suppose now that we replace A's action set by Ine In E Z,IncI < 1), where E

is small and positive,but leave the game otherwiseunchanged.As before, C and D will choose R if a > 0, L if a <0, and R with probabilities Yb and ab respectively when a = 0 and B's action is b. We write 'Tb= Yb(l - 3b) + (1 - Yb)8b for the probabilitythat C and D fail to coordinate.It is also easy to see that if 13> (1 + -)/2 then A will choose a < -e with probabilityone-a contradiction-and that if ,B< (1 - 0)/2 then A will choose a > E with probability one-a second contradiction.Overall, then, the analysis of the game reduces to the problem of finding those Nash equilibriaof the reduced-form game depicted in Figure 1 in which player B picks R with probability,3 between (1 - e)/2 and (1 + c)/2.

We shall not give an exhaustivecharacterizationof the equilibriaof the game in Figure 1. Rather, we shall identify two types of equilibrium,and use these equilibriato showthat any equilibriumpath of the extendedgame of Section2.2 can be obtainedas the limit of equilibriumpaths of the finite game. In order to obtain the first type of equilibrium, suppose that YL= 0, YR= 1, aL = 82/20, and AR= 1 - E2/20. Then rlL= 77R= E2/20, and A will be indifferent among all three of his actions if 3 = 2. Suppose that A chooses -8,0 and E with

probabilitiesa -, a0, and a + respectively.Then B will be indifferentbetween L and R if a-+ ao- a+= -2a-+ 2a0 + 2a+. Since a-+ ao + a+= 1, this implies 3 = 4a0 + 6a+. We therefore obtain a range of equilibria parameterizedby ; E [0, 'I, with a+= T, ao = (3 - 6;)/4, and a-= (1 + 2T)/4. The equilibriumpath of such an equilibriumhas the propertythat c = d = R with probability ; and c = d = L with probability 1 - - (3 - 6;),2/ 80 conditional on b = L, and c = d = R with probability (3 - 2;)/4 (3 - 6)E2//80 and c = d = L with probability (1 + 2 )/4 conditional on b = R.

The limitingpath is thereforethe path of the equilibriumof the extendedgame in which AL=4 and AR=(3-20)/4. b

R

L a

-2

1 12

12

44YR

1-2yL

0 -107L

-10

?7R

-1 ?

FIGURE 1.-The

reduced-form game in stage 1.

2

2

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In order to obtain the second type of equilibrium, suppose that YL = 1, 81aL = 1 - E2/20, and AR= E2/20. Then once again f7L= 7R = E2/20, and A will be indifferent among all three of his actions if 13= 2; B will be indifferent between L and R if a-a0-a and we +a= -2a--2a0+2aa; e [2, 1], with a'= (3 - 2;)/4, obtain a range of equilibria parameterized by E a0 = (6;- 3)/4, and a= 1 - T.The equilibrium path of such an equilibrium has the property that c = d = R with probability ; - (6T - 3)e2/ 80 and c = d = L with probability 1 - ; conditional on b = L, and c= d = R with probability (3 - 2 T)/4 and c = d = L with probability (1 + 2 T)/4 - (6 T - 3)E2/80 conditional on b = R. The limiting path is therefore the path of the equilibrium of the YR =

extended game in which AL = 4; and AR = (3 - 2T)/4.

These considerations show that, in our example, every equilibrium path of the extended game can be obtained as the limit of equilibrium paths of equilibria of an appropriate sequence of finite approximations to the original game. On the other hand, a general upper-semicontinuity result holds for finite approximations of a continuous game (cf. Section 4.4). Hence, in our example, the strategic possibilities available in the extended game are precisely those which are available via all discretizations.21

2.4. Relation to Normal-form Games The result to which our example is the most relevant is that of Simon and Zame (1990). In our example, the payoff correspondence faced by players A and B in stage 1 is upper semicontinuous, but not convex valued. (As it happens, it is connected valued.) It therefore shows that the assumption of convex valuedness is needed for their theorem. Secondly, various authors, including Nikaido and Isoda (1955), Dasgupta and Maskin (1986), Simon (1987), and Robson (1994), have given existence results for games with discontinuous payoff functions. The payoff correspondence faced by players A and B in stage 1 of our example is an example of an upper semicontinuous correspondence that admits no selection satisfying the sufficient conditions of any of these papers. Given the examples of nonexistence of a value due to Sion and Wolfe (1957) and to Shapley (1964), it might be conjectured that such a correspondence could be obtained by taking the upper semicontinuous closure of the payoff functions involved there. It turns out, however, that in both cases the correspondence obtained in this way does admit a selection for which existence obtain.

21 This conclusion is rather easier to establish in the original example of Harris (1990). For in that example, the extended game had a unique equilibrium. Moreover a point-to-set mapping that is upper semicontinuous and single valued at a point is actually continuous at a point. It therefore follows at once from the general upper semicontinuity result that the set of equilibrium paths of any finite approximation converges to the unique equilibrium path of the extended game. Harris' example can also be found in Fudenberg and Tirole (1991, Exercise 13.4).

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3. THE FRAMEWORK

In this section we describe our basic game and its extension. In the basic game, play proceedsas follows.At the outset of stage 1, playersare informedof the starting point of the game. They then pick actions simultaneouslyand independentlyfrom their action sets. At the outset of stage 2, players are informed of the actions chosen in stage 1. They then simultaneouslyand independentlypick actions from their action sets for stage 2. At the outset of stage 3, playersare informedof the actions chosen in stage 2. And so on. In the extended game, play proceeds almost exactly as it did in the basic game. The only differencesare that: (i) at the same time as playersare choosing their actions for stage t, a public signal is drawnfrom [0,11 accordingto the and (ii) at the outset of stage t + 1, playersare informed uniformdistribution22 not only of the actions chosen in stage t but also of the value of the public signal.The public signals are independentof one anotherand of the actionsof the players. Before describingthe basic game and the extended game in more detail, it will be helpful to introducesome notation.

3.1. Notation

Supposethat X is a Polish space. That is, X is homeomorphicto a nonempty complete separablemetric space. Then: (i) 9(X) denotes the set of nonemptysubsetsof X, and 4.Y(X) denotes the set of all subsets of X. (ii) e(X) denotes the set of nonempty closed subsets of X, and e4(X) denotes the set of all closed subsetsof X. Both e(X) and e4 (X) are endowed with the Hausdorfftopology.23 (iii) X(X) denotes the set of all nonempty compact subsets of X, and X (X) denotes the set of all compactsubsetsof X. Both >'(X) and X,, (X) are endowedwith the Hausdorfftopology. (iv) A(X) denotes the set of Borel probabilitymeasureson X, endowedwith the topologyof weak convergence.24 Suppose further that AEuA(X), that Y is a second Polish space, that v: X - A(Y) is a Borel measurablemapping,and that A E A(X x Y). Then: 22 The significanceof the uniformdistributionis that it is the richestpossibledistributionin the followingsense: for any nonemptycomplete separablemetric space X and any Borel probability measure A on X there exists a Borel measurablemapping f: [0,11-] X such that A is the distributionof f(i.e., such that A(A) = l(f-'(A)) for all Borel measurableA cX, where 1 denotes Lebesguemeasureon [0, 1]). 23The sequence {C'} c -e.,(X) convergesto CE e* (X) in the Hausdorfftopology iff: (i) (uppersemicontinuity)for all open U such that C' c U we have C' c U for n sufficientlylarge;(ii) (lower semicontinuity) for all open U such that C' n U $ 0 we have Cn n U # 0 for all n sufficientlylarge. 24FollowingParthasarathy(1967), we say that a sequence {An}c A(X) convergesin the weak topologyto AXiff fXdAnconvergesto fxdAr for all boundedcontinuousX: X -0 R.

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EQUILIBRIUM

(v) Support (,u) denotes the smallest closed subset C of X such that p(C) = 1. (vi) ,u 0 v E A(X x Y) denotes the joint distributionover X x Y obtainedby

combiningthe marginal,u over X with the conditionalv over Y. (vii) ,Ix (A) E A(X) denotes the marginalof A over X. Supposefinallythat there is a mappingA: X -+ 9I(Y) such that graph(A) is a closed subset of Xx Y, that support(v( Ix))cA(x) for all x c X, and that K E A(graph(A)).Then, in a slight abuse of notation: (viii) ,u

i vEc A(graph(A)) will also denote the restriction of ,u

vEc A(X x Y)

to graph(A).25 (ix)

.KX(K)

will also denote the marginalover X of the extension of

K

to

X X y.26

3.2. TheData The data of the basic model are as follows: (i) There is a finite nonemptyset , of active players,indexedby i or j.27 (ii) There is a passive player,whom we shall refer to as player 0 or Nature. The overallset of playersis A>= OU {0).28 (iii) Time is discrete,and indexedby s or t E (0, 1,2,...). (iv) There is a nonemptycomplete separablemetricspace YO.YOis the set of startingpoints for the game. (v) For each t > 1 and each i E AX, there is a Polish space Yti.Player i's action in stage t will be chosen from a subset of Yti.We set Y,= XiE ,O,OYtiI (vi) For each t > 0, there is a nonemptyclosed subset Xt c Xt=OYs.Xt is the set of historiesthat can occur up to and includingstage t. (vii) For each t > 1 there are continuousmappingsA ti: Xt_1 - (Y) for all i e , and a continuous mapping A 0: Xt-1 -- e(Yto). The set of actions available to player i in stage t following history xt_1 is Ati(xt_1). We put At=

X i (--,,

At i

(viii) Since the set of histories that can occur up to stage t is the set of (Xt_1,Yd such that xt_l eXt-1 and yt EAt(xt-.1), we take it that Xt = graph(Ad)for all t > 1. We also take it that XO= Yo. (ix) For each t > 1 there is a continuousmapping ft0: Xt-1 - A(Yt0) such that support (ft0 (- Ixt_1)) cA0 (xt_1) for all xt-1 E Xt1. Nature's action followinghistoryxt1 is chosen from Yt0 accordingto the probabilitydistribution ft0 ( Ixt1).

25 A

vEA(graph(A))

is well defined since the assumption that v(A(x x)=1

implies that

,u ? v E A(X x Y) givesprobabilityone to graph(A). 26The extensionK^of K to X X Y is definedby the formulaK^(B)= K(B n graph(A))for all Borel measurableB cXX Y. 27Our resultsare actuallyvalid for a countableset _f of activeplayers. 28We assumethat 0

518

C. HARRIS,

P. RENY, AND A. ROBSON

(x) We define Z to be the set of (y,It>O>e X 1OY, such that (y IO< s < t > E Xt for all t > 0. Z is the set of histories that can occur in the game as a whole. (xi) There is a boundedcontinuousmappingu: Z -1 RIIJFor all i E 'J and all z E Z, ui(z) is the payoffto player i when the historyof the game is z. We let U be any compactsubset of lRsuch that the range of u is containedin U{ 3.3. Strategies

A (behavior)strategyfor an active player must specify, for all t > 1 and all xt-1 E Xt- 1, the mixed action that the playerwill use at stage t when the prior historyof the game is xt 1. Moreoverit must do so measurably. DEFINITION

1: Suppose that i E . Then a strategy fi for player i is a

sequence Kfti It > 1> of Borel measurablemappingsfti: Xt- 1 - A(Yti)such that support(fti-

IXt_1l)) cAti( xt-1l)

for all t > 1 and all xt- 1 E Xt- 1.A strategycombinationf is a vector f Ii Eof strategies,one for each active player. A strategy combination induces a probabilitydistributionover the set of historiespossiblein the subgame.We refer to this probabilitydistributionas the path inducedby the strategycombinationin the subgame. DEFINITION 2: Suppose that a strategycombinationf, a stage t > 1, and a history xt1 e Xt1, are given. Let ,ttl eA(Xt-1) be bxt l, the unit mass concentratedat xt- 1.If T > t, and if aT-1 E A(XT_ ) has alreadybeen defined, then let AT = AT- 1 ? (?E fTi) E A(XT).Finally,let A E A(Z) be the unique

measure such that

/xTI1A)

=

/T

1 for all T> t. Then A is the path induced by

f in the subgamext-1 and, for all i E XJ, fuidA is the payoff obtainedby i in that subgame. DEFINITION 3: A subgame-perfect equilibrium is a strategy combination f such that, for all t > 1, xt1 e Xt1 and i E , player i cannot improve his payoffin subgamext-1 by a unilateralchange in his strategy. As the counterexampleof Section 2 shows,subgame-perfectequilibriumneed not exist. We thereforeconsiderextended games.

3.4. The Extended Game

Let us denote an entity in the extended game by the symbol obtained by placinga caret over the symbolfor the correspondingentity in the basic game. Then: (i) The space of startingpoints, and the embeddingspaces for the action sets of active players,remainunchanged:YO= YO,and Yti= Yti for all t > 1 and all iE

.

519

SUBGAME-PERFECT EQUILIBRIUM

(ii) The embeddingspace Y for Nature's actions in stage t is YtoX [0,1], for all t > 1. and the embeddingspace Y, for outcomesof stage t is X i E (iii) The set of histories Xt thatA can occurup to and includingstage t consists X t>0, hat 1y,IO e Xs=O 0Y suc uh tha yIOs >Xt for allt>0 where ys denotes the projectionof 9y onto Ys.(In other words, a historyxt is completelydescribedby a historyxt and a vector of t publicsignals.) (iv) The action set Ati t(1)

for player i in stage t is Ati(xt_1) for all t > 1,

i E O and xt- 1E Xt-1, where xt-1 denotes the projectionof xt-1 onto Xt-1. The action set At0(x t1) for Nature is Ato(x t1) x [O,1]. We put At = X i E4,,Ati* (Note that Xt = graphGAt)by construction.) A for Nature in stage t is fto( Ixt-1) X 1 for (v) The mixed action fto It) all t > 1 and xt_1 eXt- 1, where I denotes the uniformdistributionon [0, 1]. (vi) The set Z of historiesof the game as a whole is the set of K 9At1 < t E t 0. all t > for A0s that such (y eXt Xt=0Yt (vii) The payoffvector a(z) when the historyof the game is 2 is u(z) for all z E Z, where z denotes the projectionof z onto Z. (In other words, the public signalsare payoffirrelevant.) Since the extended game belongs to the same class of games as the basic game, we do not need new definitionsfor strategies,strategycombinations,and so forth. 4. EXISTENCE

The purposeof the present section is to establishthe followingresult. THEOREM

4: The extended game possesses a subgame-perfectequilibrium.

We establishthis result in four main steps. In the first, or backward,step we We associate x U-)) J*(J(Yt

consider an upper semicontinuous mapping Ct+ 1: Xt -* X(U').

with Ct+1 upper semicontinuousmappings 1ICt+1: Xt-1 -> and IFCt+1: Xt1 -- X*J(U-). For all xt-1 e Xt1, it can be shown that V1Ct+1(xt-d)is precisely the set of joint distributionsover actions and payoff vectors that can occur in Nash equilibriaof the stage game when continuation payoffvectors are chosen from Ct+1 and public randomizationis allowed, and that VICt+1(xt_ ) is the set of payoff vectors of these equilibria.It is more economical,however,to define VICt+1 and VICt+1somewhatdifferently.29 J is a Borel In the second, or forward,step we show that, if ct: Xt1* U )E measurable random selection from 'PCt1 (in other words, if ct( l!Ct+1(xx1) for all -t1 eXV_1,where xt-1 denotes the projectionof xA-1on Xt4 1), then there exist Borel measurable mappings fti: Xt1 -* A(Y -) for all 29Adoptingthe differentdefinitionsavoids duplication,in the first or backwardstep, of argumentsthat haveto be madein a moregeneralwayin the secondor forwardstep. That WCAt1(x,t1) reallyis the set of joint distributionsover actionsand payoffsthat can occurin Nash equilibriaof the stage game followsat once on combiningthe first,second, and thirdsteps.

520

C. HARRIS,

P. RENY, AND A. ROBSON

i E X, and a Borel measurablerandom selection ct+1: Xt U- from Ct+1, ^ such -i) fromI s that, t for faall Xt-t_1 cXt_ equilibriumof eX_ 1: (i)fti E-> is a NashU' Ix_l)i the stage game when payoffvectors are given by c+1(x_1, Ax( ): - 1) -> R;

and (ii) ctxt 1) is the payoffvector of this equilibrium. The mapping ItCt+1 plays a centralrole in establishingthat we can associate mappings fti and ct+1 with ct in the manner described in the previous paragraph.This role can be sketchedas follows.We begin by associatingwith ct a Borel measurable random selection At: Xt -1

A(YtX U-') from ICt+ 1 such

that ct(x^t1) is the payoff vector of At(-I t1) for all tx_ EXt_1. Next, we define fti( Jxt_l) to be the marginal of At( Jx-t_1) over Yti for all i E >f and all x_t- 1 Xt. We also construct a Borel measurable mapping vt+ 1:

Y Xt-1 x Yt ->A(U-') such that vt+1( k-x1, Yt): At+1(* [t-

1) over U-' for all x t- 1 cXt-

A(U-) is the conditionalof

1. (The existence of vt+ 1 is an instance

of a general phenomena:a measurablefamilyof joint distributionspossesses a conditional that is jointly measurable in the conditioningvariable and the parameter.)Lastly,we apply Skorokhod'stheorem to vt+1 to obtain a Borel measurablemapping +i : Xt_1 x Yt x [0,1] Ir R such that vt+lQ [xt-1, Yt) is the distributionof the random variable Q+1(xk_1,yt,*): [0, 1] -->R-' for all xt- E Xt-1 and yt E Yt.A few minor adjustmentsaside, the restrictionof t+1 to Xt is the mappingct+1 that we require. (U ') for all t> 1. In the third step we introduce mappings Et: Xt1 -* For all xt-1 EcXt_1,Et(xt_1) is the set of payoff vectors of subgame-perfect equilibriaof the extensionof subgamext- 1 of the basic game. We also consider the closure of graph(Et) cXt-1 x U- Because .Ur is compact,this set is the graph of an upper semicontinuous mapping C : Xt1

X* (U -). We use the

mappingsEt and Ct to show that E1 is an upper semicontinuousmappingfrom X0 to X*(U -). Since E1 c C1 and C1 is uppersemicontinuouswe mayassumewithoutloss of generalitythat C1 is, in addition, nonemptyvalued. In other words, we may assume that C1 is an upper semicontinuousmapping from XO to X(U '). Proceedingon this basis we establishfirst that Ct is an upper semicontinuous mappingfrom Xt-1 to X(U- ') for all t > 2. This is straightforward. We show secondlythat Et C vICt+1. This is what one would expect given the interpretation of IFCt+1:for all xt1 EXt-1, Et(xt-1) is preciselythe set of payoffvectors of the Nash equilibriathat can occurin the stage gamewhen continuousvectors are chosen from Et+1 and publicrandomizationis allowed,and hence Et(xt-1) must be a subset of the set of payoffvectors that can occur when continuation payoff vectors are chosen from the larger mapping Ct+ . Since 1PCt+1 is upper semicontinuous,Ct C 1'Ct+1 is too. Thirdly,we show that tIC2 cE1. Indeed,

suppose that c1 is a Borel measurableselection from VIC2.Then we may apply the second step recursivelyto obtain, for all t > 1, Borel measurablemappings Lt: X_1 -> A(Y -) for all i EXf, and a Borel measurablerandomselection c: Xt tU' from C + . Standard dynamic-programming considerations show that the strategycombinationf constructedin this way is a subgame-perfectequilib-

SUBGAME-PERFECr

EQUILIBRIUM

521

riumof the extendedgame such that, for all x0 E X0, c1(x0)is the payoffvector from subgame x0. In particular,c1(x0)E E1(xo). Overall, then, we have C1c IkC2 cE1 c Cl. Since C1 is an upper semicontinuousmapping from X0 to XWr(U), this concludesthe third step. In the fourthand final step we show that E1 is nonemptyvalued.To this end, we fix x0 and show that subgame x0 can be approximatedcontinuouslyby a sequence of finite games.Since everyfinite game has a subgame-perfectequilibrium, it follows at once from the conclusion of the third step that E1(xo) is nonempty.We have therefore arrivedat the followingresult. 5: E1 is an upper semicontinuous mapping from XO to X(U'- ).

THEOREM

This being the case, E1 admits a Borel measurableselection c1. Recursive application of the second step to c1 therefore generates a subgame-perfect equilibriumfor the extended game, exactlyas in the third step. 4.1. The First Step: Backwards

In this subsectionwe consider an upper semicontinuousmappingCt+1 from W ). We associate mappingsIFCt+1: Xt 1 -* Xt to '(U *(A(Yt x U')) and IFCt+1: Xt

-

-) with Ct+1, and show that these mappings are, in fact, b&I(U *

upper semicontinuous.

DEFINITION 6: For all (x, y) E Xt-1 x Yt, let pi(x, y) = min{ciIciE Ct+1(x, y)}.30 Then, for all xEeXt_1,WCt+l(x) is the set of AEA(YtXU )

such that: (i) support(A) c graph(Ct+1(x, *A (ii) the marginal,t of A over Yt is a productmeasure; (iii) the marginal p0 (iv) frjyj(yj)dA(y, r) bounded continuous (v) fpi(x, y\ai)dA(y,

of A over Yt0 is ft0(j |x); = [ JridA(y, r)][ JfX(yj)dA(y, r)] for all i E # and for all i:Yi -R; r) < fridA(y, r) for all i E-# and for all ai EAti(X).31

This definition requires some explanation. Note first that the domain of Ct+1(x, ) is At(x). Condition(i) therefore tells us that the actions chosen by the players are all feasible, and that the continuation payoff vectors which follow those actions are all chosen-albeit on a random basis-from Ct+1. Condition(ii) means that players'actions are chosen independently.Condition (iii) means that Nature chooses her action accordingto the prescribeddistribution. Condition(iv) means that player i's payoffis independentof his action. In 30By convention,pi(x, y) = o? if Ct+(x, y) = 0. Since pi may take the value oo, jpi(x, y \ai)dA(y, r) mustbe interpretedas an extended-value

31

integral.

522

C. HARRIS,

P. RENY, AND A. ROBSON

other words, all the actions that i takes yield the same payoff vi = fridA(y, r). Condition(v) means that there is no action ai EAti(x) such that i's expected payoff exceeds vi when: (a) he chooses ai with probabilityone; (b) for all j

$

i, j chooses his or her action according to the marginal ,aj of A over Ytj;and

(c) the continuationpayoffvector followingoutcome y is the worst possible in Ct+1(x,y) from i's point of view for all y eAt(x). In other words, the remaining actions of player i can all be deterred. DEFINITION

7: For all x eXt-1, ICt+1(x) = {frdA(y, r)IAE

t + A)}

That is, VICt+1(x) consistsof the payoffvectorsof the equilibriain Ct+1(x). LEMMA

8:

ICt+t1

is an upper semicontinuous mapping from

Xt-1

to

X*f(A(Yt x U-)). PROOF: We show first that the graphOlCt+?) is closed. To this end, suppose that (xn, An)E graphO"tC +1) for all n E RN,and suppose that (xn, An) >(xX,A)

as n -> oo. Conditions (i)-(iv) for n < oo directly imply the corresponding conditions for n = oo.As for condition (v), let a' EA(xA ) be arbitrary.Since Ati is continuous, we may find ai EeAti(Xn) such that a --- a'. But fpi(xn, y\ a7)dAn(y, r) < fridAn(y, r) for n < oo by condition(v). Since pi is lower semicontinuous, it follows that fpi(x', y \a'T)dA(y, r) < fridAr(y, r).

We show secondlythat VICt + 1(J) is precompactfor all compact J cXt 1. It sufficesto show that, for all 8 > 0, there exists a compactG c Ytx U"' such that V In other words, it suffices to show that A(G) > 1 - 8 for all A eCt1(x). 1(J) is uniformly tight. To this end, let VI'Ct+

8

> 0 be arbitrary. Let KI be a

compact subset of Yt0 such that ft0 (KAIx)> 1 - E for all x E J, let K = ( X i E _Ati(x)) XKo,and let G be the graphof the restrictionof Ct+1 to J x K. Q.E.D. ClearlyG possessesthe requiredproperties. PROPOSITION9:

1PCt+l

is an upper semicontinuous mapping from

K * (U - ).

Xt-,

to

PROOF:Consider the mapping H: A(Y x U") - ) U-' defined by the formula H(A) = frdA(y, r). This mapping is continuous. Indeed. frdA(y, r) = (fridA(y,

r)li

E-

),

and each ri is bounded and continuouson Ytx U{ uppersemicontinuityfrom VICt+1.

I+Ct+1 therefore inherits

Q.E.D.

4.2. The Second Step: Forwards

In this subsectionwe continue to workwith the mappingsC Ct+1, and in the and we the introduced establish previous subsection, following 1fCt+1 result.

SUBGAME-PERFECT

523

EQUILIBRIUM

-1 -* U- is a Borel measurable random 10: Suppose that ct: exist Borel measurable mappings fti: ;Xft -> I' there 1. Then selection from Ct E random selection ct+1: Xt Ua measurable all i and Borel Xf, A(Yti) for from Ct+1, such that, for all x X _1: (i) Kft(*Ix)Ii E > is a Nash equilibriumof the stage game when continuation payoff vectors are given by cU+1(k, ): At(X) - ; and PROPOSITION

(ii) ct(k)

is thepayoffvectorof thisNash equilibrium.

Condition(i) means, more explicitly,that support (ftJ(x ))cA 1(x) for all E .Y and that, if 4i denotes the restriction of fti( x) to A 1(x) for all i E AYJ, then ((p Ii E-f> is a Nash equilibriumof the normal-formgame in i

which player i has action set Atix() and payoff function c(t+1)i(k, ): At(x)

-

U

for all i E .YO,and in which Nature chooses her action according to 40. Condition(ii) means that ct(k) = fcx+1(k, 9)db(9), where = i The proof of this propositionwill be based on three lemmas. The first of these is a measurableimplicitfunctionresult. LEMMA11: Suppose that X, Y, and Z are Polish spaces, that A: X -* X(Y) is upper semicontinuous, and that b: Y-- Z is continuous. Let C = b o A, and let c be a Borel measurable selection from C. Then there exists a Borel measurable selection a: X -- Yfrom A such that c = b - a. PROOF: Considerthe set of (x, z, y) E X x Z x Y such that (x, y) E graph(A) and z = b(y). By constructionthis set is the graphof an upper semicontinuous mapping D from graph (C) to J(Y). Let d: graph (C) -> Y be any Borel

measurable selection from D. Then we may put a(x) = d(x, c(x)) for all x E X. Q.E.D.

The second lemma shows that a measurablefamily of probabilitymeasures has a family of conditional distributionsthat is jointly measurable in the parameterand the conditioningvariable. LEMMA 12: Suppose that X, Y, and Z are Polish spaces. Let A: X -, A(Y X Z) be Borel measurable, and let A(. Ix) be the marginal of A(- Ix) over Y for all x E X. Then there exists a Borel measurable mapping v: X X Y -- A(Z) such that A(CIx)= ,u Ix) ( v( Ix, *) for all x E X.

Note that we alreadyknow from the standardresult on the existence of an r.c.p.d.(regularconditionalprobabilitydistribution)that there exists a mapping v: Xx Y-*A(Z) such that, for all x eX, v( Ix, ) is a Borel measurable mapping from Y to A(Z) and A( Ix) = tW Ix) ? v( Ix, - ) (see Dellacherie and

What Lemma12 tells us is that v can be chosen to Meyer(1978, 111.70-111.73)). be Borel measurablejointly in its arguments.Since we have not been able to find preciselythis result in the literature,we sketch the proof (see Dellacherie and Meyer(1982, V.58) for a related argument).

524

C. HARRIS, P. RENY, AND A. ROBSON

PROOF: Let {F InEe NR)be a sequence of sets that generates the Borel a-algebra on Y. Let YN denote the a-algebra generated by the sets {F,n1 < n < N), and let 5K denote the Borel a-algebra on Y. Let FN(y) denote the smallestset in YN that contains y, and let v be an arbitraryelement of 4(Z). Then we may define a measure VN(- IX,Y) on Z by setting

A(FN(y) x Glx) VN(GIX,Y) =(FN(Y)IX) if ,t(FN(y)Ix)> 0, and VN(GIx, y) = i otherwise. It is easy to check that the mapping VN:X x Y -* 4(Z) constructed in this way is Borel measurable, and that VN(. IX,*) is an r.c.p.d. of A(- Ix) given EN for all x eX. Now consider the random variable VN( IX,*): Y -> 4(Z). It is a version of the

conditionalexpectationof 8, given $N. (Here 8, denotes the Dirac measure concentrated at z.) Hence the sequence {VN(*IX,*)IN E-N} is a martingale on the filtered probabilityspace (Y, {NIN ER N), t& Ix)), and it converges t& Ix)a.s. to a version of the conditionalexpectationof 8, given .K. That is, if we set v(- Ix,y) = limN-oo VN(- Ix, y) where this limit exists, and v(QIx,y) = v otherwise, then A( Ix) = A Ix) v( Ix,). But the mapping v: X x Y -- 4(Z) is Q.E.D. Borel measurableby construction. The third lemma is the "measurable"version of Skorokhod'srepresentation theorem. It is essentially Lemma 1.2 on page 9 of Gihman and Skorokhod (1979). We thereforeomit the proof. LEMMA 13: Suppose that X and Y are Polish spaces, and that v:X -* 4(Y) is Borel measurable. Then there exists a Borel measurable mapping c: X x [0, 1] - Y such that v(* Ix) is the distributionof c(x, ): [0, 1] - Yfor all x E X when [0, 1] is Q.E.D. endowed with the uniform distribution.

We are now in a position to prove Proposition10. We divide it into parts. In PROOF: The proof is lengthybut straightforward. the firstpartwe constructthe fti. By Lemma11, there exists a Borel measurable x U"-) random selection A: X_1->(Yt such that c X -Yt-1 A from iC t+I'C1 suhtAt X for all eX_1. For all x eX=-1 and all i E X, let i X be the frdA(y, rlx) marginal of A(*|x) over Yi. By definition of ICt+1 (part (i)), support (fti(- |X)) C Ati x). By construction, fti: Xt-1 -> (Yti) is Borel measurable.

For all x eXt1, let ,ut( lx) be the marginalof A(Xx) over Yt.In the second

part we show that there exists a Borel measurable mapping v: X, -1 X Yt

W(U ) such that:(i)XA Xx) u

X)?v4

x,

for all X^eX1;

(ii) support

( 4.+,y))cC By X +1(x,y) for all (X,y)eXt1 XYt such that (x,y)eX,. Lemma 12, there exists a Borel measurable mapping v': Xt- 1 X Yt-' 4(U-)

SUBGAME-PERFECT

EQUILIBRIUM

525

such that A( Jx)=A( |JxI)v1( lx, ) for all xeX. Let c2: X,-*U-' be a Borel measurableselection from Ct+1;let v2(. Ix, Y) = 6c2(x, y)-the unit mass concentrated at c2(x, y)-for all x e Xt1; and let v(KlxI,y)=v1( l,y) if or support (v1(-1,y))cCt+1(x,y), and v2( Ix,y) otherwise. By (x,y)EXt construction,v is Borel measurableand satisfies(ii). As for (i), it follows from the definition of VfCt+1 (part (i)) that support (A Xc))cgraph(Ct+1(x, . This implies that support (vl'( lx, )) c Ct+1(x, ) /ui( kx)-a.s.Hence u( Lx)? 1k,

(

)

=p(

* 1k)

?

'(_

Ik,

.

)

=

A(.

lx).

In the third part we show that there exists a Borel measurablemapping c: Xt1 x Yt-> U-` such that: (i) v(- lx, y) is the distributionof J(x, y, *): [0,1] > U' for all (x, y) eE1 x Y,; and (ii) J(x, 9) E C,+1(x, y) for all (x, 9) E Xt-1 X Yt such that (x, y) eXt. By Lemma 13, there exists a Borel measurable lx, y) is the distribution of c'(X, y, ): mapping c1: X_1 x -t U-' such that v( [0,1] U" for all (x, y) eXt_1 x Yt. Let C2be the Borel measurableselection from the previous paragraph and let c(x, 9) = c'(X, 9) if (x, y) eXt or c'(X^,9) E Ct+1(x, y), and c2(x, y) otherwise. By construction,c is Borel measurable and satisfies (ii). As for (i), it follows from the constructionof v that _C U_ is c C+1(x, y) 1-a.s.Hence the distributionofc(x, y, ): [0,1] -> the same as that of c'(X, y, *): [0, 1] -- U{ This completesthe thirdpart. In the fourthpart we adjustc to obtain a Borel measurablerandomselection ct+1: X -> UO' from C,+ . The remaining parts of the proof will then be devoted to verifyingthat ct+1 has the other propertiesrequiredof it. To this ) for all i E X,x EX-1, and E end, let J(x, Y) = f (x, 9 )dft( )(9i Y Ix); let pY() fRi(k, Yi)dfti(Yil) for all where ft(-i)(~ {x) ?IE ,,\{i)ft. i E= and x E Xt - 1;and let Di(x) ={Yi1Yi EAC1(k) J-(, and yi) >P(x)}. (Here is i's from the all player payoff stage game when players play accordingto pii(x) X

=

the ftj(-

|X);

Ji(x, yi) is i's payoff if he deviates to yi; and Di(C) is the set of

actionsto whichdeviationis both feasible and profitable.)And, for all i E Xf, let

qi: Xt

-+

UO' be a Borel measurable selection from Ct+4 such that qi(x, y)

=

min{r Ir e Ct+1(x,y)} for all (x, y) eXt. (The existence of such selections is guaranteedby Lemma 13.) Then we may define a mapping c: Xt-1 X Y-* Uo by putting c(x, 9) = c(x, y) if yi E Yti\Di(x) for all i E X, c(x, 9) = qi(x, y) if YiED(X) and yj E Ytj\D,(x) for all jEY- \{i), and c(x, 9) = c(x, 9) otherwise. (In other words, where the continuationsspecified by c are such as to cause i to deviate, we replace them with the worst possible continuationfrom i's point of view.) Finally,let ct+1 be the restrictionof c to X,. By construction, ct+1 is a Borel measurableselection from Ct+1. In the remaining three parts the subgame in question will be fixed. We thereforesuppressall dependenceon x and x. In the fifth partwe show that Di has fti-measurezero for all i E XY.We begin with some bookkeeping.By definition:Au=.4f(A); fti= 'y (A) for all i E ; By definition of 'ICt+41 (parts (ii) and (iii)): ,u ft0 =ft0 ? 1; and ft = is a product measure; and ft0 = 0yto (A). Since #yi(A) = .,(#ytGy(A)) for all E-E-A-Afti.

526

P. RENY, AND A. ROBSON

C. HARRIS,

E i sf,

we conclude that ft = /i ?D1. Next, let Xi: Yi R be bounded and continuous.Then Ri(yi)Xi( yi)dfti( yi) -

~~~A A A df(-i,( 9i) ]Xi( Yi) tiY) c(Y)

Ti( j 9)Xi(yi)dL(9) = |if -

Y, ()dl(6)

f [fridv(rly)

A)=

i(9)xi(Yi)dO'

Xi( yi)dA( Y)

Xi(yi)dIL(y)

= friXi(yi)dA(y,r)

?1)(9)

=

(by constructionof Ci(y,'))

[fridA(y, r) [fXi( yi)dA( y, r)j

(by definition of lpct+1, part (iv)). Since xi was arbitrary,we conclude that Ri = fridA(y, r) fti-a.s. This in turn implies that Pi = fRi(yi)dfti(yi)

=

fridA(y,r). So Di has fti-measurezero, as required. In the sixth part we show that ft _jiE .) is a Nash equilibriumof the stage game when continuationpaths are given by ct+1. To this end, let Ri and pi be defined in terms of c in exactlythe same way that Ri and Pi were defined in terms of J. Because Dj has ft1-measurezero for all i 0 i, we have: (i) Ri(yi) = Rii(yi) for all YiE Yti\Di; (ii) Ri(a ) = fpi(y \ai)dft( y)(9i) for all a1E D. Moreover,because in addition Di has fti-measurezero: (iii) Pi = fi. A Suppose that ai eAtA\Di. Then it follows at once from (i) that Ri(ai)
A

(by Y)(b(i)

(ii))

= fPi(y\ai)dA(y,r)

(since pi(y\ai) depends only on yi, and the marginal of ft(-i) over Yti coincideswith that of A) < ridA( y, r)

(by definition of I1C Ct+ , part (v)) =

Pi

(as in partfive).We concludethat Ri(ai) < i for all ai _A11.But Pi = Pi by (iii). In other words, fti yields i a payoffof pi, and no feasible deviationyields i a higherpayoff.

527

SUBGAME-PERFECT EQUILIBRIUM

In the seventh and final part we show that ct is the payoff vector from We have

Kfti1i E .>.

rdA(y, r)

ct=

= [c(y,() =

c(9)dft(

=

f[frdv(rly)j

dA(y)

dl(()] d/l(y) = |c(

)d(1

'$ 1)( 9)

9) Q.E.D.

(cf. part five) 4.3. The Third Step: UpperSemicontinuity In this subsection we consider mappings Et: Xt_ -I * (U -) , UEt(xUEt) ') defined as follows. For all t > 1 and all xt_ eXtXt_l -(-

and Ct:

is the set of payoff vectors of subgame-perfectequilibriaof the extension of subgamext- 1 of the basic game. For all t > 1, Ct is the mappingwhose graphis the closure of graph(Et) cXt-1 x UK And we establishthe followingresult. PROPOSITION

14: El is an uppersemicontinuous mappingfrom XO to Xl* (U - ).

The proof of this propositionwill be based on severallemmas. LEMMA15: For all t > 1, Ct is an upper semicontinuous mappingfrom Xt_ ,* (Us-). PROOF:

This follows at once fromthe fact that U-' is compact.

to

Q.E.D.

Because C, is an upper semicontinuousmappingfrom XO to X* (U-'), to prove the propositionit sufficesto show that El = C1. This is what we shall do. Since E c C,, we must have E = C, on Cj '(0). We thereforeneed only show that E = C1 on Xo\ C1 '(0). But this latter set is closed. We may therefore assume without loss of generalitythat C, is an upper semicontinuousmapping from XOto XW(Uo). We assumethis for the remainderof the subsection. LEMMA

16: For all t >2, Ct is an upper semicontinuous mappingfrom Xt,

to

PROOF:The proof is by induction.Note firstthat C, is an uppersemicontinuous mappingfrom XOto X(U -) by assumption.Suppose therefore that Ct is an upper semicontinuousmappingfrom X,-1 to y(U.-). We need only show that Ct+ is nonempty valued. To this end, let xt = (x t-, yt) eXt be arbitrary. Since C,(x,_) # 0, we may find et E C,(x,_), and a sequence {(xn 1,en)} c graph(Et), such that (xn 1, en) -> (xt 1, et). Since At is continuous, we may find ynEfAt(Xn 1) such that yn -- yt. Now the extension of subgame xn 1 has a

528

C. HARRIS,

P. RENY, AND A. ROBSON

subgame-perfect equilibrium by construction. Hence the extension of subgame

x n = (xUn1, yn) must have a subgame-perfect equilibria too. Let en+1E?E, (x7n) be the payoff vector of this equilibrium. Moving to a subsequence if necessary, n et1 for some et+1 E U{ Clearly e E1EC we may assume that e7n+ Q.E.D. LEMMA

17: For allt >

Et C TCt+C1.

1,

PROOF: Fix x E Xt- 1 and ct E Et(x). Let f be a subgame-perfect equilibrium of the extension of subgame x of the basic game, the payoff of which is ct. Let Xi be the restriction of fti( Ix) to Ati(x) for all i E X, let 00 be the restriction of fto ( Ix) x l to Ato(x) x [0,1], and let ct+1(A) be the payoff vector induced by f in subgame (x, 9) of the game under consideration for all 9 eA=(x) X [0,1]. Then standard dynamic-programming considerations show that: (i) K IiiEE ) is a Nash equilibrium of the normal-form game in which player i has action set Ati(x) and payoff function c(t+ 1)i:At(x) x [0, 1] - U for all i E X, and in which Nature chooses her action according to 00; (ii) ct is the payoff vector of this equilibrium. Since x plays no further role in what follows, we suppress it. Now we may regard (y,ct+1) as a random variable from A tx [0, 1] to Ytx U{ Let A be the distribution of (y, ct+ 1) when At x [0, 1] is endowed with the measure 4 = 0 i Oi. Clearly A satisfies parts (i), (ii), and (iii) of the definition of TCt+ 1. In order to tackle parts (iv) and (v) let Ri(yi) = and yiEAtiq where 4-i = ?Ie>\{i}4 fc(t+ )A(_)d+Li( _i) for all ie-f Standard normal-form considerations show that Ri = cti Oi-a.s. Let Xi be an arbitrary bounded continuous mapping from Yi to R. Then E=.

riXi( yi)dA( y, r) -

fc(t +)(

=|[C(t

9A)Xi(yi) do ( 9)

A)d( 9)

_i(

A_i)

(by construction of A) ]Xi( yi)doi( yi)

= Rif yi)Xif yi)doi( yi) yi)i)d)i( yi)

=

CtiXi(

=

ctifXi(yi)do( 9)

=

ctifxi(yi)dA(y

(because Ri = c,i Xi-a.s.)

(because Xidoes not dependon 9-i)

r)

(by construction of A again). In particular, fridA(y, r) = cti on putting Xi

1.

SUBGAME-PERFECT

529

EQUILIBRIUM

This establishespart (iv). Also, let ai E Ati be arbitrary.We have Pi(y\ai)dA

=

fPi(y\ai)d4(

s

C+l)i( f(t

-

f[fc(t+lAi(9\ai)

-

fRi(yi)d)i(yi)

9)

9\ai)d4i(

9) d&-i(

(by constructionof pi) A_i)j d4i(y1)

= cti-

This establishespart (v). Finally, we have already shown in the course of establishingpart (iv) that = fr1dA(y, r). Since A E ~TCt+1,it follows at once that ct E 'Ct+1. Q.E.D. Since 'Ci+C1 is upper semicontinuous, and graph(Ct) is the closure of graph(Et),it follows that Ct too is upper semicontinuous. LEMMA 18: TIC2 cE1. PROOF: It suffices to show that, if c1 is an arbitraryBorel measurable selection from TC2, then c1 is also a selection from E1. Let c1 be such a Borel measurable selection. Then, since C c 'PC1 for all t > 2, we may apply Proposition 10 recursivelyto obtain Borel measurablemappings fti: X t*l A(Yti)for all t > 1 and i E X., with the propertiesstated in that proposition. Standarddynamic-programming considerationsshow that the strategyf constructedin this way is a subgame-perfectequilibrium,and that, for all xo E XO, Q.E.D. c1(xo)is the payoffvectorfromsubgamexo.

We turn now to the proof of Proposition14. PROOF: As noted above, it suffices to show that E1 = C1, moreoverwe may assume without loss of generality that C1 is nonempty valued. Combining

c Lemmas 4.12 and 4.13 we have 'PC2 F1 c C1lTC2.

required.

Hence E1 = C1, as

Q.E.D.

4.4. The Fourth Step: Nonemptiness In this section we show that E is nonempty valued. We thereby arrive at the

followingproposition. PROPOSITION

19: E1 is an upper semicontinuous mappingfrom XQ to .f(U- ').

We shall prove this result by showingthat any subgame xo can be approximated in a continuousway by a sequence of finite games.

530 DEFINITION

C. HARRIS,

P. RENY, AND A. ROBSON

20: Suppose that X0 = {oo}. Then a finite approximation for the

basic game is a new game in which: (i) X0=NU{oo};

(ii) if xo < oothen subgamex0 of the new game is a finite game; (iii) if xo = oo then subgame io of the new game is the basic game.

We emphasizethat the new game is requiredto satisfyour standingassumptions. LEMMA 21: Suppose that XO= {oo(. Then the basic game possesses a finite approximation.

PROOF: Note first that Theorem 111.9of Castaingand Valadier (1977) implies that, for all t> 1 and i eY , there exists a sequence {(anIn ef I} of is dense in Ati(xt-1) measurableselectionsfrom Ati such that {(an(xt_1)In E NJI} for all x E E Xt1. Secondly,for all t > 1 and n E Fk,let on be a measurable function from X,-1 to A(Yto) such that for all xt-1 e Xt-1: support t-) Sm An}; and tot(x _1) is as close to ft0 (xt-1) as to {to any other measure in A(Yto), the support of which is contained in this set. Then we may define a new game as follows: (i) We put Y0= NU {oo1,Yti=Ytj for all t2 1 and all i E.YJc , and Yt=

X i e,0,Ytj for all t > 1. (ii) We put X0= Y.

(iii) If t > 1 and Xt1 has alreadybeen defined, then for all i E J we let Ati be the mapping from Xt-1 to e(Yti) obtained by putting for all -t_l =(n,y1,* ,y-1) such Atit l) = ?ti t_dIm 1 andAt has alreadybeen defined, then we put Xt = graph(At).

(v) For all t > 1, we let ft0 be the mappingfrom Xt- 1 to A(Y0 ) obtainedby such that n< putting f0( Ix1) = maxx{O,n-t} for all -t-i =(n,y1,...,y-1) and fto (*Ixt1)=fto(*Ix for all oo and xt_1 = (??,Y1,... ,yt-d1)EXt1, Xt_1 EXt-l-

(Here we denote entities in the new game by placing a tilde over the symbol for the correspondingentity in the basic game.) It is straightforwardto verify Q.E.D. that the game constructedin this way is a finite approximation. Notice that the extensive forms of the approximatinggames constructedin the proof of Lemma 21 are all contained in the extensiveform of the original game. We turn now to the proof of Proposition19. PROOF: In view of Lemma21 we may assumewithout loss of generalitythat Xo = N U {oo}, and that subgame xo (of the basic game) is a finite game for all

x0 < oo. By Proposition14, E1 is an upper semicontinuousmappingfrom X0 to

SUBGAME-PERFECr

XA*(UW). It follows that E'

(X(U

531

EQUILIBRIUM

)) is a closed subset of XO. But subgame-

perfect equilibriumexists in finite games. So N'cEGE'(G(U ')). We conclude Q.E.D.

that ET (GY(U-')) =XO. 5. UPPER SEMICONTINUITY

OF EQUILIBRIUM

PATHS

In Section 4 we defined El(xo) to be the set of payoffvectorsof subgame-perfect equilibriaof the extension of subgame xo of the basic game, and showed that EA(xd)was nonempty and compact, and varied upper semicontinuously with xo. In particular,if a sequence of finite games convergedto an infinite game, and if a subgame-perfectequilibriumis specified for each finite game, then the associated sequence of payoffvectors has limit points, and any such limit point is the payoffvector of a subgame-perfectequilibriumof the extension of the infinitegame. In the present section we prove a much more refined version of this result. E Xt_ and a strategycombinationf E for the extended game are given. Then the reducedpath induced by f in subgamex't is the marginalover Z of the path inducedby f in this subgame. DEFINITION

22: Supposethat t

> 1,

A

For all t > 1, define a mapping Et: Xt_ ->

9*?(A(Z)) by letting Et(x t-) be

the set of reduced paths of subgame-perfectequilibria of the extension of subgame xt-, of the basic game for all xt, e Xt-1. The aim of the present section is to show that El is in fact an upper semicontinuousmappingfrom Xt-, to XY(A(Z)).The proof of this result closely parallels the proof of the correspondingresult for payoffvectors given in Section 4. We shall therefore indicateonly the essential modifications. Suppose initially that t > 1 and an upper semicontinuouscorrespondence Ct+1: Xt -* JY(A(Z)) are fixed. And, for all i E>,f and q EeA(Z), let ui(Cq) denote fuid-q. Then we have the followinganaloguesof Definitions6 and 7. DEFINITION

23 (cf. Definition 6): For all (x, y) eXt-1 X Yt, let pi(x, y)

min{ui(t7)177E Ct+,(x, y)}. Then, for all x eX-t1,

tICt+,(x)

=

is the set of

A E A(Ytx A(Z)) such that: (i) support(A) c graph(Ct+ (x, (ii) the marginal,u of A over Yt is a productmeasure; (iii) the marginal AOyof A over Yt0 is ft0 (- |x); (iv) fui(?)xi(yi)dA(y, 71)= [fui(,)dA(y, -q)][fXi(yi)dA(y, -q)]for all i E >Yand

all boundedcontinuousXi: Yi [FR; (v) fpi(x, y/ai)dA(y, -q)< fui(G)dA(y,'q)for all i E DEFINITION

X

and all ai eAti(x).

24 (cf. Definition 7): For all xt_ E Xt_, Ct+ (Xt_1)}

{f77dA(y,i7)IAE

And we have the followinganalogueof Proposition9.

TICt+ (xt-)

532

C. HARRIS, P. RENY, AND A. ROBSON

PROPOSITION

ping from

Xt1

25 (cf. Proposition9): to X'*( (Z)).

TfCt+

1 is an uppersemicontinuousmap-

PROOF: It can be shown exactly as in Lemma 8 that TCt+1 is an upper semicontinuousmappingfrom Xt- 1 to X* (A(Ytx A(Z))). We thereforeneed only show that the mapping H: A (Yt x A(Z)) -* A(Z) defined by H(A) = f'qdA(y,rq) is continuous. To this end, let X: Z -* DRbe bounded and continu-

ous. Then standard considerations show that fXd(H(A))=f[fxdqjIdA(y,

q).

But fXdr is a boundedcontinuousfunctionof (y, ,q).The resultfollows. Q.E.D. This completesthe backwardstep. Turningto the forwardstep, we have the followingproposition. PROPOSITION 26 (cf. Proposition10): Supposethat ct: Xt-1 -> (Z) is a Borel measurable random selection from TICt + 1. Then there exist Borel measurable mappings fti: Xt- ^A(Yti) for all i E>Y, and a Borel measurable random selection c t+1Xt -A(Z) from Ct+ 1, such that, if we put rt +(x y)= , for all (x, 9)Xt, then, for alx eXt_1: u(ct+ 1 x,9)) > is a Nash equilibriumof the stage game whencontinua(i)
(ii) ct( IxA)=

fct+?(

X, 9)dfI(AIA)A

Condition(ii) means,more precisely,that, if Xiis the restrictionof f x( Ix)to x x y)d y A(ti ) for all iE X, and if p = C E -A, i, then E

PROOF: Proceedingexactly as in the proof of Proposition10, we construct c:(1(Z)), A: Xt-A_ (Yt x A(Z)), ,u: X1t A(Yt), v: Xt -x x >* A(Z) and Next, we R_(k, define -* yx ) = Xt-1 Yt ti: Xt1 A(Yti). = fui(8(~ Ix, y))d (9I),_Ix pi(x) = and dfti(yix) Di(x) {yilyi E > c and we alter on the to obtain c: R1(x, yi) pi(x)}; Atix() Di Xt- 1 x Yt (Z); and we let ct+1 be the restrictionof c to Xt. Finally,suppressingx, we note that dA(y9i7)

ct=

=

f[fc(

=

c(.

=

f[f77dv(7 Iy)] dA(y)

*Iy, ) dl( )j d/l(y) IA)dt(

=

fc(. *9)d(l

?l)(

A)

9).

(Note that Lemmas11-13 were stated in enough generalityto allow them to be Q.E.D. used in the presentsetting.) This completes the forward step. Our next goal is to prove the following proposition.

SUBGAME-PERFECr PROPOSITION

fromXo to

27

EQUILIBRIUM

533

(cf. Proposition14): E1 is an upper semicontinuous mapping

'*(A(Z)).

For all t > 1 let Ct: X1 -> 4a*(A(Z)) be the mappingwhose graph is the closure of graph(E,) cXt-1 x A(Z). The main difficultyarises in showingthat Ct is upper semicontinuous.We surmount this difficultywith the help of a definitionand a lemma. 28: A joint strategyf is a sequence ft It> 1) of Borel measurDEFINITION able mappings ft: Xt-1 -*>A(Yt) such that, for all xt_1 E-X1, support (ft(*Ixt-1)) c At(xt-1) and 4ytSft(* Ixt-1)) =fto0( Ixt-1). A joint strategyinduces a path in a subgamein the obviousway. X(A(Y)) LEMMA29: Suppose that X, Y, and Z are Polish spaces, that M: X is upper semicontinuous and that C: X x Y -* X(A(Z)) is upper semicontinuous and convex valued. For all xEeX, let B(x) be the set of x ? ve A(Yx Z) obtained as ,.t varies over M(x) and v varies over the Borel measurable selections from C(x, *): Y - X(YA(Z)). Then B is an upper semicontinuous mappingfrom X to X(A(YxZ)). Let W denote A(Z), and let B(x) be the set of A E A(Yx W) such that the marginal ,x of A over Y lies in M(x) and A is carried by graph (C(x, )). We show first that B is an upper semicontinuousmappingfrom X into x((Y xZ)). ObviouslyB is nonemptyvalued.We thereforeneed only show that:(i) graph (B) is closed; (ii) for all compact JcX, B(J) is compact. That graph (A) is closed follows from the closureof graph(M) and graph(C). To show that B(J) is compact, let e > 0 be arbitrary.Since M is upper semicontinuous,M(J) is compact. So there exists compact Kc Y such that bt(K) > 1 - e for all ,u E M(J). Since C is upper semicontinuous,graph(C(x,* )IK)is compact.But any B satisfies A(graph(C(x, K)) > 1 - E. We conclude that B(J) is relaA B(K) compact. tively Now suppose that A E A(Yx W), let .t be the marginalof A over Y, and let v: Y -*A(W) be any r.c.p.d of A over W. For each y E Y, let v( IY)= Jwdv(wly) denote the expectation of v( ly), and let H(A) E A(Yx Z) denote ,u ? v. Standardconsiderationsshow that H(A) is well defined(i.e. it is independentof (YxZ) is continuous. To the r.c.p.d. v chosen), and that H: A(Yx W) show that B has the required properties, then, we need only show that B(x) = H(B(x)) for all xEE X. Supposefirst that A eB(x). Let ,u be the marginalof A over Y, and let v be an r.c.p.d.of A over Z. For each y E Y let vl( ly) be the expectationof v( lY), and let Y be the set of y such that v( IY)is carriedby C(x, y). Let v2 be an arbitraryBorel measurableselection from C(x,*), and let V3 coincide with v, on Y and with v2 on Y\ Y. By definition,A is carriedby graph(C(x, )). Hence, PROOF:

534

C. HARRIS,

P. RENY, AND A. ROBSON

1. Hence ,u 0 v1 = ,u ? V3 by a standardpropertyof an r.c.p.d.,Y has ,tu-measure eE B(x). We conclude that 7(A) = ,u v1 E B(x) as well. But ,u Suppose then that A E B(x). By definition, A = ,u 0 v, where v is a Borel measurableselection from C(x, - ). Let v(Qly) = 5V( Y)'i.e. the Dirac measure V Then clearly A E B(x), and A = H(A). at v(- lY) for all y E Y, and let A = ,u v. Q.E.D.

We can now prove the followinglemma. LEMMA 30 (cf. Lemma 15): For all t > 1, Ct is an upper semicontinuous mapping from Xt1 to Y(A(Z)).

PROOF:We maysupposewithoutloss of generalitythat t = 1. For all xo E XO, let FxO) be the set of paths of joint strategiesfor the extensionof subgamexo of the basic game; and for all T>0 and all x0e X0, let FT(xO)denote the marginalsover XT of the paths in FAxO).Recursiveapplicationof Lemma 29 shows that FT is an upper semicontinuousmappingfrom XOto X(A(XT)) for all T. Since a sequence of measures in a(Z) convergesif the corresponding sequence of marginalsin A(XT) convergesfor all T, we conclude that F is an upper semicontinuousmappingfrom XOto X(A(Z)). For all xo EX0, let F(xO)denote the marginalsover Z of the paths in Fx0). Because the operation of taking marginals is continuous, F is an upper semicontinuousmapping from XO to X(A(Z)). And obviously E1 cF. It follows at once that C1 is an upper semicontinuousmapping from XO to X* (AWz).

Q.E.D.

Notice that the proof of Lemma 30 would be greatly simplifiedif the A 0 were compactvalued. We did not make this strongerassumptionbecause, in applications,it is entirely conceivable that Nature's mixed actions might not have compactsupport.For example,her actions might be normallydistributed. In view of Lemma 30, we may assume for the remainderof the proof of Proposition27 that C1 is nonemptyvalued. LEMMA31 (cf. Lemma 16): For all t >2, Ct is an upper semicontinuous mapping from Xt -1 to X(A(Z)).

We omit the proof, since it is identicalto that of Lemma 16. LEMMA

32 (cf. Lemma 17): For all t > 1, Et C TCt .

PROOF:Fix x E Xt-1 and ct E EE(x). Let f be a subgame-perfectequilibrium

for the extensionof subgamex of the basic game, the reducedpath of which is ct. Let fi be the restrictionof fti( Ix) to Ati(x) for all i E X, let b0 be the restrictionof ft0 (. Ix) x I to At0(x) x [0, 11,let c^ and rt be the path and the payoff vector of f,and let c 1Y) and rt+1(9) be the path and the payoff vector induced by f in subgame(x, 9) for all 9 eAt(x) x [0,1]. Then standard

SUBGAME-PERFECT

535

EQUILIBRIUM

dynamic-programming considerationsshow that: (i) Koili E-Y is a Nash equilibriumof the normal-formgame in which player i has action set Ati(x) and payoff function r(+ 1)j: At(x) x [0, 1] - U for all i E X., and in which Nature 19)d4(9), where 4 = chooses her action according to 00; (ii) c^,= f+ , Oi. Since x plays no furtherrole in what follows,we suppressit. i e,_ Now let c1?1Q9)=ckzG+ 19)) for all y^eA x[0,1], and let A be the distribution of (y, ct+ d: At x [0, 1] -* Yt x A(Z). Clearly A satisfies parts (i), (ii), for all and (iii) of the definition of Ct+C1.Let Ri(yi) = fr(1+l)y( )d(y() i E-Y and all yi eAti, where +-i = 1eE={i}4j. Standardnormal-formconj

siderations show that Ri = rti Oi-a.s. Let Xi: Yi-R

be bounded and continuous.

Then

|ui(-q)Xif yi)d,k( Y7' = fui(ct+

(.

A))Xi( Yi)d(

=

fi2s(ct?( I9))A'i(yi)d4(9)

=

|9[Ir(t+l)i()

=

)

r(t+l)i( )Xj(Yj)d4(9)

d&d-i(9-i) Xi(yi)doi( yi)

i yj)Xj(yi)dj(yj)

= rti Xi( yj)diE( yi) =rti Xj( yi)dA( y, r1)

In particular,fui(,&)dA(y,'7q)=rti.This establishes part (iv). Part (v) can be establishedsimilarly. It remainsto establishthat ct = J'qdA(y,'7). We have ct=

azdet(z),

where 8z is the unit mass at z (because ct = 1Z(c^d)) =

f [f8zdet+1(zI9)j

=

fct+l(. 19)d4( 9)

d(

A9)

(because .4(ct+ ( 9)) =ct+ ( 9))

-=fdA(Y,-q)

Q.E.D.

(by constructionof A). Since I'Ct+1 is upper semicontinuous,Ct c

Ct'+I is too.

536

P. RENY, AND A. ROBSON

C. HARRIS,

LEMMA33 (cf. Lemma 18): TC2 cE1. PROOF: Suppose that c1 is a Borel measurableselection from TC2. Then we may apply Proposition26 recursivelyto obtain Borel measurablemappingsfi: -* A(Z) for all -* A(Y -) for all t > 1, Borel measurable selections c ?1: t > 1, and Borel measurablefunctions rt+1: Xt -* U-J for all t > 1, with the propertiesstated in that proposition.Now to verify that the strategycombination f obtainedin this way is a subgame-perfectequilibrium,it sufficesto show that, fof all t > 1 and all x2 E Xt, ct+1(*Ixt) is the reduced path induced by f in subgame xt. For in that case rt+1(k) is the payoff vector induced by f in considerationstherefore apply. subgame xt. Standard dynamic-programming And to verifythat c1 is a selection from E1, we must show that, for all x2 E Xo c1( Ix)is the reducedpath induced by f in subgamex0. Overall,then, we may without loss of generalitytreat the case t = 0. for all t > 1, and EA(Xt) Fix g E X0. Let % = EgEA(X0), let a =A let A E A(Z) be the path inducedby f in subgameg. We show by inductionthat c1(- Ig)= fct+1(-Ixt)dA(xt) for all t > 1. To this end, note that c1(-Ig) is lft

at

obviously equal to fcl(- JxO)dA0(k0). Moreover fCt+l(.

= f[fC,+2( t2dft+lY.+l IX,,

,

X

dax

(by Proposition26 (ii)) =~|C+(*It)At+ ct+2(

=

~~ Xtld-

1(

At+l)

(by constructionof At+?).It follows that .Ix',(cl(* Ig))

=

f[f8zt

dct+

lkxt)

dat(k

)

(where zt denotes the projectionof z E Z onto Xt) =

f8

d/it(xt)

(where xt is the projection of =

&t

onto Xt, since z'=xtct+

( I2t)-a.s.)

f8ztdA( )

(where zt denotes the projectionof 2 onto Xt, because =XX,(A)

A

= k

= X,ZA)),

againfor all t > 1. But this in turnimpliesthat c1(Zg) = X4(A).

Q.E.D.

CombiningLemmas 30-32, we obtained the conclusion of Proposition27. This completes the third step. The fourth step is carried throughprecisely as before. We thereforehave the followingproposition.

SUBGAME-PERFECT EQUILIBRIUM

537

PROPOSITION34 (cf. Proposition19): E1 is an upper semicontinuous mapping from X0 to X(A(Z)). 6. EXISTENCE WITHANDWITHOUTPUBLICRANDOMIZATION

In Section 4 we showed that subgame-perfect equilibrium exists in the

extension of the basic game. The potential drawbackof this result is that, in order to obtain the existence of equilibriumin what is ostensibly a purely noncooperative situation, it introduces an extraneous element into the model,

namely public randomization.It is therefore of some interest to elucidate the circumstancesin which equilibriumexists even without public randomization. We begin with a recapitulationof the main result of Section 4. 6.1. Existence with Public Randomization

For all xo E XO,let El(xo) denote the set of payoffvectors of subgame-perfect equilibriaof the extensionof subgamexo of the basic game. PROPOSITION35

(cf. Proposition19): E1 is an upper semicontinuous mapping

from XO to

In particular,subgame-perfectequilibriumexists in the extensionof subgame xo of the basic game for all xo E XO. 6.2. Existence without Public Signals: Perfect Information

The basic game is a game of perfect informationif, for all t > 1 and all 1 xt- E Xt- 1, there is at most one i E Xf such that Ati(xt -1) containsmore than one element. Suppose that the basic game is a game of perfect information,and suppose that, for all xo e Xo, E1(xo) is the set of payoff vectors of subgame-perfect equilibria of subgame xo of the basic game. Then we have the following

proposition. PROPOSITION36: E1 is an upper semicontinuous mappingfrom XO to X(U '). Moreover E1 is convex valued. PROOF: The proof follows the same format as those in Sections 4 and 5: for all t > 1 and all xt1 eXt-1, we let Et(xt-1) be the set of payoff vectors of subgame-perfectequilibria of subgame xt-1 of the basic game; and, for all t > 1, we introduce mappings Ct, 'ICt+H, and 'ICt+ exactly as before. The main changes are as follows. First, in the forwardstep, we associatewith each

Borel measurable selection ct: Xt1 -* UX' from ICt+1 Borel measurable -* A(Yti) for all i E-,, and a Borel measurableselection

mappings fti: Xt1

538

C. HARRIS,

P. RENY, AND A. ROBSON

ct+1: Xt -- U-' from conv(Ct+1) instead of Ct+1. To obtain ctr1 we put j(x, y) = frdv(rlx, y) for all (x, y) E Xt- 1x Yt, eliminate any viable deviations to obtain c: Xt 1 x Yt- U-0, and let ct+1 be the restriction of c to X,. Secondly,we note that because there is perfect information,WCt+1and 1Ct+1

are both convexvalued. Hence, startingfrom any c1 E WC1,we may apply the forwardstep recursivelyto obtain, for all t > 1, Borel measurablemappingsfti: A(Yti) for all i e- ,, from conv(Ct+1) C 1ICt+2.

Xt--

and Borel measurable selections ct+ : Xt -U Q.E.D.

Notice that Proposition36 does use randomizationon the part of the active players,even thoughthe game is a game of perfect information.This is entirely appropriate.For, in the presence of Nature, pure-strategysubgame-perfect equilibriumneed not exist. To see this, consider the following four-stage game.32In stage 1 player 1 chooses a1 E [0, 1]. In stage 2 player 2 chooses a2 E [0, 1]. In stage 3 Nature chooses a3, randomizinguniformlyover the interval[ -(a, + a2),(al + a2)]. In stage 4 player4 picks a4 E {-1, 11.Player1 gets a payoffof 2 - a1 if a4 = 1, and a payoffof -a1 if a4 = - 1. Player2 gets -a2 if a4 = 1 and 2 - a2 if a4 = -1. Player4 gets a3a4.

Now suppose that playersare restrictedto pure actions.Note firstthat player

4 will choose a4 =1 if a3 >0, a4= -1 if a3 < 0, and will be indifferentbetween 1 and - 1 if a3 = 0. Hence, as long as a1 + a2 > 0, the payoffvector for players1 and 2 will be (1 - a1,1 - a2). If, on the other hand, a1 = a2 = 0, then the payoff vector for players 1 and 2 must either be (2, 0) or (0, 2). It is therefore easy to see that pure-strategysubgame-perfectequilibriumcannot exist: each player i E {1,21 can guarantee herself a payoff arbitrarilyclose to 1 by choosing ai arbitrarilyclose to 0; but if both i e {1,2) put ai = 0, then one or other receives a payoffof zero, and would thereforeswitchto some small positive ai. Suppose therefore that we eliminate Nature from the game, and suppose that, for all x0e X0, E1(xo) is the set of payoff vectors of subgame-perfect equilibriain pure actions of subgamex0 of the basic game. Then we have the followingproposition. PROPOSITION

37: E1 is an upper semicontinuous mapping from Xo to X(U -).

PROOF: The proof follows the usual format, but is radicallysimplified.The main change is in the definitionof 1'Ct+1. We now define I'PCt+1(x) to be the set of (y, r) E Ytx U-' such that: (i) r E Ct+ 1(x, y); and (ii) pi(y \ai) ri for all i cE and ai eAti(x). In other words,we summarizethe stage game in termsof an outcome and a payoffvector, instead of in terms of a joint distributionover outcomes and payoff vectors. Other parts of the proof are correspondingly Q.E.D. simplified. 32 This game is a simplifiedversionof Harris'(1990)originalcounterexample to the existenceof subgame-perfectequilibriumin gameswith simultaneousmoves.

SUBGAME-PERFECT

EQUILIBRIUM

539

For the original proof see Harris (1985a). For alternative proofs, see Hellwig and Leininger (1987) and Hellwig et al. (1990). 6.3. Existence without Public Randomization: Finite Action The basic game is a finite action game if Yti is finite for all t > 1 and all i Ef. (Note that we specifically exclude any assumption about the finiteness of YO. Otherwise it would not be possible to take finite approximations in the fourth step.) Suppose that the basic game is a finite-action game, and suppose that, for all x0 eX0, E1(xo) is the set of payoff vectors of subgame-perfect equilibria of subgame x0 of the basic game. Then we have the following proposition. PROPOSITION

38: E1 is an upper semicontinuous mappingfrom XO to X(U - ).

PROOF: Once again the proof follows the usual format. The main change is in the definition of t Ct+ 1. We now define it Ct+ l(x) to be the set of (,, v) E A(Yt) x (U')Yt such that: (i) v(y) E Ct+1(x, y) for all y eAt(x); (ii) ,u is the distribution of a Nash equilibrium of the normal-form game in which player i has action set Ati(x) and payoff function vilAt(x) for all i E Xf, and Nature chooses her

action accordingto fto (- Ix). In other words,we summarizethe stage game in Q.E.D. termsof a payoff-vectorfunctionand a distributionover outcomes. See Fudenbergand Levine (1983) for the originalproof. 6.4. Existence without Public Randomization: Zero Sum The basic game is a zero-sum game if there are precisely two players, and if the sum of their payoff functions is identically zero. Suppose that the basic game is a zero-sum game, and suppose that, for all x0 EX0, E1(xo) is the set of payoff vectors of subgame-perfect equilibria of the

basic game. Then we have the followingproposition. PROPOSITION

39: E1 may be regardedas a continuous function from XO to U-

Define mappings Et, Ct, TCt+1, and TCt exactly as in Section 4. By the resultsof that section, Et = Ct= tIC+1, for all t > 1. Because the extension PROOF:

of a zero-sumgame is still zero-sum,Et is single valued for all t > 1. Now suppose that, for all t > 1, ct: Xt-1 - U-' is a Borel measurable selection from tCt+1. Then we may adapt the forwardstep as in the proof of Proposition36 in such a way that we associatewith each ct Borel measurable mappingsfti: Xt1 -> A(Ytj)for both i E X, and a Borel measurableselection

ct+1: Xt -* UJ' from conv(Ct+1). But Ct+1 is single valued. It follows that

1 - ct+. The strategycombinationf constructedin this way is therefore a subgame-perfectequilibriumof the basic game. Moreover, the payoff vector

540

C. HARRIS,

P. RENY, AND A. ROBSON

inducedby f in subgamexo of the basic game is c1(xo)for all xo E XO. That is, c1 is a selection from E1. Because WC1is singlevalued and upper semicontinuous,c1 must be continuous. Because the basic game is zero sum, E1 must be single valued. The result Q.E.D. follows. It shouldbe stressedthat, unlike the set of payoffvectors of subgame-perfect equilibria,the set of paths of subgame-perfectequilibriadoes not vary upper semicontinuously. 7. ON THE MINIMALITY OF THE EXTENSION

We have seen, in Section 6, that there are at least three subclassesof the class of games studied in this paper in which it is not necessaryto introducepublic randomizationin order to obtain the existence of subgame-perfectequilibrium: games of perfect information,finite-actiongames and zero-sumgames. Hence, while it is certainly necessary to introduce public randomizationinto some games in order to obtain existence, there are other games for which such an extension appears to be redundant.This raises the question of whether the extensionis minimaland, if so, then in what sense. In the present subsectionwe argue that the extensionwe consideris the minimalextensionwith the property that the set of equilibriumpaths is robust to small changes in the extensive form. In particular,given any equilibriumpath of the extensionof any game,we can find a sequence of equilibriumpaths of nearbygames that convergesto the equilibriumpath of the extension of the reference game. To make this idea precise,we shall need the concept of an embedding. 7.1. Embeddings

Supposethat we are given two games, the basic game G and a new game G, and suppose that each entity of G is denoted by placinga tilde over the symbol for the correspondingentity of the basic game. DEFINITION 40: Supposethat there is a homeomorphismt that maps Yj into a subset of Yi for all t > 1 and i E A Supposefurtherthat: (i) tXt cX, for all t>O; (ii) Ati is an extensionof tAti for all t > 1 and i Ec-;

(iii) fto is an extension of tfto for all t > 1; (iv) uii is an extensionof ui for all i EE. Then we say that t is an embedding of G into G, that G is a topological

subgameof G, and we write G " G. If G is a topologicalsubgameof G and G is a topological subgame of G, then we say that G and G are topologically equal. In particular,if t is an embeddingof G into G then G is topologically equal to tG.

SUBGAME-PERFECT

EQUILIBRIUM

541

= Here (i) is self-explanatory; (ii) means, more explicitly, that iai(tx,1) tAti(xt_) for all xt_1 eXt-1; (iii) means that fto ( ltxt-) = tfto( Ixt-1) for all xt-1 EXt-1; tfto ( Ixt-1) is simply the distribution of 1: Yt - Yt when Yto is given the probability measure fto ( Ixt_l); and (iv) means that ui(Lz) = ui(z) for all z E Z.

Notice that we have alreadyencountereda special case of an embedding:the infinitegame consideredin Section4.4 can be embeddedin its finite approximation. 7.2. Robustness

Let a9 be the subclassof games studied in this paper, the games of which have a unique startingpoint. 41: An equilibrium concept associateswith each game G E DEFINITION set of paths of G.

a

Two examples of equilibriumconcepts are ,/, which associates with each G E E9 the set ,Y(G) of paths of subgame-perfectequilibriaof G, and /, which associateswith each G the set 91(G) of reducedpaths of subgame-perfect equilibriaof the extensionof G. From the counterexampleof Section 2, we know that ,/ may be emptyvalued. From the existence theorem of Section 4, we know that

v/

is nonempty valued.

In order to give a precise definition of what it means for an equilibrium concept 6' to be robust,we shall need the concept of topologicalconvergence. 42: Suppose that we are given a sequence {GnIn E=N}c a9 and a X E9. Then we say that {GnInE R1}converges topologically to G' iff there exists a game G with XO= N U {oo}such that Gn is topologicallyequal to subgamen of G for all n E N U {oo},and we write Gn -* G'. Suppose that, in addition, we are given AnE 6'(Gn) for all n E N U {oo}. Then we say that {AnInE N} converges topologically to AXiff {AkInE R1}convergesto AX,where An is the distributionof Anunder the embeddingof Gn into G for all n E N U {oo}, DEFINITION

and we write An-A r.

We can now define robustness. DEFINITION

43: An equilibrium concept 6' is robust if, whenever G

G',

AnE 4'(Gn) for all n E N and An A X, AE' (G'). In other words,for all referencegames G', if Gn is close to G' and Anis an equilibriumpath of Gn, then there is an equilibriumpath Arof G' such that An is close to AC.

542

C. HARRIS,

P. RENY, AND A. ROBSON

7.3. Minimality

In this subsection we shall show that / is the minimal robust extension of

/'.

44: Suppose that we are given equilibriumconcepts 41 and 4,2. DEFINITION Then 492 extends 49 iff 41 c 492. 42 is the minimal robust extension of 49 if 492 is a robust extension of 41, and 492 C 493 for any other robust extension 43 of 49. THEOREM

45:

v

is the minimal robust extension of

J?.

E and all PROOF: To prove this we need only show that, for all G 'E9 RJ} E1 e'(GW), we can find {GnIn E N and AnE (Y(Gn) such that Gn _G cJ AX and An_- A. Suppose without loss of generalitythat, in G', the set of starting points is YO= {oo}, and consider the game G for which: (i) the set of starting points is YO= RJU {oo};

(ii) Nature'sambientaction space for stage t is Yt? = Yt? x [0, 1]; (iii) in subgamen EcX0= Y0,Nature'saction set for stage t is the productof her action set for stage t of G with the interval[0, e-n], and her mixed action is the product of her mixed action for stage t of G with In, the uniform distributionon [0, e-n]; but which is otherwiseunchangedfrom G'. For all finite n E XO,subgamen of G is topologicallyequal to the extension of G', and subgameooE XO of G is topologicallyequal to G'. Fix a subgame-perfectequilibriumof the extensionof G'0,and suppose that its reduced path is A. For all finite h EXO, this equilibrium induces a subgame-perfectequilibriumin subgamen of G. Let Anbe the path of this equilibrium.Also, the reducedpath A inducesa path Arof subgame ooE Xo of G. Clearly An-* A".

Q.E.D.

It should be stressedthat, althoughthe proof of Theorem45 exploitsNature, the result itself holds even if Nature is excludedfrom the model. There are two because basic possibilities. If f contains only one player, then /= a single player can always achieve whatever can be achieved with the aid of a public signal without its aid, simply by randomizing.(Cf. the proof of Theorem36.) If .' contains two or more players, then more work is required.The basic idea is to choose two players,say players1 and 2, and extend their actionsets by adding an extra dimensionjust as Nature's action sets were extended in the proof of Theorem 45. These two players can then be made to play a jointly controlled lottery over their extra, dummy,actions.33In order to understand how this works, consider the mapping H: [0,1] x [0,1] -* [0,1] constructedas follows. Let /3: [0,1] -> {0, 1} map a to its representationas a binarysequence 33 Cf. Aumannand Maschler(1966).

SUBGAME-PERFECT

EQUILIBRIUM

543

b = < b In E N > . Let 0: {0, 1}N x {0, 1}' -> {0, 1}' map (b,, b2) to B according to the formula:Bn = 1 if bn = -b and Bn = 0 if bn * bn. And let H(a , a) =

,1-1(0(,3(a,), 13(a2))).Then it may be checked that, if player 1 chooses a, accordingto the uniformdistributionover [0, 1], then the distributionof H will be uniform too, irrespectiveof the manner in which player 2 chooses a2, and vice versa. So two or more players can achieve whatever can be achieved with the aid of a publicsignalwithoutits aid, providedthat their action sets are rich enough. 8. CONCLUSION

In this paper we have given an example of a continuous game of almost perfect informationin which subgame-perfectequilibriumdoes not exist. Existence could, however,be restoredin this game if the playersin stage two were permittedto coordinatetheir choices on the basis of a toss of a coin. Moreover allowingthem to use a coin in this waywas entirelyconsistentwith the spiritof the game. This example was the first part of our case for the use of public randomizationin continuousgames of almost perfect information. We have also shown that subgame-perfectequilibriumalways exists in a continuousgame of almost-perfectinformationif playersare allowedto observe a sufficientlyrich public signal at the end of each stage of play. This existence result constitutesthe second part of our case for the use of public randomization in continuousgames of almostperfect information. Finally, we have shown that the extension of subgame-perfectequilibrium obtainedby allowingfor public randomizationis the minimalrobust extension. This constitutes the third and last part of our case for the use of public randomizationin continuousgames of almostperfect information. King's College, Cambridge CB2 1ST, U.K, Dept. of Economics, Universityof Pittsburgh, Pittsburgh, PA 15260, U.S.A., and Dept. of Economics, Universityof WesternOntario, London, Ontario N6A 5C2, Canada Manuscriptreceived December, 1990; final revision receivedJune, 1994.

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