Endogenizing the cost parameter in Cournot oligopoly^{1}^{1}1This research has been cofinanced by the European Union (European Social Fund â ESF) and Greek national funds through the Operational Program ”Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)  Research Funding Program ”ARISTEIA II: Optimization of Stochastic Systems Under Partial Information and Application, Investing in knowledge society through the European Social Fund”.
Stefanos Leonardos gratefully acknowledges support by a scholarship of the Alexander S. Onassis Public Benefit Foundation.
Abstract
We study a single product market with affine inverse demand function in which two producers/retailers, producing under capacity constraints, may order additional quantities from a single profit maximizing supplier. We express this chain as a two stage game and show that it has a unique subgame perfect Nash equilibrium, which we explicitly calculate. If the supplier faces uncertainty about the demand intercept and whenever a Bayesian Nash equilibrium exists, the supplier’s profit margin at equilibrium is a fixed point of a translation of the MRL function of his belief under the assumption that his belief is nonatomic and the retailers are identical. If the supplier’s belief is of Decreasing Mean Residual Lifetime (DMRL), the game has a unique subgame perfect Bayesian Nash equilibrium. Various properties of the equilibrium solution are established, inefficiencies at equilibrium generated by the lack of information of the supplier are investigated, and examples are provided for various interesting cases. Finally, the main results are generalized for more than two identical producers/retailers.
Keywords: Cournot Nash, Nash Equilibrium, Duopoly, Incomplete Information, Existence, Uniqueness
Mathematics Subject Classification (2000): 91A10, 91A40
1 Introduction
The main objective of this paper is to examine equilibria in a classic Cournot market with affine inverse demand function, when the competing firms’ cost is affected exogenously by a supplier, who may even be uncertain about the demand parameter of the market. As Marx and Schaffer (2015) point out, although Cournot’s model has become the workhorse model of oligopoly theory, there has been very little discussion about the origin of the competing firms costs in the quite vast Cournot literature. As they underline, if these costs represent purchases from a third party, then important strategic considerations come into play, which are not addressed by the prevailing approach. Marx and Schaffer question the robustness of the results obtained thus far in the Cournot oligopoly theory due to this negligence.
The key in studying the effects of an exogenous source of supply for Cournot oligopolists is the relation between demand and the various costs. If the demand is high enough, the competing firms may have incentive to place orders to an external supplier, depending of course on the price he asks. In the classic original Cournot model, the crucial parameter concerning demand is the yintercept of the affine inverse demand function, which implies its important role in studying Cournot markets with exogenous supply. Next, in such a study, various questions have to be addressed: Do the firms have production capacities of their own? If yes, then these capacities have to be assumed bounded (as they are in reality), so that need may arise for additional quantities. Does the supplier know the actual demand the oligopolists face? If no, then he may ask for a price that the competing firms will not accept, even if it is to his advantage not to do so. The latter question implies an incomplete information game setup.
It is the aim of this paper to address the questions raised above and contribute to extending the classic Cournot theory to oligopolists that may purchase additional quantities from an external supplier. By viewing the interaction of the competing Cournot oligopolists with their supplier as a two stage game, the cost parameter of the classic Cournot model is endogenized and answers can be worked out.
As we shall see, obtaining the equilibrium strategies for the Cournot firms in such a market may be very tedious, but nevertheless manageable after one formulates the setup. However, the case for the supplier, who is also a player in this game, is somehow more demanding. In the duopoly case, if the supplier knows the true value of the demand parameter, a unique subgame perfect equilibrium always exists, although its description is quite complicated (see Propositions A.1 and A.2 in the Appendix). In all other cases (three or more oligopolists and/or incomplete information), the study of equilibria is possible only if we assume that the competing firms are identical. So, assuming that the oligopolists are identical, we obtain concrete results concerning the price the supplier will ask in both the complete and the incomplete information case under equilibrium. In the latter, when the supplier does not know the demand parameter, we give necessary and sufficient conditions for the existence of a subgame perfect equilibrium via a fixed point equation involving a translation of the Mean Residual Lifetime (MRL) of his belief, provided the measure this belief induces on the demand parameter space is nonatomic. If they exist, the evaluation of all subgame perfect equilibria is possible by using this equation, i.e. our approach is constructive. The MRL function is well known among actuarials, reliability engineers, and applied probabilists but, to our knowledge, has never arisen before in a Cournot context.
In detail, inspired by the classic Cournot oligopoly where producers/retailers compete over quantity, we study the market of a homogenous good differing from the classic model in the following aspects. Each producer/retailer may produce only a limited quantity of the good up to a specified and commonly known production capacity. If needed, the retailers may refer to a single supplier to order additional quantities. The supplier may produce unlimited quantities but at a higher cost than the retailers, making it best for the retailers to exhaust their production capacities before placing additional orders. The market clears at a price that is determined by an affine inverse demand function. The demand parameter or demand intercept is considered to be a random variable with a commonly known nonatomic distribution having a finite expectation.
Depending on the time of the demand realization, two variations of this market structure are examined, which correspond to a complete and an incomplete information twostage game played as follows. In the first stage the supplier fixes a price by deciding his profit margin and then in the second stage, the retailers, knowing this price and the true value of the demand parameter, decide their production quantity (up to capacity) and also place their orders (if any). The decisions of the producers/retailers are simultaneous. If the supplier knows the true value of the demand parameter before making his decision as well, this is an ordinary two stage game (and the demand random variable is just a formality). On the other hand, if the random demand parameter is realized after the supplier has set his price, then this is a two stage game of incomplete information where the demand distribution is the belief of the supplier about this variable. So, in both variations the demand uncertainty is resolved prior to the decision of the retailers, who always know the true value of the demand intercept. In this respect, our setup differs from the usual incomplete information models of Cournot markets, where demand uncertainty involves the producers (e.g. see Einy et al. (2010) and references given therein).
The limited production capacities of the producers/retailers may equivalently be viewed as inventory quantities that may be drawn at a fixed cost per unit. The latter setting corresponds to the operation of a retailer’s cooperative (hence the characterization of the ‘producers’ as retailers’ also). In case the production capacities are 0, this is a classic Cournot model with the cost input determined exogenously. Marx and Schaffer (2015) point out the scarceness of research along this line and the need of endogenizing the cost parameter in the Cournot model. To do so, they study the game that arises when competing Cournot firms purchase their inputs from a common supplier based on contracts, with the competing firms having the bargaining power.
We study the equilibrium behavior of the twostage game by first determining the unique equilibrium solution of the second stage game, which concerns the quantities that the producers/retailers will produce and the additional quantities they will order from the supplier. This stage is the same in both the complete and the incomplete information case and is solved completely. Then, we examine the first stage game, which concerns the price the supplier will ask for the product. In the case of the complete information duopoly (two producers/retailerssupplier knows the demand parameter), a unique subgameperfect Nash equilibrium always exists, which we give explicitly in the Appendix. Its description is quite complicated as it is very sensitive to the relationship of all parameters of the problem. Otherwise, we proceed by considering identical producers/retailers. Under this assumption, the complete information case simplifies significantly. However, our focus is mainly on the incomplete information case. For that case, we show that if a subgame perfect BayesianNash equilibrium exists, then it will necessarily be a fixed point of a translation of the Mean Residual Lifetime (MRL) function of the supplier’s belief. If the support of the supplier’s belief is bounded, then such an equilibrium exists. If the MRL function is decreasing (as in most “well behaved” distributions), then a subgame perfect equilibrium always exists and is unique, irrespective of the supplier’s belief support. In addition, we investigate the behavior of this subgame perfect equilibrium in various interesting cases.
A well known byproduct of incomplete information is the appearance of inefficiencies in the sense that transactions that would have been beneficial for all market participants are excluded at equilibrium under incomplete information, while this wouldn’t have been the case under complete information. In the present paper, an upper bound on a measure of inefficiency over all DMRL distributions is derived, it is shown that it is tight, and the implications of the structure of the supplier’s belief on the efficiency of the incomplete information equilibrium are investigated.
Under incomplete information, the DMRL property is sufficient for the existence of a unique optimal strategy for the supplier, but not necessary. Possible relaxations of this condition are examined, basically in terms of the increasing generalized failure rate (IGFR) condition, see Lariviere (2006). As far as necessary conditions are concerned, we don’t have a characterization of the distribution when an equilibrium exists, although we have a characterization of the equilibrium itself through the MRL function. Finally, as far as the inverse demand function is concerned, its affinity assumption may not be dropped if one still wants to express the supplier’s payoff in terms of the MRL function. Hence, a generalization in models with nonaffine inverse demand function is open.
As a dynamic twostage model, the current work combines elements from various active research areas. Cournot competition with an external supplier having uncertainty about the demand intercept may be viewed as an application of game theoretic tools and concepts in supply chain management. However, we are interested in the equilibrium behavior of the incomplete information two stage game rather than to building coordination mechanisms. The incomplete information of the supplier relates our work to incomplete information Cournot models, although the lack of information refers to the producers in most papers in this area. Having capacity constraints for the producers/consumers, as it is necessary for our model, has also attracted attention in the Cournot literature, see eg. Bischi et al. (2009). Finally, the application of the concept of the mean residual lifetime, which originates from reliability theory, to a purely economic setting is another research area to which our paper is related.
Cachon and Netessine (2004) survey the applications of game theoretic concepts in supply chain analysis in detail and provide a thorough literature review up to 2004. They state that the concept that is mostly used is that of static noncooperative games and highlight the need for a game theoretic analysis of more dynamic settings. At the time of their survey, they report of only two papers that apply the solution concept of the subgame perfect equilibrium (in settings quite different to ours). Bernstein and Federgruen (2005) investigate the equilibrium behavior of a decentralized supply chain with competing retailers under demand uncertainty. Their model accounts for demand uncertainty from the retailers’ point of view also and focuses mainly on the design of contracts that will coordinate the supply chain. Based on whether the uncertainty of the demand is resolved prior to or after the retailers’ decisions, they identify two main streams of literature. For the first stream, in which the present paper may be placed, they refer to Vives (2001) for an extensive survey.
Einy et al. (2010) examine Cournot competition under incomplete producers’ information about the demand and production costs. They provide examples of such games without a Bayesian Cournot equilibrium in pure strategies, they discuss the implication of not allowing negative prices and provide additional sufficient conditions that will guarantee existence and uniqueness of equilibrium. Richter (2013) discusses Cournot competition under incomplete producers’ information about their production capacities and proves the existence of equilibrium under the assumption of stochastic independence of the unknown capacities. He also discusses simplifications of the inverse demand function that result to symmetric equilibria and implications of information sharing among the producers.
Capacity constrained duopolies are mostly studied in the view of price rather than quantity competition. Osborne and Pitchik (1986) and the references therein are among the classics in this field. Equally common is the study of models where the capacity constraints are viewed as inventories kept by the retailers at a lower cost. Papers in this direction focus mainly on determining optimal policies in building the inventory over more than one periods. Hartwig et al. (2015) and the references therein are indicative of this field of research.
The condition of decreasing mean residual lifetime that arises in our treatment, is closely related to the more general concept of logconcave probability. In an inspiring survey, Bagnoli and Bergstrom (2005) examine a series of theorems relating logconcavity and/or logconvexity of probability density functions, distribution functions, reliability functions, and their integrals and point out numerous applications of logconcave probability that arise, among other areas, in mechanism design, monopoly theory and in the analysis of auctions. Lagerlöf (2006) examines a Cournot duopoly where the producers do not know the demand intercept but have a common belief about it. Under the condition that the distribution of the demand intercept has monotone hazard rate and an additional mild technical assumption, he proves uniqueness of equilibrium. Guess and Proschan (1988) survey earlier advances about the mean residual lifetime function, its properties and its applications. Among many other fields, they mention a single economic application in a setting about optimal inventory policies for perishable items with variable supply which is however rather distinct from ours.
1.1 Outline
The rest of the paper is structured as follows. In Section 2 we build up the formal setting for a duopoly. In Section 3 we present some preliminary results and treat the complete information case. In Section 4 we analyze the model of incomplete information, state our main results and highlight the special case with no production capacities for the retailers. In Section 5 we investigate inefficiencies caused by the lack of information on the true value of the demand parameter by the supplier. The use of results exhibited in the previous sections is highlighted by various examples and calculations of the subgame perfect equilibrium in Section 6. Finally, in Section 7 we generalize our results to the case of  identical retailers. The proofs of Section 3 and Section 7 proceed by an extensive case discrimination and are presented in Appendix A.
Throughout the rest of this paper we drop the double name “producers/retailers” and keep the term “retailers” only.
2 The Model
2.1 Notation and preliminaries
We consider the market of a homogenous good that consists of two producers/retailers ( and ) that compete over quantity ( places quantity ) and a supplier (or wholesaler) under the following assumptions

The retailers may produce quantities and up to a capacities and respectively at a common fixed cost per unit normalized to zero^{2}^{2}2See Section 1 for an alternative interpretation of these quantities as inventories..

Additionally, they may order quantities from the supplier at a price set prior to and independently of their orders. The total quantity that each retailer releases to the market is equal to the sum
or shortly , where, for , the variable is the quantity that retailer produces by himself or draws from his inventory (at normalized zero cost) and is the quantity that the retailer orders from the supplier at price .

The supplier may produce unlimited quantity of the good at a cost per unit. We assume that the retailers are more efficient in the production of the good or equivalently that . After the normalization of the retailers’ production cost to , the rest of the parameters i.e. and are also normalized. Thus, represents a normalized price, i.e. the initial price that was set by the supplier minus the retailers’ production cost and a normalized cost, i.e. the supplier’s initial cost minus the retailers’ production cost. The supplier’s profit margin is not affected by the normalization and is equal to

After the total quantity that will be released to the market is set by the retailers, the market clears at a price that is determined by an inverse demand function, which we assume to be affine^{3}^{3}3After normalization of the slope parameter (initially denoted with ) of the inverse demand function to , all variables in (1) are expressed in the units of quantity and not in monetary units and therefore any interpretations or comparisons of the subsequent results should be done with caution.
(1) 
The demand parameter is a nonnegative random variable with finite expectation and a continuous cumulative distribution function (cdf) (i.e. the measure induced on the space of is nonatomic–singular distribution functions, like the Cantor function, are acceptable). We will assume that for all values of , i.e. that the demand parameter is greater or equal to the retailers’ production cost. The latter assumption is consistent with the classic Cournot duopoly model, which is resembled by the second stage of the game (however, the second stage game is not a classic Cournot duopoly due to the capacity constraints and and to the possibility of ). After normalization, in what follows we will use the term (resp. ) to denote the lower upper bound (resp. the upper lower bound) of the support of .

The capacities and the distribution of the random demand parameter are common knowledge among the three participants of the market (the retailers and the supplier).
Based on these assumptions, a strategy for retailer is a vector valued function or shortly a pair for . Equation (1) implies that may not exceed and hence the strategy set of will satisfy
(2) 
Denoting by a strategy profile, the payoff of retailer will be given by
(3) 
Whenever confusion may not arise we will write instead of . A strategy for the supplier is the price he charges to the retailers or equivalently his profit margin . From (3) we see that may not exceed , otherwise the retailers will not order for sure. Additionally, it may not be lower than , since in that case his payoff will become negative. Hence, in terms of his profit margin , the strategy set of the supplier satisfies
(4) 
Consequently, a reasonable assumption is that , otherwise the problem becomes trivial from the supplier’s perspective. For a given value of , the supplier’s payoff function, stated in terms of rather than , is given by
(5) 
On the other hand, for the retailers it is not necessary to know the exact values of and and hence, from their point of view (2nd stage game), we keep the notation . If the supplier doesn’t know (incomplete information case), then his payoff function will be
(6) 
the expectation taken with respect to the distribution of .
To proceed with the formal twostage game model, we recall that both the production/inventory capacities and of the retailers and the distribution of the demand parameter are common knowledge of the three market participants. We then have,

The demand parameter is realized and observed by both the supplier and the retailers^{4}^{4}4Of course, this means that there is no randomness in , so the description of as a random variable is redundant in the complete information case. We use it just to give a common formal description of both the complete and the incomplete information case.. Then, at stage 1, the supplier fixes his profit margin and hence his price . His strategy set and payoff function are given by (4) and (5). At stage 2, based on the value of , each competing retailer chooses the quantity that he will release to the market by determining how much quantity he will draw (at zero cost) from his inventory and how much additional quantity he will order (at price ) from the supplier. The strategy sets and payoff functions of the retailers are given by (2) and (3) respectively.

At stage 1, the supplier chooses without knowing the true value of . His strategy set remains the same but his payoff function is now given by (6). After and hence is fixed, the demand parameter is realized and along with the price is observed by the retailers. Then we proceed to stage 2, which is identical to that of the Complete Information Case.

the above are assumed to be common knowledge of the three players.
3 Subgameperfect equilibria under complete information
At first we treat the case with no uncertainty on the side of the supplier about the demand parameter . In subsections 3.1 and 3.2 we determine the subgame perfect equilibria of this two stage game.
3.1 Equilibrium strategies of the second stage
We begin with the intuitive observation that it is best for the retailers to produce up to their capacity constraints or equivalently to exhaust their inventories before ordering additional quantitites from the supplier at unit price . For simplicity in the notation of Lemma 3.1 and Lemma 3.2 fix and let . As above, .
Lemma 3.1.
Any strategy with and is strictly dominated by a strategy with
or equivalently by .
Proof.
Accordingly, we restrict attention to the strategies in
Figure 0(a) depicts the set when and Figure 0(b) when . As shown below, Lemma 3.1 considerably simplifies the maximization of , since for any strategy of retailer , the maximum of will be attained at the bottom or right hand side boundary of the region .
Moreover, Lemma 3.1 implies that when , retailer will order no additional quantity from the supplier^{5}^{5}5In Proposition 3.3 we will see that in equilibirum retailer will order no additional quantity if .. Although trivial, we may not exclude this case in general, since in Section 4 we consider to be varying.
Restricting attention to for , we obtain the best reply correspondences and of retailers and respectively. To proceed, we notice that the payoff of retailer depends on the total quantity that retailer releases to the market and not on the explicit values of , cf. (3).
Lemma 3.2.
The best reply correspondence of retailer for is given by
The enumeration of the different parts of the best reply correspondence will be used for a more clear case discrimination in the subsequent equilibrium analysis, see Appendix A.
Proof.
See Appendix A. ∎
A generic graph of the best reply correspondence of retailer to the total quantity that retailer releases to the market is given in Figure 2.
The equilibrium analysis of the second stage of the game proceeds in the standard way, i.e. with the identification of the Nash equilibria through the intersection of the best reply correspondences and . For a better exposition of the results we restrict attention from now on to the case of identical retailers, i.e. . The general case is treated in Appendix A, where the complete statements and the proofs of Proposition 3.3 and Proposition 3.4 are provided.
If the retailers are identical, i.e. only symmetric equilibria may occur in the second stage. The equilibrium strategies depend on the value of and its relative position to .
Proposition 3.3.
If , then for all values of the second stage equilibrium strategies between retailers and are symmetric and for they are given by , or shortly , with
The quantity is the total quantity of the good that each retailer releases to the market in equilibrium. If then is equal to the equilibrium quantity of a classic Cournot duopolist who faces linear inverse demand with intercept equal to and has cost per unit. If , then the equilibrium quantities of the retailers depend on the supplier’s price . If is low enough, i.e. if , they will be willing to order additional quantities from the supplier. In this case, , which is equal to the equilibrium quantity of a Cournot duopolist who faces linear demand with intercept equal to and has cost per unit for all product units, despite the fact that in our model the retailers face a cost of 0 for the first units and for the rest. Finally, for the retailers will avoid ordering and will release their inventories to the market.
The total quantity that is released to the market has a direct impact on the consumers welfare as expressed in terms of the consumers surplus^{6}^{6}6We remind that the consumers surplus is proportional to the square of the total quantity that is released to the market.. For lower values of the demand parameter the consumers benefit from the “low cost” inventories or production capacities of the retailers, since otherwise no goods would have been released to the market. On the other hand, for higher values of the total quantity that is released to the market does not depend directly on and thus the benefits from keeping inventory are eliminated for the consumers.
3.2 Equilibrium strategies of the first stage
The general form of the supplier’s payoff function is given in (5) and his strategy set in (4). Obviously, the supplier will not be willing to charge prices lower than his cost . Based on the discussion of Proposition 3.3 above and the constraint (i.e. ), we conclude that a transaction will take place for values of and for . As we shall see, in that case, the optimal profit margin of the supplier is at the midpoint of the interval.
To see this, let denote the total quantity that the supplier will receive as an order from the retailers when they respond optimally. By Proposition 3.3, . Hence, on the equilibrium path and for , the payoff of the supplier is
(7) 
We then have
Proposition 3.4.
For and for all values of , the subgame perfect equilibrium strategy of the supplier is given by
Proposition 3.4 implies that if , then the optimal profit of the supplier is equal to , i.e. he will set a price equal to his cost. Actually, he is indifferent between any price since in that case he knows that the retailers will order no additional quantity.
In sum, Proposition 3.3 and Proposition 3.4 provide the subgame perfect equilibrium of the two stage game in the case of identical (i.e. ) retailers.
Theorem 3.5.
If the capacities of the producers (retailers) are identical, i.e. if , then the complete information two stage game has a unique subgame perfect Nash equilibrium, under which the supplier sells with profit margin and each of the producers (retailers) orders quantity and produces (releases from his inventory) quantity .
4 Subgameperfect equilibrium under demand uncertainty
We investigate the equilibrium behavior of the supply chain assuming that the supplier has no information about the true value of the demand parameter when he sets his price, while the retailers know it when they place their orders (if any). In the two stage game context, we assume that the demand is realized after the first stage (i.e. after the supplier sets his price) but prior to the second stage of the game (i.e. prior to the decision of the retailers about the quantity that they will release to the market). The case exhibits significant computational difficulties and we will restrict our attention to the symmetric case .
Under these assumptions the equilibrium analysis of the second stage (as presented in subsection 3.1) remains unaffected: the retailers observe the actual demand parameter and the price set by the supplier and choose the quantities and for . However, in the first stage, the actual payoff of the supplier depends on the unknown parameter for which he has a belief: the distribution of , which induces a nonatomic measure on with finite expectation, as assumed. Given the value of and taking as granted that the retailers respond to the supplier’s choice with their unique equilibrium strategies, the supplier’s actual payoff is provided by equation (7). Taking the expectation with respect to his belief (see also (6)), we derive the payoff function of the supplier when is unknown and distributed according to
(8) 
Let and . If , then for all , i.e. then, and the problem is trivial. Hence, in order to proceed we assume that . Then, since for , we may restrict the domain of and take it to be the interval in (8). Next, observe that since is nonnegative, , (e.g. see Billingsley, 1986) and hence, by a simple change of variable, which implies that
(9) 
We are now able to prove the following lemma.
Lemma 4.1.
The supplier’s payoff function is continuously differentiable on and
(10) 
Proof.
It suffices to show that . Then, equation (10) as well as continuity are implied. So, let
(11) 
and take . Then, and therefore
Since for all , the dominated convergence theorem implies that
In a similar fashion, one may show that . Since the distribution of is nonatomic, and hence, . By Equation 11, , which concludes the proof.
∎
At this point, we need to introduce the Mean Residual Lifetime function (MRL) of . For a more thorough discussion of the definition and properties of one is referred to applied probability books and papers–Hall and Wellner (1981) or Guess and Proschan (1988) are considered classic. It is given by
(12) 
Then, the following holds.
Lemma 4.2.
For , the supplier’s payoff function and its derivative are expressed through the MRL function by and
(13) 
respectively. In addition, the roots of in (if any) satisfy the fixed point equation
(14) 
Proof.
The formulas for and are immediate using eq. 9, eq. 10, and eq. 12. As an interesting observation, notice that the two terms of , as given in the lemma, may not be separately differentiable. Finally, if , then is positive. Hence, equation (13) implies that the critical points of (if any) satisfy . ∎
We remark that since , one may be tempted to use the product rule to derive its derivative and hence show that is differentiable. The problem is that the product rule does not apply, since the two terms in this expression of may both be nondifferentiable, even if has a density and its support is connected (e.g. consider the point in case ).
Due to Lemma 4.2, if a nonzero optimal response of the supplier exists at equilibrium, it will be a critical point of , i.e. it will satisfy (14). It is easy to see that such a response always exists when the support of is bounded, i.e. when . However, this is not the case when . So, let us determine conditions under which such a critical point exists, is unique and corresponds to a global maximum of the supplier’s payoff function. To this end, we study the first term of (13), namely .
Clearly, is continuous on . We first show that by considering cases. If , then for , . Hence, . If , then we use Proposition 1f of Hall and Wellner (1981), according to which with equality if and only if or . So, take first , which implies . Hence, and (by Proposition 1f) . Hence, if . Finally, if , then , which again implies that .
We then examine the behavior of near . If , then and by the intermediate value theorem an exists such that . For , we also notice that to get uniqueness of the critical point , it suffices to assume that the Mean Residual Lifetime (MRL) of the distribution of is decreasing^{7}^{7}7We use the terms “decreasing” in the sense of “nonincreasing” (i.e. flat spots are permitted), as they do in the pertinent literature, where this use of the term has been established., in short that has the DMRL property. On the other hand, if , then the limiting behavior of as increases to infinity may vary, see Bradley and Gupta (2003), and an optimal solution may not exist, see Example 4 below. But, if we assume as before that is decreasing, then will eventually become negative and stay negative as increases and hence, existence along with uniqueness of an such that is again established.
Now, since and hence , i.e. starts increasing on . Assuming that has the DMRL property, the first term of (13) is negative in a neighborhood of while the second term goes to 0 from positive values. Hence, in a neighborhood of , i.e is decreasing as approaches . Clearly, for sufficiently small, will take a maximum in the interior of the interval if or a maximum in the interior of the interval if . Since is differentiable, the maximum will be attained at a critical point of , i.e. at the unique given implicitly by (14).
Equation 14 actually characterizes as the fixed point of a translation of the MRL function , namely of . Its evaluation sometimes has to be numeric, but in one interesting case it may be evaluated explicitly: If , then for all , which implies that . Then, if
(15) 
we get . To see this, notice that (15) is equivalent to , i.e. to
Then, by the DMRL property for all . This implies that or equivalently that . In this case and hence will be given explicitly by
Intuitively, this special case occurs under the conditions that (a) the lower bound of the demand exceeds the particular threshold , (i.e. or ), and (b) the expected excess of over its lower bound is at most equal to the excess of over (i.e. ). Of course, since , condition (b) suffices. In that case, compare with the optimal of the complete information case (Proposition 3.4).
Finally, if and (i.e, if ) we get that , for if , then , i.e. . The latter implies that then which contradicts the assumption.
To sum up, we obtained that
Theorem 4.3 (Necessary and sufficient conditions for Bayesian Nash equilibria).
Under incomplete information with identical producers/retailers (i.e. for ) for the nontrivial case and assuming the supplier’s belief induces a nonatomic measure on the demand parameter space:

(necessary condition) If the optimal profit margin of the supplier exists when the producers/ retailers follow their equilibrium strategies in the second stage, then it satisfies the fixed point equation .

(sufficient condition) If the mean residual lifetime of the demand parameter is decreasing, then the optimal profit margin of the supplier exists under equilibrium and it is the unique solution of the equation . In that case, if , then is given explicitly by . Moreover, if , then .
Proposition 3.3 and Theorem 4.3 lead to
Corollary 4.4.
If the capacities of the producers (retailers) are identical, i.e. if , and if the distribution of the demand intercept is of decreasing mean residual lifetime , then the incomplete information two stage game has a unique subgame perfect Bayesian Nash equilibrium for the nontrivial case . At equilibrium, the supplier sells with profit margin , which is the unique solution of the fixed point equation and each of the producers (retailers) orders quantity and produces (releases from his inventory) quantity .
Theorem 4.3 enables one to show the intuitive result that under the DMRL assumption, if everything else stays the same but inventories of the retailers are rising (but staying below because of the nontriviality assumption), the supplier will decrease strictly or keep constant the price he charges at equilibrium, because his profit margin will be nonincreasing. The reason is that as increases, the graph of shifts to the left and therefore its intercept with the line bisecting the first quadrant decreases^{8}^{8}8To avoid confusion, we use the terms “decreases” in the sense “decreases nonstrictly” or “does not increase”, as we had to do before.. By Theorem 4.3, this intercept is and hence the price the supplier asks at equilibrium will be decreasing. By the same argument, if we take the supplier’s cost to be increasing (but staying below ) while everything else stays fixed, then the supplier’s profit margin will again be decreasing. However, this time the price the supplier asks at equilibrium will be increasing. To see this, let . Then, since is decreasing, . If , then . If , by Theorem 4.3, . The DMRL property then implies that , i.e. . So, we proved that
Corollary 4.5.
Under the DMRL property: {enumerate*}[label=()]
If everything else stays the same and the inventory quantity of the retailers increases in the interval , then at equilibrium, the supplier’s profit margin , and hence the price he asks, both decrease.
If everything else stays the same and the cost of the supplier increases in the interval , then at equilibrium, the supplier’s profit margin decreases while the price he asks increases.
We close this section with some remarks and observations.

By the DMRL property, Theorem 4.3 implies that , since
(16) 
A sufficient condition for the mean residual lifetime to be decreasing is that is absolutely continuous (i.e. has a density) and is of increasing failure rate (IFR). The fact that the IFR assumption on yields the desired properties for the critical point can also be seen by taking the second derivative of , which one may check that is given by
(17) and studying the behavior of through . If the IFR property applies, then finiteness of every moment of , in particular of the expectation is implied, so that this does not have to be explicitly assumed (cf. assumption 5 in Section 2).

If has a density, then by the explicit form of the second derivative, the weaker assumption that is increasing would suffice under the additional assumption that the term exceeds . This is always the case if , since then , but has to be assumed if . Lariviere and Porteus (2001) and Lariviere (2006) introduce and examine a similar condition, the increasing generalized failure rate (IGFR). Specifically, has the IGFR property if is increasing. However, the assumption that is increasing does not imply in general that , for some constant , is also increasing and therefore, assuming that is of IGFR would not suffice in our model to ensure existence of a solution in the unbounded support case.

Finally, we note that a necessary and sufficient condition for to be of DMRL is that the integral
is logconcave, see Bagnoli and Bergstrom (2005).
4.1 Classic Cournot with no production capacities and a single supplier
If , then our model corresponds to a classic Cournot duopoly at which the retailers’ cost equals the price that is set by a single profit maximizing supplier. In this case, we may normalize to (now the nonnegative assumption on implies that ) and the first derivative of , see (14), simplifies to
Thus, cf. (14), the unique  under the DMRL assumption  critical point of now satisfies , i.e. the optimal profit margin of the supplier is the unique fixed point of the mean residual lifetime function. So we have,
Corollary 4.6.
[A Cournot supply chain with demand uncertainty for the supplier] Consider a market with a linear inverse demand function for a single product distributed by two retailers without inventories, who order the quantities they will release to the market from a single supplier, who sells at a common price. Assume that the supplier doesn’t know the actual value of the demand intercept but has a belief about it, expressed by a probability distribution function . Then, under the assumption that is of decreasing mean residual lifetime and the supplier’s cost is below all values of , the supply chain formed by the supplier and the retailers has a unique Bayesian Nash equilibrium. At equilibrium and after normalizing the supplier’s cost to 0, the supplier sells at a price , which is the unique fixed point of and each of the producers (retailers) orders quantity .
Remark 1.
In case , the second derivative of the supplier’s payoff, see (17), is simplified to
so that now the assumption that has the IGFR property, i.e. that is increasing, is sufficient to ensure existence of a unique solution, if one additionally assumes that the term exceeds . Note that in that case, will still be a fixed point of , as it will have to be a root of . Example 4 of Section 6 deals with such a case, where is not a decreasing function.
5 Inefficiency of equilibrium in the incomplete information case
It is well known that markets with incomplete information may be inefficient in equilibrium in the sense that trades that would be beneficial for all players may not occur under Bayesian Nash equilibrium, eg. see Myerson and Satterthwaite (1983). In this section, we discuss the inefficiency exhibited by the Bayesian Nash equilibrium we derived for the incomplete information case of our chain. By inefficiency we mean that, under equilibrium, there exist values of for which a transaction would have occurred in the complete information case but will not occur in the incomplete information case. Inefficiency should not be interpreted as a comparison between the actual payments of market participants in the complete and in the incomplete information case for various realizations of .
So, for the incomplete information case and for a particular distribution of , let be the event that a transaction would have occurred under equilibrium, if we had been in the complete information case, and let be the event that a transaction does not occur in the incomplete information case under equilibrium. Our aim is to measure the inefficiency of our market by studying and .
By Proposition 3.3, a transaction will take place under equilibrium if and only if
(18) 
where stands for the profit margin of the supplier in the incomplete information case under equilibrium. Using (18) and Proposition 3.4, we conclude that a necessary and sufficient condition for a transaction to take place under equilibrium in the complete information case is . Hence, using for the support of , we get and . So, , and therefore