A characterisation of the perfect equilibria of infinite horizon games

JOURNAL OF ECONOMIC THEORY 37, 99-125 ( 1985)

A Characterisation of the Perfect Equilibria of Infinite Horizon Games

CHRISTOPHER HARRIS *

Received March 2, 1984; revised April 24, 1985

This paper builds on the work of Fudenberg and Levine (J. Gon. Theory 31 (1983). 251-268). It shows that the perfect equilibria of any game in which events become uniformly unimportant as their distance into the future increases can be characterised as limits of sequences of perfect approximate equilibrium points of finite horizon approximations to the game. The result holds both for a strong and for a weak topology. The topologies are tractable. and the nature of convergence relative to them is transparent. Finally, the weak topology is probably the weakest tractable topology in which the result holds. Journal of Economic Liferarure Classification Number: 022. 1’ 1985 Academic Press. Inc.

1. INTRODUCTION

This paper offers a characterisation of the perfect equilibrium points of infinite horizon games in which events become uniformly unimportant as their distance into the future increases. Such a characterisation is of interest of two reasons.

First, adequate representation of situations involving strategic interac- tion frequently requires an infinite horizon game. The assumption that a game takes place over only a finite number of periods is often artificial. Moreover, if players have complete and perfect information then the natural solution concept is Selten’s [lo] perfect equilibrium. The perfect equilibrium points of an infinite horizon game are, however, very elusive, essentially because the absence of a fixed endpoint precludes an approach using dynamic programming. It is therefore desirable to find alternative approaches.

Secondly, when games such as the prisoner’s dilemma or the Cournot oligopoly are repeated, it would seem intuitively likely that there should be an equilibrium in which players coordinate on the Pareto superior option

* I should like to thank Jim Mirrlees for his help and encouragement. I should also like to thank an anonymous referee for his careful and helpful suggestions, many of which I have incorporated in the present version of this paper. Finally I should like to thank Franklin Allen, Ken Binmore, Terence Gorman, Kevin Roberts, and Joel Sobel.

99 0022-0531/85 $3.00

Copyright t 1985 by Academic Press. Inc All rights of reproductmn m any form reserved

100 CHRISTOPHER HARRIS

of cooperation. This is not the case for finite repetitions, a result that has been called paradoxical (see Radner [S] and Selten [ 111). There are two ways in which the paradox may be avoided. One is to relax the rationality assumption to that of bounded rationality (this approach is used by Rad- ner [8], for example). The other is to repeat the game an infinite number of times. Both lead to qualitatively similar results. It is therefore interesting to make precise the nature of the connection between the approximate equilibria of the finitely repeated games and the exact equilibria of the infinitely repeated game.

Fudenberg and Levine [ 1 ] proposed that the perfect equilibrium points of an infinite horizon game could be approximated by perfect approximate equilibrium points of finite horizon approximations to the original game. I follow their proposal here. The problem divides into two parts. The first is to find tractable finite horizon approximations to the original game. Fudenberg and Levine achieved this by considering the truncations of the original game obtained when players’ freedom of action is taken away after a given period. The finite horizon games which result are then amenable to the techniques of dynamic programming.

The second part of the problem is to find a suitable sense in which one equilibrium point can be said to approximate another. This amounts to finding a tractable notion of the convergence of strategy combinations. Such a notion is represented by a metric, or, more generally, a topology on the space of strategy combinations for the game. Unfortunately, with the exception of the class of finite action games, the metric provided by Fuden- berg and Levine is unsatisfactory. It is difficult to verify convergence in their metric. Also, their metric gives little direct insight into the connection between an infinite horizon exact equilibrium and the finite horizon approximate equilibria that approximate it. That their metric is satisfactory for finite action games is a consequence of the fact that there is only one metrisable product topology on the space of strategy combinations of such a game, namely the product discrete topology. Thus it is not immediately clear how tractability can be achieved for more general games.

I provide two topologies. one strong and one weak, both of which are more tractable than the metric of Fudenberg and Levine. They are based on the concept of finite convergence. The perfect equilibrium points of an infinite horizon game can be characterized as limits, relative to the strong topology 9, of perfect approximate equilibrium points of finite horizon truncations of the original game.’ The same is true for the weak topology dl‘.” Because convergence in 9 is more restrictive, the former characterisation is likely to be more helpful for addressing questions of uni-

’ This is actually a corollary of Proposition 7 ’ This IS a corollary of Propwtcon 7.

PERFECT EQUILIBRIUM 101

queness. The latter characterisation is more likely to be helpful in the construction of equilibria, since less work is required to verify convergence in a weaker topology. The two topologies can be illustrated as follows.

Suppose that an identical game is to be played N times. Players’ payoffs are the sum of their discounted single period payoffs. For each II, let J”” be a perfect a,,-equilibrium point of the II period game. According to .Y the sequence { .f’“’ ) converges iff, for any given horizon t,,, there exists IZ,~ > t,, such that, for each rb t,,, the contingent strategies employed by players in period f are identical for all H 3 II,,.

Next, each strategy combination,f”” gives rise to a sequence of outcomes on any given subgame. According to H’ the sequence [,f”“‘j converges iff, for any given horizon I,, and any given subgame, there exists some II,, 3 t,, such that, for each t < t,,, the outcome in period f is identical for all II > II,~.

Given the general aim of characterising perfect equilibrium points as limits of equilibria of trzrncorrd games, .‘/’ is as strong as one can hope for. By contrast, %l’ is not particularly weak. This raises the question of whether a substantially weaker topology in which the characterisation holds can be found. In particular, is the topology 9 induced by Fudenberg and Levine’s metric weaker than Y/ ?

The principal factor limiting the weakness of topologies in which the characterisation holds is the need to ensure that the i:-equilibrium set is closed for all C. Ideally one should investigate the weakest topology with this property. I have unfortnately been unable to obtain any results about this topology. I therefore consider instead a very weak topology .f which is defined with a requirement that the perfect equilibrium constraints be continuous in mind. The topology .B closely parallels Fudenberg and Levine’s i/. which was defined in relation to a requirement that the perfect equilibrium constraints be equicontinuous. In particular, .f is. in general, strictly weaker than 9. Like ‘9, however, .B is not tractable as it stands.

I show that, in a wide class of games, .9 actually coincides with N ‘.I This result suggests that it may be difficult to find a topology which is both weaker than Yf and tractable. and in which the characterisation holds. It also has the corollary that .f is, in general, strictly weaker than ‘/ in such games. Indeed. one can show further that. in a slightly narrower class of games. V coincides with :/‘. Finally, it can be shown that, in a third class of games, the weakest topology in which the perfect equilibrium constraints are continuous coincides with Y/ . . Overall. the characterisation in terms of Yf appears to be the best that is attainable.

Section 3 presents the framework for the analysis, and describes the truncation procedure. Section 3 contains a brief account of finite convergence. Section 4 defines perfect I:-equilibrium. describes .Y’ and Yf . . and states the


characterisation4 The proof of the characterisation is deferred until Sec- tion 6. Before proceeding to it, the problem of topologising the space of strategy combinations is discussed in Section 5. In Section 5 the topologies .a and 9 are introduced. It is shown that 9 is potentially weaker than H ‘5 but that in a wide variety of circumstances merely coincides with W ‘.” Next, since part of the motivation for the results of this paper lies in their possible use in the analysis of infinite horizon games, it is interesting to have examples of their use. Fudenberg and Levine have already analysed a special case of Rubinstein’s bargaining model as a non-trivial example of how such results may be used to show uniqueness.’ In Section 7 I present a non-trivial example of how they may be used to find an equilibrium. Sec- tion 8 concludes.

2. THE FRAMEWORK

In this section I present the framework that I shall employ for the remainder of the paper. It is almost identical to that employed by Fuden- berg and Levine [ 11, except that I have adapted their notation to the con- structions of this paper.

Tlw Ambient Product Spare

The extensive form of a game can be described by the set H of possible histories. For mathematical convenience I shall embed this set in an “ambient” product space S. S is described as follows.

There are N players, indexed by i. Time is divided into discrete intervals, indexed by t. In each period (t ) each player (i) chooses independently some action which can be represented by an element of S,,. The outcome in any period can therefore be represented by an element of

s, = >i s,,. I=1

Play begins in period 1. The history of the game consists of the outcomes of each period and can thus be represented by an element of

S={O} x x s, r=l

( { 0 ) is included for notational convenience 1.

J Proposition 7. 5 Proposition 1. b Propositions 2 and 3. ’ Fudenberg and Levine [I. p. 265; Example 5.11.


For any sequence of outcomes s E S, I shall write

.Y = (.I-(,, s, )..., s, )... )

were x, E S, is the outcome in period t. I shall also write

s, = (9 ,, , x,-j ,..., s,v)

where .xr, E S,, is the action taken by player i in period t. Finally I denote by

i.,.u = (xc,, s, ,..., x,)

the sequence of outcomes up to period t.

Histories and Actions

Let H denote the set of histories that can arise in the game. H s S and Hf 0. Then

is the set of initial histories that can occur over the first t periods. In general the outcomes possible in period t will depend on the initial

history over the first t - 1 periods. If j”, , s is such an initial history let

A,(& ,s)= [.1’,/ JE H, jL, I y=i,, ,x)

denote the set of outcomes possible in period t. 1 assume that players choose their actions independently. Thus A,(]., ,.u) factorises as

,v A ,O., ,s)= x A,,(L ,x1

,=I

where A,,(i,- 1.r) denotes the set of actions feasible for player i given the initial history I, ,x.

Strategirs

Each player must specify in advance what action he will take contingent on any initial history. Thus player i must choose, for each t 3 1, a function

.I;, : j., , H + S,i

which satisfies

.r;,c j., ,.Y)E A,,()., ,.u) for all s E H. That is, the action specified must always be feasible.


If each player has chosen such a strategy for each period then write

f; = u;, ? f;23..., .I;,\ 1

for the strategy combination for period t, and

.f’= (.A 1 .f2.-.9 ./; ,... 1

for the strategy combination for the entire game. Let F(H) be the set of strategy combinations. The notation reflects the

fact that these have been defined naturally in terms of H. Denote by F’(H) the set of strategies of player i, and by 11’ a typical element of F’(H). If ,/‘E F( H) and h’ E F’( H) let ,f’\lt’ be the strategy combination obtained by replacing player i’s strategy I“ by 1~‘.

Outconles

Suppose that the strategy combination ,f‘ is employed from period t + I onwards after an initial history i,,s. Denote by u[,L X, t] the resulting sequence of outcomes. In order to guarantee that %[A s, r] E H for all 5 .Y and t, I must place a simple requirement on H. I require that, for any x E S, if 2.,x E ;l, H for all t 2 0 then .X E H. In other words, if every initial segment of s is an initial history, then x itself must be a history. This requirement is satisfied iff M[,L s, t] E H for all ,f’~ F(H), .Y E H, and t 3 0.

Player i’s payoff is a function.

P,: H-R.

P,(x) is player i’s payoff from the history SE H. Let P= (PI, P?,..., PN) denote the vector of players’ payoffs.

E.utensive Form

A game in extensive form is a pair (H, P) where H and P satisfy the above description. The strategy combination space F(H) is defined naturally from H. H can be thought of as the game tree. One may identify an initial history I, ].Y with a node or vertex of this tree, and the outcomes possible after 2, , X, A,(2, ,.u), with the branches leading from node i ., , s. Subgames may be identified with initial histories in a natural way. The initial history R, ,x is identified with the subgame (G,(A,- ,x), plCl(j.r I\.) ) of the original game (H, P).

Truncated Games

A truncated game is one in which players’ freedom of action is removed after a certain period. For instance, if the zero vector lies in A ,(I., ,x) for


all x and t, then “do nothing” has a reasonable interpretation, and (H, P) can be truncated by requiring that all players do nothing after period t (i.e., fY=O for all s> t). This is the procedure employed by Fudenberg and Levine [ 11.

More generally, in order to truncate (H, P) at period t it is necessary to specify the action that players will be forced to take after period t. These actions will depend on the history of the game up to period t.

DEFINITION 1. Suppose that, for each f, n,: H + H is a mapping satisfying:

(i) A,(x,.Y) = 1.,x for all .YE H;

(ii) Z,.Y = 71, y for all s, ~3 E H such that 3.,s = A, J*:

(iii) 7r,+, 7-c, = 71,.

Then {rcl),YO is a truncation procedure for (H. P).

For any given X, 71, specifies the path that play will be constrained to follow it if the initial history of the game is 1+,x. The first condition requires that it do so in such a way that this initial history is preserved. The second requires that n,.u depend only on this initial history, and the third ensures that the method of truncation is consistent across periods. The existence of a truncation procedure for (H, P) is guaranteed by the axiom of choice.

DEFINITION 2. Suppose that in, 1 jYO is a truncation procedure for (H, P). Then (7c, H, Prn,H) is the truncation of (H, P) at period f.

If 14 is a truncation procedure for (H, P) and s 3 t then x,HE~,HE H and F(~,H)~F(Tc,H)EF(H), by the third condition. That is, the history and strategy spaces of later truncations nest those of earlier truncations.

3. FINITE CONVERGENCE

This section gives a brief account of the concept of finite convergence, or convergence in the discrete topology. Consider a set A’, a sequence ( .Y,,};:= 0 E X, and a point X.

DEFINITION 3. .(.Y,, ) converges ,fhitefv to x iff there exists n, such that x,, =x for all n > n,,.

That is, the sequence (xn} converges finitely to x iff it ultimately coincides with .Y.


The usual concept of pointwise convergence of functions has a simple analog for the concept of finite convergence. Specifically, let {g,};=” be a sequence of functions,

g,, : y -+ x

and let g be a function, g: Y .+ X.

DEFINITION 4. { gll} converges pointwise finitely to g iff {g,(y)} converges finitely to g(v) for all y E Y.

In particular, a sequence of X is a function,

h: N-+X,

where N denotes the natural numbers. Thus we obtain a notion of convergence for sequences of sequences. In fact, if {h,}FCO is a sequence of sequences then {A,,} converges pontwise finitely to h iff h, ultimately coincides with h over any given initial segment. The pointwise convergence of sequences of sequences will play an important role in my analysis.

4. THE CHARACTERISATION

In this section I state the main result of the paper (Proposition 7). This proposition is concerned with games in which events become uniformly unimportant as their distance into the future increases. Such games are said to be continuous at infinity. It shows that the perfect equilibrium points of such games can be characterised as limits of sequences of perfect approximate equilibrium points of truncations of the original game.

According to Selten’s [lo] concept of perfect equilibrium, a strategy combinationf’is in perfect equilibrium iff its restriction to any subgame of the original game is in Nash equilibrium. In what follows I shall need a more general definition.

DEFINITION 5. A strategy combination f is a perfect E-equilibrium point iff

P;(Ct[f‘\h’, -Y, t])-Pi(a[.fi Mx, f] GE

for all I < i d N, h, E F’(H), x E H, and t 2 0.

In other words there is no subgame on which any player can increase his payoff by more than E by changing his strategy unilaterally. If E = 0 then f is a perfect equilibrium point of the game.


DEFINITION 6. Suppose that, for each n, ,f”” is a perfect &,,-equilibrium point of (H, P) truncated at period t,,, and that, moreover, E,, + 0 and t,, -+ cc as n + co. If ,f”“.c f then .(f”“} is an approximating sequence for f relative to F.

Suppose that {f’“‘j . IS such a sequence. The larger n becomes, the more closely the truncation at t, of (H, P) would appear to approximate (H, P). Also, as n becomes large, so E,, becomes small, and f’“’ comes closer to being a perfect exact equilibrium point. It therefore seems likely that if F is sufIiciently strong to ensure that the perfect equilibrium constraints are continuous, then f will be perfect.

DEFINITION 7. %” is the topology with basis consisting of the sets

obtained as x varies over H, and t and s vary over all periods.

Each f E F( H) induces a sequence of outcomes, or history, .V = a[J X, t] on any given subgame 1,x. Thus ,f““’ 5 ,f’ iff, on any given subgame, the sequence of histories induced by {,f”“} converges pointwise finitely to the history induced byf:

Provided that the Pi satisfy a mild continuity condition,’ the limit of any approximating sequence relative to % 1 is perfect.” In Section 5 below I shall argue that W‘ is actually the weakest tractable topology with this property. I turn now to the converse problem of finding an approximating sequence for a given perfect equilibrium point.

Suppose that f is a perfect equilibrium point of (H, P). One way of obtaining an approximation to ,f is by truncating it at some period t. This can be done without undue loss of precision provided that the variation in player’s payoffs that results from changes in the history of the game after period t is not too large. Thus, for each t 3 0, let

w, = sup (IP,(*y) - P,(y)1 ). I<,<N V.VEH

i,r = n, i’

w, bounds the variation in players’ payoffs that can result from differences in the history of the game that occur after period t. w, may be infinite. Cer- tainly { w, I;“= 0 is decreasing.

” Continuity in the product discrete topology. ‘) Proposition 5.


DEFINITION 8. (H, P) is continuous at infinity iff 0, -+ 0 as t -+ CG.“’

In other words, a game is continuous at infinity if events become uniformly unimportant as their distance into the future increases. Many games satisfy this requirement. Suppose, for example, that a game in which players’ payoffs are bounded is repeated an infinite number of times. Sup- pose further that their payoffs in the repeated game are the sums of their discounted single period payoffs. Then this repeated game is continuous at infinity. The special case of Rubinstein’s [9] bargaining model in which a player’s payoff is the share in the pie he obtains, discounted for the period in which he obtains it, is continuous at infinity. So too is the model of a race of Harris and Vickers [4]. Repeated games without discounting do not satisfy the requirement of continuity at infinity. Finally, for (H, P) to be continuous at infinity it is sufficient that H be compact and P continuous. ’ ’

The assumption of continuity at infinity is not a particularly strong one. It is, however, possible to obtain very accurate approximations to the perfect equilibrium points of any game that satisfies it, simply by truncating them. That is, it is possible to obtain approximations relative to a strong topology. It is this topology that I now introduce.

DEFINITION 9. ,!Y is the topology with basis consisting of the sets

(81 gEF(H), hg=A,f’l

obtained as f varies over F(H) and t varies over all periods.

An element of F( H) is a sequence (fi, f2,...) of functions, so that Y is the topology of pointwise convergence: fCn) -% f iff, for all t, there exists n, such that i,f(‘) = i,f for all n 3 n,. .40 is, essentially, a uniform version of W. In the case of w‘, f”‘) 5 fiff, for any given subgame, the sequence of histories induced by (J”“‘S converges pointwise finitely to the history induced by f: In the case of ,Y, f’“‘-% f iff, for all subgames simultaneously, the sequences of histories induced by (fen)} converge pointwise finitely to the histories induced by f: I can now state:

PROPOSITION 7. Suppose that (H, P) is continuous at infinity. Let Y be any topology intermediate between W and Y’, ,%‘” E Y c Y. Then f is a perfect equilibrium point of (H, P) iff it has an approximating sequence relative to Y.

Section 6 is devoted to a proof of Proposition 7. Most of the interest of the proposition centres on the two extremes, W and Y. If one is trying to

lo The notion of continuity at infinity is due to Fudenberg and Levine [ 11. It amounts to a requirement that P is uniformly continuous in the product discrete topology.

” Harris [3, pp. 91-92; Proposition 41.


construct a perfect eqilibrium point of a game it may be easier to show that an approximating sequences converges in W, since in this case pointwise finite convergence need only be verified for representative subgames. If, on the other hand, one seeks to demonstrate that a particular perfect equilibrium point is unique, or at any rate that the perfect equilibrium points belong to some restricted class, then it may be more fruitful to use .40. For ,Y imposes more restrictive conditions on an approximating sequence, which are more easily violated. Finally, whether one is interested in uniqueness or existence, a suitable choice of truncation procedure may simplify the analysis. My purpose in making an essentially trivial generalisation of Fudenberg and Levine’s truncation procedure was simply to emphasize this point.

5. A WEAK TOPOLOGY FOR THE SPACE OF STRATEGY COMBINATIONS

In general, a minimal requirement of a topology Y on F(H) is that, for all E > 0, the s-equilibrium set be closed in Y. This requirement would tend to make Y strong. From other points of view, however, it is desirable that Y be as weak as possible. The weaker Y, the more likely is it that F(H) will be compact in Y. More relevantly in the present context, the weaker Y the less the work that will be involved in verifying that a sequence of strategy combinations converges in F. This is a consideration if, for instance, one seeks to construct a perfect equilibrium point as the limit of an approximating sequence. Finally, F should be tractable. Verifying convergence in a weaker topology may require less work in principle, but this is of little help if, in practice, it is diffrcuit to do so directly.

One approach to the problem of finding a weak topology in which the e-equilibrium sets are closed is to define topologies that satisfy this requirement more or less directly. The definition of such topologies is naturally implicit, and they tend to be intractable as defined. Hence it is important to discover what convergence in them actually entails.

I consider one such topology, 3, which is defined in such a way as to ensure that the perfect equilibrium constraints are continuous when payoffs are continuous. I show that, in a wide variety of games, .a actualy coincides with W. In such games W resolves the intractability of .a. For it is relatively straightforward both to verify and to disprove convergence relative to ~5’~. The same finding also suggests strongly that W- is about as weak a topology as is consistent with the s-equilibrium sets remaining closed. A fortiori, #” is probably about the weakest tractable topology that is consistent with this requirement. Y& is, unfortunately, clearly very strong. This is well illustrated by the observation that F(H) is compact in ‘N. only if (H, P) is a finite action game.


Before I can accomplish the definition of 9, I need to construct a topology for H. Suppose that, for each i and t, S,, is endowed with a topology. Let S, = X,“=, Sti be endowed with the product topology. Similarly, let S = (0) x XT=, S, be endowed with the product topology. Lastly, let H inherit its topology from S in the usual way.

I can now define 3. For each .Y and t. there is a mapping 4, : F(H) -+ H given by d,(f) = c~[,f, x, t] for all f E F(H). Let @, be the set of such mappings. Similarly, for each i, h’ and t, there is a mapping &: F(H) -+ H given by C& = cr[ .\h’, ,Y, t]. Let Qz be the set of such mappings. Let @ = @, u Q2.

DEFINITION 10. 9 is the weakest topology on F(H) in which all the mappings in @ are continuous.

A sequence of strategy combinations {f 'n'} converges in 9 to f iff for each subgame i,x: (i) the history induced on I,x by f@' converges to that induced by f; and (ii) for each player i and each possible deviation h’ by player i on the subgame 2,x, the history induced by ,f(“)\hi converges to that induced by f\h'. If two strategy combinations are to be close in 9, then, not only must the histories they induce on any given subgame be close, but also the histories they induce when a given deviations is played against them must be close.

If the Pi are continuous then all mappings of the form Pi0 4 (where CJ~ E @), and so the perfect equilibrium constraints, are continuous in 9. Hence the s-equilibrium set is closed in 9, for all E 2 0. This is the motivation for the choice of 9.

There are other ways of defining very weak topologies in which the s-equilibrium set is closed for all E. Let 9 ** be the weakest such topology. Let 9* be the weakest topology in which all the composition mappings of the form Pi0 4, are continuous. Y** is always weaker than Y*. If, furthermore, each Pi is continuous, then .a** and .a* are both potentially weaker than Y.If, on the other hand, the Pi are relatively wild, .a may be weaker even than 9**. I shall argue below that there is nothing to be gained by using $;* rather than 9. .a*, too, coincides with W in a wide class of games. About Y** I do not have any results, however.

The topology Y parallels the topology used by Fudenberg and Levine [ 11, which I shall call 9.

Fudenberg and Levine took each S,i to be a copy of lR”, for fixed M. They endowed each S,! with the standard Euclidean metric. They then endowed S (and H) with a product metric, d, chosen in such a way that the distance between any two points could not exceed unity. They defined a metric, D, on F(H) by the formula.


for all J g E F(H). Thus 9 is simply the topology indued on F(H) by D. It is the weakest topology in which the mappings in Q, are equicontinuous. and as such can only be defined when the S,, have sufficient structure to allow a suitable definition of uniform convergence. It must therefore be understood that, when I refer to the topology f%l, I am implicitly restricting attention to such cases. This should not cause any confusion.

The relationship between 9 and ,f is analogous to that between .4a and Y#.‘. In effect D is a uniform version of 9. More precisely, in the case of .(/: there is an additional requirement that the mappings in @J be equicontinuous. This requirement arises only because Fudenberg and Levine wan- ted to define a metric on F(H). It is, nonetheless. a strong one.

Like 2, the topology .a is not tractable. The nature of convergence in .P is unclear. Furthermore, in order to verify or disprove convergence in 9, it is necessary to consider all subgames and all possible deviations in all subgames. One indirect way of verifying convergence in .f would be to demonstrate convergence in some stronger, more tractable, topology.

PROPOSITION 1. #“is stronger that1 .f.

Proposition 1 can be understood as follows. Y4. is the weakest topology in which all the mappings of @, are continuous when H is endowed with the product discrete topology. Hence, if #‘* is the weakest topology in which all the mappings in @, are continuous when H is endowed with some other product topology, 1IL’ is stronger than Y+‘*. More importantly, since all the mappings in @, are continuous in YY. when H is endowed with the product discrete topology, so too are the mappings in @,. Hence, if I‘* is instead the weakest topology in which the mappings in both CD, and Qz are continuous when H is endowed with some other topology, then X‘ is stronger than %‘*. In particular, Y!. is stronger than 9.

Proof: In view of the foregoing discussion. I need show only that all the mappings in Q2 are continuous in %. when H is endowed with the product discrete topology. To this end, let CJ~ = t( [. \/I’, X, r] be an arbitrary element of CD?. The sets

obtained as J’ varies over H and r 3 0 constitute a basis for the product discrete topology restricted to H. Fix J and r. I need to show that

4~ ‘(U)= {.I‘I.~‘EF(H), i,.(x[.f’\h’,x, r])=&y)

is open. Suppose first that r d t. Then 4 ‘(U) = F(H) and 4 i(U) = @ when 2,~ = A, JJ and I,.u # A, JJ, respectively. In either case 4 ‘(U) is open. Suppose then that r > t. If %,x # 2, J’ then once again 4 -l(U) = @ and is


therefore open. Suppose that I,x = jury. I show that, in W‘, C$ ‘(U) is a neighbourhood of each of its points. Let ,f~ d- ‘( U) be arbitrary, let r = ct[,f\hj, I, r], let

W.,= (slj..,+,(~Cg,=,sl)=~,+,=~

for each t,<s< r, and let r- I

W= n w,,. .s = I

By construction, W is open in W. Also

gc Wog,+,(j~,,z)=z,,+, for all t<s< Y,

*gv+lVA= (f\hiL+, (A4 for all t d s < r,

* k\h’),+ l (44 = (f\W,y+I (44 for all t 6 s < r,

ai,(cx[g\h’, x, t])=l,(cx[f\h’, x, r]).

ThusfE Wcq5 ‘(U), as required. 1

It can be shown in much the same way that 9 ~9’. Thus W” and Y resolve, to some extent, the problems of verifying convergence relative to 9 and 9, respectively. It can also be shown that if the Pi are continuous in the product discrete topology, then .a* E 0’.

It is so far unclear whether ,%lk’ resolves the intractability of 9 at the expense of a considerable gain in strength. The next proposition shows that this is not the case when the S,; are endowed with the discrete topology.

PROPOSITION 2. Suppose that each S,; is endowed with the discrete topology. Then .a and @- coincide.

Proof Suppose that each Sri is endowed with the discrete topology. Then S will be endowed with the product discrete topology. By construction -w‘ is the weakest topology that makes all the mappings in Qi, continuous in this topology. Since .a makes all the mappings of @ continuous and, moreover, @, c @, we have that W 5 4. But Proposition 1 shows that 4 E W, whatever the topologies on the S,;. Thus 9 = w‘. 1

Notice that, in the case where each S,; is endowed with the discrete topology, W also resolves the problems of disproving convergence in 9.

If each S,; is endowed with a discrete metric in which the distances between distinct points are uniformly bounded away from zero, then 9 and 9 coincide. In this latter case Y resolves the intractability of 9.

There is an implicit trade-off in the definition of 9. The stronger the topology on S, the weaker the continuity requirement on the Pi, but the

PERFECT EQUrLlBRlUM 113

stronger the requirement that each mapping in 0 be continuous. Proposition 2 shows that when this continuity requirement is at its strongest, 9 becomes so strong as actually to coincide with W. There are other circumstances under which 9 coincides with -II’“.

DEFINITION Il. Suppose that, for each i and for each initial history i, y, there exist T > t and a closed set Cc S,,, , ,, such that:

(i) for all z E H such that %,z #i,, I’,

A(?-+ ,,,(j.,z)n cz $3;

(ii) for all z E H such that 1,~ = A, ~1,

A cl-+ t,;(~.z)\Cf @a.

Then (H, P) is free.

If (H, P) is free then there is a closed set C from which player i can choose his action in period T + 1 when the initial history up to period t differs from A, y, and an action xfT+ i,;, not contained in C, that he can choose when the initial history up to period t is i,,y.

Any game in which, for every period and every player, there is a later period in which that player has two actions that he can take whatever the previous history of the game, is free.” Examples of such games include any repeated game (provided that each player’s strategy set for the constituent game is non-trivial), and the model of a race of Harris and Vickers [4]. An example of a game that does not satisfy this more stringent requirement, but is nonetheless free, is the game in which two players take turns removing an object from a countable set. ”

PROPOSITION 3. Suppose that (H, P) is free and that N 3 2. Then .Y=W.

I begin with an informal account of the how proof of Proposition 3 works. Both in this account and in the formal proof, I shall assume that the Tin Definition 11 can always be chosen to equal t. It will be clear that the extension to the general case is immediate. Suppose that the sequence of strategy combinations {f “‘I} convrges to f relative to 9. In order to show that it converges relative to We too, consider the subgame I,x and the initial segment of the game up to and including period s (s > t). I need to

lz Strictly speaking, a requirement that points of S,, be closed would also have to be imposed.

I’ I am grateful to Andrew Postlethwaite for this example.


show that L,s(~[,f”“’ , X, t]), the play in this initial segment associated with f”“, ultimately coincides with n,(~[f, X, t] ).

To this end consider a particular deviation, hi, by player i fromf’“‘. Sup- pose that up to periods a player i follows fi rather than (f’“‘)‘, so that hi =f: for t < I’ d s. Thus, if ~,~(~~[f~“‘\,h’, X, t]) differs from 2,(a[J; ,Y, r]) at all, it must be because the strategy (,f’“‘)’ of some player j other than i involves deviation from the latter path. Next, there exists CC S,, + , ,, such that i can choose an action h: + , (A,( cc[f”“\h’, I, t]) from C whenever j..,(a[f”“\h’, X, t]) # i..,(a[f, s, t] ), and such that i can choose an action 4, ,(&(dx .5 fl)) not in C when 2,(~[f”“\h’, .Y, t]) = i,,(a[f; X, t]). In effect, i can shout “nay” in period s + 1 whenever play up to period s does not follow the path A,(c([,i .Y, t]), and “aye” when it does.

Now convergence in 9 requires that player i’s responses in period s + 1 converge to a shout of aye. Since any limit point of any sequence of nays is a nay, this can be the case only if ultimately i shouts only aye. This in turn can be the case only if L,(a[,f’“‘\h’, X, t]) = JV,,(~[f”“\.fi, X, t]) ultimately coincides with l,(a[f, X, t]). That is, only iff’“’ does not involve deviation from the path L,,(a[f, x, t]) by any j# i. Finally, considering an analogous deviation h’ by some such j, it can be seen that f”” cannot involve deviation by i either. Thus 3.,(~[f““’ , X. t]) does ultimately coincide with UrC,L -y, fl).

A more concrete instance of this kind of reasoning is the following.‘4 Suppose that a seller must first choose a price in the range [0, 11 which a buyer will then accept or reject. Suppose that, inf’“‘, the seller chooses the price l/n and the buyer accepts whatever price he is offered. It is certainly not the case that {.f’“‘) converges in PV‘ to the strategy combination ,f’in which the seller chooses the price 0 and the buyer accepts whatever price he is offered, but it might appear as though (,f’“‘) does converge to fin 9. This is not the case, however. For suppose that the buyer deviates to a strategy lzR in which he rejects any price greater than zero and accepts only the price zero. Then the history of the game when .f’“‘\hB is followed is (l/n, reject), which does not converge to (0, accept), the history of the game whenf\,hB is followed.

Proof of Proposition 3. In order to simplify notation, I shall assume that the T in Definition 11 can always be chosen to be t. The extension to the general case is immediate. The sets

obtained as s, t, X, and ,f’vary from sub-basis for W. Fix s, t, x, and f, and let U be the set thus obtained. I show that U is open in 9.

I4 This example is based on one used by the referee.


If s 6 t then, trivially, U is open. Suppose that s > t and let y = M[,J X, t]. For each i there exists CcS,,+,), such that A,,, ,,,(jL,r)nC#O for all z such that 2,; # A,Vy, and such that A,,+ ,,;(l, p,)\C# 0. Hence, by the axiom of choice, there exists a mapping hi,, : A,H+ S,,, ,), such that ~:,+,(A,z)EC for all A,z#~.,J~, and such that Iz:+,(~,~)EA,,+,,~(~,,;)\C. Setting hi = f;; for all r, r # s + 1, I obtain a strategy It’= (111, hi,...) E F’(H). The strategy h’ induces a mapping 4, E @ given by di= X[ .\h’, x, t].

Now, for each i and j, let

Further, let

VI, = s,, + I ,,\c if j=i,

=s (3 + I I/ if ,j # i.

vi=*,-:, i v, nff, t 1 ,=I

where Ic/,V+ , denotes the projection from S to S,, + , . Finally, let

w= fi (by’( V,). ,= I

W is open in .a by construction. I claim that WEE U. For suppose that gE W. We have

gEq3; ‘( V,)oct[g\h’, x, t] E v,

oh:., ,(k(cc[g\h’, 5, tl))~ V,,

Oh;.+ ,(j.,,(~Cg\,f’, .T fl))~ V,,

-~,(tx[g\.f’, x, t]) = 3.,!, u~,(cc[g\f’,.u, t])=&(a[f;x, f]).

However, this last inequality holds for all i, so

UgE u.

This establishes my claim. Thus an arbitrary set of a sub-basis for W is open in 9. Hence ?V ~4. But Proposition 1 shows that .a c w‘. The proposition follows. 1

Four points are worthy of note. First, the proof of Proposition 3 exploits the strong requirement for convergence in 9, namely that the history resulting from anp deviation h’ from,f ‘()I’ by player i should converge to that


resulting from the same deviation against jI In particular, consideration of particular discontinuous deviations shows that (fen)} must converge in 9‘ if it is to converge .a. In the case of a one player game, however, this requirement is trivial. Hence 9 may be strictly weaker than W in one player games. I5 Even so, convergence in 4 in such games is considerably more demanding than, say, pointwise convergence of the {fi”) > to the f,. For .a still requires that sequences such as (fln,‘,(f)“‘(A,_ ]x))} converge.

Secondly, suppose that the requirement that (H, P) be free is strengthened as follows. Suppose that, for all t and i, there exists E > 0 such that for all .V there exists a subset C of St,+ l,i for which

(i) for all z E H such that 1,~ #A, r’,

A r,+I,i(~,=)~cZ12(;

(ii) there exists x~, + , )i E A,, + , ,;(L, V) such that

4x (r+l),7 C)3&.

(Here d is understood to be the metric on S,,, , ,i.) Then 9 = Y provided that N> 2.” Once again 9 may be strictly weaker than Y in one-player games, but will generally be substantially stronger than uniform convergence of the {ff”)) to the f,.

Thirdly, if the condition that (H, P) be free were reformulated in terms of payoffs rather than actions, then a proof almost identical to that of Proposition 3 would show that for all games for which the resulting condition held, and in which the Pi were continuous in the product discrete topology, 9* and W coincide. The additional continuity requirement on the Pi is needed to ensure that 9* E I-.

Finally, there are games that are not free, but for which nonetheless 9 = W. One such game is Rubinstein’s [9] bargaining model. The feature of Rubinstein’s model that ensures this is the fact that there is only one action that either player can take to terminate the game, namely accep- tance of an offer from the other player, and this action is discrete from any other action that he can take.

Proposition 3 and its analog for 9 resolve the intractability of 9 and 9. They also suggest that in most games of interest, %‘” will be the weakest tractable topology in which the s-equilibrium sets are closed. W is nonetheless very strong, as Proposition 4 below illustrates. Before giving

Is I am grateful to the referee for these observations. I6 The conditions given can be weakened to allow for the possibility that a suitable subset C

might be available only in some later period T. Since the required conditions are notationally complex, they are not given here.

” V. Fudenberg and Levine [I; Lemma 2.11.


Proposition 4, I describe a class of games for which Fudenberg and Levine’s analysis did resolve the intractability of W. It happens to be the same class as that to which Proposition 4 applies.

DEFINITION 12. The game (H, P) is finite action iff, for all t, i, and s, A ,i( i “I , s) is finite.

A game is finite action if no player is ever faced with more than a finite number of actions from which to choose.

Fudenberg and Levine [l, p. 2611 observed that, in any finite action game, Y and Y coinide. This is as one would expect. F(H) is essentially a product of finite spaces, and there is only one metrisable product topology on such a space. Hence their observation gives no direct clue as to how one might resolve the intractability of 9 (or J) in games that are not finite action.

Fudenberg and Levine [ 1, p. 261; Lemma 4.21 also showed that F(H) is compact in Y if (H, P) is finite action. Now, gN‘ and Y coincide in finite action games (as do 9 and 9). Hence F(H) is compact in Y# if (H, P) is finite action. That the converse holds too is a measure of the strength of $4”.

PROPOSITION 4. F(H) is compact in V’ {ff (H, P) is ,finitr action.

Proof: That F(H) is compact in ti. when (H, P) in finite action follows directly from Lemma 4.2 in Fudenberg and Levine [ 1, p. 2611. For the converse, suppose that F(H) is compact in %I‘. Then F,(H) is compact in the topology of pointwise finite convergence. Equivalently, the product space

is compact in the product discrete topology. Hence A,(z) is compact in the discrete topology for all : E jti, , H. But then A,(z) is finite. Since this is the case for all t and all z E 3., , H, I have in fact shown that (H, P) is finite action. 1

The converse should not be regarded as surprising. The topology W simply allows precise expression to an obvious hunch. Proposition 4 concludes my analysis of the problem of finding a weak topology for F(H).

6. PRWF OF THE CHARACTERISATION

I turn now to the proof of Theorem 1. For this I shall need two propositions, the first of which establishes conditions under which the limit


of an approximating sequence is perfect (Proposition 5), and the second of which establishes conditions under which a perfect equilibrium point admits an approximating sequence (Proposition 6). I state Proposition 5 in terms of 9 in order to illustrate the kind of simplification that can be achieved by applying Proposition 3.

PROPOSITION 5. Suppose that each Pi is continuous, and let f E F(H). [f there is an approximating sequence for f relative to 4 then f is a perfect equilibrium point of (H, P).

Proof Suppose that {f In’} is an approximating sequence for f relative to 9. I need to show that f is perfect. To this end let i, h’, x, and t be arbitrary. For any n and s 3 0,

Pi(a[f\h’, ~3 tl)-P,(a[J ~3 t])

= P,(a[f\h’, -\-, tl)- P,(a[f\nyh’, x, tl) + P;(a[f\z,h’, x, t]) - Pi(E[f’n)\x,yhi, x, t])

+Pi(rx[fcR1\,,hi, x, t])-Pi(t-x[f(n’,x, t])

+ Pj(cl[.f(“‘. x, t]) - P,(a[f, x, t]).

I consider each of the last three differences in turn and show that its limit infimum as n -+ co is zero.

Consider the second difference. By hypothesis f (n) 4, f: By construction of 9 the mapping cx[.\~,~h~, x, t] is continuous. By hypothesis Pi is continuous. Thus the second difference tends to zero as n + 00. Similarly the fourth difference tends to zero. In the case of the third difference, t, 2 s for n sufficiently large. For such n, I have that n,h’EFi(x,n H), and so, since f “) is a perfect &,-equilibrium point of the game truncated at period t,, this difference cannot exceed E,. In particular, its limit inlimum cannot exceed zero.

Taking the limit inlimum of each side as n -+ co, I obtain accordingly that

Pi(a[f\h’, ~3 tl)-Pi(uC.L ~7 tl)

<P,(~[f\h’Tx, t])-Pi(~[f\~,h’,x> tI)’ But f\n,h’z f\h’ as s--t co, and 9 G 9’. Hence f\n,h’z f\h’. Note once again that Pi and a[ ., X, t] are continuous. The right-hand side therefore tends to zero as s -+ co. Thus

P;(aC,f\h’, x, tl) - Pi(cU x> tl) G 0

and, since i, hi, x, and t were arbitrary, this shows that fis indeed a perfect equilibrium point of (H, P). 1


Almost exactly the same proof shows that if f has an approximating sequence relative to .a*, and if the P, are continuous in the product discrete topology, then f is perfect. (Once again the continuity requirement on the P, ensures that .a* is weaker than W‘, and so Y.) Again, if .f has an approximating sequence relative to .a** and (H, P) is continuous at infinity, then f is perfect. For if (H, P) is continuous at infinity andJ”“’ is a perfect &,-equilibrium point of (H, P) truncated at n, then fen' is a perfect (E, + o,)-equilibrium point of (H, P) itself.”

At first sight it appears that there is a trade-off between two of the hypotheses of Proposition 5. The weaker the topology on H the weaker the requirement of convergence in .a, but the stronger the requirement that the Pi be continuous. Proposition 3 suggests, however, that .P is likely to coincide with W, irrespectively of the topology chosen for H. Hence this topology may as well be taken to be as strong as possible, to be the product discrete topology, thereby reducing the continuity requirement on the P, to a minimum.

If this is done then, by Proposition 2, 9 reduces to W, whether or not (H, P) is free. Thus attention may as well be restricted to the special case of Proposition 5 according to which the limit of any approximating sequence relative to “IL” is perfect if the Pi are continuous in the product discrete topology.

In order to obtain a converse to this special case of Proposition 5, I need to find some topology Y which is stronger than W, and relative to which any perfect equilibrium point has an approximating sequence. In principle this might be difficult-the stronger a topology, the more accurate are approximations in it. Fortunately, all that is necessary in practice to obtain approximations relative to the substantially stronger topology Y is a strengthening of the continuity requirement on the P, to continuity at infinity.

PROPOSITION 6. Suppose that (H, P) is continuous at infinity, and that ,f is a perfect equilibrium point qf (H, P). Then there exists an approximating sequence for f relative to 9.

Proof: Suppose that f is a perfect equilibrium point of (H, P), and that (H, P) is continuous at infinity. Consider the sequence {f’(nlj given by f (‘I’ = n,f. I shall show that f 01’ is a perfect 2o,-equilibrium point of (H, P) truncated at n. To do so I need to consider, for each i and each i,x, what the effect of a unilateral deviation from f (n) by player i in the subgame 2, of the truncation at n of (H, P) will be.

‘* Fudenberg and Levine [ 1. p. 260; Theorem 3.31. More recently, in independent work, Fudenberg and Levine [2. p. 14; Proposition 6.11 have shown that only continuity at infinity is necessary to their result.


To this end, let h’ E F’(n,H) be arbitrary. Now

P’\h’= (%f)\(%h’) = n,,(f‘\h’)

and so

The point is that it is possible to move from rt,,(f\lz’) to rr,f’in three steps, about each of which there is adequate information. The first difference cannot exceed w,, since rc,(f\h’) andf\h’ differ only after period n. Similarly, the third difference cannot exceed w,. Finally, the second difference cannot exceed zero sincefis perfect. Collecting these facts together I have

P,(~Lf”“\h’, .Y, t]) - P;(cY[,f’“‘, .y, t]) < 2q.

But i, h’, X, and t were arbitrary. Thus f““’ is a perfect 2o,-equilibrium point of (x~H, P rnnH), as claimed. Since (H, P) is continuous at infinity, and f@) 2 f by construction, I have shown that if’“‘} is an approximating sequence for ,f relative to 9. m

Propositions 5 and 6 now combine to yield Proposition 7, which I restate for convenience.

PROPOSITION 7. Suppose that (H, P) is continuous at infinity. Let Y he any topology intermediate between ~9~ and Y, $4’ G F G 9’. Then f is a perfect equilibrium point qf (H, P) iff it has an approximating sequence relative to Y-.

Proof: Suppose that f E F(H). If f is a perfect equilibrium point of (H, P) then, by Proposition 6, f possesses an approximating sequence relative to Y. Since Y is stronger than r-, this sequence is also an approximating sequence relative to Y.

For the converse, suppose that f possesses an approximating sequence relative to Y. This sequence is also an approximating sequence relative to W’. Since (H, P) is continuous at infinity, P must be continuous in the product discrete topology. Also, by Proposition 2, 4 and W coincide when S is endowed with the product discrete topology. It follows from Proposition 5 that f is a perfect equilibrium point of (H, P). 1


Fudenberg and Levine showed that if each Pi is uniformly continuous then the perfect equilibrium points of (H, P) can be characterised as limits, relative to 9, of approximating sequences.‘” They also noted that uniform continuity implies continuity at intinity.20 Since 9 E 9 5 Y, and since 9 = %I for any free game, it follows that Proposition 7 nests their result in the case of free games. Proposition 7 should, moreover, be easier to use in all cases, given the relative tractability of W’ and 9. Finally, an exact analogue of Proposition 7 holds for strong perfect equilibrium-no alteration to the topologies % - and .4p is necessary.

7. AN EXAMPLE

In this section I give an example to show how Proposition 4.3 can be used as a heuristic device to find the equilibrium of a simplified version of the model of a race of Harris and Vickers [4]. The simplification is made solely for expositional reasons-the analysis carries over directly to the full model. In this example the pattern required by the strong topology arises naturally. The reader is also referred to Fudenberg and Levine [ 1, pp. 262-2651 for two examples of how a characterisation of perfect equilibrium can be used to demonstrate uniqueness.

A race takes place between two players, A and E, who begin the race at distances x0 and y, respectively from the finishing line. A and B move alter- nately. Progress towards the finishing line is a function of the effort, or bid, that a player makes at his turn. Thus a bid of I moves a player a distance M(Z) closer to the finishing line, whre w is taken to be strictly increasing and continuous, and it is assumed that W(O) = 0. Once made a bid is irrecoverable. The first player to each the finishing line is awarded a prize of value I/. Players apply a discount factor p both to expenditures and to rewards.

Let us suppose that, after a certain period t, both players are constrained to bid zero. They must win the prize by this period or not at all. Suppose that it is A’s move in period T. If he can move to the finishing line and obtain a positive payoff, then he will do so. That is, A moves to the finishing line with one bid iff his distance x from the finishing line satisfies x < W( V). Similarly, at his bid in period T- 1, B moves to the finishing line with one bid iff his distance y from the line satisfies y < w(V). Figures 1 (i) and (ii) illustrate the strategies for periods T and T- 1.

Let us focus attention on the area {(s, y) ( .Y f MI(V) or y d W( I’)}. Con- sider A’s move in period T- 2. Suppose that y 6 w(V). A knows that in

I9 Fudenberg and Levine [l, p. 258; Lemma 3.11 ?” Property 3 in Harris and Vickers [4].


A does nothmg

IO Pemd T 11 Perlod T-l

12 Period T-2 13 Pemd T-3

FIGURE 1

period T- 1, B will finish with one move. Thus A must finish at once or give up. If x < w( V) then A finishes. If x > w( V) then A gives up. Suppose now that y > w( V). A knows that B will do nothing in period T- 1. A therefore finishes in move one or two, whichever is optimal for him.

Continuing to focus attention on the area ((x, .Y) 1 x 6 w(V) or y 6 w( I’) ), consider B’s move in period T - 3. Suppose that x 6 w( V). If also y < w(V) then B knows that A will finish in one move in period T- 2. B therefore finishes in one move. If instead y > w(V) then B faces a dilemma. He cannot win in one move without making a loss in the process, while in order to win in two moves he must move into the area ((x, y)Jx < w(V) and J’ d w( V)}. If he did move into this area, then A would finish in one move in period T - 2. B would therefore have done better to stay put. Overall, B cannot win and so bids zero. Suppose now that x > W(V). B knows that A will give up in period T- 2. Hence B finishes in one move or two, whichever is optimal. Figures l(iii) and (iv) illustrate the strategies for periods T- 2 and T - 3.

This line of argument can be continued. In period T- 4, if x < w( l’) and v > w(V) then A will have the choice of finishing in one, two, or three moves. Similarly, in period T- 5, if x > w)(V) and y Q w(V) then B will have the choice of finishing in one, two, or three moves. A point will even- tually be reached at which both A and B are effectively unconstrained in the number of moves that they can take to reach the finishing line. For-


mally, there exists k, such that, for all t 6 T-k, , players behave as follows in (and after) period t:

(i) if x 6 w( I’) and y 6 W(V) then the player whose move it is wins in one more;

(ii) if x < w( V) and y > w(V) then A wins in whatever number of moves is optimal for him and B always bids zero;

(iii) if x > W(V) and y 6 w(V) then B wins in whatever number of moves is optimal for him and A always bids zero.

Before proceeding I introduce some notation. Let C, = 0 and let C, = w(V). Then, if C,-, has been defined, let C, be the greatest distance that a player can cover, subject only to covering at least C,, - C,, ~, with his first bid and obtaining a non-negative payoff overall.

Arguing as before one can then establish that there exists some k2 3 k, such that, for all t < T- kz, players behave as follows in (and after) period t:

(i) If C,+,<x<C, and C,-,<ydC, for some m~(1,2) then the player whose turn it is to move wins. His bids are those he would make if, in isolation, he were subject to moving within C,,_ , of the finishing line with his first bid. The other player always bids zero.

(ii) Ifx<C,andy>C,forsomemE{1,2}thenAwins.Hisbids are those he would make in the absence of rivalry from B. B always bids zero.

(iii) If x > C, and y < C, for some m E { 1, 2} then B wins. His bids are those that he would make in the absence of rivalry from A. A always bids zero.

If, for some m 3 1, (x, y) is as in (i), then I shall say that (x, JJ) belongs to a “trigger zone.” If, for some m, (x, y) is as in (ii), then (x. y) belongs to A’s “safety zone.” If (x, .r) is as in (iii) then (x, J) belongs to B’s safety zone.

Let us assume that (x,, y,) is such that earlier player could, in isolation, cover the distance to the finishing line and obtain a non-negative payoff overall. Then there is some M such that x,, < C, and y0 6 C,. Iterating the argument that established the existence of k, and k,, one can show that there exists k, such that, for all t d T-k M, players behave as follows in (and after) period t:

(i) If (x, y) lies in a trigger zone then the players whose turn it is to move wins. He jumps immediately to his safety zone and then proceeds in whatever fashion is optimal for him. The other player always bids zero.

(ii) If (x, ~1) lies in the safety zone of one of the players then that


Trigger zones

player wins. He does so in whatever fashion is otimal for him. The other player always bids zero.

It is now simple to see that, as T -+ W, the perfect equilibrium point that we have found for the game truncated at period T converges, relative to topology Y, to a perfect equilibrium point of the race as a whole. In this equilibrium players use the strategies deicted in Fig. 2 in all periods.

The argument given here is certainly less elegant than that contained in Harris and Vickers [4]. For this example I contend only that Proposition 7 might act as a heuristic device by way of which the final result could be obtained. In other examples Proposition 7 could play an integral role in the analysis of perfect equilibrium.

8. CONCLUSION

In this paper I considered a class of infinite horizon games in which events in the distant future are of little importance, in the sense that the games are continuous at infinity. I showed that the perfect equilibrium points of such games could be characterised as limits, relative to either a strong or a weak topology, of sequences of perfect approximate equilibrium points of truncations of the game. I also argued that this characterisation was, in a sense, the best possible.

The characterisation provides a tool for analysing perfect equilibrium. The weak topology is better adapted to demonstrating the existence of perfect equilibrium points. For it is easier to verify pointwise finite convergence on a particular subgame than on all subgames simultaneously.


The strong topology is more useful for showing that a game possesses a unique perfect equilibrium point, or that its perfect equilibrium points belong to a restricted class. For it may be possible to prove that the equilibria of an approximating sequence can only conform to the detailed pattern imposed upon them by the strong topology under special conditions. It is worth emphasising that, whether one is trying to find or classify perfect equilibrium points, a good choice of truncation procedure can simplify the analysis considerably.

The pattern among approximate equilibria which is required by my topologies is precisely that found by Radner [S] in his work on a finitely repeated Cournot oligopoly. An equilibrium configuration in which players employ stable strategies for most of the game, departures from which occur only near the endpoint, is also found in models of reputation (see Kreps, Milgrom, Roberts, and Wilson [5]; Kreps and Wilson [6]; Milgrom and Roberts [7]). Of course, these observations are suggestive rather than con- clusive, since none of the models mentioned uses discounting, and therefore none tits directly into the present framework.

Finally, given the objections to approximate equilibria as an economic construct (rather than as a mathematical one), the findings of this paper and of Fudenberg and Levine [I ] could be taken to suggest that whenever arguments based on approximate equilibria are successful in resolving problems such as the chain-store paradox, more acceptable arguments based on exact equilibrium will be equally so.

REFERENCES

1. D. FUDENBERG AND D. LEVINE, Subgame-perfect equilibria of finite and infinite horizon games, J. Gon. Theo~r 31 (1983). 251-268.

2. D. FUDENBERG AND D. LEVINE, “Limit Games and Limit Equilibria.” UCLA Department of Economics Working Paper 289. April 1983.

3. C. HARRIS. “Essays in Perfect Equilibrium,” M. Ph. thesis. Oxford University, 1983. 4. C. HARRIS AND J. VICKERS, Perfect equilibrium in a model of a race, Reviews of Economic

Studies (1985). 5. D. KREPS, P. MILGROM. J. ROBERTS, ANU R. WILSON. Rational cooperation in the finitely

repeated prisoner’s dilemma, .I. Econ. Theq, 27 ( 1982). 245-252. 6. D. KREPS AND R. WILSON, Reputation and imperfect information, J. Econ. Theory 27

(1982) 2533279. 7. P. MILGR~M AND J. ROHERTS. Predation, reputation and entry deterrence, J. Ewn. Theor)

27 t 1982). 280-312. 8. R. RADNER, Collusive behaviour in non-cooperative epsilon equilibria of oligopolies with

long but finite lives, J. &on. Theory 22 (1980). 121-154. 9. A. RUBINSTEIN, Perfect equilibrium in a bargaining model, Ecmometricu 50 (1982),

97-109. 10. R. SELTEN. Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit. Z.

p.s. Stuut.W.w. 121 ( 1965). 301-324, 667-689. I I. R. SELTEN. The chain-store paradox, Theory und Decision 9 (l979), 1277159.

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A characterisation of the perfect equilibria of infinite horizon games