Cournot Oligopoly and the Theory of Supermodular Games

GAMES AND ECONOMIC BEHAVIOR 15, 111–131 (1996)ARTICLE NO. 0061Continuous Stochastic Games of Capital Accumulation withConvex TransitionsRabah Amir ∗WZB (IV/2), Reichpietschufer 50, 10785 Berlin, GermanyReceived October 23, 1992We consider a discounted stochastic game of common-property capital accumulationwith nonsymmetric players, bounded one-period extraction capacities, and a transition lawsatisfying a general strong convexity condition. We show that the infinite-horizon problemhas a Markov-stationary (subgame-perfect) equilibrium and that every finite-horizontruncation has a unique Markovian equilibrium, both in consumption functions which arecontinuous and nondecreasing and have all slopes bounded above by 1. Unlike previousresults in strategic dynamic models, these properties are reminiscent of the correspondingoptimal growth model. Journal of Economic Literature Classification Codes: C73, O41,Q20. © 1996 Academic Press, Inc.1. INTRODUCTIONOne of the emerging trends in the expanding relationship between game theoryand economics is the application of dynamic/stochastic games in a varietyof settings. Early studies initiated this development via specific exampleswhere equilibria are computable in a more or less straightforward manner. Avery partial list of papers along these lines includes those of Shubik and Whitt(1973), Reinganum and Stokey (1988), and Cyert and DeGroot (1970). For athorough treatment of the widespread linear quadratic case, see Basar and Olsder(1982).More recently, several studies of strategic dynamics have been conductedalong general lines, i.e., with the reward function and the transition law satisfyingonly common structural properties derived from economic considerations.∗ This work was initiated while the author was visiting C.O.R.E., Belgium and completed while theauthor was visiting the Department of Economics, University of Dortmund, Germany. The author isgrateful to both institutions for providing an extremely congenial and stimulating work environment.The author also thanks Wolfgang Leininger, Jean-Francois Mertens, Abraham Neyman, and Matt Sobelfor helpful conversations and comments. Financial support by the D.F.G. is gratefully acknowledged.1110899-8256/96 $18.00Copyright © 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

112 RABAH AMIRThis strand of the literature may be divided into two groups, according to oneimportant aspect of the information structure of the dynamic game under consideration,namely, perfect information vs simultaneous moves.In the former category, at least three separate areas figure prominently. A notionof consistent planning/altruistic growth, introduced by Strotz and Pollack, hasbeen thoroughly investigated by Leininger (1986) and Bernheim and Ray (1983).See also Lane and Leininger (1986) and Bernheim and Ray (1987). A modelof dynamic duopoly has been the object of several studies: Cyert and DeGroot(1970), Maskin and Tirole (1988a,b). Finally various racing models have beenanalyzed; see, e.g., Harris and Vickers (1987) and Budd et al. (1992).Most notably, very general pure-strategy existence results have been derivedfor dynamic games of perfect information: Harris (1985) and Hellwigand Leininger (1987) consider history-dependent equilibria while Hellwig andLeininger (1988) deal with Markovian equilibria. A common feature of allthe studies in this class of dynamic games is the deterministic nature of thedynamics.For games with simultaneous moves, the dominant setting has been that ofstrategic capital accumulation/resource extraction, initiated by the pioneeringwork of Levhari and Mirman (1980). Existence of a stationary equilibrium forthe deterministic version of this class of games is established in Sundaram (1989)and Amir (1990). See also Amir (1989). An extension to the stochastic case isgiven in Majumdar and Sundaram (1991) and Dutta and Sundaram (1990).It should come as no surprise that no general pure-strategy existence resultsare available for such games. Indeed, the simultaneity of moves in a dynamiccontext often implies that quasi-concavity, which is a basic requirement for theapplication of most fixed-point theorems, is not preserved, even under mostfavorable assumptions on the data of the game. Note that such difficulties arealso encountered when dealing with stationary equilibria of games with perfectinformation.Somewhat more surprisingly, it turns out that the available abstract subgameperfectequilibrium existence results in mixed (or behavioral) strategies forstochastic games (see Raghavan et al., 1991, and references therein) are notapplicable to most economic models. The primary reason is that these results involvethe finiteness of the action spaces or the continuity in the variation norm ofthe transition law with respect to the actions. In this context, the basic difficulty isthe absence of upper hemi-continuity of the best response correspondence in theweak-star topology on the strategy spaces (consisting of transition probabilitiesfrom the history or state space to the action spaces).Another related strand of the literature on dynamic games builds on the workof Boldrin and Montrucchio (1986) on the indeterminancy of (optimal) capitalaccumulation paths, and extends it to various dynamic games with sequential aswell as simultaneous moves, including Cournot adjustment processes. See Danaand Montrucchio (1991) for a survey.

STOCHASTIC CAPITAL ACCUMULATION GAMES 113The present paper reconsiders strategic capital accumulation/resource extractionwith bounded one-period capacities and convex transitions as a discountedstochastic game. While previous work on this problem extended the existence resultof the deterministic counterpart under a symmetry assumption on the players(Dutta and Sundaram, 1990), our analysis goes further in a variety of directions:(i) the assumption of symmetric players is removed, (ii) the stationary equilibriumstrategies are continuous and nondecreasing, in addition to having, as inthe deterministic case, all slopes bounded above by one, and (iii) uniqueness ofMarkovian equilibrium for every finite-horizon truncation is established. Whilepresented in the context of two players only, our results do extend to the n-playercase, n > 2.These surprising conclusions are reached at the expense of a new naturalconvexity assumption on the stochastic technology, represented as a transitionprobability mapping today’s joint investment into tomorrow’s random output.This key assumption may be simply stated as follows: The probability that thenext output is at or below any given level is a convex function of current investment.A class of examples given in the next section suggests that this convexitycondition is rather general; in particular it allows for atoms in the productionprocess.Nevertheless, this notion of convexity is intrinsically stochastic and has nomeaningful deterministic analog. Its convexifying effect may be expressed by thefollowing elementary but potent observation: The integral of any nondecreasingfunction with respect to a transition probability which is convex in its parameteris a concave function of that parameter. In the context at hand, think of the typicalvalue function as the integrand and of the joint investment as the parameter, inthe above statement (cf. Lemma 1.2).It is worthwhile to observe that many of the qualitative divergences betweenthe one-player (optimal growth) and the multiplayer cases prevailing in the deterministicframework are no longer present in the stochastic convex setting.The equilibrium strategies have marginal propensity of consumption between 0and 1, as does the (one-player) optimal policy, cf. Brock and Mirman (1972).Thus the classical properties of consumption functions are restored. Under theassumption of bounded extractions capacities, Markovian equilibria are evenunique for any finite horizon. Elsewhere, similar results are shown to hold forthe consistent planning problem under the same technology.It is instructive to point out that our key assumption on the stochastic technologymay also be interpreted along lattice-theoretical lines, cf. the proof ofLemma 1.3. Furthermore, Topkis’ theorem is conveniently invoked a numberof times in our analysis. Finally, it is shown that an N-period horizon problemmay be analyzed, via the dynamic programming recursion, as a sequence of Nparametrized one-shot supermodular games, cf. Topkis (1979), Vives (1990),Milgrom and Roberts (1990), and Sobel (1988). Finally, it is our belief that thestochastic technology introduced here will turn out to be of crucial importance in

114 RABAH AMIRtackling hitherto unsolved problems in strategic dynamics and multidimensionaloptimal dynamics.The paper is organized as follows. Section 2 describes the game together withall assumptions and their interpretation, and a statement of our main results. Allproofs are contained in Section 3.2. THE MODEL AND THE MAIN RESULTThe economic interaction under consideration may be simply described asfollows: Two players jointly own a productive asset characterized by a stochastictwo-period input–output technology. Each player, distinguished by his utilityfunction, his discount factor, and his one-period consumption capacity, has ashis objective the maximization of the discounted sum of utilities from his ownconsumption, over a finite or an infinite horizon, as will be specified. At eachperiod, moves are simultaneous.More specifically, consider the following stochastic game: Player i’s payoffis given by, with T ∈ N ∪ {+∞} to be specified,T∑E δi t u i(ct i ),t=0where E denotes the expectation operation over the induced probability measureon all histories, described below.The (technological) stochastic transition law is described bywithx t+1 ∼ q(· |x t −c 1 t −c 2 t ),c i t ∈ [0, K i (x t )], t = 0, 1,...,T; i =1,2Here, δ i , with 0 ≤ δ i < 1, is Player i’s discount factor, u i his utility function,and K i (x) his one-period consumption/extraction capacity, as a function of theavailable stock x. The feasibility constraint says that each player may consumeany nonnegative amount up to his one-period capacity. Finally, q is a transitionprobability from the state space (i.e., the set of all possible capital stocks) toitself, specifying a probability distribution over the next stock x t+1 , given thecurrent joint investment y t = x t − c 1 t − c 2 t .Throughout this paper, any implicit reference to a σ -algebra will tacitly beto the Borel σ -algebra, and thus measurability will always mean Borel measurability.Our model here will easily be seen to represent a discounted Markovstationarystochastic game with uncountable state and action spaces (all realsubsets) and state-constrained actions. Thus, the measure-theoretic structure, on

STOCHASTIC CAPITAL ACCUMULATION GAMES 115which we do not expand here, may simply be adapted from this general classof games to our specific model. See Raghavan et al. (1991). In fact, we willmake strong structural assumptions on u i , q, and K i , motivated by economicconsiderations, which would make it possible to define our game using only theRiemann integral.Before listing the model assumptions, we complete the formulation of thegame by defining strategies, expected payoffs and equilibria. Denote by S thestate space for this game (S = [0, +∞) as will be described below) and byA i (x) Player i’s action space (A i (x) = [0, K i (x)]). Let A i = ⋃ x∈S A i(x) =[0, K i (∞)] (see Assumption A.3 below regarding the K i ’s).Define the sequence of possible histories for this game as follows. LetH 0 = S 0 , H 1 = S 1 × A 1 1 × A1 2 ,..., H n = H n−1 ×S n × A n 1 × An 2 ,....Note here that all S n ’s and all Ai n ’s are the same. The index n is only addedto indicate time for clarity. A general strategy for Player i is a sequence ofmeasurable maps (σ 0 ,σ 1 ,...)with σ n : H n → Ai n, such that σ n(h n ) ∈ A i (x n ).Inthis paper, we deal only with more restricted classes of strategies, described next.With T finite, we will consider only Markovian strategies defined as above butwith H n = S, n = 0, 1,...,T. With T =+∞, we restrict attention to stationary(Markov) strategies, defined as Markovian strategies for which (σ 0 = σ 1 =···).Given an initial state x and a pair of strategies (σ 1 ,σ 2 ) = ((σ1 n), (σ 2 n )), thereis a unique probability distribution on the space of all histories, m(x,σ 1 ,σ 2 ), accordingto the Ionescu–Tulcea theorem. All expectations defined in our analysisare tacitly w.r. to the measure m. Let m i be the marginal of m on Ai 1×A2 i ×···×AT iand min the marginal of m i on Ai n , n = 1, 2,...,T. The expected discountedpayoff to player i when the strategy pair (σ 1 ,σ 2 )is used, and x is the initial state,is given by∫ T∑ i (σ 1 ,σ 2 )(x) = δi t u i(ci t ) dm i(x,σ 1 ,σ 2 ).t=0The tth stage expected utility of player i is given by∫U i (σ 1 ,σ 2 )(x) = u i (ci t ) dmt i (x,σ 1,σ 2 ),It is easily shown that i (σ 1 ,σ 2 )(x) =T∑δi t U i(σ 1 ,σ 2 )(x).t=0t = 0, 1,...,T.When T is finite (infinite), a pair (σ1 ∗,σ∗ 2)of Markov (stationary) strategies is aMarkovian (stationary) Nash equilibrium if 1 (σ ∗ 1 ,σ∗ 2 )(x) ≥ 1(σ 1 ,σ ∗ 2 )(x)

116 RABAH AMIRand 2 (σ ∗ 1 ,σ∗ 2 )(x) ≥ 2(σ ∗ 1 ,σ 2)(x),for all feasible Markov (stationary) strategies σ 1 , σ 2 , and all initial states x.In both cases, it is well known, from standard dynamic programming arguments,that an equilibrium from the proposed subclass of strategies remains anequilibrium when one or both players are allowed to use more general strategies.Furthermore, Nash equilibria based on these two restricted classes of strategiessatisfy a strong version of subgame perfection.We now provide a list of structural assumptions on the data of the game underconsideration which hold, without further reference, throughout the paper:A.1. u i :[0,+∞) → [0, +∞) is a strictly increasing strictly concave function,satisfying u i (·) ≥ 0A.2. q is a transition probability from [0, +∞), with its Borel σ -algebra,to itself. Let F(· |y)denote the cumulative distribution function associatedwith q(· |y),i.e., for x ≥ 0, F(x | y) = q([0, x] | y). It is assumed, that(interpretation of the following conditions is given later):(i) For each x ∈ [0, +∞), F(x |·)is a strictly decreasing function, i.e., Fis (first-order) stochastically increasing in y.(ii) For each x ∈ [0, +∞), F(x |·)is a convex function.(iii) F(0 | 0) = 1 and F(x |·)is continuous at y = 0, ∀x ∈ [0, +∞).A.3. K i (·) is continuous, strictly increasing, uniformly bounded above bysome constant C i ∈ R + , and satisfies K i (0) = 0 andK i (x 1 ) − K i (x 2 )≤ 1, ∀x 1 , x 2 ∈ [0, +∞), x 1 ≠ x 2 .x 1 − x 2Furthermore K 1 (x) + K 2 (x)

STOCHASTIC CAPITAL ACCUMULATION GAMES 117Assumptions A.2 and A.3 are crucial to our analysis of the game at hand.Some direct consequences of these assumptions are described below and invokedthroughout the paper. A.2 (ii), together with the continuity part of A.2 (iii),implies that for each x ∈ [0, +∞), F(x |·)is a continuous function. Hence, asy n → y,wehaveF(x|y n )→F(x|y) for each x ∈ [0, +∞). (2.1)This pointwise convergence is clearly stronger than the more common weak (orvague) convergence of distribution functions.A.2 (i) and A.2 (ii) together require that the state space S—the set of all possiblecapital stocks—of this game be unbounded above. This may easily be seen froma graph of F(x |·), and is thus not formally proved here. Note that while thisrequires arbitrarily large stocks to be reachable with positive probability fromany given small stock, this probability can be taken to be as small as desired.As a consequence, all integrals in this paper are taken from 0 to +∞ withoutfurther indication.As an example of a transition probability satisfying all the requirements ofAssumption A.2, consider the following (uncountable) family, for y ∈ [0, +∞),{[1 − g(y)]G(x) + g(y) if x ≥ 0F(x | y) =0 if x < 0,where G(·) is any distribution function satisfying G(x) = 0ifx ≤0, and g(·) isany strictly decreasing convex function from [0, +∞) to [0, 1] with g(0) = 1.We leave to the reader the computational steps establishing that F is indeed atransition probability and that it satisfies A.2. For the sake of further illustration,here is a specific example, obtained by letting g(y) = e −y , y ≥ 0 and G(x) =1 − e −2x , x ≥ 0:{1 − eF(x | y) =−2x + e −y e −2x if x ≥ 00 if x < 0.Finally, Assumption A.3 turns out to be crucial for some aspects of our results,in particular, the uniqueness part. Its main effect is that it allows the model athand to be treated as an ordinary (stochastic) game, rather than as a generalizedgame, in Debreu’s (1952) terminology, i.e., a game in which the joint actionspace is not a cartesian product of the individual players’ action spaces. Animmediate consequence of this hypothesis is that the stock cannot be driven to 0by exhaustive consumption by the players. Hence the game will effectively lastforever, with probability one provided q does not have an atom at 0 for all y > 0.Such an assumption of nonexhaustive capacities has previously been made in asimilar context, Benhabib and Radner (1988).The level of realism of such an assumption obviously depends on the specificeconomic context represented by the described dynamic game. In the most

118 RABAH AMIRprominent setting of noncooperative common-property resource extraction, AssumptionA.3 is undoubtedly appropriate and is easily interpreted in terms ofcapital and labor (bounded) capacities.In the following, it is convenient to refer to the (n + 1)-period horizon game(including the 0th period) as G n , with G ∞ being the infinite horizon game.Another related (auxiliary game) is defined in Section 3.We are now in a position to state theMAIN THEOREM. The infinite-horizon game G ∞ has a stationary equilibrium.Furthermore, every finite-horizon game G n has a unique Markovian equilibrium.In both cases, equilibrium strategies consist of (consumption) decisionfunctions which are continuous and nondecreasing and have all slopes below 1,and corresponding value functions are continuous and nondecreasing.3. PROOFSThe proof of the Main Theorem proceeds via two intermediate propositions,each of which requires, in turn, a few lemmas. Many of these results are ofindependent interest. Note that we assume the reader is familiar with Topkis’theorem (Topkis, 1978).We start with two lemmas which establish useful facts invoked repeatedlythroughout our analysis.LEMMA 0.1. Let F 1 and F 2 be probability distributions over [0, +∞), withits Borel subsets. Then F 2 first-order stochastically dominates F 1 , or∫ ∫F 1 (x) ≥ F 2 (x), ∀x if and only if v dF 1 ≤ vdF 2 ,for all real-valued nondecreasing functions v on [0, +∞).Proof. This result is well known. See, e.g., Stoyan (1983).LEMMA 0.2. Let U: [0,+∞) → [0, +∞) be a bounded concave function.Let h: [0,+∞) → [0, +∞) satisfy h(0) = 0 and [h(x 1 ) − h(x 2 )]/(x 1 −x 2 ) ≤ 1, for all distinct x 1 , x 2 ∈ [0, +∞). Then (i) the set ϕ ={(x,y):x∈[0, +∞) and y ∈ [0, x − h(x)]} is a lattice, and (ii) U[x − y − h(x)] is supermodularon ϕ.Proof. (i) This follows directly from the fact that x − h(x) is nondecreasingin x. (ii) We show the equivalent fact that U[x − y − h(x)] has nondecreasingdifferences in (x, y) ∈ ϕ. Let x 2 ≥ x 1 , and y 2 ≥ y 1 be fixed with 0 ≤ y i ≤x i − h(x i ), i = 1, 2. Then, clearly for i = 1, 2, we havex 2 − y 1 − h(x 2 ) ≥ x i − y i − h(x i ) ≥ x 1 − y 2 − h(x 1 ),

STOCHASTIC CAPITAL ACCUMULATION GAMES 119with the sum of the two outer terms being equal to that of the two inner terms.Hence, there exists λ, with 0 ≤ λ ≤ 1, such thatandx 2 − y 2 − h(x 2 ) = λ[x 2 − y 1 − h(x 2 )] + (1 − λ)[x 1 − y 2 − h(x 1 )],x 1 − y 1 − h(x 1 ) = (1 − λ)[x 2 − y 1 − h(x 2 )] + λ[x 1 − y 2 − h(x 1 )].(3.1a)(3.1b)Now, using the concavity of U twice, with (3.1a) and then with (3.1b), yieldsU[x 2 − y 2 − h(x 2 )] + U[x 1 − y 1 − h(x 1 )]≥ λU[x 2 − y 1 − h(x 2 )] + (1 − λ)U[x 1 − y 2 − h(x 1 )]+ (1 − λ)U[x 2 − y 1 − h(x 2 )] + λU[x 1 − y 2 − h(x 1 )]= U[x 2 − y 1 − h(x 2 )] + U[x 1 − y 2 − h(x 1 )]which says that U[x − y − h(x)] has nondecreasing differences in (x, y) ∈ ϕ.This completes the proof of Lemma 0.2.LEMMA 0.3. Let f n , f :[0,+∞) → R. If f nu−→ f , i.e., uniformly on anycompact subset of [0, +∞), x n ∈ arg max f n , and x is a limit point of (x n ), thesubsequence (x m ) being convergent to x, thenf (x) = sup f = limm→∞ (sup f m).Proof. See Kall (1986). Note here that this result only requires f nh−→ f ,or hypoconvergence, which is weaker than uniform convergence.Additional notation is introduced as the need for it arises. The effective strategyspace for Player i in our stochastic game is the following space of Lipschitzcontinuousmonotone nondecreasing functions (recall that C i is the maximalextraction level, cf. Assumption A.3):LCM i ={γ:[0,+∞) −→ [0, C i ],γ(x)∈[0, K i (x)], ∀x > 0 and0 ≤ γ(x 1)−γ(x 2 )≤1 for all distinct x 1 , x 2 in [0, +∞)}.x 1 −x 2LCM i is the subset of the space of all stationary strategies, which is used inour analysis of the infinite-horizon problem. For finite horizons, we deal withthe subset of all Markovian strategies consisting of sequences of elements inLCM i .The corresponding space of possible value functions when players use strategiesin LCM i will be shown to be (cf. Lemma 1.1) the following space ofcontinuous monotone nondecreasing functions:CM i ={v:[0,+∞) −→ [0, +∞) such that 0 ≤ v ≤ u i (C i )1−δ i,v continuous and nondecreasing}.

120 RABAH AMIRHere, the upper bound on the possible values that Player i can get follows fromthe definition of payoffs and Assumption A.3. It is also useful to define theclosure of CM i under pointwise convergence:We are now ready forM i : {v: [0,+∞) → [0, +∞) such that 0 ≤ v ≤ u i (C i )1−δ iand v is nondecreasing}.PROPOSITION 1. The infinite-horizon game has a Nash equilibrium in stationarystrategies which are elements of LC M 1 × LCM 2 .It is convenient to break down the proof of Proposition 1 into three distinctlemmas, labeled 1.1–1.3. We first define the associated best-response optimizationproblem.Suppose that Player II, say, uses a stationary strategy h ∈ LCM 2 . Then PlayerI’s value function for optimally responding to h (in the infinite-horizon game),to be denoted V h , is defined byV h (x) = sup E ∑ ∞t=0 δ 1u 1 (c 1 t )subject to x t+1 ∼ q(· |x t −c 1 t −h(x t )), t = 0, 1,...(3.2)with x 0 = x,where the expectation is over the unique probability measure induced by x, hand a stationary strategy by Player I (see Section 1), and the supremum may betaken over the space of all stationary policies (see, e.g., Bertsekas and Shreve,1978, Stokey et al., 1989).LEMMA 1.1. In the optimization problem (3.2), assume that h ∈ LCM 2 .Then V h ∈ CM 1 and V h is the unique solution to the functional equation∫V h (x) = max{u 1 (c) + δ 1 V h (x ′ ) dF(x ′ | x −c−h(x)): c ∈ [0, K 1 (x)]}.(3.3)Proof. Define the map T : CM 1 →CM 1 by∫T (v)(x) = sup{u 1 (c) + δ 1 v(x ′ ) dF(x ′ | x −c−h(x)): c ∈ [0, K 1 (x)]}.(3.4)First, we show that T indeed maps CM 1 into itself. To this end, we start by provingthat the supremand in (3.4) is continuous in (c, x). Let c n → c and x n → x.Then, since h ∈ LCM 2 , x n − c n − h(x n ) → x − c − h(x). By Assumption A.2(see also (2.1)), we have F(· |x n −c n −h(x n )) → F(· |x−c−h(x)), and

STOCHASTIC CAPITAL ACCUMULATION GAMES 121hence, since v ∈ CM 1 , by a well-known characterization of weak convergenceof measures,∫∫v(x ′ )F(dx ′ | x n −c n − h(x n )) −→ v(x ′ ) dF(x ′ | x −c−h(x)). (3.5)Continuity of the supremand of (3.4) follows from (3.5). Furthermore, the feasibleset [0, K 1 (x)] clearly represents a continuous correspondence, in view ofA.3. Hence, by the Maximum theorem, T (v) is continuous.Next, we show that T (v) is nondecreasing. Let x 1 ≥ x 2 . Then, by AssumptionA.2 (i), Lemma 0.1, and the fact that v ∈ CM 1 , we have, since x 1 − h(x 1 ) ≥x 2 − h(x 2 ), for every c,∫u 1 (c) + δ 1 v(x ′ ) dF(x ′ | x 1 −c−h(x 1 ))∫≥ u 1 (c) + δ 1 v(x ′ ) dF(x ′ | x 2 −c−h(x 2 )). (3.6)Since T (v)(x 1 ) is the sup of the LHS of (3.6) over c ∈ [0, K 1 (x 1 )], and T (v)(x 2 )is the sup of the RHS of (3.6) over c ∈ [0, K 1 (x 2 )], and since [0, K 1 (x 2 )] ⊂[0, K 1 (x 1 )] by A.3, we have T (v)(x 1 ) ≥ T (v)(x 2 ). This shows that T mapsCM 1 into itself.Next, observe that CM 1 , endowed with the uniform distance, defined byd(γ 1 ,γ 2 ) = sup |γ 1 (x) − γ 2 (x)|, is a closed subset of the Banach space C,consisting of bounded continuous functions on [0, +∞), endowed with the supnorm. Hence CM 1 is a complete metric space.A standard argument in discounted dynamic programming theory shows thatT is a contraction with unique fixed-point V h ∈ CM 1 which thus satisfies(3.3).A best response of Player I to h is defined as any argmax of (3.3), for x ∈[0, +∞). For the next results, it is convenient to define two alternative choicevariables for Player I. Instead of choosing c ∈ [0, K 1 (x)], Player I may bethought of as choosing z = x − c, z ∈ [x − K 1 (x), x]. Then (3.3) may berewritten in the equivalent form∫V h (x) = max{u 1 (x − z) + δ 1 V h (x ′ ) dF(x ′ | z−h(x)): z ∈ [x − K 1 (x), x]}.(3.7)Likewise, Player I may be viewed as choosing joint investmenty = x − c − h(x), with y ∈ [x − K 1 (x) − h(x), x − h(x)].Then (3.3) becomes∫V h (x) = max{u 1 [x − y − h(x)] + δ 1 V h (x ′ ) dF(x ′ | y):y ∈ [x − K 1 (x)−h(x), x − h(x)]} (3.8)

122 RABAH AMIRThe next result shows that, if h ∈ LCM 2 , there is a unique best response g.LEMMA 1.2.Player I.If h ∈ LCM 2 , there is a unique stationary best response g byProof. We show that the maximand in (3.8) is a strictly concave function ofy, for each x ∈ [0, +∞). Since u 1 is strictly concave, it suffices to show that theintegral term is concave in y. To this end, first note that F(x |·)being conveximplies that for any y 1 , y 2 ∈ [x − K 1 (x) − h(x), x − h(x)], any λ ∈ [0, 1], andany x ′ ∈ [0, +∞)F[x ′ | λy 1 + (1 − λ)y 2 ] ≤ λF(x ′ | y 1 ) + (1 − λ)F(x ′ | y 2 ). (3.9)Then, consider∫λ∫V h (x ′ ) dF(x ′ | y 1 )+(1−λ)∫=∫≤V h (x ′ ) dF(x ′ | y 2 )V h (x ′ )d[λF(x ′ | y 1 )+(1−λ)F(x ′ | y 2 )]V h (x ′ ) dF[x ′ | λy 1 +(1−λ)y 2 ],as a consequence of (3.9), Lemma 0.1, and the fact that V h is nondecreasing (infact V h ∈ CM 1 , by Lemma 1.1).This shows that the integral term is concave in y. Since the feasible set [x −K 1 (x) − h(x), x − h(x)] is a convex interval for fixed x ∈ [0, +∞), there isa unique argmax in (3.8) and hence also in (3.3). This completes the proof ofLemma 1.2.LEMMA 1.3.LCM 1 .If h ∈ LCM 2 , the unique best response g by Player I is also inProof. From Lemma 1.2, we know that g is single-valued. We now show thatg is nondecreasing. To this end, observe that the concavity of the integral termof (3.8) implies the supermodularity of the integral term of (3.3) in (c, x), asaconsequence of Lemma 0.2. Furthermore, the correspondence x → [0, K 1 (x)]is clearly ascending since K 1 is nondecreasing. Hence, by Topkis’ theorem, g isnondecreasing.Next, we show that no slope of g can exceed one. Both terms of the maximandof (3.7) are supermodular in (x, z), again as a consequence of Lemma 0.2 andthe fact that h is nondecreasing (since h ∈ LCM 2 ). Also, the correspondence[x − K 1 (x), x] is ascending in x, since the function x − K 1 (x) is increasing,as a consequence of Assumption A.3. Hence, by Topkis’ theorem, the argmaxin (3.7) is nondecreasing, which is equivalent to saying that the slopes of g (theargmax in (3.3)) are all less than one, since z = x − c.

STOCHASTIC CAPITAL ACCUMULATION GAMES 123We have thus shown that the slopes of g are all in the interval [0, 1]. Sog ∈ LCM 1 . This completes the proof of Lemma 1.2.We are now in a position to formally define B, the best response map for theinfinite-horizon game, which is thus single-valued by Lemma 1.2:B: LCM 1 × LCM 2 −→ LCM 1 × LCM 2(g,h) −→ (g ′ , h ′ ),whereV h (x) = u 1 [g ′ (x)] + δ 1∫V h (x ′ ) dF(x ′ | x −g ′ (x)−h(x))andV g (x) = u 2 [h ′ (x)] + δ 2∫V g (x ′ ) dF(x ′ | x −g(x)−h ′ (x)).Endow LCM i with the topology of uniform convergence on compact subsetsuof [0, +∞), to be denoted −→ for short. We write LCM i for the resultingtopological space as well.LEMMA 1.4.B is a continuous map from LC M 1 × LCM 2 to itself.Proof. We show continuity along one coordinate, say h → g ′ , or equivalently,h → y ′ , where y ′ (x) = x − g ′ (x) − h(x), cf. (3.8). Note that as h variesin LCM 2 , the possible y ′ ’s that can arise also form an equicontinuous family(since their slopes are bounded). Hence, pointwise and uniform convergence areequivalent for the y’s.uLet h n −→ h and suppose (passing to a subsequence if necessary, which isjustified since the range of B is compact by the Arzela–Ascoli theorem) thaty n′ u−→ y ′ . We must show that y ′ is the best response to h, or, in other words,that y ′ is the argmax in (3.8).We have, with V n denoting V hn ,V n (x) = u 1 [x − y n ′ (x) − h ∫n(x)] + δ 1 Vn (x ′ ) dF(x ′ | y n ′ (x)) (3.10a)∫= max{u 1 [x − y − h n (x)] + δ 1 Vn (x ′ ) dF(x ′ | y):y ∈ [x − K 1 (x)−h n (x), x − h n (x)]}. (3.10b)Since h n ∈ LCM 2 , V n ∈ CM 1 by Lemma 1.1. By Helly’s theorem (see Billingsley,1968), we conclude that (V n ) has a subsequence, itself relabeled V n w.l.o.g.,converging pointwise to a nondecreasing (but possibly discontinuous) limit V .

124 RABAH AMIRThus V may, a priori, fail to be in CM 1 . To complete this proof, we need toestablish that V must satisfy∫V (x) = u 1 [x − y ′ (x) − h(x)] + δ 1 V (x ′ ) dF(x ′ | y ′ (x))∫= max{u 1 [x − y − h(x)] + δ 1 V (x ′ ) dF(x ′ | y):y ∈ [x − K 1 (x)−h(x), x − h(x)]},(3.11a)(3.11b)from which we also conclude by Lemma 1.1, since h ∈ LCM 2 , that V ∈ CM 1 .To this end, we would like to show that the maximand in (3.10b) converges uniformlyin y, for fixed x, to the maximand in (3.11b), and then invoke Lemma 0.3to conclude that (3.11) holds. However, because of the dependence of the feasibleset in (3.10b) on n, we need an intermediate step. Define, for each fixedx ∈ [0, +∞), W x n , W x :[0,x]→ R, by (note here that one can think of W x nand W x as representing the maximands of (3.10b) and (3.11b), respectively, butdefined on the common domain [0, x] without changing their maxima)∫u 1 [K 1 (x)] + δ 1 Vn (x⎧⎪ ′ ) dF×(x ′ | x − K 1 (x)−h n (x))− p(y − x + K 1 (x) + h n (x)) if y ∈ [0, x − K 1 (x) − h n (x)]⎨Wn x (y) = u 1 [x −∫y − h n (x)]if y ∈ [x − K 1 (x)−h n (x),+ δ 1 Vn (x ′ ) dF(x ′ | y) x −h n (x)]∫u 1 (0)+δ 1 Vn (x ′ )dF⎪ ×(x ′ | x −h ⎩ n (x))− p(x − h n (x) − y) if y ∈ [x − h n (x), x]where p is any Lipschitz-continuous strictly decreasing (penalty) function withp(0) = 0, and⎧u 1 [K 1 (x)] ∫+ δ 1 V (x ′ ) dF×(x ′ | x − K 1 (x)−h(x))⎪⎨− p(y − x + K 1 (x) + h(x)) if y ∈ [0, x − K 1 (x) − h(x)]W x (y) = u 1 [x −∫y − h(x)]if y ∈ [x − K 1 (x)−h(x),+ δ 1 V (x ′ ) dF(x ′ | y) x −h(x)]∫u 1 (0)+δ 1 V(x ′ )dF×(x ⎪⎩′ | x −h(x))− p(x − h(x) − y) if y ∈ [x − h(x), x]For extra clarity, note that for each x ∈ [0, +∞), Wn x and W x are continuousin y and that, in view of the properties of p(·), they achieve their maximumwithin the feasible set of (3.10b) and (3.11b), respectively.uNow, for each fixed x, u 1 [x − y − h n (x)] −→ u 1 [x − y − h (x )]iny.Also, by Lebesgues’s dominated convergence theorem, we have for every y,

STOCHASTIC CAPITAL ACCUMULATION GAMES 125∫Vn (x ′ ) dF(x ′ | y) → ∫ V(x ′ )dF(x ′ | y). Since V is nondecreasing, Lemma0.1 and Assumption A.2(i) imply that ∫ V (x ′ ) dF(x ′ | y)and ∫ V n (x ′ ) dF(x ′ |y) are nondecreasing in y. Furthermore, Assumptions A.2(ii)–(iii) imply (notethat Lemma 1.2 also holds for V ∈ M 1 even if V ∉ CM 1 ) that ∫ V (x ′ ) dF(x ′ | y)is also continuous in y (see (2.1)). Hence, by Polya’s theorem (see, e.g., Bartle,1976), we have∫∫V n (x ′ ) dF(x ′ u| y) −→ V (x ′ ) dF(x ′ | y).Then, by construction, we have, for each fixed x ∈ [0, +∞), Wn x(y)u−→ W x ( y ),in y ∈ [0, x]. By Lemma 0.3, together with the observation thaty ′ n (x) = arg max W x n (y) and y′ (x) = arg max W x (y),and the hypothesis y n ′ (x)u−→ y ′ (x ), the desired conclusion, i.e., (3.11a–b),follows with V ∈ CM 1 as mentioned earlier. Thus y ′ is the best response to h.The proof of Lemma 1.4 is now complete.We are now ready for the proof of Proposition 1.Proof of Proposition 1. By the Arzela–Ascoli theorem, LCM i is a normcompactsubset of the Banach space of bounded continuous functions on [0, +∞)with sup norm. By Lemma 1.4, B is norm-continuous from LCM 1 × LCM 2 toitself. Finally, LCM i is clearly a convex subset. Hence, all the hypotheses ofthe Schauder fixed-point theorem are satisfied, and B has a fixed-point, whichis easily seen to be a stationary equilibrium of the infinite-horizon game.Next, we state and prove Proposition 2.PROPOSITION 2. For every n ∈{0,1,...},the finite-horizon (n + 1)-periodgame G n has a unique Nash equilibrium in Markovian strategies.As before, the proof of Proposition 2 is broken down into three intermediateresults: Lemmas 2.1–2.3. It is convenient to define the following auxiliaryparametrized family of one-shot games: For a given pair v = (v 1 ,v 2 ) ∈CM 1 ×CM 2 let the game G v be characterized by the action spaces [0, K i (x)],i = 1, 2, and the payoffsu i (c i ) + δ i∫v i (x ′ ) dF(x ′ | x −c 1 −c 2 ). (3.12)Here, x ∈ [0, +∞) is to be viewed as a parameter. A strategy for Player iin G v may be thought of as a Borel-measurable function γ from [0, +∞) to[0, +∞) satisfying 0 ≤ γ i (x) ≤ K i (x), i = 1, 2.

126 RABAH AMIRLEMMA 2.1. For every v = (v 1 ,v 2 )∈CM 1 ×CM 2 ,the game G v has a Nashequilibrium in LC M 1 × LCM 2 .Proof. Repeat the argument for proving Proposition 1 step for step uponreplacing V h by v 1 (and similarly for Player II, V g by v 2 ). Observe that the proofhere is a special case of that of Proposition 1 since (v 1 ,v 2 )is exogenously fixed.In particular, for the continuity step (Lemma 1.4) v 1 does not depend on n whileV 1 does.The actual details of this proof are left to the reader.Define the best response map Bv x for the game Gx v (which is best thought ofas the game G v with x fixed)asBv x :[0,K 1(x)]×[0, K 2 (x)] −→ [0, K 1 (x)] × [0, K 2 (x)](c 1 , c 2 ) −→ (c1 ∗ , c 2 ∗ ),wherec ∗ 1 = arg max {u 1 (c 1 ) + δ 1∫c ∗ 2 = arg max {u 2 (c 2 ) + δ 2∫}v 1 (x ′ ) dF(x ′ | x −c 1 −c 2 ): c 1 ∈ [0, K 1 (x)](3.13a)}v 2 (x ′ ) dF(x ′ | x −c 1 −c 2 ): c 2 ∈ [0, K 2 (x)] .(3.13b)Here, the single-valuedness of Bv x above is due to the fact that the maximandsin (3.13) are strictly concave functions of c 1 and c 2 , respectively; see the proofof Lemma 1.2.LEMMA 2.2. For each fixed x ∈ [0, +∞) and (v 1 ,v 2 ) ∈ CM 1 ×CM 2 , themap Bv x is nonexpansive and monotone nonincreasing.Proof. We show the desired conclusion for the map c 2 → c1 ∗ . From Lemma1.2, we know that ∫ v 1 (x ′ ) dF(x ′ |·)is a concave function. Hence, by Lemma 0.2,the integral term in (3.13a) is supermodular in (c 1 , −c 2 ). The feasible set [0, K 1 (x)]is independent of c 2 , thus ascending in c 2 . Therefore, by Topkis’ theorem, c1 ∗ isnondecreasing in (−c 2 ), or nonincreasing in c 2 .As in the proof of Lemma 1.3, it is convenient here to rewrite (3.13a) with thedecision variable being joint investment for Player I, i.e., y = x − c 1 − c 2 ,as∫y ∗ = arg max{u 1 (x − y − c 2 ) + δ 1 v 1 (x ′ ) dF(x ′ | y):}y ∈ [x − K 1 (x)−c 2 ,x −c 2 ] . (3.14)

STOCHASTIC CAPITAL ACCUMULATION GAMES 127The maximand in (3.14) is supermodular in (y, −c 2 ) by Lemma 0.2. Moreover,the feasible set [x − K 1 (x) − c 2 , x − c 2 ] is ascending in (−c 2 ), for fixed x ∈[0, +∞). Hence, by Topkis’ theorem once more, y ∗ is nondecreasing in (−c 2 ),or nonincreasing in c 2 . Note here that y ∗ (c 2 ) is single-valued since the maximandin (3.14) is strictly concave in y. Now, since y ∗ (c 2 ) = x − c1 ∗(c 2) − c 2 , the factthat y ∗ is nonincreasing in c 2 means thatc1 ∗(c 2) − c1 ∗(c′ 2 )c 2 − c2′ ≥−1, ∀c 2 ,c 2 ′ ∈[0, K 2(x)].Together with the fact that c1 ∗ is nonincreasing in c 2 (which is the first step in thepresent proof), this yields− 1 ≤ c∗ 1 (c 2) − c1 ∗(c′ 2 )c 2 − c2′ ≤ 0. (3.15)Obviously, a similar conclusion holds for c2 ∗(c 1). The desired conclusion is easilyseen to follow from (3.15).LEMMA 2.3. For each fixed x ∈ [0, +∞) and (v 1 ,v 2 ) ∈ CM 1 ×CM 2 , thegame Gv x has a unique Nash equilibrium in pure strategies.Proof. Existence of a Nash equilibrium for the game Gv x follows fromTarski’s fixed-point theorem applied to the composite map Bv x ◦ Bx v , which isnondecreasing. A fixed-point of the composition is easily seen to be a Nashequilibrium. This argument is due to Vives (1990).Alternatively, one may also use Brouwer’s fixed-point theorem since Bv x is obviouslycontinuous, as a consequence of (3.15), and the action spaces [0, K i (x)]are compact. To establish uniqueness of Nash equilibrium for Gv x , assume, foreventual contradiction, that Gv x has two distinct Nash equilibria (a 1, b 1 ) and(a 2 , b 2 ) with, say a 1 > a 2 and hence, b 1 < b 2 (since the best response functionis nonincreasing). By (3.15), we have− 1 ≤ c∗ 1 (b 1) − c1 ∗(b 2)b 1 − b 2= a 1 − a 2≤ 0b 1 − b 2(3.16a)and similarly for Player II− 1 ≤ c∗ 2 (a 1) − c2 ∗(a 2)a 1 − a 2= b 1 − b 2≤ 0.a 1 − a 2(3.16b)It follows from the first inequalities in (3.16a) and (3.16b) that a 1 − a 2 =−(b 1 − b 2 ), i.e., that the first inequalities in (3.16a) and (3.16b) are actuallyequalities. This in turn, implies that, say for Player I,c1 ∗(c 2) − c1 ∗(c′ 2 )c 2 − c2′ =−1, for all distinct c 2 , c 2 ′ in [b 1, b 2 ]. (3.17)

128 RABAH AMIRFIG. 1. Reaction functions for G x v .This is because the existence of slopes of c ∗ 1 strictly larger than −1 within [b 1, b 2 ]would imply the presence of slopes less than −1 also, for a 1 − a 2 =−(b 1 −b 2 )to hold. Furthermore, c ∗ 1 is clearly interior on (b 1, b 2 ) as a result of (3.17), or inother words (see Fig. 1)0 < c ∗ 1 (c 2)

STOCHASTIC CAPITAL ACCUMULATION GAMES 129[b 1 , b 2 ], (3.17) implies that c1 ∗(c 2)+c 2 is a constant, say C. We now claim that theintegral term in (3.19) is differentiable at x − c1 ∗(c 2) − c 2 = x − C. Suppose not.Then, from (3.19), we conclude that Vv x(c2) − u[c1 ∗(c 2)] is not differentiable atany c 2 ∈ [b 1 , b 2 ], a contradiction, since both Vv x(·)and u[c∗ 1(·)] are differentiablefor almost all c 2 , the latter being so in view of (3.17). Therefore, invoking also(3.18), the following first order condition must hold, for almost all c 2 ∈ [b 1 , b 2 ],for the maximization in (3.13a),{∫}∣u ′ 1 [c∗ 1 (c d∣∣∣z=x−c2)] = δ 1 v 1 (x ′ )dF(x ′ | z). (3.20)dz∗1 (c 2)−c 2Let ¯c 2 and ĉ 2 be two distinct points in [b 1 , b 2 ] for which (3.20) holds. Sincec1 ∗(¯c 2) +¯c 2 =c1 ∗(ĉ 2)+ĉ 2 =C, the RHS of (3.20) is the same when c 2 =ĉ 2 aswhen c 2 =¯c 2 . Hence, u ′ 1 [c∗ 1 (¯c 2)] = u ′ 1 [c∗ 1 (ĉ 2)], or, in view of the monotonicityof u ′ 1 , c∗ 1 (¯c 2) = c1 ∗(ĉ 2), a contradiction to (3.17). Hence there cannot be morethan one equilibrium for Gv x , and this completes the proof of Lemma 2.3.Now we are ready for theProof of Proposition 2. We present an argument by induction on n. Forn=0, i.e., for the one-period game G o , Player i solves the following optimizationproblem max{u i (c i ): c i ∈ [0, K i (x)]}. Therefore, the unique equilibriumconsumption pair is (K 1 (x), K 2 (x)), with corresponding value functions(V1 1,V 2 1)= (u 1[K 1 (x)], u 2 [K 2 (x)]).Then, the two-period problem (or n = 1) has payoffs given by (3.12) withv i replaced by Vi 1 , i = 1, 2. The action spaces are, of course, independent ofn. Since Vi1 ∈ CM i , Lemma 2.1 guarantees the existence of an equilibriumin LCM 1 × LCM 2 for the game G 1 = G V 1. But, by Lemma 2.3 we haveuniqueness of the equilibrium actions of the game G x , for every x ∈ [0, +∞).V 1Therefore G V 1 has a unique equilibrium strategy pair, and furthermore, this pairis in LCM 1 × LCM 2 .Let the equilibrium payoffs of G 1 be (V1 2,V 2 2 ) and repeat the above argumentfor n = 2, 3,.... This process clearly yields the desired conclusion.REFERENCESAmir, R. (1989). “On the Continuity of the Best-Response Map in Some Dynamic Games,” unpublishednote.Amir, R. (1990). “Strategic Common Property Capital Accumulation: Existence of Nash Equilibria,”preprint.Bartle, R. (1976). The Elements of Real Analysis. New York: Wiley.Basar, T., and Olsder, G. J. (1982). Dynamic Non-cooperative Game Theory. New York: AcademicPress.

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