Competitive Pricing & Game Theory

When two or more firms sell substitutable products, each firm’s optimal price depends on the prices set by its rivals. Pricing is no longer an optimization problem with a fixed environment—it is a game, and the solution concept shifts from “optimum” to “equilibrium.”

The Pricing Game

Every pricing model considered so far on this site treats the firm as a monopolist facing a demand curve it cannot influence except through its own price. In practice, most markets contain competitors whose pricing decisions shift demand toward or away from a firm’s product. A price cut by a rival draws customers away; a price increase by a rival sends customers toward you. The result is strategic interdependence: each firm’s profit depends not only on its own price but on the prices chosen by others.

This interdependence transforms the pricing problem from a single-agent optimization into a game in the formal sense of game theory. The players are the firms, the strategies are prices, and the payoffs are profits. The natural solution concept is the Nash equilibrium—a set of prices from which no firm has a unilateral incentive to deviate.

An important subtlety is that competitive pricing is not necessarily zero-sum. When products are differentiated, both firms can earn positive profits in equilibrium. Indeed, greater differentiation tends to raise equilibrium prices and profits for everyone—a key strategic insight emphasized by Nagle and Müller (2018). The destructive case arises when products are nearly identical: prices are driven toward marginal cost and profits evaporate, the classic result of undifferentiated Bertrand competition described by Tirole (1988).

Differentiated Bertrand Model

The Demand System

Consider two firms, each selling a differentiated product. The demand for firm $i$ depends on both its own price $p_i$ and the rival’s price $p_j$ :

Definition — Differentiated Bertrand Demand

The differentiated Bertrand demand system specifies quantity demanded for each firm as:

q_i = a - b \, p_i + d \, p_j

where:

$a > 0$ is the base demand (intercept).
$b > 0$ is the own-price sensitivity: demand lost per dollar of own-price increase.
$0 \le d < b$ is the cross-price sensitivity: demand gained when the rival raises price by one dollar. The condition $d < b$ ensures that own-price effects dominate.
$c_i \ge 0$ is firm $i$ ’s constant marginal cost.

When $d = 0$ , the two firms are independent monopolists—each faces a demand curve that depends only on its own price. When $d \to b$ , the products become perfect substitutes and the model approaches the homogeneous Bertrand game, in which equilibrium prices collapse to marginal cost.

Best Response Derivation

Each firm maximizes its own profit taking the rival’s price as given:

\max_{p_i} \; \pi_i = (p_i - c_i)(a - b \, p_i + d \, p_j)

(1)

This is a concave quadratic in $p_i$ . Setting the first-order condition to zero:

\frac{\partial \pi_i}{\partial p_i} = a - 2b \, p_i + d \, p_j + b \, c_i = 0

(2)

Solving for $p_i$ yields the best response function:

p_i^*(p_j) = \frac{a + d \, p_j + b \, c_i}{2b}

(3)

This is a linear, upward-sloping function of the rival’s price. When the rival charges more, the best response is to raise your own price as well—prices are strategic complements in differentiated Bertrand competition.

Nash Equilibrium

Definition — Nash Equilibrium

A pair of prices $(p_1^*, p_2^*)$ is a Nash equilibrium if each price is a best response to the other: $p_1^* = p_1^*(p_2^*)$ and $p_2^* = p_2^*(p_1^*)$ . At the equilibrium, neither firm can increase its profit by unilaterally changing its price.

Substituting each best response into the other produces a system of two linear equations in two unknowns. Solving yields the closed-form equilibrium:

Bertrand–Nash Equilibrium (Differentiated Duopoly)

Under the linear differentiated demand system with $d < b$ , the unique Nash equilibrium prices are:

p_1^* = \frac{2b(a + b \, c_1) + d(a + b \, c_2)}{4b^2 - d^2}

(4)

p_2^* = \frac{2b(a + b \, c_2) + d(a + b \, c_1)}{4b^2 - d^2}

(5)

In the symmetric case where $c_1 = c_2 = c$ , these simplify to:

p^* = \frac{a + b \, c}{2b - d}

(6)

Symmetric Duopoly

Let $a = 100$ , $b = 2$ , $d = 0.5$ , $c = 10$ .

Nash price: $p^* = \frac{100 + 2 \times 10}{2 \times 2 - 0.5} = \frac{120}{3.5} \approx \$34.29$
Nash demand: $q^* = 100 - 2(34.29) + 0.5(34.29) \approx 48.57$ units
Nash profit per firm: $\pi^* = (34.29 - 10)(48.57) \approx \$1{,}179$

Now consider what would happen if both firms could coordinate on the single-firm monopoly price. A monopolist facing $q = 100 - 2p + 0.5p = 100 - 1.5p$ (treating the rival’s price as its own) would set $p_m = \frac{100 + 1.5 \times 10}{2 \times 1.5} = \frac{115}{3} \approx \$38.33$ . At this price each firm would earn approximately $1,344—about 14% more than the Nash profit. But neither firm would stick to this agreement: given that the rival charges $38.33, the best response is only $33.93, capturing more demand while free-riding on the rival’s high price. This is the prisoner’s dilemma of pricing.

Application: National Brand vs. Private Label

The differentiated Bertrand model finds a natural application in the competition between national brands and private labels (store brands). This asymmetric game captures a market structure that accounts for roughly 20% of grocery sales in the U.S. and over 40% in several European markets (Pauwels and Srinivasan (2004)).

The Asymmetric Game

Let firm $N$ be the national brand and firm $S$ be the store (private label) brand. The store brand has two structural advantages: lower costs (no advertising, simpler packaging) and shelf control (the retailer decides placement, facings, and promotion). The national brand has a quality/brand premium—consumers perceive it as higher quality and are willing to pay more.

We model this as asymmetric Bertrand with:

q_N = a_N - b \, p_N + d \, p_S, \qquad q_S = a_S - b \, p_S + d \, p_N

where $a_N > a_S$ (brand premium gives the national brand higher base demand) and $c_N > c_S$ (the store brand has a cost advantage). The best responses follow directly from the general model derived above.

Strategic Role of Private Labels

Definition — Private Label as Outside Option

The retailer uses its store brand as a credible outside option in negotiations with national brand manufacturers. By offering a competent private label alternative, the retailer can extract price concessions from the national brand: “match our store brand’s margin contribution, or we’ll allocate your shelf space to our own product.” The store brand need not be maximally profitable on its own—its value includes the bargaining leverage it provides (Raju, Sethuraman, and Dhar (1995)).

Quality Positioning and the Umbrella Effect

How should the store brand position its quality relative to the national brand? The equilibrium analysis reveals an umbrella effect: as store brand quality increases ( $a_S$ rises toward $a_N$ ), Nash equilibrium prices for BOTH brands fall. The national brand must cut price to retain customers, and the store brand’s best response follows suit.

The Umbrella Effect

In the asymmetric Bertrand model, as the store brand’s base demand $a_S$ increases (improving quality positioning):

The Nash equilibrium price of the national brand falls.
The Nash equilibrium price of the store brand also falls.
The national brand’s equilibrium profit falls unambiguously.
The store brand’s profit may rise or fall—it gains volume but at lower margins.

The optimal store brand quality positions just below the national brand, maximizing the sum of direct profit and bargaining leverage.

Pass-Through Asymmetry

Cost shocks pass through asymmetrically in this model. A cost increase hitting both brands raises both equilibrium prices, but the national brand’s price rises more because its higher-WTP customer base can absorb more of the increase. Conversely, an industry-wide cost decrease benefits the store brand disproportionately—its lower price attracts price-sensitive customers who switch away from the national brand.

Interactive Explorer

Adjust the demand and cost parameters to see how best response curves shift and the Nash equilibrium moves. Drag the red dot on the best response diagram to explore off-equilibrium price combinations and compare the resulting profits to those at the Nash equilibrium.

The “Best Responses” view plots each firm’s best response function; the intersection is the Nash equilibrium. The “Profit Landscape” view shows Firm 1’s profit as a function of its price, holding Firm 2 at the Nash price, confirming that deviating from $p_1^*$ reduces Firm 1’s profit.

Key Insights

1. Differentiation Is a Shield Against Price Wars

Slide the cross-price sensitivity $d$ toward zero and observe that each firm’s Nash price approaches the monopoly optimum $(a + bc)/(2b)$ . The firms are effectively independent monopolists, each earning the full monopoly profit on its own demand. Now slide $d$ toward $b$ : the products become near-perfect substitutes, the best response curves flatten, and equilibrium prices collapse toward marginal cost. Differentiation—whether through branding, features, or service—raises the equilibrium price and profit for all firms in the market.

2. Asymmetric Costs Shift the Equilibrium but Do Not Eliminate It

Raise Firm 1’s cost while keeping Firm 2’s constant. Firm 1’s Nash price rises (it passes part of the cost through), but Firm 2’s Nash price also rises—because prices are strategic complements, a cost increase for one firm relaxes competitive pressure on the other. The equilibrium continues to exist; only the profit split between the firms changes.

3. The Prisoner’s Dilemma of Pricing

As the numerical example above demonstrates, both firms would earn higher profits if they could jointly commit to prices above the Nash equilibrium. But such coordination is unstable: given the rival’s high price, each firm has an incentive to undercut, capturing additional demand at the rival’s expense. The Nash equilibrium is the only self-enforcing outcome—no explicit or implicit agreement is needed, because neither side wants to deviate. This tension between collective and individual rationality is the defining feature of oligopolistic pricing, and it explains why price competition in undifferentiated markets is so fierce.

References

Nagle, T. T. & Müller, G. (2018). The Strategy and Tactics of Pricing: A Guide to Growing More Profitably, 6th ed.. Routledge.
Pauwels, K. & Srinivasan, S. (2004). “Who Benefits from Store Brand Entry?.” Marketing Science, 23(3), 364–390.
Raju, J. S., Sethuraman, R. & Dhar, S. K. (1995). “The Introduction and Performance of Store Brands.” Management Science, 41(6), 957–978.
Tirole, J. (1988). The Theory of Industrial Organization. MIT Press.

Pricing Against Competitors