A general model of Bertrand-Edgeworth duopoly

Allison, Blake A.; Lepore, Jason J. Article A general model of Bertrand-Edgeworth duopoly Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Allison, Blake A.; Lepore, Jason J. (2025) : A general model of Bertrand-Edgeworth duopoly, Games, ISSN 2073-4336, MDPI, Basel, Vol. 16, Iss. 3, pp. 1-37, https://doi.org/10.3390/g16030026 This Version is available at: https://hdl.handle.net/10419/330140 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Academic Editor: Konstantinos Serfes Received: 25 February 2025 Revised: 18 April 2025 Accepted: 30 April 2025 Published: 19 May 2025 Citation: Allison, B. A., & Lepore, J. J. (2025). A General Model of Bertrand– Edgeworth Duopoly. Games,16(3), 26. https://doi.org/10.3390/g16030026 Copyright: © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/). Article A General Model of Bertrand–Edgeworth Duopoly Blake A. Allison 1and Jason J. Lepore 2,* 1Department of Economics, Emory University, Atlanta, GA 30322, USA; [email protected] 2Department of Economics, California Polytechnic State University, San Luis Obispo, CA 93407, USA *Correspondence: [email protected] Abstract: This paper studies a class of two-player all-pay contests with externalities that encompass a general version of duopoly price competition. This all-pay contest formulation puts little restriction on production technologies, demand, and demand rationing. There are two types of possible equilibria: In the first type of equilibrium, the lower bound to pricing is the same for each firm, and the probability of any pricing tie above this price is zero. Each firm’s equilibrium expected profit is their monopoly profit at the lower bound price. In the second type of equilibrium, one firm prices at the lower bound of the other firm’s average cost and other firm prices according to a non-degenerate mixed strategy. This type of equilibrium can only occur if production technologies are sufficiently different across firms. We derive necessary and sufficient conditions for the existence of pure strategy equilibrium and use these conditions to demonstrate the fragility of deterministic outcomes in pricing games. Keywords: price competition; contest; demand rationing; capacity constraints 1. Introduction The determination of prices in markets with very few sellers has been a central subject of inquiry since the inception of mathematical economics . Edgeworth (1925) moved the understanding of this subject forward by appreciating the impact of consumer rationing and the prominence of price indeterminacy, or pricing cycles . 1 While the conceptual origins of the Bertrandâ e “Edgeworth (hereafter BE) model can be traced back to Edgeworth, his basic insights were first formalized into a game theoretic model by Shubik (1959). 2 Shubik focused on understanding the range of pricing in mixed strategy equilibrium and the character of pure strategy equilibria when they exist. These pricing games have been widely studied since Shubik’s formalization. The standard BE model in the literature has the following features: firms possess constant marginal costs up to capacity (an absolute limit on production), and consumers are rationed according to either the efficient or proportional rationing rule. 3,4 This BE model has been used to understand fundamental issues in price determination, including duopoly pricing and capacity investment (Allen & Hellwig,1993;Davidson & Deneckere,1986;Deneckere & Kovenock,1996;Kreps & Scheinkman,1983;Lepore,2009;Levitan & Shubik,1972;Osborne & Pitchik,1986), sequential pricing (Allen,1993;Allen et al.,2000;Deneckere & Kovenock, 1992), large markets (Allen & Hellwig,1986;Dixon,1987,1992;Vives,1986), oligopoly (De Francesco & Salvadori,2010;Hirata,2009), and uncertainty (de Frutos & Fabra,2011; Lepore,2008,2012;Reynolds & Wilson,2000). Only a few papers have investigated BE models with cost structures outside the constant marginal cost case. 5 Dixon (1987) considers a model of BE oligopoly with strictly convex costs, showing the non-existence of pure strategy equilibrium for the case of efficient and proportional rationing. Yoshida (2006) Games 2025,16, 26 https://doi.org/10.3390/g16030026 Games 2025,16, 26 2 of 37 characterizes equilibrium pricing in symmetric duopoly with convex cost and efficient rationing. 6 While the literature has produced interesting results, the models are restrictive in terms of production technologies and demand rationing of consumers and asymmetries across firms. The purpose of this paper is to analyze the properties of equilibria in a BE model with a broad range of production technology, minimal restriction demand rationing, and asymmetries across firms. Our approach to the analysis is based on modeling price competition as a particular extension of an all-pay contest (Siegel,2009,2010,2014). In order to contextualize our analysis, we identify some of the abstract properties of the BE model with those of a standard all-pay auction. 7 In the BE model, firms place bids in the form of a price in an attempt to win the larger share of demand, which goes to the firm with the highest bid (lowest price). There are two fundamental distinctions between the BE model and the all-pay contest or traditional all-pay auction. First, the payoff of the losing player (the firm with the highest price) may depend on the price of the winner through the rationing of residual demand, while, traditionally, the losing player’s payoff depends only on her committed bid. Second, the payoff of both the winner and loser can be non-monotonic in her bid, as a reduction in price increases the quantity demanded, possibly raising profits, while an increased bid in traditional contests merely commits the winner or loser to a lower payoff.8 In Section 2, we present the general model and introduce key notation. The model is defined based on abstract properties of the front-side profit of the lower-priced winner firm and the residual profit of the higher-priced loser firm. The abstract contest formulation allows us to analyze a model with a broad range of underlying specifications: general production technology including the case of U -shaped average cost of production, minimal restriction on demand allowing for a wide swath of demand rationing (including rationing outcomes that are equilibria from consumer search), and asymmetries across firms. 9 In order to establish the bounds on equilibrium prices and payoffs, we define the following preliminary objects. First, define the critical judo price as the highest price either firm can set to guarantee that the other firm would rather maximize its residual profit than undercut. This terminology is based on the sequential pricing model of judo economics by Gelman and Salop (1983). 10 The second important price we define is the critical safe price, which is the infimum of all prices at which it earns at least its minâ e “max profit if its rival undercuts. The general results on the properties of all equilibria are presented in Section 3. After establishing generic equilibrium existence, 11 we show that there are only two types of possible equilibria. The first type of equilibrium is such that both firms’ pricing distributions have the same lower bound, and ties at prices greater than the lower bound occur with probability zero. This type of equilibrium includes the possibility of pure strategy equilibrium in which firms play the lower bound price with certainty. The second type of equilibrium can only occur if the firms have sufficiently different production technology, and are such that one firm plays a pure strategy price while the other plays a nondegenerate mixed strategy. Since a model specification can have multiple non-payoff equivalent equilibria, we show that, in all equilibria, the expected profits of each firm are bounded between its monopoly profit at the critical safe price and the critical judo price. In the process of establishing the payoff bounds, we provide abstract bounds for the range of equilibrium pricing. The results particular to pure strategy equilibrium are presented in Section 4. We provide necessary and sufficient conditions for the existence of pure strategy equilibrium. All pure strategy equilibria must be symmetric and only exist under two circumstances. The first case is a symmetric pricing profile at which each firm’s residual profit is maximized and equal to the monopoly profit at the same price. The second case is a symmetric pricing Games 2025,16, 26 3 of 37 profile at which exactly one firm’s monopoly profit exceeds its residual profit (weakly exceeding its maximum residual profit), and the sharing rule is such that this firm receives its monopoly profit with certainty. In this second case, the price must maximize the other firm’s residual profit. The existence of a pure strategy equilibrium does not guarantee uniqueness, as there may be additional mixed strategy equilibria that exist concurrently. We show that a pure strategy equilibrium price x∗ is unique as the pure strategy equilibrium price x∗ is unique if, for each firm, this price is the unique maximizer of residual profit when the other firm prices at x∗ , and each firm’s residual profit is nonincreasing in the other firm’s price. We present a special case model using a standard BE construction with all assumption made directly on each firm’s production technology and demand in Section 5. First, we explore additional properties of pure strategy equilibrium for this special case model, identifying the necessary and sufficient conditions on the underlying primitives of production technology and demand. Second, we examine the impacts of demand and supply shifts on the bounds of equilibrium prices and profits. These shifts can accommodate changes in rationing, production cost, or capacity. We demonstrate that an increase in residual demand will weakly increase the bounds on the lowest equilibrium price along with the bounds on profits; however, by example, we show that the upper bound on pricing may be reduced. An increase in a firm’s supply weakly decreases the bounds on the lowest equilibrium price along with the bounds on the other firm’s profits. A general prediction cannot be made for the bounds on the profit of the firm with the supply increase, as there are countervailing effects: a direct effect through which lower costs or higher capacities enhance profitability, and an indirect competitive effect through which those changes increase the level of competition, driving down prices and profits.12 Finally, all proofs of lemmas and propositions are located in Appendix A. 2. The Model In this section, we lay out the general model and then provide a subsection of examples to illustrate the scope of the model. All assumptions stated in this section are maintained throughout the remainder of the paper. Consider a homogeneous product industry with two firms i= 1, 2. We will use j= 1, 2, to refer to the firm other than i . The firms simultaneously and independently announce prices. We denote by pi the price of firm i and by p the vector of both firms’ prices. Since p is the vector of prices (p1 , p2) , we will use x to unambiguously denote a single price when it is not associated with a particular firm. The profit that each firm receives depends on whether it has a lower price than the other firm. The front-side profit of the firm i with a lower price than firm j is φi(pj) , while the residual profit of the firm i with a higher price than firm j is ψi(p) . The domain of residual profit ψi is {(pi , pj)∈R2 + : pi≥pj} , as it need not be defined for prices such that pi<pj since the residual profit cannot be obtained at such prices. We make the following assumptions on the profit functions φiand ψi. Assumption 1. φi(x)≥ψi(x,x′)≥0for all x ≥x′. The assumption that the front-side profit is at least as large as the residual profit is consistent with the notion that customers prefer lower prices, and thus, the firm with the lowest price has weakly greater potential to sell. This assumption also implies that the lower-price firm is not required to sell units that decrease profit. Assumption 2. For each firm i , there exists the largest ai such that φi(x) = ψi(x , pj) = 0for all pj≤x≤ai. Further, ψi(pi,x) = φi(pi)for all pi≥x such that x <aj. Games 2025,16, 26 4 of 37 The price ai is the minimal price for which a firm is willing to produce, typically the infimum of the average cost of production when explicitly modeled. As such, when a firm i prices below ai , firm j ’s residual profit is equal to its front-side profit because firm i produces nothing. Assumption 3. φi has a unique maximizer b pi>ai with φi(b pi)> 0. On the interval (ai , b pi) , φi is positive valued, and strictly increasing. Further, ψi(pi , pj) = 0for all pi≥pj≥b pj , and pi>b pj. This assumption on the front-side profit is weaker than assuming the strict quasiconcavity of φi as it does not restrict behavior at prices pi>b pi . The third part of the assumption is that there is no residual profit for firm i if firm i prices above b pj . This assumption fits the two cases that demand is continuous at b pj, or that there is zero quantity demanded above the price b pj. The following assumption is important for our characterization. Assumption 4. For each firm i, there exists a price ρi∈[ai, min{b pi,b pj}]such that φi(x)>ψi(x,x)for all x ∈ρi,b pii, φi(x) = ψix,x′for all x′≤x<ρi. Further, if ai>aj, then ρi≤ρj. The maximum of the two firms’ prices ρi is a primary object used in the analysis that follows; as such, we denote ρ=max{ρ1,ρ2}. Remark 1. The existence of the prices ρi is a natural consequence of traditional constructions of the BE model. It is common that ρi corresponds to the price at which total industry supply is equal to market demand, as below such a price, the residual demand would exceed the supply of the high-priced firm. Alternatively, in that case that the marginal cost of i is less than the marginal cost of j , ρi can correspond to the (constant) marginal cost of firm j , as below that price, firm j does not produce, leaving the market demand to the residual, while at or above that price, firm j produces up to its capacity, potentially limiting residual profit below the front side. Assumption 4accommodates either of these scenarios and generally allows more variety of market structures. For example, it allows for situations in which unsatiated demand is not rationed to other firms, as may be the case with directed search models. We make the following assumption to rule out the possibility that one firm has a sufficiently competitive advantage to act as a monopoly. Assumption 5. For each i and j, b pi>aj.13 Each firm i’s profit is specified as follows: ui(p) =      φi(pi)pi<pj αi(p)φi(pi) + (1−αi(p))ψi(p)pi=pj ψi(p)pi>pj , (1) where αi(p)∈[ 0, 1 ] and α1(p) + α2(p)∈( 0, 2 ) . If we instead assume that α1+α2= 1, then this restricts attention to sharing rules that assign one firm its front-side profit and the other its residual profit, with some randomization over the assignment. By permitting the sum of Games 2025,16, 26 5 of 37 the shares to be greater (or less) than one, the model captures any share of demand at ties, which can naturally result in each firm receiving a (non-stochastic) profit strictly between its front-side and residual profits. We denote the set of maximizers of ψi at any pj by e Pi(pj) . Denote the maximized residual profit by e ψi(pj), that is, e ψi(pj) = max pi≥pj ψipi,pj. Assumption 6. There exists the lowest price b x such that φi(b x)>φi(x) and ψi(b x , pj)≥ψi(x , pj) for all prices x and pjwith x >b x≥pj. Note that b x≥b pi for each firm i . Given Assumption 6, the price b x weakly dominates all prices x>b x. Assumption 7. Both φi and ψi are continuous in pi on [ 0, b x) and left continuous at b x . ψi is right upper semicontinuous in pj , that is, lim supkψi(pi , xk)≤ψi(pi , x) for any sequence {xk} such that pi≥xk>x and xk→x. These continuity assumptions are satisfied in most BE models previously studied. The right upper semicontinuity captures the notion that a firm does not drastically decrease its production when the other firm’s price increases. The potential for discontinuities at prices above b xallows the model to accommodate settings with box demand. Define ri to be firm i ’s judo price, which is the lower bound such that the front-side profit of firm i is greater than the maximal residual profit of firm i when firm j uses any price weakly greater. Formally, ri=inf{x|φi(x)>supz≥xe ψi(z)}. Define ri to be firm i ’s safe price, which is the lower bound of price such that the frontside profit of firm i is greater than the highest profit that firm i can guarantee itself. Formally, ri=inf{x|φi(x)>ui}, where ui=infpjsuppiui(pi,pj). Define the larger of the two firms’ judo prices to be critical judo price, denoted by r=max ri . Similarly, define the larger of the two firms’ safe prices to be the critical safe price, denoted by r=max ri . Based on the fact that ui≤supz≥xe ψi(z) and that φi is strictly increasing when positive, the judo price is always weakly greater than the safe price, that is, r≥r. Note further that r≤b x. Define firm i ’s judo profit to be the front-side profit of firm i at the critical judo price, denoted by φi≡φi(r) . Similarly, define firm i ’s safe profit to be the front-side profit of firm i at the critical safe price, φi≡φi(r). For equilibrium strategies µ=(µ1,µ2) , we use xi and xi to denote the infimum and supremum of the support of firm i ’s strategy, respectively. We will use x to denote the minimum of x1 and x2 , and x to denote the maximum of x1 and x2 . 14 Further, we define Fi to be the distribution function (CDF) of firm i ’s mixed strategies on [x,x] , with F= (F1 , F2) . Additionally, let u∗ idenote firm i’s equilibrium expected profit. Before proceeding with the analysis of the model, we discuss some underlying specifications that our model contains in the following subsection. Games 2025,16, 26 6 of 37 Examples Within Our Framework To provide context for the abstract model, we present three examples nested within our framework. The framework of our model puts very little restriction on the underlying market demand except continuity on [ 0, b x) , even allowing for increasing demand. As such, the demand structure is easily understood. To help convey the generality of our model, we present the following examples to demonstrate the range of production technologies and demand rationing rules that can be accommodated in our model. With this purpose in mind, we use rectangular unit demand in these numerical examples. Formally, the market demand is D(x) = (0 if x>1 1 if x∈[0, 1]. The first two examples exhibit production technologies included in our specification. The third example exhibits a residual demand rationing scheme based on directed search. In each example, we specify each firm’s cost of production as a function of quantity produced and use that to derive the supply correspondence of quantities that maximize the direct profit function πi(x , q) = xq −ci(q) . 15 We then use these to derive the corresponding front-side and residual profit functions as well as the prices b x,b pi,ai, and ρi. Example 1 (Discontinuous supply).Firm 1’s cost of production is c1(q) = (1 2q−1 8if q ≥1/2 1 4q if q ∈[0, 1/2], while firm 2’s cost of production is c2(q) = 1 3q . Neither firm faces a capacity constraint. The supply correspondence of each firm is thus ϑ1(p1) =                {∞}if p1>1/2 [1 2,∞]if p1=1/2 n1 2oif p1∈(1/4, 1/2) [0, 1 2]if p1=1/4 {0}if p1∈[0, 1/4) , and ϑ2(p2) =      {∞}if p2≥1/3 [0, ∞]if p2=1/3 {0}if p2∈[0, 1/3) . The front-side profits are φ1(p1) =            0if p1>1 p1−1 41 2+p1−1 21 2if p1∈[1/2, 1] p1−1 41 2if p1∈[1/4, 1/2) 0if p1∈[0, 1/4) , and φ2(p2) =      0if p2>1 p2−1 3if p2∈[1/3, 1] 0if p2∈[0, 1/3) . As the supply correspondence is not single-valued, the residual profits of the firms may depend on the particular quantity chosen. Rather than present all possibilities, we use the convention that the firms produce the largest quantity possible in their supply correspondence. With this convention, Games 2025,16, 26 7 of 37 the residual profits of the firms are defined with demand rationed according to the efficient (or equivalently proportional) rule ψ1(p1,p2) = (0if p2≥1/3 φ1(p1)if p2<1/3 , ψ2(p2,p1) =            0if p2>1, or p1≥1/2, or p2<1/3, or p1∈[1/4, 1/2) p2−1 31 2if p2≥1/3, p1∈[1/4, 1/2) φ2(p2)if p1<1/4 . The key model parameters for this example are b x=b pi= 1, a1= 1 / 4, a2= 1 / 3, and ρi= ρ=1/3. The next example includes firms with a U-shaped average and marginal costs. Example 2 (U-shaped average and marginal costs).Each firm i’s cost of production is ci(q) = (2 3q3−1 4q2+1 16 q+1 20 if q >0 0if q =0. Neither firm faces a capacity constraint. The supply correspondence of each firm i can be expressed as the function ϑ(pi) = (1+√32pi−1 8if pi≥0.194 0if pi∈[0, 0.194]. Thus, the front-side profit of firm i is φi(pi) =      0if pi>1 piϑ(pi) + ci(ϑ(pi)) if pi∈[0.194, 1] 0if pi∈[0, 0.194) , and the residual profit of firm i (pi≥pj) is (again using efficient rationing)16 ψi(pi,pj) =      0if pj≥0.397 pimin{ϑ(pi), 1 −ϑ(pj)}+ci(min{ϑ(pi), 1 −ϑ(pj)})if pj∈(0.194, 0.397) φi(pi)if pj<0.194 . The key model parameters for this example are b x=b pi=1, ai=0.194, and ρi=ρ=0.3125. The final example has demand rationing determined by a directed consumer search game. Example 3 (Search).Each firm has zero cost of production. Both firms have the same capacity k∈( 1 / 2, 1 ) , with each firm limited to producing a quantity of at most k . Demand rationing is determined by the equilibrium of a directed consumer search game. There is a unit mass of consumers, each of which demands a single unit of the good, which they value at 1. The consumers observe the prices of the firms, and then simultaneously choose a firm to visit. If a mass Wi≤k of consumers visit firm i , each of those consumers receives a good at price pi . If a mass Wi>k of consumers visits firm i , then each of those consumers receives a good at price pi with probability k/Wi . Consumers that do not receive goods obtain a payoff of zero. In any pure strategy equilibrium of the consumer Games 2025,16, 26 8 of 37 game, if pi<pj , then either all consumers shop at firm i , or the mass Wi of consumers that shop at firm i satisfies k Wi (1−pi)=1−pj. Therefore, we can write Wi=(0if pi>1 minn1, k1−pi 1−pjoif pi∈[0, 1], and the mass of consumers who go to firm j are Wj= 1 −Wi . Notice that Wi≥k for all prices pi<pj. Thus, we can write the front-side and residual profit of firm i as follows: φi(pi) = (0if pi>1 pik if pi∈[0, 1], and ψi(pi,pj) =    0if pi≥1 pimax0, 1 −k(1−pj) 1−piif pi∈[pj, 1). The key model parameters for this example are b x=b pi=1, ai=ρi=ρ=0. 3. Mixed Strategy Equilibria In this section, we establish some abstract properties of all equilibria. We begin the analysis by dealing with the problem of existence of equilibrium. As long as each firm’s residual profit is lower semicontinuous in the other firm’s price, we are able to show that an equilibrium exists if ρ1=ρ2 . When there is firm i such that ρi<ρ , our proof requires an additional condition that this firm i receives its front-side profit with certainty at ties below ρ. Proposition 1. Assume that (1) αi(x , x) = 1for any x such that φi(x)>ψi(x , x) and φj(x) = ψj(x , x) and (2) that ψi is lower semicontinuous in pj . Then a mixed strategy equilibrium of the BE game exists. If the residual profits ψi are continuous, then the existence of equilibrium for the BE duopoly follows directly from Proposition 2 in Allen and Lepore (2014). A generalization of this proposition is presented in Appendix A.1, which applies to the case in which the residual profits are not continuous. The requirement that ψi is lower semicontinuous is added not out of necessity for the existence of equilibrium, but out of necessity for the abstract verification of the existence of equilibrium without explicit calculation.17. We now turn to the analysis of the set of equilibria of the BE game. The following proposition partitions the set of equilibria into two possible types and provides a partial characterization of each type. Proposition 2. There are two types of equilibria: symmetric lower bound and asymmetric lower bound. Symmetric lower bound equilibria are such that •x1=x2=x≥ρ; •The probability of an atom at any price x >ρis zero; •The probability of a tie at x is positive only if φi(x) = ψi(x,x)for each firm i; and •At most one firm can price higher than b x. Games 2025,16, 26 15 of 37 It will be useful to denote the lower bound of all prices such that firm i ’s supply is at least as big as the demand by τi. Formally, τi=inf{x:si(x)≥D(x)}. Condition 7. piD(pi)−ci(D(pi)) is strictly quasiconcave on [τi,pc]. The restriction of the strict quasiconcavity to only the interval [τi , pc] allows some additional freedom for the demand function at prices below τi . The properties of piD(pi)− ci(D(pi)) at prices pi<τi are irrelevant, as the the profit of the firm does not correspond to this expression at such prices. An immediate consequence of Condition 7is that there is a unique maximizer b pi of the front-side profit for each firm i . The next condition is a restatement of Assumption 5, which we still need to guarantee that no single firm will monopolize the market. Condition 8. b pi>ajfor each firm i. The following lemma shows that Conditions 1–8 imply Assumptions 1–7 of the general model. Lemma 6. If Conditions 1–8 are satisfied, then Assumptions 1–7 are satisfied. In this special case model, we can define the judo and safe price of each firm i in a slightly simplified way. The judo price of firm iis ri=inf{x|φi(x)>e ψi(x)}. Note that, based on the assumptions of this section, either φi(ri) = e ψi(ri) , or ri=aj . The safe price of firm iis ri=min{x|φi(x)≥ui}. 5.1. Special Results for Pure Strategy Equilibrium As we described in the general model section on pure strategy equilibrium, there is an intuitive way to classify the pure strategy equilibrium into two types. Particularly, Type B with x∗=max{a1 , a2} and Type C with x∗>max{a1 , a2} . The structure added in this section allows us to say more about these types of equilibrium. Particularly, in the following proposition, we present the necessary and sufficient conditions on supply, demand, and residual demand for the existence of pure strategy equilibrium. Proposition 6. The price x∗=ρ is a pure strategy Nash equilibrium if and only if one of the following three conditions holds: B.1 ρ=a1=a2and di(x′,ρ) = 0for each firm i, and all x′>ρ. B.2 ρ=ai>aj , di(x , ρ) = 0for all x≥ρ , uj(ρ , ρ) = φj(ρ) , and ψj(x , ρ)≤φj(ρ) for all x ≥ρ. C ρ∈(max{a1 , a2} , min{b p1 , b p2}] , ρ∈e Pi(ρ) for each firm i , and si(ρ) + sj(ρ)≤ D(ρ)for any firm i with αi(ρ,ρ)<1, where si(x) = min ϑi(x). The specificity of the model in this section allows us to break Type B equilibrium into two categories. Type B.1 requires that each firm’s infimum of average cost is the same, a1=a2 . In all Type B.1 pure strategy equilibrium, both firms make zero profit. There are two different possibilities for B.1 pure strategy pricing. The first case, akin to Classical Bertrand marginal cost pricing, is the case in which min{s1(ρ) , s2(ρ)} ≥ D(ρ) ,, and thus by Condition 6, di(ρ , ρ) = 0 for both i . The second case allows for a firm’s supply to not cover all of demand si(ρ)<D(ρ) as long as there is no residual demand for the other firm Games 2025,16, 26 16 of 37 dj(x′ , ρ) = 0 for all x′>ρ . This case relies on no consumers going to the higher-priced firm jwhen firm iprices at ρ. Type B.2 is pure strategy pricing that can only occur in the case that the two firms have different infima of their average costs. In such an equilibrium, both firms price at the higher of the two infimum average costs, and it must be that the firm with the lower infimum average cost obtains its full front-side profit. This means that the lower-cost firm must obtain a large-enough share of demand at this price to achieve its front-side profit, while the higher-cost firm makes zero profit. This type of pure strategy equilibrium was shown in Theorem 2 of Deneckere and Kovenock (1996) for a model of firms with asymmetric constant marginal costs. In all Type C equilibria, both firms make positive profits. Like B.1, Type C pricing has two different possibilities. The first case is similar to Classical Cournot market clearing prices ( s1(ρ) + s2(ρ) = D(ρ) ) above the infimum of average cost. The second case is such that total supply is less than demand s1(ρ) + s2(ρ)<D(ρ) and residual demand rationing does not permit a profit increase by pricing higher than ρ . This case of Type C pricing requires that each firm’s residual demand is sufficiently low at prices greater than ρ. Next, we establish a condition such that the only possible pure strategy equilibrium is Type C. Proposition 7. Index the firms such that a1≥a2 . If s1(a1) = 0, then only Type C pure strategy pricing equilibria are possible. From this proposition, in order to have a Type B equilibrium, we see that the minimum average cost of the higher-cost firm must be achieved by at least one positive production quantity (it could be equal to the infimum of the average cost at zero at well). This necessitates either a flat section of this firm’s marginal cost or a U-shaped marginal cost curve. Now we make a remark regarding the character of Type C pricing in a differentiable model. Remark 4. By adding standard differentiability assumptions, it becomes clear that the existence of a pure strategy equilibrium is remarkably fragile. If all key components of the model are differentiable (demand, residual demand, cost, and supply), then any Type C equilibrium must be such that s1(ρ) + s2(ρ) = D(ρ) . This basic insight on the non-existence of pure strategy equilibrium is first found by Shubik (1959). 5.2. Some Comparative Statics on Equilibrium Bounds We examine the effects of changes in residual demand rationing and individual supply on the bounds of equilibrium prices and payoffs. It should be clear that, for small changes to these components, the actual equilibrium payoffs need not follow the bounds (though perhaps they are likely to). However, for sufficiently large changes, when the new range of prices or profits does not intersect the old, we are able to precisely conclude how the actual equilibrium profits are affected. We begin by examining the role of changes in the demand side of the market. Specifically, we consider an increase in residual demand rationing of consumers, whereby at least one firm has an increase in their residual demand. We use residual demands di and d′ i ; let φ= (φ1 , φ2) and φ′ denote the corresponding front-side profits and ψ= (ψ1 , ψ2) and ψ′ denote the corresponding residual profits. Our first result establishes that an increase in the residual demand of either firm weakly increases the bounds on equilibrium payoffs. Proposition 8. If d′ i≥di, then r′≥r, r′≥r, φ′≥φand φ′≥φ. Games 2025,16, 26 17 of 37 Simply put, this proposition establishes that a shift to a more generous rationing scheme increases the bounds on the profits of each firm. Now we turn our attention to understanding the impact of changes in production technology. The following proposition shows conditions such that an increase in a firm’s supply, which could result from an increase in capacity or reduction in costs, will weakly reduce its judo and safe price and weakly reduce the profit of the other firm. Given supply functions s= (s1 , s2) and s′ , let φ= (φ1 , φ2) and φ′ denote the corresponding front-side profits and ψ= (ψ1,ψ2)and ψ′denote the corresponding residual profits. Proposition 9. Consider a weak cost reduction for firm i that results in a weak supply increase so that s′ i≥si . Suppose further that this results in a weak reduction in residual demand for firm j , dj(x , y)≥d′j(x , y) for all x≥y . Lastly, suppose that ci(q)−c′ i(q) is nondecreasing in q . Then r′≤r, r′≤r, φ′j≥φj, and φ′j≥φj. Note that this proposition does not make any statements regarding the profits of the firm whose supply shifts. The reason is that the effect is ambiguous. That is, a technological increase for a firm does not necessarily imply an increase in equilibrium profits for that firm. The direction of the change in profits is instead determined by the nature of the shift and market conditions. Two extreme examples illustrate this point. Consider a duopoly in which identical firms have constant marginal costs and capacities equal to half the monopoly quantity. In such a setting, pure strategy pricing can be sustained with each firm earning half the monopoly profit. Now consider a technology shock that increases the capacity of both firms so that their capacity is nonbinding at any price. This technological increase actually lowers each firm’s profit from something strictly positive to zero. The previous example involved an industry-wide capacity shock; however, the same result may occur as a result of a cost reduction for a single firm. Consider a duopoly in which firm 1 has constant marginal cost c= 0 while firm 2 has a strictly convex cost of production with supply s2(x)> 0 for all x> 0 and s2( 0 ) = 0. Suppose that the firms are not capacity constrained. It follows that ρ= 0. Note that by choosing a price x arbitrarily close to zero, the right continuity of s2 guarantees that ψ1(x , 0 )> 0. Thus, p1=p2= 0 cannot be an equilibrium, and any equilibrium must be in mixed strategies. Therefore, it must be that firm 2 receives its front-side profit in equilibrium with positive probability, in which case, it earns positive profits. Consider a technology increase of firm 2 that reduces its cost to zero. Then the game becomes the classic Bertrand duopoly with zero profits. Thus, a reduction in one firm’s cost may actually reduce its profits. These examples highlight that there are countervailing effects associated with a change in technology. There is a primary cost effect or capacity effect that allows a firm to earn a higher profit margin or produce more at any given price, both of which increase the profits of that firm. Alternatively, there is a secondary competition effect, whereby the change in cost or capacity alters the strategic environment and incentivizes the other firm to price more competitively, driving the prices of both firms down and thereby reducing profits. Whether the net change in profits is positive or negative depends on the relative strength of these two effects. 6. Concluding Remarks Reformulating price competition as an all-pay contest with externalities has allowed us to derive results on the nature of equilibrium for a class of BE pricing games far more general than using conventional methods. The broad range of underlying specifications includes many new specifications (as U-shaped average cost of production, minimal restriction on demand, demand rationing based on consumer search, and technology asymmetries Games 2025,16, 26 18 of 37 across firms) that expand the possible policy applications of the BE model. Further, we have presented a methodology that can be extended to analyze BE oligopoly including the possibility of incomplete information. Author Contributions: Conceptualization, B.A.A. and J.J.L.; Formal analysis, B.A.A. and J.J.L.; Writing—original draft, B.A.A. and J.J.L. Both authors have contributed to all phases of the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: There was no funding recieved for this project. Data Availability Statement: No new data were created or analyzed in this study. Conflicts of Interest: The authors declare no conflicts of interest. Appendix A Appendix A.1. Existence of Equilibrium (Proof of Proposition 1) We prove the existence of equilibrium using the following result based on the works of Reny (1999) and Bagh and Jofre (2006). Fact A1. If the mixed extension of a compact game is payoff secure and satisfies weak reciprocal upper semicontinuity (WRUSC), then the game has a mixed strategy Nash equilibrium. Verifying payoff security and WRUSC in the mixed extension of a game is burdensome, and so we rely on the recent results of Allen and Lepore (2014) and Allison et al. (2018), which provide easily verifiable conditions for games that imply that these properties are satisfied in the mixed extension. The following definition is from Allen and Lepore (2014). Let Xi and ui denote player i ’s strategy set and utility function, respectively. Define the discontinuity mapping Di:Xi→X−isuch that Di(xi) = {x−i∈X−i:ui(xi,x−i)is discontinuous in x−iat (xi,x−i)}. Definition A1. A game satisfies disjoint payoff matching (DPM) if, for each player i and all xi∈Xi, there exists a sequence {xk i} ⊂ Xisuch that (1) lim infkui(xk i,x−i)≥ui(xi,x−i)for all x−i∈X−i; and (2) lim supkDi(xk i) = ∅.22 Fact A2 (Allen and Lepore (2014)).If a compact game satisfies DPM, then the mixed extension of the game is payoff secure. The problem in using this definition of DPM to verify the existence of equilibrium in our model is that the payoff function ui can be discontinuous in pj through ψj at some prices p regardless of the choice of pi . 23 As such, it may be impossible to satisfy part 2 of the definition. A trivial modification is sufficient to generalize the existence result. Define the discontinuity map D′ i:Xi→X−isuch that D′ i(xi) = {x−i∈Di(xi):ui(xi,x−i)is not lower semicontinuous in x−iat (xi,x−i)}. By replacing Di with D′ i in the definition of DPM, the proof of the main result of Allen and Lepore (2014) is unaffected. Since ψi is assumed to be lower semicontinuous in the statement of Proposition 1, it follows that the discontinuity sets Di(xi) and D′ i(xi) coincide, and so we will be able to use this modified definition of DPM in our model. Games 2025,16, 26 19 of 37 Verifying that the BE game satisfies DPM is quite simple: for any price pi> 0, the sequence of deviations pk i=pi− 1 /k satisfies the definition, as these deviations result in either the same profit in the limit or a higher profit by guaranteeing the front-side profit if there would be a tie at pi . Further, the only points of discontinuity at which the payoffs are not lower semicontinuous are ties, and lim supkDi(xk i) = ∅ since Di(pk i) = {pk i} , and thus, Di(pk i)∩Di(pk′) = ∅ for all k=k′ . If pi= 0, then ui(pi , pj) = 0 for all pj and Di(pi) = ∅ , so pk i= 0 for all k trivially satisfies the definition. Thus, the mixed extension of the BE game is payoff secure. We now verify that the mixed extension of the BE game satisfies WRUSC. It will be useful to define the object ui(x) = lim supx′→xui(x′) and u to be the vector valued functions whose individual components are each ui. Fact A3 (Allison et al. (2018)).Let G= (N , X , u) be a compact game. Suppose that, (1) for each player i , there exists a sequence of Borel measurable functions Tk i : Xi→Xi such that, for all x∈X , lim infkui(Tk i(xi) , x−i)≥ui(x) , and (2) for any strategy profile x∈X , if there is some sequence {xk} with limku(xk) = u(x) , then u(x) = u(x) . Then the mixed extension of the game satisfies WRUSC. These two conditions intuitively state that (1) each player can deviate from any strategy so that, given any strategy profile of the other players, the deviating player obtains the highest feasible payoff near that strategy profile, and (2) if it is feasible for all players to simultaneously obtain their highest feasible payoff near a strategy profile, then the payoffs specify that they all receive such a payoff at that strategy. In the context of our BE model, (1) is satisfied by the same deviations as with DPM: Tk i(pi) = max{pi− 1 /k , 0 } . This sequence of deviations maximizes the firm’s chances of obtaining the front-side payoff, which corresponds to ui . For (2), observe that since ψi is lower semicontinuous in pj , then ψi=ψi , where ψi is derived from ψi as ui is derived from ui . Thus, ui(pi , pj) = ψi(pi , pj) = ui(pi , pj) for any pi>pj . If pi<pj , then ui(pi , pj) = φi(pi) = ui(pi , pj) . Any violation of condition (2) can thus only be at ties. Note that if φi(x)>ψi(x , x) for both firms i , it is not feasible that both firms i simultaneously obtain ui . If φi(x) = ψi(x , x) for both firms i , then ui(x , x) = ui(x , x) . Lastly, if φi(x)>ψi(x , x) and φj(x) = ψj(x , x) , then by the assumption in the statement of the proposition, αi(x , x) = 1, and so ui(x , x) = φi(x) = ui(x , x) and uj(x , x) = φj(x) = uj(x , x) . Thus, condition (2) is satisfied. We conclude that the mixed extension of the BE game satisfies WRUSC. Appendix A.2. Proof of Lemmas and Propositions Lemma A1. suppj∈[x0,x]ψi(x,pj)is right upper semicontinuous in x at x =x0. Proof of Lemma A1. Let x0≥ 0 and observe that suppj∈[x0,x]ψi(x , pj) = ψi(x0 , x0) at x=x0 . Let ε> 0 and xn→x0 be such that xn>x for each n . For each n , let yk n be a sequence in k such that limkψi(xn , yk n) = suppj∈[x0,xn]ψi(xn , pj) . For each n , let K(n) be such that ψi(xn,yk n)−suppj∈[x0,xn]ψi(xn,pj)<ε/ 3 for all k>K(n) and choose yn=yk n for some k>K(n). Then note that sup pj∈[x0,xn] ψi(xn,pj)−ψi(x0,x0)≤sup pj∈[x0,xn] ψi(xn,pj)−ψi(xn,yn) + ψi(xn,yn)−ψi(x0,x0) <ε 3+ψi(xn,yn)−ψi(xn,x0) + ψi(xn,x0)−ψi(x0,x0). Since yn∈[x0 , xn] , it follows that yn→x0 . Thus, by the right upper semicontinuity of ψi in pj , there exists an N1 such that ψi(xn , yn)−ψi(xn , x0)<ε/ 3 for all n>N1 . Similarly, by Games 2025,16, 26 20 of 37 the continuity of ψi in pi , there exists an N2 such that |ψi(xn,x0)−ψi(x0,x0)|<ε/ 3 for all n>N2. Thus, for all n>max{N1,N2}, it must be that sup pj∈[x0,xn] ψi(xn,pj)−ψi(x0,x0)<ε 3+ψi(xn,yn)−ψi(xn,x0) + ψi(xn,x0)−ψi(x0,x0)<ε. Therefore, sup pj∈[x0,xn] ψi(xn,pj)<ψi(x0,x0) + ε, so by definition, suppj∈[x0,x]ψi(x,pj)is right upper semicontinuous in xat x=x0. Proof of Lemma 1. Suppose to the contrary that µi([ 0, b x]) < 1 for each firm i . Then for any firm i and any x>b x in support of µi , Assumption 6guarantees that φi(b x)>φi(x) and ψi(b x,pj)≥ψi(x,pj)for all pj≤b x. Observe that Zui(b x,pj)dµj≥(1−Fj(b x))φi(b x) + Z[0,b x]ψi(b x,pj)dFj ≥(1−Fj(b x))φi(b x) + Z[0,b x]ψi(x,pj)dFj and (1−Fj(b x)) >0 since µj([0, b x]) <1. Thus, we have Zui(b x,pj)dµj≥(1−Fj(b x))φi(b x) + Z[0,b x]ψi(x,pj)dFj >(1−Fj(b x))φi(x) + Z[0,b x]ψi(x,pj)dFj ≥(1−Fj(x) + µj({x}))φi(x) + Z[0,x)ψi(x,pj)dFj =Zui(x,pj)dµj. The last inequality follows from the fact that Fj is a CDF and thus nondecreasing and from Assumption 1guaranteeing that φi(x)≥ψi(x , x′) for x′≤x . This contradicts all x>b x as equilibrium strategies. Proof of Lemma 2. Let µ be an equilibrium and x be such that φi(x)>ψi(x , x) and µi({x})> 0. Suppose that µj({x})> 0 and αi(x , x)< 1. Consider a sequence of deviations by firm ito e µn idefined by e µn i(E) = (µi(E∪{x})if x−δn∈E µi(E∖{x})otherwise , where each δn is chosen so that 0 <δn< 1 /n and µj({x−δn}) = 0. That is, e µn i is the measure created from µi by shifting all mass from the price x to the price x−δn . Then note that Zui(p)de µn i×µj=Zui(p)dµ+µi({x})Zui(x−δn,pj)−ui(x,pj)dµj. We will show that limnRui(x−δn,pj)−ui(x,pj)dµj> 0 for sufficiently large n , which will guarantee a profitable deviation for firm i , violating µi as an equilibrium strategy. Note that Games 2025,16, 26 21 of 37 ui(x−δn,pj)−ui(x,pj) =              ψi(x−δn,pj)−ψi(x,pj)if pj<x−δn φi(x−δn)−ψi(x,pj)if x−δn≤pj<x φi(x−δn)−αi(x,x)φi(x) −(1−αi(x,x))ψi(x,x)if pj=x φi(x−δn)−φi(x)if pj>x . It follows that the pointwise limit as n→∞is lim nui(x−δn,pj)−ui(x,pj)=     0 if pj<x (1−αi(x,x))(φi(x)−ψi(x,x))if pj=x 0 if pj>x . Thus , since |ui|≤φi(b pi), then by the Lebesgue dominated convergence theorem, lim nZui(x−δn,pj)−ui(x,pj)dµj=Zlim nui(x−δn,pj)−ui(x,pj)dµj =µj({x})(1−αi(x,x))(φi(x)−ψi(x,x)). Since φi(x)>ψi(x , x) and αi(x , x)< 1, µn i is a profitable deviation for firm i for sufficiently large n , violating µ as an equilibrium. We conclude that either αi(x , x) = 1 or µj({x}) = 0. From Lemma 1, we know that µi([ 0, b x]) = 1. Observe that if x∈(ρ , b x] , it must be that µ({(x , x)}) = 0 since φi(x)>ψi(x , x) for each firm i at any price x∈(ρ , b x] and αi(x , x)< 1 for some firm iat any price x. Next, we show that the equilibrium is invariant to the choice of α at prices x∈(ρ , b x] . Let µ be an equilibrium given the sharing rule α with expected profits v= (v1 , v2) and consider another sharing rule α′ such that α(x , x) = α′(x , x) for all x≤ρ . Let ui(x , µj) denote firm i ’s expected payoff when choosing a price x given α and u′ i(x , µj) the corresponding payoff given α′ . To show that µ is an equilibrium for the game with sharing rule α′ , it will suffice to show that, for each player i , (i) u′ i(x , µj) = viµi -almost everywhere and (ii) u′ i(x,µj)≤vifor all prices x. (i) Note that the sharing rule does not influence the payoffs at any price x such that µj({x}) = 0, and so ui(x , µj) = u′ i(x , µj) at all such prices. Further, at all prices x≤ρ , ui(x , µj) = u′ i(x , µj) since α(x , x) = α′(x , x) . The first part of this lemma demonstrates that µi({x}) = 0 for all x∈(ρ , b x] such that µj({x})> 0. Since µj has at most countably many atoms, then µi({x : µj({x})> 0 }) = 0. It follows that u′ i(x , µj) = viµi -almost everywhere. (ii) As we have shown in part (i), ui(x , µj) = u′ i(x , µj) except possibly at prices x∈(ρ , b x] such that µj({x})> 0. Since price above b x is weakly dominated by b x , it is sufficient to examine prices in [ 0, b x] . Consider any such price x and let {xk} be a sequence such that xk→x , xk<x for all k , and µj({xk}) = 0 for all k . Then note that the continuity of φi and ψi in pi on [ 0, b x] from Assumption 7implies that limku′ i(xk , µj)≥u′ i(x , µj) . Since µj({xk}) = 0 for all k , then ui(xk , µj) = u′ i(xk , µj) for all k . If u′ i(x , µj)>vi , then ui(xk , µj)>vi for sufficiently large k , violating µi as an equilibrium strategy with the sharing rule α. Therefore, u′ i(x,µj)≤vifor all x. We conclude that µis an equilibrium given the sharing rule α′. Proof of Lemma 3. We first argue that, in any equilibrium, xi≥ai for at least one firm i . Suppose to the contrary that xi<ai for each firm i . Then Assumption 2implies that Rui(x , pj)dµj=φi(x) = 0 and ψj(pj , x) = φj(pj) for all pj≥x and all x∈[xi , ai) . It follows that either player i could choose a price of b pi and receive a payoff of φi(b pi)> 0 with positive probability since µj([xj , aj)) > 0. This contradicts prices in [xi , ai) as equilibrium strategies. We conclude that xi≥aifor at least one firm i. Games 2025,16, 26 22 of 37 Let µ be an equilibrium with xi<xj . Note that Rui(x , pj)dµj=φi(x) for all x∈[xi , xj) . If xj>ai , then Assumptions 2and 3imply that φi , and thus Rui(x , pj)dµj is strictly increasing on [ai , xj) , violating prices in [xi , xj) as equilibrium strategies for firm i . Thus, xj≤ai and so xi<ai . Assumption 2thus implies that φi(x) = 0 for all x∈[xi , xj) , so firm i ’s equilibrium profit must be zero. From the result proved immediately above, it must be that xj≥aj . Suppose that xj<ai . Then Assumption 2implies that Ruj(x , pi)dµi=φj(x) for all x∈[xj , ai) . Therefore, since xj≥aj , Assumption 3guarantees that φj , and thus, uj is strictly increasing on [xj , ai) , violating these prices as equilibrium strategies for firm j . Thus, it must be that xj=ai . If µj is nondegenerate, then there is some x>ai such that firm j prices strictly higher than x with positive probability. If firm i sets a price of this x , then with positive probability, firm i will receive φi(x) , which is strictly positive by Assumption 3, contradicting zero as its equilibrium profit. Therefore, µj is degenerate with µj({ai}) = 1. Since there is a positive probability that firm i chooses a price x<ai , Assumption 2implies that there is a positive probability that firm j will obtain a profit φj(b pj) if it sets its price at b pj . Consequently, firm j ’s equilibrium profit must be positive, and thus aj<ai. Suppose that ai<ρ=ρj , since aj<ai by Assumption 4. Assumptions 3and 4 guarantee that Ruj(x , pj)dµj=φj(x) for all x∈(ai , ρ) and that φj is strictly increasing on this interval. Thus, it must be that ai=ρ. Finally, suppose that µi([ρ , ρ+ε)) = 0 for some ε> 0. Then firm j could set any price x∈[ρ , ρ+ε) and still receive φj(x) with certainty. Since φj is strictly increasing, this would violate ρ as an equilibrium strategy for firm j . We conclude that µi([ρ , ρ+ε)) > 0 for any ε>0. Proof of Lemma 4. Lemma 3implies that any nondegenerate mixed strategy equilibrium requires that x1=x2=x . Let µ be an equilibrium with x1=x2=x . Recall from the proof of Lemma 3that at least one firm imust have xi≥ai. First, we show that x≥max ai . Suppose to the contrary that x<ai for some firm i . Then it must be that x≥aj . By Assumption 2, firm j can set any price x∈[x , ai) and obtain a profit of φj(x) with certainty. From Assumption 3, φj is strictly increasing on this interval, contradicting these as equilibrium strategies. Therefore, it must be that x≥max ai. Second, we show that x≥ρ . Suppose to the contrary that x<ρ . Then by Assumption 4, ψi(ρ , pj) = φi(ρ) for all pj<ρ . Consequently, either firm i could choose any price x∈(x , ρ) and earn Rui(x , pj)dµj=φi(x) . Since x≥ai , Assumption 3guarantees that φi(x) is strictly increasing on this interval, violating these prices as equilibrium strategies. Therefore, it must be that x≥ρ. Third, we show that neither firm i can have an atom at an x if φi(x)>ψi(x , x) . Suppose to the contrary that firm j has an atom at x , µj({x})> 0, noting that x≤b x since it is in support of each firm’s strategy. From Lemma 2, we know that µi({x}) = 0; however, as we have just shown, xi=x . Thus, µi((x , x+δ)) > 0 for all δ> 0. We will show that there is some x′<x and neighborhood (x , x+δ) such that Rui(x′ , pj)dµj>Rui(x , pj)dµj for all x∈(x,x+δ). Define β=µj({x})> 0 and let ε> 0 be such that ε<β(φi(x)−ψi(x,x)) . If firm i sets a price x<x , then its profit will be φi(x) with certainty. Since x≤b x , Assumption 7guarantees that φi is left continuous, so there exists a δ1> 0 such that |φi(x)−φi(x)|<ε/ 2 for all x∈(x−δ1 , x) . Note that Rui(x , pj)dµj≤( 1 −β)φi(x) + βsuppj∈[x,x]ψi(x , pj) at any price x>x and observe that suppj∈[x,x]ψi(x , pj) = ψi(x , x) at x=x . Note that suppj∈[x,x]ψi(x , pj) is right upper semicontinuous by Lemma A1. If x=b x , then Assumption 6 guarantees that φi(x)<φi(x) for all x>x , and thus, φi is right upper semicontinuous at x . Alternatively, if x<b x , then Assumption 7guarantees that Games 2025,16, 26 23 of 37 φi is continuous and thus right upper semicontinuous. In either case, the function ( 1 −β)φi(x) + βsuppj∈[x,x]ψi(x , pj) is right upper semicontinuous in x at x , so there exists δ2>0 such that (1−β)φi(x) + βsup pj∈[x,x] ψi(x,pj)<(1−β)φi(x) + βψi(x,x) + ε 2 for all x∈(x,x+δ2). Let x′∈(x−δ1,x)and note that φi(x′)>φi(x)−ε/2. Thus, φi(x′)>(1−β)φi(x) + βφi(x)−ε 2 = (1−β)φi(x) + βψi(x,x) + β[φi(x)−ψi(x,x)] −ε 2. For all x∈(x,x+δ2), we have (1−β)φi(x) + βψi(x,x) + β[φi(x)−ψi(x,x)] −ε 2 >(1−β)φi(x) + βsup pj∈[x,x] ψi(x,pj) + β[φi(x)−ψi(x,x)] −ε and therefore, φi(x′)>(1−β)φi(x) + βsup pj∈[x,x] ψi(x,pj) + β[φi(x)−ψi(x,x)] −ε. Since ε<β(φi(x)−ψi(x , x)) , this implies that Rui(x′ , pj)dµj>Rui(x , pj)dµj for all x∈(x , x+δ2) . This violates such prices as equilibrium strategies for firm i . We conclude that µj({x}) = 0, so neither firm ican have an atom at xif φi(x)>ψi(x,x). Proof of Proposition 3. We will prove the proposition for the two types of equilibria from Proposition 2separately. Case 1: Let µbe a symmetric lower bound equilibrium. For this case, we first show that x∈[r , r] in two parts. First, we argue that x≤r . Suppose to the contrary that x>r . From Lemma 2, at most one firm can have an atom at x . Without loss of generality, let firm i be such that that x is in support of µi and µj({x}) = 0. Choose {xk i}in support of µisuch that xk i→x; then note that lim k→∞Zx xui(xk i,pj)dµj=Zx xψi(x,pj)dµj since ψi is continuous in pi from Assumption 7. Next, since each xk i is in support of µi , it must be that each Rx xui(xk i,pj)dµj=u∗ i. Note that Zx xψi(x,pj)dµj≤Zx xe ψi(pj)dµj by definition of e ψi . By definition of ri and the fact that x>ri , φi(ri)>e ψi(x) for all x≥ri for each firm i . Further, Assumptions 2and 3imply that φi is nondecreasing, so φi(x)>e ψi(x) for all x≥x . This implies that Rx xe ψi(pj)dµj<φi(x) , and thus u∗ i<φi(x) . This violates µi as an equilibrium strategy since firm i has a profitable deviation to x−ε for sufficiently small εthat would guarantee a payoff of φi(x). We conclude that x≤r. Second, we argue that x≥r . Suppose to the contrary that x<r . Let firm i be such that ri=r . By definition of ri and the continuity of φi from Assumption 7, it must be that ui=φi(r) . From Lemma 4, x≥ρ , and since ρ≥ai , Assumption 2implies that φi(x)>φi(x) for all x>x . Thus, φi(x)<φi(r) = ui . The continuity of φi and ψi in pi Games 2025,16, 26 24 of 37 thus guarantee that φi(x)<ui and ψi(x , x′)<ui for all prices x∈[x , r) with x′≤x , so Rui(x , pj)dµj<ui for all such x . This violates all x∈[x , r) as equilibrium strategies. We conclude that x≥r. It follows from Lemma 4that each firm i ’s equilibrium expected profit is u∗ i=φi(x) . The statement of the proposition thus follows from x∈[r , r] and the facts that φi is strictly increasing on [ρ,b pi]and that ρ≤r. Case 2: Let µbe an asymmetric lower bound equilibrium. First, consider player i . The lower bound payoff ui= 0, which implies that φi(r) = 0. Further, we know from Lemma 3that u∗ i= 0. By construction, φi(r)≥ 0; therefore, u∗ i∈[φi , φi] . Next, consider player j . From Lemma 3, u∗ j=φj(ρ) = φj(ai) . This payoff is possible for player j for any pricing by player i . Thus, u∗ j=uj=φj(r) . By construction, φj(r)≥0; therefore, u∗ j∈[φj,φj]. Proof of Lemma 5. We use pi(Fj) and pi(Fj) to denote the smallest and largest conditional residual maximizer, respectively. That is, pi(Fj) = min e Pi(Fj) and pi(Fj) = sup e Pi(Fj) , where the right continuity of Fj ensures that e Pi(Fj) contains a minimal element, while it need not contain a maximal element. Let µ be an equilibrium with the corresponding CDF’s F . If x=ρ , then from Lemmas 3 and 4, either the equilibrium is degenerate or xi<xj for some firm i . If the equilibrium is degenerate, then the statement of Proposition 4(in Section 4) guarantees that ρ∈e Pi(ρ) for some firm i . It trivially follows that min{p1(F2) , p2(F1)} ≤ x≤max{p1(F2) , p2(F1)} . Alternatively, xi<xj for some firm i ; then Lemma 3guarantees that µj({ρ}) = 1 and u∗ i= 0. In order for µ to be an equilibrium, firm i cannot have any profitable deviations, so it must be that ψi(x , ρ) = 0 for all x≥ρ . Thus, by definition, e Pi(ρ) = [ρ , ∞) , so min{p1(F2),p2(F1)} ≤ x≤max{p1(F2),p2(F1)}. Finally, let x>ρ and suppose that either x<min{p1(F2) , p2(F1)} or x>max{p1(F2) , p2(F1)} . From Lemma 2, at most one firm may have an atom at x . Let firm i be such that xi=x and µj({x}) = 0. Since x is in support of firm i ’s strategy, we may choose {xk i}in support of µisuch that xk i→x; then note that lim k→∞Zx xui(xk i,pj)dµj=Zx xψi(x,pj)dµj =EFj[ψi(x,pj)|pj≤x]. Since each xk i is a best response for firm i , this implies that x is also a best response for firm i. It follows from our supposition that x/∈e Pi(Fj). Note that, for any price x, ui(x,Fj)≥(1−Fj(x))φi(x) + Z[x,x]ψi(x,pj)dFj = (1−Fj(x))φi(x) + Fj(x)EFj[ψi(x,pj)|pj≤x]. By definition of e Pi(Fj) , EFj[ψi(x , pj)|pj≤x]>EFj[ψi(x , pj)|pj≤x] for all x∈e Pi(Fj) . Thus, ui(x , Fj)>ui(x , Fj) for all x∈e Pi(Fj) since φi(x)≥ψi(x , pj) for all pj . This contradicts x as a best response. We conclude that min{p1(F2),p2(F1)} ≤ x≤max{p1(F2),p2(F1)}. Proof of Proposition 4. First, observe that, in any equilibrium (p∗ 1 , p∗ 2) with corresponding profits (v∗ 1 , v∗ 2) , it must be that v∗ i≥φi(min{p∗ j , b x}) . To see why, note that Assumption 7guarantees that φi is left continuous at min{p∗ j , b x} . Thus, firm i can guarantee itself a payoff of φi(min{p∗ j , b x}−ε) by deviating to pi=min{p∗ j , b x}−ε , with the guarantee that φi(min{p∗ j,b x}−ε)→φi(min{p∗ j,b x}). Games 2025,16, 26 31 of 37 follows that ψi(x′ , ρ) = πi(x′ , 0 ) = 0 for all x′>ρ . Thus, neither firm possesses a profitable deviation, so each firm pricing at x∗=ρis an equilibrium. B.2 Consider firm i with ai>aj . As noted in the (B.1) case, firm i has no profitable deviations. From the assumptions of B.2, observe that uj(ρ , ρ) = φj(ρ) . From Assumptions 2and 3, φj is nondecreasing on [ 0, b pj] , and so φj(ρ)≥φj(x) for all x<ρ . Lastly, the assumption in B.2 that ψj(x , ρ)≤φj(ρ) for all x≥ρ guarantees that there are no profitable deviations for firm j to prices higher than ρ . Thus, neither firm possesses a profitable deviation, so each firm pricing at x∗=ρis an equilibrium. C. From Condition 6, di(ρ , ρ) = D(ρ)−Qj(ρ) . Since Qj(ρ)≤sj(ρ) , it follows that di(ρ , ρ)≥D(ρ)−sj(ρ) . Observe that if αi(ρ , ρ) = 1, then by definition ui(ρ , ρ) = φi(ρ , ρ) . Suppose that αi(ρ , ρ)< 1. Then from the assumption of C, si(ρ) + sj(ρ)≤D(ρ) , so D(ρ)−sj(ρ)≥si(ρ) . It follows that πi(ρ , Qi(ρ)) = πi(ρ , min{si(ρ) , di(ρ , ρ)) for each firm i , and so ui(ρ , ρ) = φi(ρ) . As noted above, φi is nondecreasing on [ 0, b pi] , so there are no profitable deviations to prices x<ρ . Further, since ρ∈e Pi(ρ) , there are no profitable deviations to prices x>ρ . Thus, neither firm possesses a profitable deviation, so each firm pricing at x∗=ρis an equilibrium. Next, we prove that any pure strategy equilibrium must satisfy either B.1, B.2, or C. Proposition 4implies that any pure strategy equilibrium must be symmetric with x∗=ρ. Further, by definition, it must be that ρ≥aiand ρ<b pifor each firm i. B.1 Suppose that x∗=ρ=a1=a2 and that di(x , ρ)> 0 for some x>ρ for some firm i . Then since x>ai , from Condition 3, there is some quantity z∈( 0, ρ) such that x>ci(z) z , so πi(x , z)> 0. Since ψi(x , ρ)≥πi(x , z) , it follows that ψi(x′ , ρ)> 0. This contradicts x∗ as an equilibrium since φi(x∗) = 0. B.2 Suppose that x∗=ρ=ai>aj . If di(x , ρ)> 0 for some x>ρ , then the preceding argument for the (B.1) case applies and rules out x∗ as an equilibrium. If uj(ρ , ρ)<φj(ρ) , then by continuity of φj by Assumption 7, there exists a price x<ρ such that uj(x , ρ) = φj(x)>uj(ρ , ρ) . This contradicts x∗ as an equilibrium. Lastly, suppose that ψj(x , ρ)> φj(ρ) for some x≥ρ . Then since ui(x , ρ)≥ψi(x , ρ) , then x is a profitable deviation from x∗, violating x∗as an equilibrium. C. Suppose that ρ∈(max{a1 , a2} , min{b p1 , b p2}] . If ρ/∈e Pi(ρ) for some firm i , then by definition, any price x∈e Pi(ρ) is a profitable deviation for firm i , violating x∗ as an equilibrium. Lastly, suppose that si(ρ) + sj(ρ)>D(ρ) for some firm i with αi(ρ , ρ)< 1. Then since di(ρ , ρ) = D(ρ)−sj(ρ) by Condition 6, it must be that di(ρ , ρ)<si(ρ) , so qi(ρ , ρ)/∈arg maxzπi(ρ , z) . Therefore, since ψi(ρ , ρ) = πi(ρ , qi(ρ , ρ)) and αi(ρ , ρ)< 1, it must be that ui(ρ , ρ) = αi(ρ , ρ)φi(ρ)+ ( 1 −αi(ρ , ρ))ψi(ρ , ρ)<φi(ρ) . Then, since φi is continuous by Assumption 7, there exists a price x<ρ such that ui(x , ρ) = φi(x)>ui(ρ , ρ) . This violates x∗as an equilibrium. Proof of Proposition 7. Without loss of generality, we assume a1≥a2 . Suppose to the contrary that there is an equilibrium price x∗=a1 with s1(x∗) = 0. Then Q1(x∗) = 0, so Condition 6guarantees that d2(x , a1) = D(x) for all x>a1 . It follows immediately that ψ2(x , a1) = φ2(x) for all x≥a1 . From Condition 8, b p2>a1 . Therefore, by definition of b p2 , we have ψ2(b p2,a1) = φ2(b p2)>φ2(a1), contradicting x∗=a1as an equilibrium price. Proof of Proposition 8. Since D and ci are unchanged, then φi(x) = φ′ i(x) for all x . Note that ψi(pi,pj) = max z∈[0,min{ki,di(pi,pj)}]xz −ci(z)and ψ′ i(pi,pj) = max z∈[0,min{ki,d′ i(pi,pj)}]xz −ci(z). Games 2025,16, 26 32 of 37 It follows immediately that ψ′ i≥ψi . The fact that r′ i≤ri and r′ i≤ri follows immediately from their definitions. Proof of Proposition 9. The fact that r′j≤rj and r′j≤rj follows directly from Proposition 8since any change in firm i ’s supply has no impact on the front-side profit of firm j , so φj(x) = φ′j(x) for all x . It remains to show that r′ i≤ri and r′ i≤ri . Observe that di(x,y) = d′ i(x,y)for all prices x≥y. Define si(x) = sup ϑi(x) , with s′ i defined analogously for π′ i . Let Qi(x) = min{si(x) , D(x)} and qi(x , y) = min{si(x) , di(x , y)} , with Q′ i and q′ i defined analogously. Then note that φi(x) = πi(x,Qi(x)). Note that siis nondecreasing. Part 1: r′ i≤ri The proof that r′ i≤ri is conducted in four steps. In Step 1, we argue that πi(b pi , z)< πi(b pi , D(b pi)) for all z<D(b pi) and then use that fact to argue that we show that ri<b pi . In Step 2, we show that qi(x , y)≤Qi(y) for all x≥y>ri . In Step 3, we argue that qi(x , y)≥ q′ i(x,y)for all x≥y>ri, implying that q′ i(x,y)≤Qi(ri)for all x≥y>ri. Finally, in Step 4, we show that if r′ i>ri , we can find prices x≥y>ri such that ci(Qi(x)) −c′ i(Qi(x)) < ci(q′ i(e x , x)) −c′ i(q′ i(e x , x)) , contradicting the assumption that ci(q)−c′ i(q) is nondecreasing in q. Step 1: We first show that πi(b pi , z)<πi(b pi , D(b pi)) for all z<D(b pi) . Suppose to the contrary that πi(b pi , z)≥πi(b pi , D(b pi)) for some z<D(b pi) . As noted in the proof of Proposition 6, Qi(b pi) = D(b pi) . Thus, πi(b pi , z) = φi(b pi) . Since φi is continuous by Assumption 7, there exists a price x>b pi such that z<D(x) . Observe that φi(x)≥ πi(x , z) = xz −ci(z)>b piz−ci(z) = φi(b pi) . This contradicts b pi as the maximizer of φi . We conclude that πi(b pi,z)<πi(b pi,D(b pi)). We next argue that ri<b pi . Suppose to the contrary that ri≥b pi . Since aj<b pi by Assumption 5, we may choose a strictly increasing sequence {yk} such that y0>aj and yk→b pi . By definition of ri , φi(x)≤e ψi(x) for all x<ri . Thus, since e ψi(x) is nonincreasing as noted earlier, this implies that e ψi(y0)≥φi(yk) for all k . By continuity of φi by Assumption 7, limkφi(yk) = φi(b pi) , and so e ψi(y0)≥φi(b pi) . Let e x∈e Pi(y0) and note that ψi(e x , y0)≤φi(e x) by Assumption 1. Observe that φi(b pi)≥φi(e x)≥e ψi(y0)≥φi(b pi) . Since b pi is the unique maximizer of φi , it follows that e x=b pi . As demonstrated in the proof of Proposition 6, Qi(b pi) = D(b pi) and sj(b pi)> 0 since b pi>aj . By Condition 6, this implies that di(b pi , y0)<D(b pi) , so qi(b pi , y0)<D(b pi) . From the first paragraph of this step, this implies that πi(b pi , qi(b pi , y0)) <πi(b pi , D(b pi)) and thus that ψi(b pi , y0)<φi(b pi) . A contradicts to e ψi(y0)≥φi(b pi). We conclude that ri<b pi. Step 2: We show that qi(x , y)≤Qi(y) for all x≥y>ri . Suppose to the contrary that qi(x,y)>Qi(y)for some x≥y>ri. Note that ψi(x,y) = max z∈[0,min{ki,di(x,y)}]πi(x,z). Since Qi(ri)<qi(x,y)≤di(x,y), it follows that ψi(x,y)≥xQi(y)−ci(Qi(y)) ≥yQi(y)−ci(Qi(y)) =φi(y). This contradicts the definition of ri as ri=sup{x|φi(x)≤e ψi(x)} . We conclude that qi(x,y)≤Qi(ri)for all x≥y>ri. Step 3: We argue that qi(x , y)≥q′ i(x , y) for all x≥y>ri . Let x≥y>ri and suppose to the contrary that qi(x , y)<q′ i(x , y) . Then since q′ i(x , y)≤di(x , y) , it must be that qi(x , y)<di(x , y) . It follows that qi(x , y) = si(x) , and so ψi(x , y) = πi(x , si(x)) = φi(x) . Games 2025,16, 26 33 of 37 Since si is nondecreasing and di is nonincreasing by Condition 5, it follows that si(pi)< di(pi , pj) for all pi<x such that pi≥pj>ri . Thus, ψi(pi , pj) = πi(pi , si(pi)) = φi(pi) for all pi and pj such that x>pi≥pj>ri . Thus, e ψi(pj)≥ψi(pj , pj) = φi(pj) for all pj∈(ri , x) , contradicting the definition of ri . We conclude that qi(x , y)≥q′ i(x , y) for all x≥y>ri. In summary we have now established that, since si is nondecreasing, then for any x≥y>ri, it follows that q′ i(x,y)≤qi(x,y)≤Qi(y). Step 4: We argue that r′ i≤ri . Suppose to the contrary that r′ i>ri . Then for any price x∈(ri , r′ i) , it must be that φ′ i(x)≤e ψ′ i(x) . Let x∈(ri , r′ i) and e x∈e P′ i(x) and note that from above, qi(e x,x)≤Qi(x). Note that e ψ′ i(x)is nonincreasing in xsince e ψ′ i(x) = max pi max z∈[0,min{ki,di(pi,x)}]πi(pi,z) and di(pi , x) is nonincreasing in x by Condition 5. From above, we may choose the price x∈(ri , r′ i) such that x<b pi . Since φi is strictly increasing on (ai , b pi) by Assumption 3 and from continuity of D from Condition 4, we may choose x so that φ′ i(x)≤e ψ′ i(x) and φi(x)>e ψi(x). Thus, we have φ′ i(x)−φi(x)<e ψ′ i(x)−e ψi(x). (A1) Note that φi(x) = xQi(x)−ci(Qi(x)) and φ′ i(x)≥xQi(x)−c′ i(Qi(x)) . Putting these together, we have φ′ i(x)−φi(x)≥ci(Qi(x)) −c′ i(Qi(x)). (A2) Next , note that e ψ′ i(x) = e xq′ i(e x , x)−c′ i(q′ i(e x , x)) and e ψi(x)≥e xq′ i(e x , x)−ci(q′ i(e x , x)) . Putting these together yields e ψ′ i(x)−e ψi(x)≤ci(q′ i(e x,x)) −c′ i(q′ i(e x,x)). (A3) The inequalities (A1), (A2), and (A3) together imply that ci(Qi(x)) −c′ i(Qi(x)) <ci(q′ i(e x,x)) −c′ i(q′ i(e x,x)), which contradicts the assumption that ci(z)−c′ i(z) is nondecreasing in z≥ 0. We conclude that r′ i≤ri. Part 2: r′ i≤ri The proof that r′ i≤ri is also done by contradiction. Suppose to the contrary that r′ i>ri . Recall that ui=suppiinfpjui(pi , pj) . As noted above, ψi is nonincreasing in pj . Further, by Assumption 1, we can conclude that infpjui(pi , pj) = ψi(pi , pi) , and thus ui=suppiψi(pi , pi) . Additionally, since φi is continuous by Assumption 7, it follows that φi(ri) = ui. Let {xk} be a sequence such that ψ′ i(xk , xk)→u′ i . We may without loss of generality choose this sequence such that xk→x∗ for some price x∗ . It follows that u′ i=π′ i(x∗ , q∗) , where q∗=limkq′(xk , xk) . By definition of ui , it must be that ui≥ψi(xk , xk) for all k . Further ψi(xk,xk) = πi(xk,qi(xk,xk)) ≥πi(xk,q′ i(xk,xk)), and so ui≥πi(x∗,q∗). Therefore, u′ i−ui≤π′ i(x∗,q∗)−πi(x∗,q∗) = ci(q∗)−c′ i(q∗). (A4) We briefly argue that x∗≥r′ i . To see this, suppose to the contrary that x∗<r′ i . Then note that φ′ i(r′ i) = u′ i=π′ i(x∗ , q∗)≤φ′ i(x∗) . By definition of ai , it must be that a′ i≤ai , and since ai≤ri<r′ i , it follows that r′ i>a′ i . Thus, Assumption 3implies that φ′ i is strictly Games 2025,16, 26 34 of 37 increasing on (max{a′ i , x∗} , r′ i) . Since φ′ i(x) = 0 for x<a′ i by Assumption 2, it follows that φ′ i(x∗)<φ′ i(r′ i), a contradiction. We conclude that x∗≥r′ i. We will now argue that q∗≤Qi(x)for all xin some neighborhood (ri,ri+δ). We begin by arguing that Qi(y)≥di(x , x) for all prices x>y>ri . Suppose to the contrary that Qi(y)<di(x , x) for some x>y>ri . Recall that φi(ri) = πi(ri , Qi(ri)) . Next, since πi(x , z) is quasiconcave in z by Condition 2, it follows that ψi(x , x) = xqi(x , x)− ci(qi(x,x)), where qi(x,x) = min{si(x),di(x,x)}. Note that by definition of uiand si, ui≥ψi(x,x) =xqi(x,x)−ci(qi(x,x)) ≥xQi(y)−ci(Qi(y)). Since y>ri and ri≥ai , it follows that φi(y)> 0, and so Qi(y)> 0. Thus, xQi(y)>yQi(y) . Therefore, ui≥xQi(y)−ci(Qi(y)) >yQi(y)−ci(Qi(y)) =φi(y) ≥φi(ri) =ui. This is a contradiction. We conclude Qi(y)≥di(x,x)for all prices x>y>ri. Now, suppose to the contrary that there exists a sequence {yn} with yn→ri and yn>ri such that q∗>Qi(yn) for all n . Since x∗≥r′ i>ri , we may without loss of generality assume that yn<min{xk , x∗} for all k and n . Thus, from above, Qi(yn)≥di(xk , xk) for all k and n . Since q′ i(xk , xk)≤di(xk , xk) , this implies that Qi(yn)≥q∗ , a contradiction. We conclude that q∗≤Qi(x)for all xin some neighborhood (ri,ri+δ). Choose such a δ. Now observe that by definition of r′ i , φ′ i(ri)≤u′ i for any price x∈(ri , r′ i) . Thus, for any price x∈(ri , b pi) , it must be that φi(x)>ui since φi is strictly increasing by Assumption 3. Recall that ri≤ri , and as shown above, ri<b pi , so (ri , b pi) is nonempty. Let x∈(ri , min{r′ i , b pi , ri+δ}) with δ> 0 picked such that q∗≤Qi(x) for all x in some neighborhood (ri,ri+δ). Note that φ′ i(x)−φi(x)<u′ i−ui. Observe that φi(x) = xQi(x)−ci(Qi(x)), and φ′ i(x) = xQ′ i(x)−c′ i(Q′ i(x)) ≥xQi(x)−c′ i(Qi(x)). Putting these together, we have φ′ i(x)−φi(x)≥ci(Qi(x)) −c′ i(Qi(x)), and thus ci(Qi(x)) −c′ i(Qi(x)) <u′ i−ui. Games 2025,16, 26 35 of 37 Recall that from (A4) u′ i−ui≤ci(q∗)−c′ i(q∗). It follows that ci(Qi(x)) −c′ i(Qi(x)) <ci(q∗)−c′ i(q∗), which contradictions the assumption that ci(z)−c′ i(z) is nondecreasing in z≥ 0 since Qi(x)≥q∗. We conclude that r′ i≤ri. Notes 1Vives (1986,1993) both provide excellent context for Edgeworth’s contribution to oligopoly. 2 Before Shubik (1959), Shapley (1957) published an abstract with a description of results derived from a game theoretic model of pricing. Other early contributions to BE competition were made by Beckmann and Hochstadter (1965), Shapley and Shubik (1969) , and Levitan and Shubik (1972). 3 To contextualize the different rationing schemes, imagine that demand is composed of a continuum of consumers with different levels of willingness to pay for a single unit of the good. The efficient rationing rule specifies that the low price firm serves the consumers with the highest willingness to pay. That is, all rationed consumers have a weakly lower willingness to pay than all consumers that purchase from the low price firm. The proportional rule specifies that all consumers willing to pay the low price are equally likely to be served by the low price firm, resulting in a proportion of high willingness to pay being rationed and thus a larger residual demand than the efficient rule. 4 Almost all of the BE literature also assumes that the firms have a symmetric, constant marginal cost up to capacity. Deneckere and Kovenock (1996) and Allen et al. (2000) are the notable exceptions. These papers focus on the interesting case in which firms have constant marginal costs that are asymmetric. Additionally, the bulk of this literature further restricts demand to be such that a firm’s monopoly profit is concave in its price. Our analysis is based on the considerably weaker assumption that a firm’s monopoly profit is strictly increasing in its own price up to its unique profit-maximizing monopoly price. 5 Hoernig (2007) provides a treatment of classical Bertrand price competition with general cost structure and sharing rules. In the classical Bertrand specification, any firm that does not have the lowest price receives no residual demand. 6Yoshida (2002) provides a similar treatment to Yoshida (2006) for a model with linear demand and quadratic cost. 7 The relationship between BE games and all-pay auctions is discussed in Baye et al. (1996), the first comprehensive treatment of allpay auctions. More recently, Chowdhury (2017) rely on techniques from the analysis of BE duopoly by Osborne and Pitchik (1986) and Deneckere and Kovenock (1996) to examine all-pay auctions with non-monotonic payoffs. 8 The two distinctions between a traditional all-pay auction and our BE game have been each treated individually in the all-pay auction literature. In Baye et al. (2012), the issue of externalities of bids (contingent on being a winner or a loser) has been addressed in the context of all-pay auctions. Chowdhury (2017) provides a treatment of all-pay auctions in which the winning payoff is nonmonotonic in a player’s own bid. 9 In terms of market demand restrictions, we require only that the monopoly profit be strictly increasing at prices above minimum average total cost up to its unique maximizer. 10 Gelman and Salop (1983) show that, in a two-period sequential game, a single potential entrant can use judo capacity restriction and pricing to induce an unconstrained monopolist to allow entry. The mathematical object that we have denoted as the critical judo price has played a critical role in the analysis of BE price competition since it was first used to characterize the Edgeworth range of price fluctuation in (Shubik,1959, p. 96). 11 Progress with the analysis of models with more general costs and demand rationing has been hindered by theoretical problems with the existence of equilibrium (pure or mixed) in this setting. However, we utilize advances in the literature on the existence of equilibrium in discontinuous games by Bagh (2010) and Allen and Lepore (2014) that allow for the straightforward verification of the existence of equilibrium in vast generalizations of BE oligopoly. 12 This countervailing effect of a supply increase is immediate in the existing BE literature with regard to an increase in a firm’s capacity. This is also related to the impact of an import trade quota in a duopoly with an international and domestic firm, for example Krishna (1989). 13 When this assumption fails to hold, the equilibrium is trivial: one firm charges its monopoly price, and the other firm charges any price and does not produce. While it would be easy to conduct the analysis in this paper without this assumption, it would take away from the clarity of the results and would not meaningfully contribute to the study of duopoly. 14 Here, the bounds xi and xi are inherently dependent on the equilibrium strategies, though we suppress notation indicating this for clarity as there is no ambiguity as to which strategies they correspond to. 15 The supply correspondence is taken to be a subset of the extended real line. Games 2025,16, 26 36 of 37 16 Observe that the residual profit may be zero despite the presence of residual demand as the cost of engaging in low levels of production may exceed the associated revenues, thus inducing the firms to not produce. 17 For readers familiar with Simon and Zame (1990), their result can be used to guarantee that an equilibrium exists for some sharing rule of the game. The complication in this setting is that the sharing rule in the Simon and Zame framework does not correspond only to the division αi between φi and ψi at pricing ties. The sharing rule in this setting also reflects specifications of payoffs at points of discontinuity of ψi in pj . The results of Simon and Zame (1990) do not give any way of identifying the sharing rule for which an equilibrium exists. If the sharing rule with an equilibrium fits the specifications of Proposition 1 and the corresponding equilibrium happens to place mass at price ties as in Proposition 1 or at a point of discontinuity of ψi , then this strategy profile would not be an equilibrium of a sharing rule that violated the conditions of Proposition 1 at those prices. However, if the sharing rule with an equilibrium violates the conditions of Proposition 1, then the corresponding equilibrium will also be an equilibrium for a sharing rule that satisfies the conditions of Proposition 1. Other applicable results that guarantee the existence of equilibrium typically necessitate the same conditions. We have found more general results to be infeasible for application to our model. 18 Ties at a price x<ρ are irrelevant since each firm’s front-side profit is identical to its residual profit. Ties at prices x=ρ may be relevant by this notion and are covered by the lemma. 19 The efficient and proportional rationing rules that have typically been used in studying price competition both lead to residual profit functions being nonincreasing in the rival firm’s price. 20 In this formulation of the model, demand is finite at all prices, so a firm that is not capacity-constrained can be accommodated via an arbitrarily large capacity. 21 With any continuous rationing rule, such as efficient or proportional rationing, firm i ’s residual demand will be lower semicontinuous in pjso long as sjis upper semicontinuous. 22 Here, the limit superior of the sequence of sets Akrefers to the set T∞ n=1S∞ k=nAk. 23 This occurs at prices pj such that firm j ’s cost of production is constant and equal to pj for some levels of production. 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