C&^OSf: *: IHAHES) m Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium Member Libraries http://www.archive.org/details/tacitcollusioninOOIour working paper department of economics i TACIT COLLUSION IN A DYNAMIC DUOPOLY WITH INDIVISIBLE PRODUCTION AND CUMULATIVE CAPACITY CONSTRAINTS Glenn Loury No. 55*7 May 1990 massachusetts institute of technology 50 memorial drive Cambridge, mass. 02139 TACIT COLLUSION IN A DYNAMIC DUOPOLY WITH INDIVISIBLE PRODUCTION AND CUMULATIVE CAPACITY CONSTRAINTS Glenn Loury No. 55;«7 May 1990 A Dynamic Duopoly with Indivisible Production and Cumulative Capacity Constraints Tacit Collusion in by Glenn C. Loury* J. F. of Government Harvard University** Kennedy School June 1990 Abstract This paper studies a dynamic, quantity setting duopoly game characterized as follows: Each firm produces an indivisible output over a potentially infinite horizon, facing the constraint that its cumulative production cannot exceed an initially given bound. The environment is otherwise stationary; the remaining productive capacities of the firms at any moment are common knowledge; the firms choose production plans contingent on these capacities which are mutual best responses in every contingency. The resulting Markov Perfect Equilibria are analyzed using a two—dimensional backward induction, and compared with the equilibria which emerge when precommitment to time paths of output is possible. It is shown that the ability to precommit can be disadvantageous; that collusion in Markov Equilibrium is facilitated by the symmetrical placement of the firms; and that having greater capacity confers basic strategic advantage on a firm by enabling it to credibly threaten future production. The model solves an open problem in the theory of exhaustible resource economics by imposing subgame perfection in a resource oligopoly with independent stocks. It also formalizes the intuition that, when indivisibilities are important, tacit coordination of plans so as to avoid destructive competition is facilitated by establishing a convention of "taking turns" — that is, a self—enforcing norm of mutual, alternate forebearance. * The author thanks helpful discussions; Ariel Halperin, Eric Maskin, Robert Rosenthal and John Vickers for and the Bradley Foundation, the Olin Foundation, and the Center for Energy Policy Research, MIT, ** for research support. This paper was written while the author was Department of Economics, MIT. in residence as a Visiting Scholar at the I. Introduction This paper studies tacit collusion in a dynamic model of duopoly. indivisibility in production at any moment., and an intertemporal constraint on the cumulative amount of output which is to accomodate its essential, limited production We is whether to produce opponent's action. The possible over the horizon of the game. On explores the strategic implications of these two conditions: indivisibility, a firm deciding Firms face an On at a point in analysis the one hand, due to the time has limited flexibility the other hand, due (say) to the presence of an and non-reproducible factor of production, each firm's capacity progressively diminished by decisions it for future has taken to produce the past. study the Markov Perfect Equilibria of the resulting dynamic game when each firm's strategy specifies a rate of production at acities for future any moment production remaining as of that moment. as a function of the joint cap- An equilibrium is a pair of such state—contingent rules which are optimal for each firm, given the other's strategy, from every conceivable state. In a sense our game is a "closed—loop", "multiple stock" version of the "oil"igopoly problem encountered in the exhaustible resources literature. our knowledge, the first with multiple stocks. the context of This is, to treatment of perfect equilibrium in a depletable resource model Previous work has succeeded in imposing subgame perfection only in "common The assumed 1 pool" situations, in which there is a single limiting resource. 2 indivisibility in production further distinguishes the present analysis from existing work in resource economics. Our firms decide whether or not to produce at each moment, but are unable to vary the non-zero rate at which output regulate an "on-off control variable, facing a constraint on the total ' variable can be "on" over the horizon of the game. 'See Loury (1986), and the literature cited is supplied. amount They of time this In effect, their output decisions are there. An early contribution was the study of competition in the exploitation of a common fishery by Levhari and Mirman (1980). Related analyzes include Reinganum and Stokey (1985V and the recent, more general, work of Sundaram (1989) and Dutta and Sundaram (1989). 2 qualitative choices about participation. Dilemma in The game which the capacity to play "tough" Dilemma — analogous to a dynamic Prisoner's in the future diminishes We focus frequency of having played "tough" in the past. 3 study of the Prisoner's is on the the extent to which, in a with an increased classic question in the non—cooperation equilibrium, the players can avoid dissipative competition through mutual forebearance. Characterization of closed—loop equilibrium in several state variables is non—zero sum a very formidable, largely open, problem. rare outside of the linear—quadratic context, though a number differential games with Explicit solutions are of economically interesting examples have been identified with a special structure permitting profitable study. 4 In this paper we make progress by simplifying the problem in two ways: As noted, we the set of actions from which the players can choose at any instant; and, actions, once chosen, must remain fixed for These assumptions permit explicit solutions Our ability to solve the model some is be obtained. exploited to pose a number of interesting questions for which general answers are unavailable in the existing literature on duopoly: Are firms more, or less, they can, and do, commit at the tion in which such commitment tween the the other? size is and distribution of How dynamic able to avoid mutually destructive competition initial when date to specific future actions, relative to a situa- not possible? What is the relationship in equilibrium be- payoffs, on the one hand, will the distribution of that (possibly quite short) interval of time. for perfect equilibria to explicitly we assume limit and capacities to produce, remaining capacity evolve over time? How on are A closely related model is the continuous time Prisoner's Dilemma game of Rosenthal and Spady (1989). There Markov Perfect Equilibria in strategies defined on a two dimensional 3 state space of cumulative profits are studied. A good discussion of the differential game approach to dynamic oligopoly problems is in Fudenberg and Tirole (1986). Two linear—quadratic analyses are Fershtman and Kamien (1987), who study a deterministic, one-state-variable model of duopolistic pricing, and Hansen, Epple and Roberds (1985), who solve a stochastic two-state-variable model of resource depletion, though without imposing sugbgame perfection. Other models with a structure permitting profitable study of subgame perfect equilibrium are the memoryless patent race of Reinganum (1981), the racing model of Harris and Vickers (1985), the capacity investment model of Spence (1979), extended in Fudenberg and Tirole (1983), and the duopoly pricing model with menu costs of Halperin (1990). 4 these considerations affected by the rate of discount and the elasticity of market Among the results obtained are: (1) state—contingent strategies imply equilibrium payoffs at least as great, (2) symmetry mine and often greater, than those possible with open—loop the interest rate, and to the elasticity of is demand; that, at the initial date to future actions when such commitment to their mutual advantage if is inversely related to total capacity, to (4) the ability to credibly threaten future more than Result (1) implies that the ability to can be disadvantageous! Result (2) suggests not possible, firms are better able to maintain high prices neither of vides theoretical support for the them enjoys a common large market share! Result (4) pro- intuition that a firm with a substantial share of Arabia in OPEC!) can disproportionately influence the course of play. total capacity (Saudi To is endowments consistent a basic strategic advantage which allows the appropriation of a proportionate share of the industy's total payoff. commit the extent of strategies are employed; (3) with attaining tacit collusion in perfect equilibrium production strategies; of capacities facilitates collusion in closed—loop equilibrium, but can under- when open-loop it demand? obtain these results we employ the two simplifications mentioned to transform a complete information, continuous time game of dynamic duopoly into a more tractable crete time analogue. Markov strategies is This conversion is defined in section described in section III. A II. dis- Perfect equilibrium in two-dimensional backward induction is em- ployed in section IV to solve for explicit equilibria, which are compared to the open-loop Nash equilibria of the ior in the discrete same game. In section model V we consider the limit of equilibrium behav- as period length shrinks, raising issues discussed briefly in that some technical and conceptual and the concluding section of the paper. 5 Because of the indivisibility in production, coordination of play is closely linked with the idea of "precedence" in this model. Our discrete time equilibria exploit this, and the related notion of "alternation" heavily. In continuous time, when players are free to change their actions at any instant, it is nolonger clear how to define who produced "last period". Thus, it is not clear how to interpret the limit, as period length goes to zero, of the discrete time equilibria analyzed in section IV. See the treatment of extensive form games in continuous time by Simon and Stinchcombe (1989) for a fuller discussion of these issues. 5 H. The Model We A consider the following model of competition between two independent firms, and B, facing an indivisibility in produc ion, with given limitations on their cumulative capacities to produce. Time is continuous over an infinite horizon, A flow rates of production of firms that 2 a b that q =(q ,q )e{0,l} ; Q e{0,l,2}, Vr>0. feasible to A either a firm b is 2 is of) a inactive, or and q it is b is At date r the respectively, are such producing at a unitary the industry's production rate at r; obviously a number R. (j=A,B) denoting production for that firm over the course of the game; Capacity the vector of initial capacities/stocks. active; a firm exits the is is firm's initial capacity (or stock) cumulative (resource cost Re(R a ,R r )gIR a firm a Q =q +q any moment. rate at is, and B, denoted by q re[0,oo). game when its falls at a unitary rate capacity has fallen to zero. There is if no way add to stocks during the play of the game. The firms' stocks are common knowledge. The demand ically. There is side of the market is passively modelled; buyers do not behave strateg- an inverse demand function, P(-), which only on the total rate of flow of output of the two firms. P(l)=l>e=P(2)>0. (For completeness, dustry flow revenue inelastic. If e>l/2, falls as let is time invariant and dependent Monetary units P(0) be any finite number.) the rate of production increases, so then we will refer to demand as being we say elastic. are such that If that There is e<l/2 then demand common, uniform their respective discounted flows of rate r>0. The is no other cost of production beyond the opportunity cost implied by the depletion of capacity. ceipts are discounted at the in- Future re- firms' payoffs are simply revenue received over the infinite horizon as a result of their joint paths of supply. We study the equibbria of the following discrete time analogue of the continuous time set—up introduced above: Assume that the firms cannot change their flow rate of output except at particular, exogenously given and uniformly spaced moments in time that is, except at the beginning of a "period." tain their production at the new — After any change they are bound to main- rate until the next such moment. These moments when adjustments are permitted are — length of a "period" discrete time We ize. common knowledge the interval between dates dynamic game is which emerges Suppose then that T(h)E{0, Associate with each be taken. t", Monetary flows over tions, and may be regarded Suppose that at dates V» and V R where: tel intervals as equivalent length h. ], the Vc = and lump sum equivalent initial state, s and b =max{beI|b-h<R is actions can will often refer to is t "date [h-t, h-(t+l)). worth 6(h) . at the beginning of Then dollars at M-P{Q S l tel each period. their respective payoffs are l ' )], of the flow of one dollar over an interval of Unique payoffs are implied by every path of firms' actions via h, we evolve over the course of the game. that A „-rh Given periods of length Define the We meaning the interval lump sums paid tGl for /?(h)=[ allowed. [r,r+h), reT(li), are constant given our assump- t ! t", when new the set of dates is the firms take the actions q f Va A =../3Sto.P(Q)]; t (2.1) is the whose equilibria we character- h, a dollar received on date ; h>0 be a change of action reT(h) the integer t=r/h. Define the discount factor 8(h)se Let firms. in these equilibria in the limit, as h-»0. h, 2h, 3h,...} meaning the instant r=h-t, and to "period date zero. when defined for each value of also study the behavior two to the inessential.) The (h,R)=(a R }. (We consider now how (2.1). the firms' productive capacities Let I be the set of positive integers, and IeIu{0}. (h,RAb (h,R R )), as follows: suppress explicit dependence of initial state is a pair of integers a s =max{aeI|a-h<R A }, on h and R when denoting the "number of plays" which each firm can make over the course of the game, where a "play" involves supplying output at the constant unitary rate for a period of duration by neglecting the fact that the initial stocks may denote the state space. In this way, for every h, and where we "round—off" not be evenly divisible by h. initial vector of capacities ReR 2 Let SeI«I and every . h>0, we assign an — initial state eS, representing this association by: s s. is the state of the actions during period t, then the state is apparent that IQ. Markov if al game at date date t+1 and t, satisfies: if is q. s. , -R. It the vector of firms' — s, q, Perfect Equilibrium obvious that, for any given history of play prior to date It is = , (h,R)~h s action paths subsequent to date t, t, the set of feasible and the firms' conditional valuations of these paths, depend on that history only through the state of the game "state of the game", as defined here, is a at t, s =(a,,b, "payoff relevant state" in the sense, Maskin and Tirole (1989): Conditional valuations are independent of the stationarity assumed big enough; all Feasibility in (2.1). histories prior to That is, e.g., of the history, given the depends only on whether remaining stocks are which reach the state t ). s, imply the same set of feasible subsequent actions. We employ the concept game, with h from S into MPE. Markov Perfect Equilibrium (MPE) fixed, in the analysis to follow. [0,1], reporting for each during a period when the state tegy of is seS. This restriction involves Cim In general then, strategies are functions the probability of participation in the market However, we shall restrict attention to less loss of generality since pure strategy equilibria always exist in this model, cient here. If, at some node s, A getting "stuck" there for a period with pure strategies. — may appear to be the case, and mixed strategies are ineffi- risk a source of lost industry profits which does not exist Because we are primarily concerned with understanding when a rev- Given a pair of pure Markov strategies on during the play of the point than pure stra- and B play "0" with positive probability, they enue maximizing outcome occurs in equilibrium, we tice for the discrete restrict attention to S, game may be envisioned pure strategies. the resulting motion through the lat- as exemplified by Figure 1. At each seS four continuations are possible, corresponding to the four elements qe{0,l} These outcomes result in motion of the state from s to some successor node, s'=s—q. 2 . If, at node s, both firms stay out of the market remains where it began. With Markov (i.e., q=(0,0)), then s'=s, and the state amounts strategies this to remaining at s forever, an obvious impossibility in an equilibrium unless s=(0,0). The three other continuations correspond to one or both firms being in the market, and result in motion of the state either "due South", "due West", or "Southwest", as depicted by the arrows in the Figure. Eventually the state reaches one or both of the axes, representing the exit of one or both firms. At this point, the firm with remaining capacity produces one unit in each period until the state reaches the origin. / / / I , Figure < 1 — < s i 1 1 i i / , ' 1 1 . I I ^ H -> I 1 1 Thus, with each profile of pure Markov strategies and each initial state may be associated a unique path through the lattice and, via (2.1), unique payoffs for the firms. Given the period length, these payoffs are a function of the and the parameters e and 6. Every path terminates initial in a finite node, the strategy pair, number of steps. Current period payoffs (divided by /3(h)) at each node depend on the actions taken there as follows: B 1 1 Table 1 1 We now E A E{<r precede formally, introducing some notation and definitions. Define: A :S-*{0,l}|CT A (0,b)=0,VbeI}, E A B B e{<7 (a,0)=0,VaeI}, and £eE *E b :S^{0,1} aB | A pure strategy profile (or simply, a profile) profile generates a path {s creE s (3.1) Q s = s, = s t We denote this sequence s'e{s,(s,i7)}, in (2.1) and that with V R (cr)(s), cr(s.(s,<7)), Definition 3.1: Definition 3.2: path A t-l is s, we may since ular initial state. it We say that point are not credible. s' is under a s if seS, every a successor of se{s,(s', cr)}. V s under a Identifying . (<r)(s) if q, and way, as the respective sum in in the obvious <reX, ctgS a is Nash Equilibrium from sGS A V A (flr)(s) > V A (<rA ,&B )(s) VB (a)(s) > V B (a A ,c B )(s) VatfZ 8 o*gS for every is a Va eS A , if: and . Markov Perfect Equilibrium if it is a Nash SGS. all MPE's in pure strategies. Nash Equilibrium is weaker requires strategies to be best responses to each other only from a partic- In a NE neither firm gains by deviating from its strategy at any node it anticipates the response to the deviation to be as reported by the strategy of the other firm beyond state defined inductively as follows: define the value function for each firm, along the subsequent path, given that strained, From any {s,(s,o-)}. Let S cS be the set of MPE cre£. tGl a predecessor of strategy profile Equilibrium from than ~ ^ s t-l)' strategy profile A an element and seS given profile at state (2.1) along the ,s,,...,s.,... }, {s,(s,<7)}. s' is feasibility, MPE by the Yet these responses are not themselves con- definition. Therefore, NE may depend on threats which requires that promised responses to deviations with subsequent equilibrium play. must be consistent We now we In the sequel space. a formalize the idea that a profile is MPE an on some subset of the state construct equilibna by finding particular strategies that are best responses to each other from initial states in a given region of extending the resulting S, We say profile to the entire state space so as to constitute a full equilibrium. W states is closed to deviations from W, and stay in by a\a- the Given a closed to deviations VseW, Va 6E Let S| cr'eE by replacing A A , Vcr B and a' sa on (a) W t aeE (b) g is is a a NE from implies W is a-, {0,1} , a W. Denote j=A,B. we say that W cr' eSi , v . set in question. closed to deviations from It is a. W MPE on a; B w then )}cW. if: and VseW. s, MPE on W. Obviously, aeE, if Restricted equilibria on a given set thought of as equivalence classes of strategy if generated by 2 -» {s (s,a\a t restricted denote the class of restricted W and A )}cW, closed to deviations from is a with and a function a:W {s (s,a\a : strategy profile W cS agreement on the seW the following condition holds: if B eE strategy under j's WcS set of states from a A Definition 3.4: paths from if all one firm's deviation frcm such a path also yields a point in profile obtained Definition 3.3: is if a a profile that a set of profiles, in which membership also obvious that Theorem if aeE then w and only if MPE establishes that every restricted 1 can be extended to the entire state space in a manner consistent with thus be defined by is aeE. may , Markov Perfection. Consider the following simple example which illustrates the definitions. For nel define is any S set contains.) n{s=(a,b)eS| a+b<n}. Obviously, S containing the origin, and containing Now, consider the strategy profile whenever he has a positive stock. A is all closed to deviations from any points "southwest" of any point o=(a.,(7 waits until creE B R) defined as follows: B it plays has exhausted before playing any- (as 10 Tl thing. n'=Max{nGl \6 Let Then, >e}. cannot gain from deviating; and A if — we have n'>l, e>6 will deviate only if * ae£, that kn n . But for it is 2<n<n'+l: B easy to see that * a£E . For max{a,b} large enough, A To establish that a profile will deviate. a ccnstitutes an seS, neither firm can increase his payoff the path {s,(s,<7)}, given that a is MPE it suffices to show that, for every by a single deviation from a at some point along followed by both firms after the deviation. That is, only "one—shot" deviations need be considered. This follows from the obvious fact that, a sequence of deviations which is desirable, raises the firm's payoff. it must contain a "one—shot" deviation Moreover, since no firm will deviate at alone produces, only two forms of deviation are relevant: prefers to stay out of the market, or o-(s)e{(l,0),(0,l)} Either at if some node a node where he ct(s)=(1,1) and someone and the firm not producing prefers to be active. At the node determine if s, A, and given the putative equilibrium profile for the firm a profitable deviation is possible we need compare two quantities: to a, to (i) Pe+SVj. (a)(s'), the current revenue flow when q=(l,l) plus the discounted continuation value of moving from s to s'=s—(1,1); and (ii) 6V Ja)(s' '), the discounted continuation value occasioned by the actions q=(0,l) which lead to the node <r(s)=(l,0), or if (ii) A a at s. If and a(s)=(l,l), then A deviates. never deviates from exceeds (i) erize the desirability of deviations for firm If a profile a provides no incentive actions at a given node, assuming Vi(u)(s) and V- R Definition 3.5: must (cr)(s) Given Bellman Inequalities it is at s if (i) exceeds (ii) and If <r(s)=(0,l), The analogous conditions charact- B. for either firm to deviate from its specified followed thereafter, then the firms' value functions satisfy standard strictly positive quantity s"=s—(0,1). s?S, dynamic programming conditions. Thus: we say cr(s)^(0,0), and: that a strategy profile aeS satisfies the 11 (a) (3.2) o(s)=(l,l) * and Then 1: cr£S| It is Markov > ^(^0,1)), V B (a)(s) = 0e+«VB (a)(s-{l,l)) > SV a(s)=(0,l)=> /36+iV (c) o(b)=(1,0)=» /3c+*V (a)(s-(l A B (3e+6V (a)(s-(l > > B (a)(s-{l,0)). A (a)(s-(0,l)) = V A (a)(s). l))<«V (ff)(s-{l 0)) = V (a)(s). B l))<«V B > the foregoing definitions and discussion with the following useful result: WcS let (a) W closed to deviations from (b) <7 (c) a Given W A (a)(s-(l,l)) (b) We summarize Lemma V A (a)(s) = If . is aeS the profile A (a,0)=cr B (0,b)=l, satisfies a W: a; any non—zero points and (c) on all of S, then two—dimensional backward (a,0) and (0,b)e W; V. and V R at the nodes ctgS . induction, to characterize Perfect Equilibria in pure strategies for this game. For arbitrary values of on the Bellman Inequalities at every strictly positive se W; satisfies (b) possible, using a at satisfy the following conditions This may all be seen as follows: s—(1,1), s—(1,0) and s—(0,1), there is always some pair of actions q=cr(s) which permit the Bellman Inequalities to be satisfied at s. That one of them over, is, for fails, strictly positive s in and ceE, which case one of the inequalities in (3.2)(b) or V A (o-)(n,0)=V B (a)(0,n)=/?(l-^n )/(l-^) fore, starting at either both inequalities in (3.2)(a) hold or the point (1,1) for all at given the values already assigned at to s, Bellman Inequalities are s, satisfied at each step. values for the functions V.(a)(-) and nodes s —(1,1), MPE on W a MPE at It is V R (c-)(-) s—(1,0) and s—(0,1). Employing a subset satisfying (b) of Lemma 1. MoreThere- and working systematically "backward", up and to the right in the lattice, one constructs a pure strategy a a (3.2)(c) holds. by successively assigning values s'<s, in such a way to that the possible to do this so that at each node have already been determined this induction, we prove at the that a restricted of the state space can always be extended to the entire lattice in such 12 a way that the resulting profile constitutes a Theorem Proof: we and ctg£| W) equilibrium on there exists a*eS S. such that: o*(s)=a(s), VseW. See the apppendix. Theorem turn WcS Given 1: full now 1 is an important tool which to a substantive analysis of solve the model MFE shall We be used repeatedly in the sequel. strategy profiles. The argument explicitly to find sta.te—contingent plans of action is constructive; which are mutually consistent with revenue maximization by both firms from every conceivable state. We begin by defining tacit collusion and describing open—loop equilibrium in the model. IV. On the Possibility of Collusion in Markov Perfect Equilibria A. Tacit Collusion In this simple setting, with only *r o possible actions, t In order to maximize the takes a very primitive form. arrange to avoid supplying at the same time It is must of their payoffs the firms that reduces discounted industry revenues. obvious that the industry revenue maximizing output at any strictly positive node s=(a,b) depends only on the total (a+b). behavior at seS. the if sum the notion of tacit collusion MPE under a, Regarding ueS achieves collusion at s M(a), is MPE With a from the demand , we call it initial it is node a achieves collusion. a collusive path. s iff is "large", then we say If that a path satisfies Industry revenues are maxim- a generates a collusive path from s . obvious that Q*(s)h1, s^(0,0), since supplying two units instead of one lowers both current and future revenue. and a+b be this optimal collusive cr»(s)+(7o(s)=Q*(s). The region of collusion if the set of nodes at which inelastic Q*(s)e{0,l,2} tacit collusion (or coordination, or cooperation), {s,(s,a)}cM(a), seS and ce£ ized in the Let With elastic demand, if s=(a,b) Q*(s)=2: Because of discounting, the opportunity cost of sup- plying another unit today vanishes as total capacity grows large, while the marginal rev- 13 enue of two instead of one unit of output is /?(2e— 1)>0. Q*(a,b)=l, a+b<n; and Q*(a,b)=2, a-i-b>n. The n" (4.1) That n is, is = Max{neT| ^j > 2e+f(^y- )} So critical if n e>l/2 then, is = Max{neT| for some n>l, given as follows: n l 6 > 2e-l}. the largest total stock at which the opportunity cost of an incremental unit of production, given that one unit is to be supplied in each subsequent period, is no less than the marginal revenue of raising output from one to two units. With inelastic demand, collusion in equilibrium requires the avoidance of simultaneous production everywhere. With elastic demand, such coordination B. Collusion in is only relevant when the total stock Markov consider the latter briefly. is so committed, then ize the sum (a,b)eS 2: it may There we mean that firms can bind themselves If one firm believes has a corresponding path which maximizes in these paths. Such an equilibrium achieves avoids simultaneous production of the firms' payoffs. tacit collusion Lemma if it play same game, we its payoff. to be compatible with the available capacities of the firms, Nash Equilibria the node (a,b) By "open—loop" time paths of actions which are known to both. With paths constrained sider the n. Perfect Equilibria with the outcomes achieved in the open-loop (precommitment) equilibria of the the other than Open Loop Equilibrium In order to contrast the possibilities for collusion in at the initial date to is less The following when to do so is we tacit collusion necessary to result characterizes the region in con- from maximS where be achieved using open-loop strategies. exists an open—loop Nash Equilibrium from the which avoids simultaneous production everywhere along strictly positive its path if node and only if: 14 Proof: from + jMax{a,b} (4.2) In an its last in these Min{a,b} < n' e open—loop Nash Equilibrium neither firm wishes date of positive supply to two dates be for either firm its first such a 0, librium without simultaneous production is n Max{nel| 6 >_e}. to shifts a unit of output date of zero supply. Letting the difference shift pays impossible e>6 if e>6 if . for So an open—loop equi- some firm, along every path where one or the other, but not both of them, supplies on successive dates. Consider the problem of finding a sequence of a of: difference in order between the 0(a,b). between the last "0" Thinking of and the "i's" last "1" first "1"; and b and the is first "0"; and Denote n.inimal. such that the larger "O's" (2) the difference in order this minimal difference by A sequence as the tLne path of supply for firm this in equilibrium from initial node (a,b) which avoids simultaneous production, neither firm will deviate 0(a,b) is realized 0(a,b)<n'. if by placing the The reader can lesser in number verify that the two of the an open-loop we see that minimal difference digits in the middle of the sequence and surrounding them as evenly as possible on either side by the other Assuming that a>b, and that a is even, this is illustrated in Figure 1, l,...l, 0, C, a/2 digit. 2: >| |< 1, (1) the 0,...0, 1, 1, 1....1 a/2 b Figure 2 Examination of 0(a,b) this = Min{nel|n Figure shows that this disposition cannot be improved upon, and that > ^Max(a,b)+Min(a,b)}. With open—loop d strategies each firm takes the other's path as given, and so anti- cipates no consequence of an increase in cuvrent supply besides a reduction of its acity for future production. own So the "shadow price" on a marginal unit of capacity present value of revenue from the last unit sold. There exist is cap- the endowments from which each 15 firms' initial shadow price on marginal capacity output, along every collusive path. whatever the elasticity of demand, total C. its e, for marginal revenue at zero which collusion in open—loop equilibrium. (Note: 2n'<n.) size and symmetry relative maximal industry revenues show that something in Markov Our method Perfect Equilibrium. effective at achieving coordination because, game contingent upon an open—loop equilibrium. is constructive; is we Intuitively, closed-loop strategies are by making the entire future course of the current actions, they can impose on the firms a cost of increased out- strate formally in Proposition 1 that when implying a vanishingly small shadow price symmetric and the periods key concept in and Markov We total production capacity is arbitrarily large, for the marginal unit, if capacities are relatively in a non-cooperative, perfect equilibrium. this construction is the notion of strategies, colluding to "turn—taking". With {0,1} maintain a high price amounts to specifying regions of the state space within which one firm has precedence, and the other to forebear. demon- short, the firms can always coordinate their actions tacitly so as maximize discounted industry revenue actions true regarding shall exhibit a put far greater than the foregone discounted revenue from the last unit sold. A So, of capacities can close to the opposite of the foregoing particular equilibrium profile to establish this point. to both (4.2) fails, Precedence and Alternation with Markov Strategies We will more than less From such endowments, firms produce on the initial date in every militate against achieving is Along a collusive path, one firm produces until a node is is expected reached which signals the other's "turn", the other then produces until the state passes out of its region of precedence, and so on. In equilibrium the firm without the turn must prefer forebearance to "playing out of turn." A special case of this turn-taking partition might be called "pure alternation," where the firms share the market by producing on alternate dates. To motivate the alternating strategy profile, consider a one—shot game with payoffs as given in Table 1. 16 This game return of has (1,1) e. as its strictly dominant Nash Equilibrium, providing each firm with a Suppose that the firms were to encounter of periods, with common 6 being their librium would imply substantial losses. game over an this discount factor. For small e Both firms would be much infinite sequence the one-shot equi- better-off if they could coordinate their actions so as to avoid "head to head" competition, yet each prefers to supply in the current period. Resolution of this Prisoner's that the firms reach A will agreement tacit as to who last period stays out this period". If them is is: not too low be better to wait until tomorrow for one unit of revenue, and the implied oppor- It is units of revenue every e easy to see that such an alternating convention, supported by the threatened punishment of reverting to "(1,1) forever 1 librium of the infinitely repeated game "competing forever". That and only ( in such a situation the discount factor tunity to wait yet again the day after tomorrow, than to claim day. requires and when. will supply, natural understanding which might develop between "Whoever produced it some Dilemma—like problem 43 ) is, if « This condition is <- if ' if deviation occurs, and only if a subgame perfect equi- "waiting to alternate" 2 if: is 5-(l+6 +^+...) = S/(l-6 2 ) is better than > e/{l-8). Or, lh- important in the analysis of Markov Equilibrium to follow. Note that holds with inelastic demand if it the periods are short enough, but never with elastic demand. Return now to the dynamic game with cumulative capacity constraints which we have been considering. Formally, define a profile for all strictly positive seS: <reE a(s)+cr(s— u(s))=(l,l). as an alternating strategy The path {s,(s,a)} profile if, therefore involves the firms supplying the market alternately in successive periods for as long as both have positive stocks. supplies There are exactly two such when the profiles, distinguished stocks of both are positive and their without loss of generality that firm A sum is by which of the firms an even integer. We assume produces under the alternating strategy profile 17 when the sum Notice of positive stocks how the reduction even, and denote the corresponding profile by is of the continuous game to one in discrete aeE. time with a countable state space, induced by the assumption that actions cannot be instantly changed, being is heavily exploited in defining this form of precedence. Figure 3 b+ s t (*>)} s=(a,b) ^_i {s (s,fr)} l*~ t i i i i — 1 i i , 1 1 • «- -• a (a-b,0) Take an arbitrary point a, which we call the alternating path several alternating paths. a, ahead" to s—(1,1). from s, denoted VseS. By A deviation from definition of a, to deviation if itself <r<e under These paths have a {s,(s, cr)}. to (0,0). stair- Figure 3 illustrates a means either remaining at s—(l,l)=[s— a(s)— <r(s— a(s))]=s a always leads to a node belonging to from itself.) A useful way to state this fact is 2 S, are closed to s, or "jumping (s,o-)e{s,(s,cr)}, for alternating path its any alternating that, along path, the consequence of deviation from "waiting one's turn" For s (Indeed, only an alternating strategy profile generates paths which are closed s. It is nodes to set of all successor Notice that these paths, regarded as subsets of s>>(0,0), so a deviation from through and consider the and include nodes along the axes leading step appearance, deviations from scS, is always to "lose one's turn." easy to see that the turn—taking convention represented by it must be that a (a,l)=l, VcreS A a-1 a payoff of at least /3[e+£(l-<S can get from playing "0." Since 5r )/(l-<5)] A (a,l)=0 : By playing "1" at a (a,l) is A not an MPE. guarantees which exceeds /36{l-^)/{l-S), the best when a is even, it follows that a£E . it 18 Nevertheless, there is some non-trivial subset of nodes, implicitly defined as follows, which the alternating profile a For a Definition 4.1: 3(e) = a restr icted is {s6S|Vs'6{s (s,a)}, aeE, g,i y 2: est subset of Proof: 3(e) ations from if ^c£| if W WcJ(t). That then 3(e) Therefore 1, if we can show node s£.3(e). is closed to deviations from firm If "waits j it receives it is turn" at its fie by definition, precisely the s its c) if of receiving in perpetuity , , iff itself, it V.(3-)(s)><5/?(j^). erty of a "locally," at a node partial converse of when it does Theorem 2, s and for this general rule only its V.(a)(s). must be that Thus, a does not pay. A firm "waits d its come does not exceed the value , the revenue loss to the other firm caused by Moreover, the argument is This continuation yields the payoff nodes where deviation from the value of that turn, follows are satisfied at every This result leads to the following general rule about turn—taking: turn" for one period closed to devi- payoff there waiting to have the market to set of is immediately, but loses the oppor- V(b-)(s)=6-[0+5V.(a)(s-(l,l))]. So the deviation pays . the larg- The Theorem then a. that Bellman Inequalities (3.2)(b or (3e+8V -(o)(s—(1,1))- But, because out of turn is the countable union of alternating paths, each of which tunity to have the market to itself nexl period. is, 3(e) is deviates by playing "out of turn" 3(e) is, on which a mutual agreement to "take turns" could be self—enforcing. strictly positive If it Moreover, ap')=Q, then Vfa)(a')<6-0§=$, j=A,B}. S h. Lemma from MPE: above and 0<f<l define the region of alternation, 3(e), by: as t Theorem from it having played used the alternating prop- immediate successor. So we have the following relevant whenever precedence changes hands. 19 Corollary 1: Suppose aeE . Then a(s) + cr(s— cr(s)) = and a-(s)=0 imply that (1,1), V.(a)(s)a/?(|^), for je{A,B} and seS. The next result provides n For s=(a,b)eS define n(s)E|a—b|, and k(s)=min{a,b}. Let the larger firm and functions on s the smaller firm at i(s) 2k ~2 <|>(n,k) 2: 0(e)={seS s, i,je{A,B}. e 6 n - <5 [Vkel, k<k(s): (}>(n,k) + (l-^"2 ).^), <|>(n(s)+l,k)>e if j(s) denote (Explicit dependence of these be suppressed.) Define the function will often (4.4) Corollary also introduce nota- Recall that n'EMax{n6l| S >e}. tion which will be used extensively in the sequel. Definition 4.2: We an explicit description of .0(e). for as follows: nel and kel. a-, \(s')=0, and <j)(n(s),k)>e if s aw Js)=0}. Moreover, when (4.3) holds then 0(e)={seS|Vt>O, n(s (s,a))<n'}. t See the appendix. Proof: With indivisible production forebearance, by one firm or the other, is essential if they are to share the market and maintain a high price. Theorem 2 and Corollary that such restraint is consistent with the firms' interests only if 0(e) is too large for a firm to be willing to "wait consists of nodes generated by able (in a. from which Thus, MPE), meaning if the its turn" to receive this criterion is satisfied at initial node s € it. is Now greater the region 0(e) we say that alternation is implement- that a tacit agreement to share the market through simple "turn- how the implementability demand Any every point along the path taking" could be implemented by a state—contingent, self-enforcing rule. If imply the "contingent right to produce" has been allocated in such a way that payoffs never exceed f3{jzr)return 1 of this norm depends on Corollary 2 shows the size and distribution of capacitities. sufficiently inelastic, in that (4.3) holds, then 0(e) contains any node 1 20 from which alternation never causes the stocks to differ by more than n' without regard , the interest rate or longer the period length, the less elastic to hold. With symmetry. The higher So, for £<yr~T, impleir.entability depends only on to total capacity. demand elastic (4.4) implies a demand must be for this result trade—off between the aggregate size of cap- acity and the inequality of its distribution, in determining whether alternation mentable. For e>T^-c, smaller is n, n only <j)(n,k)>e (i.e., n<n') and k the larger can k be while maintaining total capacity, the more equally must it be held Another implication of Theorem 2 limit as 6 >e if of the set h-»0 h- 3>{t), is Thus, the greater (j)(n,k)>e. the Proposition below. 1 proof examines the Its a to a to extend 1: For any fixed £6(0,1), r>0, which achieves collusion whenever and the initial Let total capacity R~, e, and r V R~eR.+R r _ | R.-Rg | of the paper. an MPE exists enough R=(R ,R R ), where . such that {s (s t whenever h<E, S. * x>0. K>0, and aeS exists , on Formally: be given, and consider the vector of capacities There MPE full relevant, given that the periods are short holdings are sufficiently symmetric. R A +R B =K. Proof: it is and the is implementable. This method foreshadows limit arguments developed more fully in section Proposition imple- not too large; the is for alternation to be and uses Theorem is o (h,R),cr)}cM(a) <x, and Q*(s (h,R))=l. See the appendix. The proof is based upon the observation that, when the periods are short and the firms equally placed private incentives to conform with the alternating profile nearly , coincide with "social" incentives to max mize i industry revenues : If each firm has kxR"/2h C units, then (4.4) implies that neither firm deviates from a other hand, coordination (4.1). But, as h-tO and is if required to maximize industry revenues <5->l, r)1, i e<(Tqrr)(l + ^ these condition both require that R~ < 1 if e<-Jl+6 )• On 2k— ), the by -ln(2e-l)/r. Thus, the notions of "alternation" and "coordination" are intrinsically linked in this model. 21 MPE Extension of the Alternating Convention D. We now is complete the program initiated in Theorem 2 by exhibiting a profile which The extension a perfect equilibrium extension of a to the entire lattice. The the following general form on S\ 3(e): while the smaller firm supplies nothing 3(e) within a given number reaches only on the smaller firm's endowment. and only if of periods. if, by The The argument Definition 4.3: For strictly positive seS, and so doing, the equilibrium path m(s) e Theorem 3: +cd, With any is if of periods depends based on a backward induction We III. On l<m(s)<m,/\, otherwise in require the following: 0<e<l, the "waiting time" m(s) if is (j)(l,k(s))>e, given by: and <Kl,k(s))<e. ueX which arbitrary sequence {m, :k>l}cl associate the profile a as follows: extends 3(e)-, ct(s)eo-(s). On S\3(e), cr.,. \(s)=l; a-, \(s)=0 if For each e£(0,l) there exists a unique sequence a-, \(s)=l. {m?:k>l} such that the associated Proof: number critical m(s) = min{meI|V;c, Kk<k(s): <J)(n(s)+l-m,k)>e}, (4.5) profile takes larger firm supplies one unit at every node, two dimensions, as suggested oy the discussion of section profile a*, determined in this way, is MPE. an See the appendix. Comparing (4.4) and time that the smaller firm (4.5), in light of Corollary 2, [i(sj] we have associated this strategy for and only if one may see that would have to wait before reaching 3(e) produce, given that the larger firm if a [j(s)] i supplies constantly. on S\ you can get there within m, .25(e): "Wait m(s) if it is the does not With every sequence {m, for 3(e) when holding k periods; otherwise, produce." be interpreted as the parameters of an "optimal waiting policy" for i, So } units {mf} may given the assumed 22 behavior of ing" by The Theorem j. together with constant production by i on asserts that, given alternation j "optimal wait- .25(f), this on S\ .2* (e) are mutual best responses. This waiting strategy extends the set of nodes at which simultaneous production avoided (and thus collusion is achieved) beyond the region of alternation, -29(e), ing interpretation of the MPE a* is the the larger firm exploits t its by giving An precedence (and therefore, enhanced payoff) to the firm with greater capacity. is interest- position so as to garner a greater than proportionate shar^ of the collusive profits by credibly threatening to "produce all the time" on S\.25 (e), thereby persuading the smaller firm that interest to forebear node To \. .< We and has the dominant play j will wait as long as it noted above that (or, credible in its outcome consider a see the basis of this such that k(s)=l, and n(s)=n>n'-l. s such a node. Thus this, whenever m(s)<mf , it is a-, _\(s)=l, threat) of "1" at s; i WeE, at knows can have the market to itself within n' periods. This reasoning, applied recursively, generates the full sequence of optimal waiting times, {mf}. When demand We found. Corollary 3: inelastic in the sense of (4.3) these waiting times e<Tj-r then the sequence If We use .25(e) a nested induction to construct and k(s)=l, then waits for .25(e) it is m , earning @6 proof of Corollary 2, firm j ^ '. obvious that So i vi k>l, determining ...,m? 1 k(s)=k. } as i's i(s) aits if a* is The proof rests on the fact that given by: is {^=1} on first s, since, point reached in i(s) best play S\ 25(e). earning m(s)<n'— lEm?, if optimal for i's 3. )>f}, kel. either produces at and only given in (4.6), k>2, and consider a* Theorem ~1 best response to produces under a at the Inductively, assume that following {m?, {mf }, m* = Max{m6l|<5m -( T^)-(l+<52k Proof: can be explicitly give the derivation in the text to illustrate the conceptual basis of (4.6) seS\ is /3e, or by the .29(e). whenever k(s)<k— 1, from seS\ m(s)<mf implies m(s—(l,l))<mf If .25(e) , for where when e<yq— i- 23 Corollary 2 implies that m(s)=n(s)— N, for Ne{n',n'— 1}, and one easily verifies from (4.6) m£e{m£_, m?_,— 1}, that "1" at If it k>2, so , s, continuing optimally from s-(l,l), ^(e) waits for better. earns: it ^' 6 Therefore, by induction, On the other hand, does better to play "1" at a* s and then "0" earned arrives at after s' s—(1,1), two -(1—8 for this true is plays "0" at i periods, but if it s for fixed for which m(s)>mw •>, we note Finally, s less if has exhausted If So a* s, coordination is i(s) plays "1" If it two periods, having at the successor node, it also along the way. 6j3e implies Clearly i if it plays "0". do will m Therefore, induction on cfi optimal for is at i any s i if m(s)<mw \, in n ,, > For s£S\3(e), which case the ensuing path forebearing until this region m(s)>mf/ satisfies: The is reached, and alternating 0<t<k(s)—l, then both firms produce until but i withdraws seM(o*) and Q*(s)=l only achieved at any node, it is i m(s—t-(l,l))=mf/ \., l<t<k(s)— 1, the if at the node s—t- (1,1) and (s,o*)}cM(o*). That {s t is, if \. while the induction hypothesis on establishes that equilibrium path begins with both producing, but waits for &(e). after s') coordinated play as capacities decline. m(s—t-(l,l))>m.w capacity. If its k, is when k(s)=k and mf<m(s)<mf+m-l, a* obeys a kind of "inverse entropy" law: and only leads directly to .2>(f), with firm play thereafter. 2 )/(l-<5 )]. l<m(s)<m£/ The induction hypothesis on k plays "1" at MPE that the achieved at is plays i completing the proof. market never evolves toward coordination 2k_2 .(l-<5 such that now having earned only followed by induction on k, s and then "1" can do no better than the latter course m, all +2 firm Simple algebra shows waiting some node (say arrives at it no worse than the former course i )/(l—6). s) when k(s)=k and m(s)=mf+m. i(s) the former course dominates the latter. implies /3-[e+^ earns: optimal from is Suppose s. alternatively, If, fie. at it if k(s)=k and m(s)=m?+l, the same algebra now shows if m>2, and consider the best play at m(s—(l,l))=m(s)<mf <m?_ .. Hence, achieved at every subsequent node. 24 E. The Limits We complete to Collusion in Perfect Equilibrium this discussion of ta collusion by showing that ;it possible to it is achieve the industry's revenue maximurr in an MPE nodes than M(o*) when demand At the same time, our constructive methods show why collusion distributed. is For the is inelastic. not possible in any equilibrium rest of this section we assume nodes seS from which there exists some Any MPE that ctgE reached is t wb en where that V-(<7)(s,)<^/?(t—4) is great at s j is at s {s (s,cr)}cM(a). )+a(s From ,)=(1,1). this way on <7.(s,)=0. But means that the this Ignoring If 1 we the difference in at s in larger firm would not any revenue maximizing in the proof of may MPE. Theorem can then use Theorems 1 and 2, is So a be determined by an induc- A 3. limit is the degree of disparity in initial holdings consistent with an MPE Corollary t must have begun with the smaller firm waiting. There argument similar to that used We the set of there will be no pt'th which maximizes industry revenue and keeps tive collusive path. M* between the firms' "regions of pre- on how long a firm would be willing to wait, which struct an Denote by t cr(s have been willing to forebear in order to arrive limit c^tt-t- the firm for which the larger firm's payoff below this bound. any such path arriving set of capacities are very unequally satisfying: t holdings when collusive path {s,(s, cr)} occasionally passes cedence", so that a date know from a considerably larger ascertainable in MPE generating and the notion of alternation, to con- generating a collusive path from any node within these limits. Markov strategies momentarily, let us study those time paths of output which avoid simultaneous production everywhere without either firm's implied payoff be- coming "too large". Call the path {q.:0<t<t} admissible from Vt; (2) q +q.+...+q = s; and any intermediate date and t, (3) the value of the sequence {q 0<t<t, does not exceed /?(t-4). s if (1) q.e{(0,l),(l,0)}, :t<t'<t}, for either firm Consider the recursion: a 25 (4.7) (i) n =n', and (ii) c = Max{cel| k HV . k + ) ) where kel, ^-V^ for - V k-1> k=l, Vo with ~= 2, ..., and < /?({=£)}, cW , S + +c,— 1, , /3(|=f 1+C, rC, ^fe ~= =n, n, #££ )• Then we have: Lemma 3: Moreover, An admissible path from for kel, ^-n^j+l c < < s exists iff n(s)<n, where {n,}, %, / kel, solves (4.7). c+1, where cEMax{ceI| f{l-S)/{l-f +1 )>e}>l. Consider the problem of finding the longest possible sequence of "l's" and "O's", Proof: given that exactly k "O's" are to be used in the sequence, such that no "tail" of the sequence ever has a present value exceeding /?(t—r), where a "1" earns the value earns nothing, and the discount factor this sequence, kel. s if and only n(s)<n, if tion problem: With c, the is \. / The number is follow n, +k be the maximal number of "l's" k— 1 to k. So, c, maximal its as the num- =(k+n, )—{k— 1+n, _,), which is (i). length, given k. So (ii) and (iii) from the Principle of Optimality and the definition of present value. c = (ii) and Max{cel| (iii): c <5 /?(^^)<V k </?(^), (l-<5)/(l-(5 c+1 )>e} kel. < < c, Substitution into c Min{cel| (5 (l-<5)/(l-<5 (ii) c yields: )<e} = c, Since tx^>£, c>1; while the definitions above imply c<c+2, and hence c<c, <c+l. over, in the Bellman equation for this optimiza- is which may be added to the sequence of "l's" a "0" obvious that an admissible path exists from it is recursion (4.7) the present value of a sequence at From (4.8) Let 6. {h, } so defined, ber of allowable "O's" increases from V, is /?, if f{l-5)/{l-f +1 )>e>f +1 (l-6)/{l-f Define N"={seS|k(s)=0 or n(s)>n, given in (4.7). Notice that if / x—c, n(s)>hw s—c, / \ +1 ), \}, / then c, where then, by =c, Vk>l. {n, } Lemma and 3 {c, }, k>l. More- d k>l, are as and Corollary 1, no col 26 lusive MPE path is possible in which the smaller firm produces at maximum from serT attaining an industry revenue n(s)— [h, path at \-Ci / / J periods before producing. Once some node s'erT for involves firm it is to collusion in sgS, or (4.7): If (b) If (c) in the Appendix we prove that if l<t<k— 1. Thus, although If n(s)=h k(s) ^ k =0 and then * j(s) ^+m k ^ path in profile then seM*. IT, a on rT as follows: 3j (b) =1; m k(s) +h k(s) >n(s)>h k(s) ^ k(s) k(s)=0, or n(s)>n k(s) an admissible path exists from to reach the only admissible s c,_. units supplied by provides a powerful tool for assessing the limits it Given any sequence {m,:k>l}cl define the 4: (a) For would wait from if i(s) Theorem an "open—loop" concept, MPE. MPE which n(5')=h,/>,—c,/>,, the ensuing path, necessarily the larger firm, followed by one unit from the smaller firm, for admissibility path waiting at least i(s) has produced on a collusive must follow the pattern of production implied by admissible, MPE Thus, any s. then , <r j(s) =l and =0; &. (s) then a(s)=(l,l). * Then: Proof: (i) 3 {m k :k>l} which o^Eivr; moreover, (ii) {seS|n(s)<h k / s+niw JcM*. See the Appendix. The kinship MPE for of Theorems and 3 Theorem 4 should be apparent. 3 constructs the a on ^(e)cS, showing that constant supply by a* from the restricted equilibrium the larger firm, and optimal waiting by the smaller one, are mutual best responses on S\.2>(e). Theorem the "boundary" of 4 constructs the restricted rT MPE j by specifying behavior on together with optimal waiting by responses, given this specified boundary behavior. some ae£ N to coincide with the unique admissible path in that set, showing that constant supply by there exists a on such that the fact that, for any s^N, o=u on we can choose ~o rT. a The second is We are mutual best a restricted part of so that the path nating play on S\rT, and hence (s (s,ff)}cM(ff). t Since i MPE Theorem {s,(s,o}} and then 4 is on IT, based on implies alter- conjecture, but have as yet not proven, 27 d completely characterizes that Theorem plies some 3 together show clearly why dynamic duopoly with low price on his on each firm's first / \+m, ^ then s£M*. , Lemmas from that node. So coordination via non-cooperative play possibility of coordination turns when they more is and difficult to use state—contingent With open-loop indivisible production. 2 strategies, the on the ability to find a path for which the discounted value last supply date is no less than the discounted value of the With Markov play what matters date of forebearance. is whether the produce can be allocated so that neither firm has to wait when the value of contin- right to on a prorated per period dination. Comparing Theorem al maximum firms precommit to paths of actions than of the high price n(s)>n, if that the existence of an admissible path from a given node im- attains an industry revenues rules, in this uing, in the sense that MPE when achieve 4(ii) states M* (4.2) and induced by a basis, exceeds the fall in price (4.7) reveals that the former is the loss of coor- more severe constraint. 4 suggests that the concepts of "alternation", "forebearance", and "optim- waiting" are sufficient to fully describe actions in this setting. As the proof how of the the firms might tacitly coordinate their Theorem reveals, the extreme admissible path implied by (4.7) constitutes a "boundary" separating the "interior" of rT from the rest of S. From any serT not do so. seS\rT the firms agree to alternate until reaching this boundary; from any on the boundary, the smaller firm waits for the boundary if it is in its interest to Play along the boundary gives the larger firm sole access to the market roughly the fraction 1— e firm knows it of the time [see (4.8)]. This tacit agreement to be in the other's interest to is self—enforcing, because each conform from any node in the state space. V. Depicting Discrete Equilibrium Dynamics in a Continuous State Space Consider a continuous formulation of this game, without the assumption that output rates are fixed for intervals of length h. The state space is all capacity pairs, 2 [R , . o Each firm's (Markov) strategy maps capacities into production The profile rates: q-:R i=A,B. -»{0,l}, q=(q»,q-n) partitions the state space into regions of production, q. (1), and 28 forebearance, (0), for q- each firm, and induces a vector the system over time: -r— =—q(R ), R gram shows motion "due South" on "Southwest" on q. (l)fiq qT i 0)f1q For R R=(R.,R R ), (1), the associated phase dia- "due West" on qT (l)nq^ and (0), Payoffs from any node are given in the obvious way, as (!)• R g-ven. determining the evolution of field An the discounted integral of instantaneous revenue flows. equilibrium consists of a stra- tegy pair from which neither firm can affect a unilateral improvement in payoff from its any node, where strategies are restricted so that the differential equation describing the evolution of the system always has a solution. q. (1) length h. let {a :h>0} be a family For h fixed, a the path {s.(s (h,R ),a R r+h-R r = -h By analogy with R , )}. - MPE of ah a unique initital state sets 1 l {o- (s h-^0 - ah ( ' S o( h :h>0} defines a vector h the limit exists. , to evolve over time in (h,R )«h s c7h = K( S o( R o)' )) -h the continuous case, r R -R o (h,R ' R r))' field Vr-h-t, on K cr* a family of MPE, {cr* defined in Theorem 3 treats h parametrically, so :h>0}, as envisioned above. But alternating profile, and {a tering" for infinitesmal h. (s tel. given by: the motion over time of would mimic closely the path of capacities and the payoffs generated by a MPE determined, r ))}, The implied phase diagram, showing The is So, g = -^(R ),-lim{a if imply that the of the discrete game, indexed by the period cause the vector of capacities, will Given the following manner: is restriction will have a relatively simple topological structure. Now, as Any such (h,R))} has no limit, as The continuous approach a* h-»0: is it , (R : r>0), for "small" h. actually defines an extension of a , the Alternation leads to "chat- outlined above could not produce behavior in equilibrium like that which emerges in the equilibria we have studied for the discrete game, with h "small". This is not necessarily a criticism of the discrete approach; one might as readily conclude that the continuous formulation it fails is flawed, to the extent that to capture the strategic possibilities inherent in a convention of "turn—taking". 29 Moreover, this "chattering" in the limit need not prevent a phase diagrammatic depiction of equilibrium dynamics; With we can simply lim{a define: (j,^) e mind, note that the equilibrium a* defined this definition in into three mutually exclusive partitions the state space S and regions: a region of alternation, 3>; a region of forebearance region of simultaneous production, Jr ={seS|0<m(s)<m?/ "small" and, Section IV (and [-(0,1)], if if s Q | explicit m(s)>mw (h,R)G (h.RJe,? as s 3 when Abusing notation, write Re explicit 3 [«?, dependence of 3) on (h,R))=A if state space K 2 &] if 3E>0 such limiting dynamics implied by the = (3U3r U<^') Re3 if RG^ if Moreover, w(x,y) = x e~ + _2ry/ -e w(x,y) < -ln(e), e<^; MPE 2ry + ^(l^T ') > w*(y); and, w(x.y) (5.1) w*(y)=i.ln[(l+e- (5.2) w*(y) = Clearly, Q (h,R)6 The 2 in , a*, for & as h-»0; results in which these that, Vh<E, s Q (h,R)6^ \&, = x lim {h-m*, K h-.0 (\.j>\\}lV n tt " e,Vye[0,y]; Reo^if w(x,y) > w*(y). + j-ln^ + ( e_2~) e ^ e<£; Max{0, l.ln[s(z)^(z)l£^] }| ' a* on the continuous (family) 2ry )/26], &\. then h-+0=>h-n(s (h,R))-»x and are as described above, where: rx < e. formulae cannot be found in the discrete case. Define w(x,y)=lim{h-m(s (h,R))}, and w*(y) = ° The 2: s [B]. h^O Proposition 3 ; ^=-(1,1), j(s . , , by the smaller firm, 3f and a xe|R.—Rg|, yEMin{R.,R R } and zEMax{R.,R R }; h-k(s (h,R))-»y. Theorem The dynamics generated by >.}. as h-»0; h->0, for in VReR collectively exhaustive their proofs) can be used to solve for the regions of K various motions obtain, even Denote Drop the <tf. and o^fseS 37=-^), h, satisfy: ^=-(1,0) >.} (h,R))}, as h->0, (s e> l, £> 2' and: ] 30 for u (y)=[^=^-^— The argument Proof: (4.6), > is that, Vh<K: <j)[n(s o e>l, w*(y)=0, y<y<y, for ye(0,y). A a straightforward consequence of taking limits in (4.4), (4.5) and except for the derivation of (5.2), which 3K>0 such When 0<y<y=-ln(2f.-l)/2r. yJ 4+(2e-l)e By Appendix. in the is Corollary 2, Re 3> if (h,R))+L,k)>e, Vkel, k<k(s (h,R)). Then (4.4) implies: cj>(n(s (h,R))+l,k(s (h,R))]^e" rx -e~ o 2ry 2ry + ^(l^e~ ), h - as 0. Moreover, from (4.5) one sees that m(s (h,R)) must "solve": <K n—m+l,k)«e. Equating the limit above with ting h^O in (5.1). substituting e, (4.6) gives: The derivation e s ^ of (5.2) is * (x—w) for )(l-^ x, and solving more co-nplex since no when OrriS equation (A. 2) in the proof of It follows from (5.1) that, when if z r 1 e --In - — x+y > r L as —It^ - y+ e h-»0, which is {mf } is explicit representation of Theorem 3 must be c<^, a coordination failure occurs r h w, gives w(x,y). Let- _2ry k_1 i -rw *-(l+e 2-e ) ), available a* for "small" for utilized. under the MPE ~ry that a coordination failure must occur in any l . The comparable , N limit in (4.2) implies J open—loop equilibrium if z > 1 --In fe- 2ry l *— •- e . -• This gives a clear measure of the superiority of closed-loop play for achieving coordination. A comparable measure can be derived from contingent play vanishes as tion 2, is firms' payoffs for various endowment and state- (5.2) converge It is easy to configurations using the results of Proposi- and so to verify our assertion that the larger firm garners a greater than propor- depict the limit phase diagrams for ihe The power MPE in (5.1) continuous in the elasticity of demand. tionate share of industry earnings from nodes 4B which shows that the advantage of Observe that the expressions e-»l. as e-»l/2, so equilibrium behavior compute (5.2), of this limit a defined in Theorem argument 4. is R£& where MPE made w(x,y)»w*(y). Figures a* in the respective cases clear by its For there the discrete game e<jr 4A and and e>^. application to the restricted is very difficult to analyze, A 30 Figure 4 / / ^45 F ieur e 4 B v - 3o b - y *.-i.Uic^^iV| F i°;u Q re 4 c / / / / 31 but its continuous analogue employed to see that is transparent. Lemma and equations (h,R)erT, for h sufficiently small, s extreme admissible path defining the boundary of (by (4,8)) c,e{c, c+1}, where c=Max{et-:r|c< line 3, rT — with slope (relative to the larger firm's axis) -r^-, and 2>-ln(e)/r + if can be (4.8), (—)-y. The "chatters" with infinitesmal h, since But -}. (4.7) as h-»0 boundary becomes a this along which the larger firm produces on average the fraction 1— e of the time. (See Figure 4C.) The proof of Theorem 4 shows that one can extend a The alternate turns. way in such a that, "above" this line, the firms agree to take a requires that "below" profile Optimal waiting constantly and the smaller firm waits optimally. boundary of cient use of an option to go to the Let us see how decide how a, may be derived. From equation problem facing the smaller firm produce the fraction and nothing for e of the time. duration 1— e any node im{h-m, ,, / (A. 3) in the s for each time j3 it effi- a there. -q\\}> an<^ hence Appendix one can which n(s)>h, move , \ to is to the ex- produces, but only gets to (Because of discounting, earning /? for duration e, better than earning @e continuously, so waiting might is For each unit of time that the firm produces, the wait increases by roughly units of time, since the firms produce has slope boundary of 1. before beginning to wait, then (5.3) n(y,t,w) = N is its — has slope Passing to the limit, the wait for the boundary of rT initially while the path along which both the smaller firm has y units of stock, if w, and if the firm produces for t |.{[l-^r rt ] + ^-^[e-rt/'W ^/ 6]}, w(y) = Max{w>0| n(y,0,w) > Max 0<t<y y>0, w>0, and 0<t<y. II(y,t,w)}. if units of time earnings are: Accordingly: (5.4) equivalent to the long to earn @e by producing each period, before waiting to treme admissible path where the firm can earn pay.) at is produces taking the payoff implied by rT, the limit of the waiting times, w(y)El the behavior implied by verify that the this line the larger firms 32 One may if and only from verify (5.3) that d"U/dwdt>0, w<w(y). Notice as well that if is II n(y,0,w)>fl(y,t,w), Vte[0,y], so that not generally a concave function of This fact has the implication that the equilibrium path under a the initial conditions, since the y. [E.g, this continuity is argmax in (5.4) crucial to the derivation of (5.2).] endowment would cause iod. A number from (5.3) and Proposition3: may be desirable for some node, and yet a slight de- by the following: (5.4), as exemplified Assume e<l/3. 1-e Then w(y)E--__ In the inequality is if -ry/<h . L 1-e z < —ln(e)/r + ry (1— e)y/e Some + MPE extension of a w(y), for h sufficiently reversed, then xhe low price obtains at every date until the smaller firm has exhausted, under any Proof: it and of interesting results along these lines can be derived with remarkable ease attains the high price continuously If Thus, w strategy to switch to producing for a protracted per- its r small. not be continuous in need not be a continuous function of the smaller firm to wait to reach the boundary of N" from crease in may t. MPE extension of a, when h is small enough. Using subscripts to denote partial derivatives, (5.3) implies: r- 1 n w) = (y,t,w) = (y,t t 2 r" n l -rt ^T -w r e rt 6 ).[(^)e- / + e- y/ r 2 rt e £ r t -w )-[(^) e- / -e- y/ +e e -e r e rt ( t ]; while, ( ]. tt Thus: -1 [r 2 n +r~ n t e<l/3. So n, VI. ] = r e ( t_w maximum rt/e 2 1 )-[(^)(- -= ^)e~ <0 3 n,,>0, and n,,<0 attains an interior for tt 3 II -2e~ ry/e ] > 0, 0<t<y, Vw>0, since >0. In particular, the function n(y,t,w) never with respect to t on the interval any y>0 there exists w(y) such that n(y,0,w)>n(y,y,w) [0,y]. iff Thus, since IT. >0, y<w. Conclusion In the model of dynamic duopoly studied here firms regulate an "on-off production ' ' 33 non—replenishable process, depleting their respective supplies of an a constant rate whenever their process If only one of "on". When them produces an any moment the moment, the remaining capacity In investigated. hi to both are producing the price price is Two paths of production which are mutual best responses. In any at notions of equilibrium are commit firms low. is Each of them knows, high. produce of the other. open—loop Nash Equilibrium the factor of production at at the initial Markov date to time Perfect Equilibrium, the firms specify state—contingent plans of actions which are mutual best responses in every The capacity contingency. of the firms, in these alternative strategic settings, to achieve coordination of their actions via non-cooperative play has been systematically investigated. By assuming firms can change their actions only on exogenously given, equally spaced moments in time, we convert the problem to a discrete time game which, due to the dichotomous production structure, can be explicitly solved. The possibility for coordin- ation to be attained via non-cooperative play, in either strategic setting, characterized (in Lemmas 2 and 3) by two relatively simple integer characterizations allow us to derive the ability to precommit undermines the strategies are The more programming problems. These main comparative results of the paper: possibility of coordination; effective at coordinating play is when that the and that state-contingent firms are symmetrically placed. analysis emphasizes a difference between discrete and continuous time dynamic games recently stressed by Simon and Stinchcombe (1989). The extensive form analyzed the discrete game, though restricted to Markov strategies, reveals a rich structure largely hidden from view in a continuous time treatment. The idea of "precedence" ing their play contingent on "who did imal coordination in this environment. what Of last period" — is — firms' mak- fundamental to achieving max- course, this finding is related to the "on—off process of production which has been assumed here, but the point would seem to be of general import. The results in section V in more on the limit behavior of our Markov Equilibria suggest the possibility of profitably using this discrete approach in other settings where continuous time formulations have been reflexively adopted. ' 34 Appendix Proof of Theorem 1: k The Define S ={(a,b)eS|a+b<k}, kel. closed to deviations E =f Now, |{^i|gK v k>l}; I se(S from any k+1 \S k ), ^E. S. clearly, one of the alternatives are nested, cover } , -, For fixed k>l, |gi | k Therefore E, ugE. profile {S sets cS for and hence ,, k>l, • any ctgE, ,, IS holds at (3. 2)(a)-{c) s. Moreover, the profile se(S k+1 <7*eE is WcS and aGS, \S )n\V, set , o*(s)=o(s). (2) At each seS k+1 \(S k 2, UW), ...: Now with the Bellman Inequalities there. 1 implies that a*eE, such k and a on S\S s, . w |W <7eEi At each node (1) o* con- * implies aeE, l , s K , Thus k>l. nW * * Lemma for every assign a value to * sistent and the following inductive procedure constructing For successive values of k=l, well defined. k w K for the continuation payoffs in (3.2)(a)—(c) do not depend on values taken by Therefore, given and are S, , at each step , Therefore u*e£ k. q . Proof of Corollary 2: Let seS be strictly positive, with n=n(s), k=k(s), i=i(s) and j=j(s). By the definition of the alternating profile we have that: 2 2k -2 n+1 2k-2 (i) V.(a)(s) = ^.[(l-^ -(l-8 it a-.(8)=0; and, )/(l-« ) + 8 ) /(1-5)}, V.(a)(s) (ii) = 8P(l-6 2k )/(l-P), a (s)=0. if i Now, by Definition a, RHS se^(e) of if: (i) (1) cr.(s exceeds the RHS if, for every of if Given n>l. for if n<n' . a strictly positive successor of s under any n>0, k>l. Therefore, when a-(s)=0 then Kk<k(s) and n=n(s); and, a.(s)=0 then se3{e) if: (1) (2) [RHS (i)]<#3(^j), for l<k<k(s). Condition (2) [RHS (ii)]<#?(^) for l<k<k(s) is for and redundant in each (4.4), condition (1) is equivalent to: (J)(n(s),k)>e, l<k<k(s), if cr.(s)=0: and, <J>(n(s)+l,k)>e, l<k<k(s), only for (ii), n=n(s)-l; and, (2) [RHS (ii)]<£,5(t4) case s' }=0, and V.(o-)(s')<^3) when 3j(-s'j=0. Notice that [RHS (i)]<$(yEj) When l<k<k(s)-l. se .25(e) when V.(a-)(s')<£(jEj) the 4.1, if cr.(s)=0. But if (4.3) holds then (j)(n,k)>e, V k>l, if and This completes the proof. Proof of Proposition 1: Suppose e<l/2. Let R=(R.,Rr>) be some vector of initial stocks such that 35 R A +R B =R. Given any h>0, Corollary |R (i) A -RB (i) imply there exists an the definition of a, < o -ln(e)/r, (ii) MPE a coinciding with a on we conclude that and R=(R A ,R B ), let R A +R B =R~, with s=s (h,R). It o follows from and define |R that C4. 4) 5(h) in (A.l) as for h 2k ( s H h-+0, MPE e~ 2 together £^(f), from s |ex, satisfying Min{R ir-, • a=a on Fix A ,R B }Ey. (s)=0 if 3){t). h>0 and only if: e-\l+5(h)]-6{h) > s ^[l + ^h)]-l s, n(-), and k(-), to see that if: 2ry > (2e-l)/(2e~ rx -l). as: (iv) will a -R b using the definitions of sufficiently small (iii) it Since -2>(e). sG^(e) when -^h)^ As long and for }{t>) (A.l) Take the limit s(h,R)e .25(c) 1 which vanishes ,a)}cM(a). {s.^s Suppose e>l/2, and continue to denote by a an Let Theorems gaurantees :hat (4.3) holds. profile if: and n(s (h))<n'(h)— 1 (neglecting rounding error, assures that small h), while condition | (h,R)e 3(i), VR">0, s < ln(^)/r. h (ii) Condition 2 implies be possible to select the analysis leading to (4.1) implies h-n(s (h,R)) -» R< -ln(2e-l)/r, x>0 and y>0 we know —ln(2e— l)/r, such that x+2y=R~, and Q*(s )=1 that as h-»0. Thus (iii) n(s)>n(s)+2k(s)sR~. iff: (iv) is satisfied Now, by holds. whenever But R is (4.1) below the threshold at which collusion becomes desirable. For such R" it is always possible to divide the total endowment such that (iii) holds. Thus, for h is small enough we can find R=(R A ,R B ), R A +R B =R~, such that {s (s t Q (h,R),a)}cM(a) whenever Q*(s (h,R))=l. Q Proof of Theorem 3: We have to show that the profile t* is well defined, and that neither firm will deviate from it at any node seS\ 3){i). This amounts to showing two things: (1) that there exists a unique sequence of integers {m?} such that on S\J?(e) the strategy "wait 3)(e) from s if you can get there within mw \ periods" is a best response for i(s) for to the strategy "always produce" for j(s); and, (2) the strategy "always produce" is a best response for j(s) to the strategy given in (1) for i(s). The proof of Corollary 3 establishes (1) in the case e<8/(l+6), when an explicit formula for {mf } can be found. The nested induction employed there may be extended to the general case e€(0,l) as follows. Consider S(k)={seS\i5(e)|k(s)<k}, k>l. Given any integers {m,, ..., m, }, the . 36 Theorem, and thus each firm's value function, is uniquely determined node in S(k). Therefore, the following induction uniquely defines a sequence of integers {mf}, k>l, and so, by the definition in the Theorem, a profile o*: profile defined in the at every m* (A. 2) m*k e = Max {m(s)} s.t. m k(s)=k, and £ ( s ) +1 if 2k -[—o 0e+8V.,Jo*)(s-(l,l)), > ] 1[s) i-r seS mt =0 where n'-l; and for k>2, the inequality always This definition fails. is meaningful because at each k>2 the profile a* and values V-(cr*)(-), i= A, B, are already defined on S(k— l)U^(e); and, k(s)=k implies s—(l,l)eS(k— l)U^(e). ment that waiting ever i(s) m(s)>mf if integers mf are defined by the require- to reach an alternating path is dominated for i(s) by producing whento show is that waiting for one period at s is optimal for What we have . and only The m(s)<mf Vs6S\^(e). For if , this we use the following nested induction. i(s) will not deviate from a* on S(k— 1), k>2; and, (b) the inequality where k(s)=k and m(s)=m>l. Then that inequality also fails at s', where k(s')=k and m(s')=m+l. By (b), waiting to reach -2>(e) from s' is worse for i(s) than waiting for one period, producing for one period, and then following o* While, by (a), producing for one period and then following a* is at least as good as producing for one period, waiting for one period, and then following a*. Thus, producing and then following o* dominates waiting to reach i?(e) from s'. So either, for all seS(k) with k(s)=k, producing for one period and then following a* is better than waiting to reach @(e), whence rn^EO; or, for some m?>0, the inequality in (A. 2) holds iff m(s)<mf For Suppose: (a) in (A. 2) fails at s, . . m follows a* on S(k) if it does so on S(k— 1), i(s) follows a* on S(l), then imply that the "optimal waiting strategy" defined in (A. 2) is firm i's best response on all of S. fixed k, this induction on k>2. The induction on We now k, implies and the i(s) fact that from constant production on a*. We do so by induction on k. Clearly j(s) will not deviate from o* at any node where i(s) is waiting, so we need only consider s tor which m(s)>mw y We argued in the text that j(s) follows a* on S(l). Assume show that the larger firm will not deviate S(k), k>l, given that the smaller firm follows this is so on S(k— 1), k>2. Take any seS(k) such that k(s)=k and m(s)>mf. Let the node reached and let if j(s) follows a*, let m=m(s'). There are three " s cases: (1) be the node reached m>mf_, ; (2) if j(s) deviates from m<m?;_, and ; s' (3) be a*, m=m?_, be playing "1" at the successor to node s, whether j(s) follows a* or not. But then j(s) cannot gain by deviating. For, given discounting, firm j does better to follow a* at s and then deviate at s', than to deviate at s and then follow o* at s", since both paths terminate after two periods at the same node. Yet, by the induction hypothesis, the payoff from the former course is a lower bound on the value of following a* at s, and the payoff to the latter course is the best that can happen if firm j deviates at s. Case (2): In this case i(s) will be waiting to reach @{e) from the successor to s, whether j(s) follows o* or not. So j(s) does better under a* from s" than it would do under the pure alternating profile. Moreover, since s£3(e), it follows that: Case (1): By the definitions, m(s' V j(s) (0(s"')>V So, were j(s) to play "0" at s, its ')=m+l. So in this case i(s) will (a)(s"')>«±3j(5) value there would exceed 6(3(1— e)/(l— S). But then the 37 reasoning of Theorem 2 and Corollary 1 implies firm j will not wait to "take its turn" under such circumstances. Hence, j follows a* at s. Case (3): In this case if j(s) follows a* it goes to the node s' where i(s) begins waiting to reach -S>(c), while by deviatirg it goes to s" where i(s) is not waiting. know from Case (2) that if i(s) were to begin waiting at s" then it would not pay for firm j to deviate at s. Thus, we can dispose of this case if we can show that firm j is worse off at s" when firm i produces there than when firm i begins waiting there. Let t' denote the first date at which firm i(s) waits along the path {s,(s' ',o*)}, 0<t'<k—1. We t'=0, we are in Case If A (2). m(s proof critical fact for this (s",a*)) < is the following: m(s"), l<t'<k-l. t/ The waiting time does not rise as one to reach 3){e) moves along the path {s (s' ',&*)} t To of n for which 2, m(s)=Min{meI|Vk, l<k<k(s), <}>(n(sV-m+l,k)>e}. 2 in the text implies that, as k(s) falls, the critical value by Corollary see this notice that, The discussion following Corollary ,(s",cr*))<k(s")=k-l, by the k(s But n(s,,(s",<7*))=n(s"), and 4>(n,k)>e, l<k<k(s), does not decline. Thus, m(s ,(s",a*))<m(s")- definitions. t t With this observation in hand, we complete the proof by showing that firm j is worse off under a* at s" if firm i produces there, than if it does not. Given that firm is about to begin waiting for .0(e), it is obviously better for firm j if the wait is longer. Define V*(t)=V./\(o )(s"), given that t'=t;let s=s.(s",a), and m=m(s,). We need J|c to show that V*(0)>V*(t), l<t<k— 1. Frcm the rt , = V*(t) where s,E3(e) favors j, is /?e(i=| the first ,=m + fitffi node reached + -V S is which firm .=m. assumed. Biu m, > pe{(^ have: t 0<t<k-l, ), implies § t _, H + p{s\^l 1 =s. —(1,1). Therefore: 1+m t t+1 ({^ )) 1+t+m + Now because s,6^(e), and {V & j is )-^V (&)(s j(s) waiting at t s\, we -(l,l))} (5^)(s j(s) t t J 0<t<k-l. have: V.(a)(s.)>0t+6V.(cr)(s.—(1,1))- i v t j j t Substituting this inequality into the expression above yields the following: V*(t)-V*(t+1) So V*(0)>V*(t), l<t<k-l. Proof of Theorem (i) We n, >ri, We p5 {l-e)(l-6 conclude that j 1+m l ) > 0, will not deviate 0<t<k-l. from a* in Case (3). 4: begin by noting that N" _-, k>l. l > is closed to deviation Thus we must show that i no greater than the value implied is 1+m t )} we Since a longer wait by waits. j f+1 H^j- (a)(s j(s) t t v*(t)-v*(t+i) at ) ,<m,, we know that V*(t+1) and since m, above when m, ) o* 1+t+m, t r , , definition of the profile from a, in view of the fact that neither firm can gain by deviating at any node i 1 1 1 38 Neither will deviate when seN". from waiting will not deviate The o-(s)=(l,l). Consider {m k }cl, when will not deviate i(s) }, nodes seN for a does not exceed the successor node under some {m, at a- \—c, / / 2 implies that, for is when n(s)=0, and neither firm will deviate / j(s) since its payoff at \, So what must be shown <5/?(t—r). structure of the argument is similar to that used for Theorem 3. firm i(s) We assert the existence of some sequence of integers first yet to be determined, nich that "a., x(s)=0 iff n A nested induction of the sort used in the proof of behavior of firm must be i —c,<n(s) Theorem for to produce, then i Let N" E{seN|k(s)<k}, kel. establishes that the optimal 3 k— k of the "optimal waiting" form: optimal it is iff <n ^ s ^- ii c + k( s )~ k(s) k(s) ™k(s)" h,/ „\-Cw >><n(s)," seN. a best response to "a-/ \(s)=l h, Theorem of which n(s)=n, is with The proof alone produces. it some seN \N I.e., if for it is also optimal for to produce i k— k s'eN \N for which n(s')>n(s). The argument turns on the fact that, due to discounting, it could never pay i to wait in order to reach a node at which i will produce, when it could produce and reach a node from which it might choose to wait. So we need to at show that the length of true for k=l, since i m^n'—c.>0. (4.7)) gives would wait a Now — \N N waiting by , total of n' periods to play its and consider takes i to a it at least is Suppose, inductively, that {in,, best play on i's optimal wait on N" i's m, . This s/ for obviously is one unit, so (consulting a ,}cl, k>2, are such that \N optimal behavior at seN his node ..., c, which n(s/)=n\ —c, , which c after causes the following pattern of play to ensue: i produces for one period, j produces for periods, i produces for one period, j produces for c, know c periods, etc. lc_i 2 We that v ;(^)( s be the sum ic ) =<^v k-l' for V k-1 given in of firms' capacities at (i) s/. k k— be optimal at seN" \N" ^Ww,K i (ii) s." if The T k E2k + n k-V=n/ + k +( c 1 +---+ c k--l) Let Therefore firm is k )/(i-6) i's payoff at - sv _ k r — on well defined and only N~ , > Pe+6-V.(~a)(s-il,l)) 1 equality in , ^ = Max {/3e(« + l ~° (ii) l< and W =c =0. s/ is given by: we conclude that waiting if: i \ where m(s)sn(s)—h, +c, at 4J )- w k e VjC&k^o = p(i-s Assuming, inductively, that V(a)(s) will ( t m(s)+c, ,+...+Ci k_1 <k The inequality in . ^-W,K 6 (ii) states: "i }, l prefers to wait uses the induction hypothesis to infer that, conditional on i — producing at s, i's optimal Kjhavior on N" takes the form: continue to produce until, with k—t (possibly zero) units left, it pays to wait for the node s/_., and associated payoff W, __. ." We need to show that the inequality in (ii) holds when m(s)=c, , so as to c 39 conclude that waits at serT i Lemma In the proof of (iii) Substituting U 1 \N and only if we noted 3 k + |=| /?[! into (iii) k Wk > ] ^- + k and rearranging, we see that (ii) k k for P{^)>V k > P(^fr) .hat > fll n(s)<h,+m,, if k>l. So by , (i): k>2. ], (ii) hold will if, for l<t<k: k k l some m,el. k l -[1-6+6-6 > e(l-6 ) + 6 .[i-6+6e-8 ]. Recalling the definition of T, and using a bit of algebra we have that (ii) holds if: 6 (iv) ] Ck (v) .(i-^^).[l-(5 (5 c Now by By Lemma may 8 (4.7)(ii), T ^f)] k+Ck >[e-6 t "].(l-^ c k_ ] ( e-[l-t W k =/?(l-£2t )/(l-<$2 )+£2t W k_ , and t W fc > [RHS ] k>l; ) if so, 6 k k we ,+...+c, _.>t+l, then c, of (v)], and _t <j9(l-^( l<t<k-l. )) {l-6+6e)>e, k>2. holds. (ii) 2 2 (4.8), 6 /(l+6+6 )<e<6/(l+6), l<m<t. Then by m =l, k_ So, "t )<-r. -j_T f > (v)] Vk_1 <(^) and ]>e c Obviously, [LHS of conclude that c .( -[l-6(l-6)V k>l. 3, c, >1, Suppose k Ck - 1+ '" +Ck -t 2 ^)/(l-<5 ). So and holds (ii) c k if, Also, <2. for l<t<k: p k 5 But p ^ [j^ ~ 0~V(l-£)W k _ ]>e, c l<m(s)<c that k The . integer best play on N* i's It m k is remains to show that firm which n(s)>hw N+m, closely parallels that n, / and employed three cases, distinguished a-, for {m,} cr(s)=(l,l). by the behavior K s , might wait. a of If it holds}— k >0. will i (ii) We holds conclude seF for sketch the argument here, as the reasoning Theorem at the nodes s' j the fact that <r(s)=(l,l) implies V-(<r)(s) is If a-(s' ')=() 3. and <x(s')=(l,l) then will produce, then it j^^- inductively defined in this manner. in this part of the proof of if j does or does not follow a- at s. that involves waiting to reach a node where definition of h\ We if Wk </?(^), firm cannot gain by deviating from a at j(s) reached reach a node from which which holds Thus we have proven that m k EMax{m(s)| (ii) given by: given by is }>e, =l, Vk>l. Since k wait at least one period to reach an alternating path. for 2 t [j^ t t>8 /(l+6), then if 6 if 2 (4.8) implies that 2k k which holds when j it Again, there are s' ', respectively will not deviate, since could produce and will not deviate, since too large for j by the to forebear at j s", where in order to reach i forebears. There remains the case in which a-(s')—0, but a(s' ')=(1,1), so by deviating j avoids reaching a node where i would have begun to wait. Argument analogous to that used for Theorem 3 disposes of this case as well. The point, as in the proof of Theorem 3, is that j never gains by delaying i's wait. Bear in mind that i is waiting to reach the unique admissible path in rT at some node s/ 40 for —c, which n(s/)=n, Vi, )/(l—6)+6 were to begin waiting i —c, +m, k(s)=k and n(s)=n, P(l—8 If . then the payoff to s— (1,1), then the implied payoff latter payoff from a at is less s, it any t>l. And yet, at j would rather follow a it (4.7), <T^ V, at s will not deviate j MPE a restricted on however, If, n~ 1 which for i by )/(l-6)+^ mV than at s' than deviate and reach s' s' ' is: i were to begin at k_1 _•</?( t—r)- Thus, begin waiting at i s as a consequence of this waiting j ^e+^(l-^ is: s strictly prefer that beginning to wait there. So is to from a node s/ ai defined in (4.7). than the former since, by would conclude that a V, _, for , for if deviates j '—t ' The . - (1,1), for with i from a under any contingency, and we may N~. * We (ii) now show that there is sequence {m, So we will {seS|n(s)<h, that some ueS such } we know that that / \+m, a=a on / N". JcM*. By {s,(s,o)}cM(a), for have proven the Theorem if aeEiTT, Theorem definition of the profile all we can show Since seN for implies a and the which n(s)<n, that, for every 1 / \+mw seS\N", there \. some is * which generates a collusive path from ctgE a which has the property that it reached, after which a. Note that it follows We do so by constructing an extension of generates an alternating path from (s,ar)}n^0,VseS\K {s s. since until s N~ is For seS\N~, (a,0)eN~ and (0,b)eN~, Va,b>0. t let s that at s. be the first n(s)<h,/-N When node reached from If s Then along {s,(s,&-)} such that seN". k(s)>0 and a(s)=a(s) we will say that a is it must be a "compatible" with at s, then redefine a so that firm A odd, instead of even. With this redefinition in a and a are not compatible produces when the sum of stocks is mind we can state: VseS\K there exists an alternating profile a which is compatible with the sense that they both have the same firm producing at the first node reached from under a that lies in N~. With a ^(s)E{s so defined, let (s,a)}llN" and let a[s] a in s be a t strategy profile defined on y£(s) so that- Vs'£^(s)\N~. In words, for given seS\N", and the compatible alternating path from N", and follows a on N\ that <r[s] It is Clearly it a[s] s, is s. c-[s](s')=a(s'), the profile defined on the union of N" it generates alternation outside of such that ^(s) obvious that generates a collusive path from showed that j[s](s')=a(s'), Vs'cN", and We is closed to deviation from now show that cr[s]eE cr[s], and WcV only be conerned with points s'6 ^(s)\N\ The proof of Theorem pays to deviate from alternation only when the payoff at the node reached we need without deviating exceeds 0{j^)- Neither firm's payoff exceeds this bound at s. At any 2 6 41 predecessor of and successor of s along the path {s.(s,a)}, each firm's payoff consists of s receiving /? on alternate dates until the node s is reached, and hence may be written: 2 2 2t 2t 2t 2t )/(l-<5 )+ <5 .( cr)(i)<^( l— )/(l— <? )+ <5 /?(^^)</?(^|), in view of the fact that /?( l-<? V So neither firm e<8/(l+6). * cr[s]eSi w By Theorem qy _ 1, 3 3, anywhere on ^(s). Hence <r[s] _ * aeS such that Derivation of Equation (5.2): We seek w*(y)Elim{h-mw in from will deviate /, ct[s]e<7 From t-,\\}. on <s6(s), and so seM*. the implicit definition of {m£ } MPE equation (A. 2), and the properties of the that m(s)=mf / \ if and only we conclude (A.3) m ^<5 ( s ) 1-62k(s)l +1 > . a* established in the proof of the following condition holds: given Theorem + ^+^-tuM.MTC*? ihxW^ Kt<k } 1-6' 1-6 >• if n >/M *(s)+2. 12" 1-6 L i(s) does better tc wait for 3(e) from s than to produce (and to continue producing until waiting becomes optimal), but that this would not be so if the wait were one period longer. In view of the fact that h-k(s (h,R))-*y, as h-»0, we have that (A.3) states that firm h-m(s)-»w*(y) as h-k(s)-»y in (A.3). in other words, (5.2) is obtained from (A.3) by w*(y) with lim{h-m(s)}, and y with lim{h-k(s)}. The key to the derivation is the observation that h-t->0 as h-»0. Assuming this is true for the moment, the limit of the RES of the first inequality in (A. 3) can be evaluated using calculus. Introduce the infinitesmais dysh-t and dw»h-[m(s)—m(s—t (1,1))] for hxO; let h-m(s)-*w* and h-k(s)-*y as h-»0. In the text we noted that, by taking limits in (4.4) letting h->0 in the latter, identifying • —ln[pd-(e—^)e the waiting time defined in (4.5) satisfies: h-m(s)-»H ry ]Ew(x,y), as h-k(s)-y and h-n(s)-tx. Therefore, simple differentiation yields the following: (i) dw s w(x,y)-w(x,y-dy) s dv. ' r . -gjxj) dy 2(2e-l)e -l+(2e-l)e 2ry where u(y)E[l-(2e-l)e ]/[l+(2£-l)e and taking hsO, (A.3) becomes: ,.., («) e -rw* l-^~ :' ry Now, using the ]. 2 ry , -[-21— ]~fdy J+ r 1 2ry- • e -rdy *-e 5r7 dy = [l-u(y)].dy, definition of 6(h) -r(w*-dw) l-e~ K '•[— r and /3(h) 2r ( y " dy -^ ) J. Note that (ii) has the intuitive interpretation that firm i, with capacity y, is indifferent between waiting w* units of time to share the market, or producing for a short interval and waiting thereafter. Letting dy->0 in (ii), and using (i) yields the following: (iii) It is easy to see that (iii) iw 2ry-2iyi = e-e--rw .g.(i-e-^). u (y)+e-^]. implies (5.2). 42 Notice that u(0)=(l-e)/e<l; u'(y)<0; and u(y)=0, for yE-ln(2e-l)/2r. — So there r ^. For y>y the value of w*(y) a unique ye(0,y) satisfying: [2e—u(y)]/[2—u(y)] = e implied by (iii) is negative, so the first inequality in (A. 3) fails when h is sufficiently small, and hence w*(y)=0. Finally, we must verify the presumption that h- 1-»0 as h-»0, upon which the above argument is based. This presumption is valid if, for hsO and firm indifferent between producing and waiting, producing for any positive interval leaves it at a point where waiting is preferred. 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