A General Model of Bertrand–Edgeworth Duopoly

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.

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 Bertrand–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) 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 min–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 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 $p_i$ the price of firm i and by p the vector of both firms’ prices. Since p is the vector of prices $(p_1,p_2)$, 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 ${\phi }_i\left(p_j\right)$, while the residual profit of the firm i with a higher price than firm j is ${\psi }_i\left(p\right)$. The domain of residual profit ${\psi }_i$ is $\{(p_i,p_j)\in {\mathbb{R}}_{+}^2:p_i\ge p_j\}$, as it need not be defined for prices such that $p_i<p_j$ since the residual profit cannot be obtained at such prices. We make the following assumptions on the profit functions ${\phi }_i$ and ${\psi }_i$.

Assumption 1. 

${\phi }_i\left(x\right)\ge {\psi }_i(x,x^{\prime })\ge 0$ for all $x\ge x^{\prime }$.

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 ${\underline{a}}_i$ such that ${\phi }_i\left(x\right)={\psi }_i(x,p_j)=0$ for all $p_j\le x\le {\underline{a}}_i$. Further, ${\psi }_i(p_i,x)={\phi }_i\left(p_i\right)$ for all $p_i\ge x$ such that $x<{\underline{a}}_j$.

The price ${\underline{a}}_i$ 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 ${\underline{a}}_i$, firm j’s residual profit is equal to its front-side profit because firm i produces nothing.

Assumption 3. 

${\phi }_i$ has a unique maximizer ${\widehat{p}}_i>{\underline{a}}_i$ with ${\phi }_i\left({\widehat{p}}_i\right)>0$. On the interval $({\underline{a}}_i,{\widehat{p}}_i)$, ${\phi }_i$ is positive valued, and strictly increasing. Further, ${\psi }_i(p_i,p_j)=0$ for all $p_i\ge p_j\ge {\widehat{p}}_j$, and $p_i>{\widehat{p}}_j$.

This assumption on the front-side profit is weaker than assuming the strict quasiconcavity of ${\phi }_i$ as it does not restrict behavior at prices $p_i>{\widehat{p}}_i$. The third part of the assumption is that there is no residual profit for firm i if firm i prices above ${\widehat{p}}_j$. This assumption fits the two cases that demand is continuous at ${\widehat{p}}_j$, or that there is zero quantity demanded above the price ${\widehat{p}}_j$.

The following assumption is important for our characterization.

Assumption 4. 

For each firm i, there exists a price ${\underline{\rho }}_i\in [{\underline{a}}_i,min\{{\widehat{p}}_i,{\widehat{p}}_j\}]$ such that

$$\begin{array}{ccc}\hfill {\phi }_i\left(x\right)& >& {\psi }_i\left(x,x\right)\mathit{for}\mathit{all}x\in \left({\underline{\rho }}_i,{\widehat{p}}_i\right],\hfill \\ \hfill {\phi }_i\left(x\right)& =& {\psi }_i\left(x,x^{\prime }\right)\mathit{for}\mathit{all}{x}^{\prime }\le x<{\underline{\rho }}_i.\hfill \end{array}$$

Further, if ${\underline{a}}_i>{\underline{a}}_j$, then ${\underline{\rho }}_i\le {\underline{\rho }}_j$.

The maximum of the two firms’ prices ${\underline{\rho }}_i$ is a primary object used in the analysis that follows; as such, we denote $\underline{\rho }=max\{{\underline{\rho }}_1,{\underline{\rho }}_2\}$.

Remark 1. 

The existence of the prices ${\underline{\rho }}_i$ is a natural consequence of traditional constructions of the BE model. It is common that ${\underline{\rho }}_i$ 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, ${\underline{\rho }}_i$ 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 4 accommodates 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, ${\widehat{p}}_i>{\underline{a}}_j$.13

Each firm i’s profit is specified as follows:

$$u_i\left(p\right)=\left\{\begin{array}{cc}{\phi }_i\left(p_i\right)& p_i<p_j\\ {\alpha }_i\left(p\right){\phi }_i\left(p_i\right)+\left(1-{\alpha }_i\left(p\right)\right){\psi }_i\left(p\right)& p_i=p_j\\ {\psi }_i\left(p\right)& p_i>p_j\end{array}\right.,$$

(1)

where ${\alpha }_i\left(p\right)\in [0,1]$ and ${\alpha }_1\left(p\right)+{\alpha }_2\left(p\right)\in (0,2)$. If we instead assume that ${\alpha }_1+{\alpha }_2=1$, 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 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 ${\psi }_i$ at any $p_j$ by ${\tilde{P}}_i\left(p_j\right)$. Denote the maximized residual profit by ${\tilde{\psi }}_i\left(p_j\right)$, that is,

$${\tilde{\psi }}_i\left(p_j\right)=\underset{p_i\ge p_j}{max}{\psi }_i\left(p_i,p_j\right).$$

Assumption 6. 

There exists the lowest price $\widehat{x}$ such that ${\phi }_i\left(\widehat{x}\right)>{\phi }_i\left(x\right)$ and ${\psi }_i(\widehat{x},p_j)\ge {\psi }_i(x,p_j)$ for all prices x and $p_j$ with $x>\widehat{x}\ge p_j$.

Note that $\widehat{x}\ge {\widehat{p}}_i$ for each firm i. Given Assumption 6, the price $\widehat{x}$ weakly dominates all prices $x>\widehat{x}$.

Assumption 7. 

Both ${\phi }_i$ and ${\psi }_i$ are continuous in $p_i$ on $[0,\widehat{x})$ and left continuous at $\widehat{x}$. ${\psi }_i$ is right upper semicontinuous in $p_j$, that is, $lim{sup}_k{\psi }_i(p_i,x^k)\le {\psi }_i(p_i,x)$ for any sequence $\left\{x^k\right\}$ such that $p_i\ge x^k>x$ and $x^k\to x$.

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 $\widehat{x}$ allows the model to accommodate settings with box demand.

Define ${\overline{r}}_i$ 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,

$${\overline{r}}_i=inf\left\{x\right|{\phi }_i\left(x\right)>{sup}_{z\ge x}{\tilde{\psi }}_i\left(z\right)\}.$$

Define ${\underline{r}}_i$ to be firm i’s safe price, which is the lower bound of price such that the front-side profit of firm i is greater than the highest profit that firm i can guarantee itself. Formally,

$${\underline{r}}_i=inf\left\{x\right|{\phi }_i\left(x\right)>{\underline{u}}_i\},$$

where ${\underline{u}}_i={inf}_{p_j}{sup}_{p_i}{u}_i(p_i,p_j).$

Define the larger of the two firms’ judo prices to be critical judo price, denoted by $\overline{r}=max{\overline{r}}_i$. Similarly, define the larger of the two firms’ safe prices to be the critical safe price, denoted by $\underline{r}=max{\underline{r}}_i$. Based on the fact that ${\underline{u}}_i\le$${sup}_{z\ge x}{\tilde{\psi }}_i\left(z\right)$ and that ${\phi }_i$ is strictly increasing when positive, the judo price is always weakly greater than the safe price, that is, $\overline{r}\ge \underline{r}$. Note further that $\overline{r}\le \widehat{x}$.

Define firm i’s judo profit to be the front-side profit of firm i at the critical judo price, denoted by ${\overline{\phi }}_i\equiv {\phi }_i\left(\overline{r}\right)$. Similarly, define firm i’s safe profit to be the front-side profit of firm i at the critical safe price, ${\underline{\phi }}_i\equiv {\phi }_i\left(\underline{r}\right)$.

For equilibrium strategies $\mu =\left({\mu }_1,{\mu }_2\right)$, we use ${\underline{x}}_i$ and ${\overline{x}}_i$ to denote the infimum and supremum of the support of firm i’s strategy, respectively. We will use $\underline{x}$ to denote the minimum of ${\underline{x}}_1$ and ${\underline{x}}_2$, and $\overline{x}$ to denote the maximum of ${\overline{x}}_1$ and ${\overline{x}}_2$.14 Further, we define $F_i$ to be the distribution function (CDF) of firm i’s mixed strategies on $\left[\underline{x},\overline{x}\right]$, with $F=(F_1,F_2)$. Additionally, let $u_i^{*}$ denote 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.

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 $[0,\widehat{x})$, 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

$$D\left(x\right)=\left\{\begin{array}{cc}0& \mathrm{if}x>1\\ 1& \mathrm{if}x\in [0,1]\end{array}\right..$$

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 ${\pi }_i(x,q)=xq-c_i\left(q\right)$.15 We then use these to derive the corresponding front-side and residual profit functions as well as the prices $\widehat{x}$, ${\widehat{p}}_i$, ${\underline{a}}_i$, and ${\underline{\rho }}_i$.

Example 1 

(Discontinuous supply). Firm 1’s cost of production is

$$c_1\left(q\right)=\left\{\begin{array}{ccc}{\displaystyle \frac{1}{2}}q-{\displaystyle \frac{1}{8}}& \mathit{if}& q\ge 1/2\\ {\displaystyle \frac{1}{4}}q& \mathit{if}& q\in [0,1/2]\end{array}\right.,$$

while firm 2’s cost of production is $c_2\left(q\right)={\displaystyle \frac{1}{3}}q$. Neither firm faces a capacity constraint. The supply correspondence of each firm is thus

$${\vartheta }_1\left(p_1\right)=\left\{\begin{array}{ccc}\left\{\infty \right\}& \mathit{if}& p_1>1/2\\ [\frac{1}{2},\infty ]& \mathit{if}& p_1=1/2\\ \left\{\frac{1}{2}\right\}& \mathit{if}& p_1\in (1/4,1/2)\\ [0,\frac{1}{2}]& \mathit{if}& p_1=1/4\\ \left\{0\right\}& \mathit{if}& p_1\in [0,1/4)\end{array}\right.,$$

and

$${\vartheta }_2\left(p_2\right)=\left\{\begin{array}{ccc}\{\infty \}& \mathit{if}& p_2\ge 1/3\\ [0,\infty ]& \mathit{if}& p_2=1/3\\ \left\{0\right\}& \mathit{if}& p_2\in [0,1/3)\end{array}\right..$$

The front-side profits are

$${\phi }_1\left(p_1\right)=\left\{\begin{array}{ccc}0& \mathit{if}& p_1>1\\ \left(p_1-{\displaystyle \frac{1}{4}}\right){\displaystyle \frac{1}{2}}+\left(p_1-{\displaystyle \frac{1}{2}}\right){\displaystyle \frac{1}{2}}& \mathit{if}& p_1\in [1/2,1]\\ \left(p_1-{\displaystyle \frac{1}{4}}\right){\displaystyle \frac{1}{2}}& \mathit{if}& p_1\in [1/4,1/2)\\ 0& \mathit{if}& p_1\in [0,1/4)\end{array}\right.,$$

and

$${\phi }_2\left(p_2\right)=\left\{\begin{array}{ccc}0& \mathit{if}& p_2>1\\ p_2-{\displaystyle \frac{1}{3}}& \mathit{if}& p_2\in [1/3,1]\\ 0& \mathit{if}& p_2\in [0,1/3)\end{array}\right..$$

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, the residual profits of the firms are defined with demand rationed according to the efficient (or equivalently proportional) rule

$${\psi }_1(p_1,p_2)=\left\{\begin{array}{ccc}0& \mathit{if}& p_2\ge 1/3\\ {\phi }_1\left(p_1\right)& \mathit{if}& p_2<1/3\end{array}\right.,$$

$${\psi }_2(p_2,p_1)=\left\{\begin{array}{ccc}0& \mathit{if}& \begin{array}{c}{p}_2>1,\mathit{or}{p}_1\ge 1/2,\\ \mathit{or}{p}_2<1/3,\mathit{or}{p}_1\in [1/4,1/2)\end{array}\\ \left(p_2-{\displaystyle \frac{1}{3}}\right){\displaystyle \frac{1}{2}}& \mathit{if}& p_2\ge 1/3,p_1\in [1/4,1/2)\\ {\phi }_2\left(p_2\right)& \mathit{if}& p_1<1/4\end{array}\right..$$

The key model parameters for this example are $\widehat{x}={\widehat{p}}_i=1$, ${\underline{a}}_1=1/4$, ${\underline{a}}_2=1/3$, and ${\underline{\rho }}_i=$$\underline{\rho }=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

$$c_i\left(q\right)=\left\{\begin{array}{ccc}{\displaystyle \frac{2}{3}}{q}^3-{\displaystyle \frac{1}{4}}{q}^2+{\displaystyle \frac{1}{16}}q+{\displaystyle \frac{1}{20}}& \mathit{if}& q>0\\ 0& \mathit{if}& q=0\end{array}\right..$$

Neither firm faces a capacity constraint. The supply correspondence of each firm i can be expressed as the function

$$\vartheta \left(p_i\right)=\left\{\begin{array}{ccc}{\displaystyle \frac{1+\sqrt{32p_i-1}}{8}}& \mathit{if}& p_i\ge 0.194\\ 0& \mathit{if}& p_i\in [0,0.194]\end{array}\right..$$

Thus, the front-side profit of firm i is

$${\phi }_i\left(p_i\right)=\left\{\begin{array}{ccc}0& \mathit{if}& p_i>1\\ p_i\vartheta \left(p_i\right)+c_i\left(\vartheta \left(p_i\right)\right)& \mathit{if}& p_i\in [0.194,1]\\ 0& \mathit{if}& p_i\in [0,0.194)\end{array}\right.,$$

and the residual profit of firm i ($p_i\ge p_j$) is (again using efficient rationing)16

$${\psi }_i(p_i,p_j)=\left\{\begin{array}{ccc}0& \mathit{if}& p_j\ge 0.397\\ p_{i}min\{\vartheta \left(p_i\right),1-\vartheta \left(p_j\right)\}+c_i(min\{\vartheta \left(p_i\right),1-\vartheta \left(p_j\right)\})& \mathit{if}& p_j\in (0.194,0.397)\\ {\phi }_i\left(p_i\right)& \mathit{if}& p_j<0.194\end{array}\right..$$

The key model parameters for this example are $\widehat{x}={\widehat{p}}_i=1$, ${\underline{a}}_i=0.194$, and ${\underline{\rho }}_i=\underline{\rho }=0.3125$.

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 $k\in (1/2,1)$, 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 $W_i\le k$ of consumers visit firm i, each of those consumers receives a good at price $p_i$. If a mass $W_i>k$ of consumers visits firm i, then each of those consumers receives a good at price $p_i$ with probability $k/W_i$. Consumers that do not receive goods obtain a payoff of zero. In any pure strategy equilibrium of the consumer game, if $p_i<p_j$, then either all consumers shop at firm i, or the mass $W_i$ of consumers that shop at firm i satisfies

$${\displaystyle \frac{k}{W_i}}\left(1-p_i\right)=1-p_j.$$

Therefore, we can write

$$W_i=\left\{\begin{array}{ccc}0& \mathit{if}& p_i>1\\ min\left\{1,k{\displaystyle \frac{1-p_i}{1-p_j}}\right\}& \mathit{if}& p_i\in [0,1]\end{array}\right.,$$

and the mass of consumers who go to firm j are $W_j=1-W_i$. Notice that $W_i\ge k$ for all prices $p_i<p_j$. Thus, we can write the front-side and residual profit of firm i as follows:

$${\phi }_i\left(p_i\right)=\left\{\begin{array}{ccc}0& \mathit{if}& p_i>1\\ p_{i}k& \mathit{if}& p_i\in [0,1]\end{array}\right.,$$

and

$${\psi }_i(p_i,p_j)=\left\{\begin{array}{ccc}0& \mathit{if}& p_i\ge 1\\ p_{i}max\left\{0,1-{\displaystyle \frac{k\left(1-p_j\right)}{1-p_i}}\right\}& \mathit{if}& p_i\in [p_j,1)\end{array}\right..$$

The key model parameters for this example are $\widehat{x}={\widehat{p}}_i=1$, ${\underline{a}}_i=$${\underline{\rho }}_i=\underline{\rho }=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 ${\underline{\rho }}_1={\underline{\rho }}_2$. When there is firm i such that ${\underline{\rho }}_i<\underline{\rho }$, our proof requires an additional condition that this firm i receives its front-side profit with certainty at ties below $\underline{\rho }$.

Proposition 1. 

Assume that (1) ${\alpha }_i(x,x)=1$ for any x such that ${\phi }_i\left(x\right)>{\psi }_i(x,x)$ and ${\phi }_j\left(x\right)={\psi }_j(x,x)$ and (2) that ${\psi }_i$ is lower semicontinuous in $p_j$. Then a mixed strategy equilibrium of the BE game exists.

If the residual profits ${\psi }_i$ 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 ${\psi }_i$ 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

• ${\underline{x}}_1={\underline{x}}_2=\underline{x}\ge \underline{\rho }$;
• The probability of an atom at any price $x>\underline{\rho }$ is zero;
• The probability of a tie at $\underline{x}$ is positive only if ${\phi }_i\left(\underline{x}\right)={\psi }_i(\underline{x},\underline{x})$ for each firm i; and
• At most one firm can price higher than $\widehat{x}$.

Asymmetric lower bound equilibria are only possible for the case that ${\underline{a}}_i>{\underline{a}}_j$ and are such that

• ${\underline{x}}_i<{\underline{x}}_j={\underline{a}}_i=\underline{\rho }$;
• ${\mu }_j\left(\left\{\underline{\rho }\right\}\right)=1$;
• ${\mu }_i\left([\underline{\rho },\underline{\rho }+\epsilon )\right)>0$ for any $\epsilon >0$; and
• $u_i^{*}=0$, $u_j^{*}>0$.

Notice that every pure strategy equilibrium must be a symmetric lower bound equilibrium. Either type of equilibrium can be mixed, although the asymmetric lower bound allows only one player to use a nondegenerate mixed strategy.

The proof of Proposition 2 is extensive and based on the series of Lemmas 1–4. The first of these lemmas provides an upper bound for at least one firm’s prices.

Lemma 1. 

In any equilibrium, at most one firm can select prices above $\widehat{x}$. That is, in any equilibrium μ, ${\mu }_i\left([0,\widehat{x}]\right)=1$ for at least one firm i.

The following lemma is instrumental in many of the proofs of this paper. This lemma establishes that relevant ties ($p_1=p_2>\underline{\rho }$) occur with probability zero in all equilibria of the BE game.18 It is worth noting that ${\mu }_i$ may have an atom ${\mu }_i\left(\left\{x\right\}\right)>0$ at a price $x>\underline{\rho }$, although ${\mu }_1\left(\left\{x\right\}\right){\mu }_2\left(\left\{x\right\}\right)=0$. That is, ties above the price such that the front-side profit equals residual profit occur with probability zero in equilibrium.

Lemma 2. 

In any equilibrium, if firm i’s strategy has mass at a price x such that ${\phi }_i\left(x\right)>{\psi }_i(x,x)$, then either ${\alpha }_i(x,x)=1$ or firm j’s strategy does not have mass at x. Consequently, in any equilibrium, both firms cannot simultaneously have an atom at a price $p_1=p_2>\underline{\rho }$, and the equilibrium strategies and payoffs are unaffected by the choice of sharing rule α at any price $x>\underline{\rho }$.

The next lemma constrains the possibilities for equilibria in which the infimum of each firm i’s strategy is not the same.

Lemma 3. 

In any equilibrium, if ${\underline{x}}_i<{\underline{x}}_j$, then firm j’s strategy is degenerate, with ${\underline{x}}_j=\underline{\rho }$ and ${\mu }_j\left(\left\{\underline{\rho }\right\}\right)=1$. In any such equilibrium, ${\underline{a}}_j<{\underline{a}}_i=\underline{\rho }$, $u_i^{*}=0$, $u_j^{*}>0$, and ${\mu }_i\left([\underline{\rho },\underline{\rho }+\epsilon )\right)>0$ for any $\epsilon >0$.

We use the following example to help illustrate the type of equilibria characterized by Lemma 3. The types of equilibria exhibited in this example are not novel observations and were comprehensively treated in Deneckere and Kovenock (1996). The purpose of this example is simply to provide a concrete illustration of the statement of Lemma 3.

Example 4. 

Consider a market in which firm 1 has zero cost of production and firm 2 has a constant marginal cost of $c>0$. Demand is given by $D\left(x\right)$, a continuous and nonincreasing function with $D\left(0\right)<\infty$ and $D\left(c\right)>0$. Given such a market, the front-side profits are ${\phi }_1\left(x\right)=D\left(x\right)x$ and ${\phi }_2\left(x\right)=D\left(x\right)(x-c)$. Residual profits are

$${\psi }_1(x,p_2)=\left\{\begin{array}{ccc}D\left(x\right)x& \mathit{if}& p_2<c\\ 0& \mathit{if}& p_2\ge c\end{array}\right.$$

and ${\psi }_2(x,p_1)=0$. Assume that $D\left(x\right)x$ is maximized at some price ${\widehat{p}}_1>c$.

In this game, ${\underline{a}}_1=0$, ${\underline{a}}_2=c$, and ${\underline{\rho }}_1={\underline{\rho }}_2=\underline{\rho }=c$. There are infinitely many equilibria in which firm 1 sets a price $p_1^{*}=c$. An example of such equilibrium strategies for firm 2 is to set ${\mu }_2\left([c,c+\epsilon ]\right)\ge 1-{\phi }_1\left(c\right)/{\phi }_1\left({\widehat{p}}_1\right)$ for all $\epsilon >0$, with the remaining mass placed at prices below c and possibly above ${\widehat{p}}_1$. To see that such strategy profiles are equilibria, note that $u_2(c,p_2)=0$ for all $p_2$, so firm 2 has no incentive to deviate. Firm 1 has no incentive to deviate to any price other than ${\widehat{p}}_1$. Such a deviation is not profitable since $[1-{\phi }_1\left(c\right)/{\phi }_1\left({\widehat{p}}_1\right)]{\phi }_1\left({\widehat{p}}_1\right)\le {\phi }_1\left(c\right)$.

The next lemma characterizes equilibria in which the infimum of the support of each firm’s strategy is identical.

Lemma 4. 

The following conditions hold at any equilibrium in which both firms use nondegenerate mixed strategies: (i) ${\underline{x}}_1={\underline{x}}_2=\underline{x}\ge \underline{\rho }$, and (ii) if there is a firm i such that ${\phi }_i\left(\underline{x}\right)>{\psi }_i(\underline{x},\underline{x})$, then neither firm’s equilibrium strategy may have an atom at $\underline{x}$.

An important implication of Lemma 4 is that, in such an equilibrium, each firm selects prices arbitrarily close to $\underline{x}$ with positive probability, receiving its front-side profit with probability arbitrarily close to one. Consequently, each firm i’s expected profit in equilibrium is $u_i^{*}={\phi }_i\left(\underline{x}\right)$.

Because of the general structure of each firm’s residual profit, there can be multiple non-payoff equivalent equilibria. The following example is a simple BE game with two non-payoff equivalent mixed strategy equilibria.

Example 5. 

Consider a market with a mass of 1 of consumers all with a maximum willingness to pay of 1 for the homogeneous product. The perceived value of the good to the consumers is based on the observed prices. When prices are low (high), consumers have a low (high) willingness to pay. The willingness to pay of any consumer is

$$v\left(p\right)=\left\{\begin{array}{ccc}1& \mathit{if}& min\{p_1,p_2\}>1/2\\ 1/2& \mathit{if}& min\{p_1,p_2\}\le 1/2\end{array}\right..$$

Thus, if both firms set prices weakly less than $1/2$, then willingness to pay is $1/2$. If both firms price above $1/2$, then willingness to pay is 1.

Each firm i has zero production cost but is limited to producing a quantity no greater than its capacity $k_i<1$, where $k_1+k_2\ge 1$. The demand system follows standard “efficient rationing”, although this is determined by Nash equilibrium play of the consumer search game.

The front-side profit of firm i is

$${\phi }_i\left(x\right)=\left\{\begin{array}{ccc}0& \mathit{if}& x>1\\ x{k}_i& \mathit{if}& x\in [0,1]\end{array}\right.,$$

and the residual profit of firm i is

$${\psi }_i(p_i,p_j)=\left\{\begin{array}{ccc}0& \mathit{if}& p_i>1,\mathit{or}{p}_i>1/2\ge p_j\\ p_i\left(1-k_j\right)& \mathit{if}& 1\ge p_i\ge p_j>1/2\mathit{or}1/2\ge p_i\ge p_j\end{array}\right..$$

Let $k_1=k_2=5/8$. There are two mixed strategy equilibria. Each equilibrium is symmetric, with both firms employing the CDF $F\left(x\right)$ or both firms employing the CDF $G\left(x\right)$ defined as follows:

$$\begin{array}{ccc}\hfill F\left(x\right)& =& \left\{\begin{array}{ccc}0& \mathit{if}& x<3/10\\ {\displaystyle \frac{10x-3}{4x}}& \mathit{if}& x\in [3/10,1/2]\\ 1& \mathit{if}& x>1/2\end{array}\right.,\hfill \\ \hfill G\left(x\right)& =& \left\{\begin{array}{ccc}0& \mathit{if}& x<3/5\\ {\displaystyle \frac{5x-3}{2x}}& \mathit{if}& x\in [3/5,1]\\ 1& \mathit{if}& x>1\end{array}\right..\hfill \end{array}$$

In the first equilibrium, each firm i has the expected payoff $u_i^{*}=3/16$, while in the second equilibrium, each firm i has the expected payoff $u_i^{*}=3/8$.

Note that, for this example, $\overline{r}=3/5$ and $\underline{r}=3/10$. Consequently, ${\overline{\phi }}_i\left(x\right)=3/8$ and ${\underline{\phi }}_i\left(x\right)=3/16$, which coincide with the two equilibrium expected payoffs.

In the following proposition, we show that the expected profits of all equilibria lie between the safe profit and the judo profit. The bounds on the equilibrium expected payoffs follow immediately from showing that the lower bound on pricing of any equilibrium must lie between the critical safe price and the critical judo price.

Proposition 3. 

All equilibria are such that $u_i^{*}\in [{\underline{\phi }}_i,{\overline{\phi }}_i]$.

These payoff bounds provide a solid foundation for understanding the properties of the equilibria of BE games in a general setting. While the literature on BE games has, in some cases, been able to provide precise payoff predictions, the bounds presented here apply to a much larger class of games than previously studied. The proposition gives precise predictions of the equilibrium profits when the judo price and safe price coincide ($\underline{r}=\overline{r}$). While it may seem restrictive to characterize only the bounds on the equilibrium profits along with the lowest equilibrium price, this result is inherently valuable as a precise computation of the equilibrium bounds and payoffs can only be obtained by first calculating the equilibrium strategies. Without very precise and often simplistic classes of functions, such a computation can be prohibitively difficult, and we are unaware of any method that would allow one to solve for the equilibrium strategies in general.

In order to bound the upper bound of equilibrium pricing, we need to formally define maximizers of the residual profit function conditional on the other firm’s mixed strategy. Given a distribution of prices $F_j$, define the set of conditional residual maximizers ${\tilde{P}}_i\left(F_j\right)\equiv arg{max}_x{E}_{F_j}\left[{\psi }_i(x,p_j)\right|p_j\le x]$. The following lemma demonstrates that the upper bound on pricing must lie between the smallest and largest of all firms’ conditional residual maximizers.

Lemma 5. 

For any equilibrium $F=(F_1,F_2)$, $\overline{x}\in {\tilde{P}}_1\left(F_2\right)\cup {\tilde{P}}_2\left(F_1\right)$.

Lemma 5 is useful for the technical analysis but has a limited predictive value for two reasons. First, this result requires knowledge of equilibrium pricing distributions, and second, it is not possible to infer payoffs from the upper bound on pricing because of the dependence of the residual profit on the lower price.

Remark 2. 

In contrast with the literature on BE duopoly, the generality of our specification introduces an additional level of pricing indeterminacy. The first level of indeterminacy, in the previous literature, is based on the equilibrium being in non-degenerate mixed strategies. The second level of indeterminacy present in our framework is driven by the fact that there can be multiple non-payoff equivalent equilibria.

4. Pure Strategy Equilibria

The objective of this section is to understand when price indeterminacy is resolved by equilibrium play. To that end, we present necessary and sufficient conditions for the existence of pure strategy equilibrium. Under additional restrictions, our conditions become necessary and sufficient for this to be the unique equilibrium. When these conditions fail to hold, all equilibria of the pricing game must be in mixed strategies.

We show that the only possible pure strategy equilibrium is both firms pricing at $\underline{\rho }$. This allows us to derive necessary and sufficient conditions for pure strategy equilibrium. There are two different sets of conditions for the two cases: ${\phi }_i\left(\underline{\rho }\right)={\psi }_i(\underline{\rho },\underline{\rho })$ for all i, and for some i, ${\phi }_i\left(\underline{\rho }\right)>{\psi }_i(\underline{\rho },\underline{\rho })$. The only condition for the case that ${\phi }_i\left(\underline{\rho }\right)={\psi }_i(\underline{\rho },\underline{\rho })$ for all i is that $\underline{\rho }$ is a residual maximizer for both firms. For the case that there is a firm i with ${\phi }_i\left(\underline{\rho }\right)>{\psi }_i(\underline{\rho },\underline{\rho })$, firm i must obtain the front-side profit for sure at the tie and cannot have a residual maximizer that is better than this front-side profit, while for firm j, it must be that ${\phi }_j\left(\underline{\rho }\right)={\psi }_j(\underline{\rho },\underline{\rho })$ and $\underline{\rho }$ is a residual maximizer for firm j.

Proposition 4. 

Any pure strategy equilibrium $(x^{*},x^{*})$ must be symmetric and $x^{*}=\underline{\rho }$. The following are two types of possible pure strategy equilibrium:

• If ${\phi }_i\left(\underline{\rho }\right)={\psi }_i(\underline{\rho },\underline{\rho })$ for all i, then $x^{*}=\underline{\rho }$ is a pure strategy equilibrium if and only if $\underline{\rho }\in {\tilde{P}}_i\left(\underline{\rho }\right)$ for all i;
• If ${\phi }_i\left(\underline{\rho }\right)>{\psi }_i(\underline{\rho },\underline{\rho })$ for some i, then $x^{*}=\underline{\rho }$ is a pure strategy equilibrium if and only if ${\phi }_j\left(\underline{\rho }\right)={\psi }_j(\underline{\rho },\underline{\rho })$, $\underline{\rho }\in {\tilde{P}}_j\left(\underline{\rho }\right)$, ${\alpha }_i(\underline{\rho },\underline{\rho })=1$, and ${\phi }_i\left(\underline{\rho }\right)\ge {\tilde{\psi }}_i\left(\underline{\rho }\right)$.

Intuitively, it is easy to understand why both firms setting a price of $\underline{\rho }$ is the only possible pure strategy equilibrium. Each firm i must price at least at ${\underline{a}}_i$, as otherwise, firm j would act as a monopoly and then firm i would have a profitable deviation to undercut the monopoly price of firm j. Next, each firm must price at least at ${\underline{\rho }}_i$, as otherwise, the firm with the lowest price could increase its price and still receive a payoff equal to their front-side profit. This is a profitable deviation since the front-side profit is increasing. Similarly, both firms must set the same price, as otherwise, the firm with the low price could improve by increasing its price. Lastly, if both firms set a higher price than $\underline{\rho }$, then each firm’s front-side profit is higher than their respective residual profit. Since both firms cannot receive their front-side profit with certainty, at least one firm has incentive to undercut.

We now proceed to the proof of Proposition 4.

Proposition 4 is important in that it specifies the exact circumstances that Edgeworth’s concerns about price indeterminacy can be alleviated. However, in this general setting, the existence of a pure strategy equilibrium does not guarantee the uniqueness of this equilibrium. Since $\underline{\rho }$ is uniquely defined, it is the only pure strategy equilibrium candidate; however, it may be that a mixed strategy equilibrium concurrently exists. The following example is a BE game with both a pure strategy equilibrium and a mixed strategy equilibrium.

Example 6. 

Consider an industry with a market with demand

$$D\left(x\right)=\left\{\begin{array}{ccc}0& \mathit{if}& x>10\\ 10& \mathit{if}& x\in [0,10]\end{array}\right.$$

Both firms have zero cost of production up to the capacity constraint $k_1=k_2=10$. The front-side profit of firm i is

$${\phi }_i\left(x\right)=\left\{\begin{array}{ccc}0& \mathit{if}& x>10\\ 10x& \mathit{if}& x\in [0,10]\end{array}\right.,$$

There is an additional mass of 8 consumers with no income for whom the government will fully subsidize the purchase of this good up to a price of 10 dollars, but only if the lowest posted price is at least 3. Based on this program, the residual profit of each firm i is

$${\psi }_i(p_i,p_j)=\left\{\begin{array}{ccc}8p_i& \mathit{if}& 10\ge p_i\ge p_j\ge 3\\ 0& \mathit{if}& p_j<3\mathrm{or}{p}_i>10\end{array}\right..$$

This game has the following two equilibria:

EQ 1 (pure strategy): Both firms set a price of zero, with corresponding equilibrium profits of zero.

EQ 2 (mixed strategy): Each firm i employs the CDF $F\left(x\right)$ defined by

$$F\left(x\right)=\left\{\begin{array}{ccc}0& \mathit{if}& x<8\\ {\displaystyle \frac{5x-40}{x}}& \mathit{if}& x\in [8,10]\\ 1& \mathit{if}& x>10\end{array}\right.$$

and earns an equilibrium profit of $u_i^{*}=80$.

The following proposition demonstrates that no other equilibrium in pure or mixed strategies may exist as long as residual profit is nonincreasing in the other firm’s price and $\underline{\rho }$ is the only residual maximizer for each firm when the other firm sets a price of $\underline{\rho }$.

Proposition 5. 

Suppose that there is a pure strategy equilibrium. If $\underline{\rho }$ is the unique maximizer of ${\psi }_i(p_i,\underline{\rho })$ and ${\psi }_i$ is nonincreasing in $p_j$ for each firm i, then both firms pricing at $\underline{\rho }$ is the unique equilibrium of the BE game.

Remark 3. 

Based on Proposition 5, in the case that residual profits are nonincreasing in the rival’s price and have the unique residual maximizer of $\underline{\rho }$ to the price $\underline{\rho }$, a non-degenerate mixed strategy and pure strategy equilibrium cannot coexist for the same parameters.19 Thus, in this environment, the necessary and sufficient condition for the existence of pure strategy equilibrium also guarantees its uniqueness. In getting back to the theme of price indeterminacy, Proposition 5 provides precise conditions for determinate pricing.

Propositions 4 and 5 provide the basic character of all pure strategy equilibria of the BE game. We turn now to a discussion of the nature of pure strategy equilibrium in our model and classifying all pure strategy equilibria of this game as one of two distinct types. The first type of pricing requires that price equals marginal cost, a la Bertrand pricing. The second type of equilibrium includes pricing above marginal cost with supply to equal demand, a la Cournot pricing. These types are defined formally as follows.

Type B: $\underline{\rho }=max{\underline{a}}_i$

Type C: $\underline{\rho }>max{\underline{a}}_i$

It is important to point out that our structure permits pure strategy equilibrium pricing not of classical Bertrand (price equals marginal cost) or classical Cournot (market clearing) character. For example, a model with underlying cost functions that have constant marginal cost c from 0 to q and then a jump in constant marginal cost to $c^{\prime }>c$ for all quantities greater than q. Although a pure strategy equilibrium with each firm pricing at marginal cost $c^{\prime }$ has more of the flavor of Bertrand pricing, it actually falls in the category of Type C. This is based on the fact that the model ${\underline{a}}_i=c$ to satisfy our abstract assumptions. It is worth noting that the conditions of Proposition 5 can only apply to Type C pure strategy equilibrium based on the unique maximizer restriction. The reason for this is that, in any Type B equilibrium, at least one firm i has an equilibrium profit equal to zero, which means that firm i’s residual profit is maximized at any $x\ge \underline{\rho }$.

Additional characterization of the set of pure strategy equilibrium is provided in Section 5 to follow for a special case model where we use a traditional construction of the market based on primitive assumptions on demand and production technology.

5. Special Case Model

In this section, we construct a BE duopoly model from assumptions on cost, demand, and residual demand. The purpose of the section is two-fold: First, show a model with underlying production technology and demand assumptions sufficient for the abstract front-side and residual profit functions to follow Assumptions 1–8. Second, provide results for this model that cannot be shown in the abstract formulation. These results pertain to the conditions for pure strategy equilibrium and how changes in residual demand rationing and individual supply change the bounds on equilibrium prices and payoffs. We now turn to establishing the basic structure of the special case model.

Each firm i has a cost of production $c_i:{\mathbb{R}}_{+}\mapsto {\mathbb{R}}_{+}$ satisfying the following properties.

Condition 1. 

Each $c_i$ is continuous and nondecreasing with $c_i\left(0\right)=0$.

Define the profit function for firm i to be ${\pi }_i(x,z)=xz-c_i\left(z\right)$. Firm i has a capacity denoted by $k_i>0$ that serves as the upper bound on the quantity that can be produced.20 The supply function of each firm i, denoted by $s_i\left(x\right)$, is a quantity that would maximize ${\pi }_i(x,z)$. The supply function $s_i\left(x\right)$ is any selection from the supply correspondence ${\vartheta }_i\left(x\right)=arg{max}_{z\in [0,k_i]}xz-c_i\left(z\right)$, which must exist due to the continuity of $c_i$ from Condition 1 and the compactness of $[0,k_i]$.

Condition 2. 

${\pi }_i(x,z)$ is quasiconcave in z.

The key parameter ${\underline{a}}_i$ is the infimum of firm i’s average cost, defined formally as ${\underline{a}}_i={inf}_{z>0}{c}_i\left(z\right)/z$. The following condition is that each firm’s lowest average cost is as the production quantity approaches zero.

Condition 3. 

${inf}_{z>0}{\displaystyle \frac{c_i\left(z\right)}{z}}=lim{inf}_{z\to 0}{\displaystyle \frac{c_i\left(z\right)}{z}}$.

The market demand is denoted by $D:{\mathbb{R}}_{+}\mapsto {\mathbb{R}}_{+}$. We make the following assumptions about the properties of market demand.

Condition 4. 

D is continuous and nonincreasing. There is a choke price ${\overline{p}}^c=inf\{x:D\left(x\right)=0\}<\infty$, with $max\{{\underline{a}}_1,{\underline{a}}_2\}<{\overline{p}}^c$.

The residual demand of firm i is the demand available to the firm i if $p_i\ge p_j$ and is denoted by $d_i:\{(p_i,p_j)\in {\mathbb{R}}_{+}^2:p_i\ge p_j\}\mapsto {\mathbb{R}}_{+}$. We make the following assumptions about the properties of each firm i’s residual demand.

Condition 5. 

$d_i(p_i,p_j)$ is continuous and nonincreasing $p_i$, and right continuous and nonincreasing in $p_j$.21 Furthermore, $d_i(p_i,p_j)\le D\left(p_i\right)$ for all $p_i\ge p_j$.

Given Conditions 1–5, at any price $p_i$, the optimal production quantity for firm i with the lower price given $D\left(p_i\right)$ is $Q_i\left(p_i\right)=min\{s_i\left(p_i\right),D\left(p_i\right)\}$. Similarly, the optimal production quantity for the firm with the higher price given residual demand $d_i(p_i,p_j)$ is $q_i(p_i,p_j)=min\{s_i\left(p_i\right),d_i(p_i,p_j)\}$. The following condition relates the quantity produced by the firm with the lower price to the residual demand.

Condition 6. 

If $Q_j\left(x\right)=D\left(x\right)$, then $d_i(x,x)=0$. If $Q_j\left(x^{\prime }\right)=0$, then $d_i(x,x^{\prime })=D\left(x\right)$ for all $x\ge x^{\prime }$. For any price x, $d_i(x,x)=D\left(x\right)-Q_j\left(x\right)$. If $Q_j\left(x\right)>0$, then $d_i(x,x)<D\left(x\right)$ for all $x<{\overline{p}}^c$.

We define the front-side and residual profit functions of each firm i, respectively, by

$$\begin{array}{ccc}\hfill {\phi }_i\left(p_i\right)& =& p_i{Q}_i\left(p_i\right)-c_i\left(Q_i\left(p_i\right)\right)\hfill \\ \hfill {\psi }_i(p_i,p_j)& =& p_i{q}_i(p_i,p_j)-c_i\left(q_i(p_i,p_j)\right)\hfill \end{array}$$

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 ${\tau }_i$. Formally, ${\tau }_i=inf\{x:s_i\left(x\right)\ge D\left(x\right)\}$.

Condition 7. 

$p_{i}D\left(p_i\right)-c_i\left(D\left(p_i\right)\right)$ is strictly quasiconcave on $[{\tau }_i,{\overline{p}}^c]$.

The restriction of the strict quasiconcavity to only the interval $[{\tau }_i,{\overline{p}}^c]$ allows some additional freedom for the demand function at prices below ${\tau }_i$. The properties of $p_{i}D\left(p_i\right)-c_i\left(D\left(p_i\right)\right)$ at prices $p_i<{\tau }_i$ are irrelevant, as the the profit of the firm does not correspond to this expression at such prices.

An immediate consequence of Condition 7 is that there is a unique maximizer ${\widehat{p}}_i$ 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. 

${\widehat{p}}_i>{\underline{a}}_j$ for 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 i is

$${\overline{r}}_i=inf\left\{x\right|{\phi }_i\left(x\right)>{\tilde{\psi }}_i\left(x\right)\}.$$

Note that, based on the assumptions of this section, either ${\phi }_i\left({\overline{r}}_i\right)={\tilde{\psi }}_i\left({\overline{r}}_i\right)$, or ${\overline{r}}_i={\underline{a}}_j$. The safe price of firm i is

$${\underline{r}}_i=min\left\{x\right|{\phi }_i\left(x\right)\ge {\underline{u}}_i\}.$$

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 $x^{*}=max\{{\underline{a}}_1,{\underline{a}}_2\}$ and Type C with $x^{*}>max\{{\underline{a}}_1,{\underline{a}}_2\}$. 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 $x^{*}=\underline{\rho }$ is a pure strategy Nash equilibrium if and only if one of the following three conditions holds:

B.1 $\underline{\rho }$$={\underline{a}}_1={\underline{a}}_2$ and $d_i(x^{\prime },\underline{\rho })=0$ for each firm i, and all $x^{\prime }>\underline{\rho }$.

B.2 $\underline{\rho }$$={\underline{a}}_i>{\underline{a}}_j$, $d_i(x,\underline{\rho })=0$ for all $x\ge \underline{\rho }$, $u_j(\underline{\rho },\underline{\rho })={\phi }_j\left(\underline{\rho }\right)$, and ${\psi }_j(x,\underline{\rho })\le$${\phi }_j\left(\underline{\rho }\right)$ for all $x\ge \underline{\rho }$.

C $\underline{\rho }\in (max\{{\underline{a}}_1,{\underline{a}}_2\},min\{{\widehat{p}}_1,{\widehat{p}}_2\}]$, $\underline{\rho }\in {\tilde{P}}_i\left(\underline{\rho }\right)$ for each firm i, and ${\underline{s}}_i\left(\underline{\rho }\right)+s_j\left(\underline{\rho }\right)\le D\left(\underline{\rho }\right)$ for any firm i with ${\alpha }_i(\underline{\rho },\underline{\rho })<1$, where ${\underline{s}}_i\left(x\right)=min{\vartheta }_i\left(x\right)$.

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, ${\underline{a}}_1={\underline{a}}_2$. 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 $min\{s_1\left(\underline{\rho }\right),s_2\left(\underline{\rho }\right)\}\ge D\left(\underline{\rho }\right)$, and thus by Condition 6, $d_i(\underline{\rho },\underline{\rho })=0$ for both i. The second case allows for a firm’s supply to not cover all of demand $s_i\left(\underline{\rho }\right)<D\left(\underline{\rho }\right)$ as long as there is no residual demand for the other firm $d_j(x^{\prime },\underline{\rho })=0$ for all $x^{\prime }>\underline{\rho }$. This case relies on no consumers going to the higher-priced firm j when firm i prices at $\underline{\rho }$.

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 ($s_1\left(\underline{\rho }\right)+s_2\left(\underline{\rho }\right)=D\left(\underline{\rho }\right)$) above the infimum of average cost. The second case is such that total supply is less than demand $s_1\left(\underline{\rho }\right)+s_2\left(\underline{\rho }\right)<D\left(\underline{\rho }\right)$ and residual demand rationing does not permit a profit increase by pricing higher than $\underline{\rho }$. This case of Type C pricing requires that each firm’s residual demand is sufficiently low at prices greater than $\underline{\rho }$.

Next, we establish a condition such that the only possible pure strategy equilibrium is Type C.

Proposition 7. 

Index the firms such that ${\underline{a}}_1\ge {\underline{a}}_2$. If $s_1\left({\underline{a}}_1\right)=0$, 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 $s_1\left(\underline{\rho }\right)+s_2\left(\underline{\rho }\right)=D\left(\underline{\rho }\right)$. 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 $d_i$ and $d_i^{\prime }$; let $\phi =({\phi }_1,{\phi }_2)$ and ${\phi }^{\prime }$ denote the corresponding front-side profits and $\psi =({\psi }_1,{\psi }_2)$ and ${\psi }^{\prime }$ 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 $d_i^{\prime }\ge d_i$, then ${\overline{r}}^{\prime }\ge \overline{r}$, ${\underline{r}}^{\prime }\ge \underline{r}$, ${\overline{\phi }}^{\prime }\ge \overline{\phi }$ and ${\underline{\phi }}^{\prime }\ge \underline{\phi }$.

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 $s=(s_1,s_2)$ and $s^{\prime }$, let $\phi =({\phi }_1,{\phi }_2)$ and ${\phi }^{\prime }$ denote the corresponding front-side profits and $\psi =({\psi }_1,{\psi }_2)$ and ${\psi }^{\prime }$ denote the corresponding residual profits.

Proposition 9. 

Consider a weak cost reduction for firm i that results in a weak supply increase so that $s_i^{\prime }\ge s_i$. Suppose further that this results in a weak reduction in residual demand for firm j, $d_j(x,y)\ge d_j^{\prime }(x,y)$ for all $x\ge y$. Lastly, suppose that $c_i\left(q\right)-c_i^{\prime }\left(q\right)$ is nondecreasing in q. Then ${\overline{r}}^{\prime }\le \overline{r}$, ${\underline{r}}^{\prime }\le \underline{r}$, ${\overline{\phi }}_j^{\prime }\ge {\overline{\phi }}_j$, and ${\underline{\phi }}_j^{\prime }\ge {\underline{\phi }}_j$.

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 $s_2\left(x\right)>0$ for all $x>0$ and $s_2\left(0\right)=0$. Suppose that the firms are not capacity constrained. It follows that $\underline{\rho }=0$. Note that by choosing a price x arbitrarily close to zero, the right continuity of $s_2$ guarantees that ${\psi }_1(x,0)>0$. Thus, $p_1=p_2=0$ 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 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.