Cournot and Bertrand Duopoly Models in Game Theory

Imagine two rival smartphone manufacturers deciding how many units to produce without knowing each other’s plans, or two competing coffee shops on the same street trying to undercut each other’s prices. These everyday business scenarios form the foundation of two influential economic models developed in the 19th century that continue to shape how economists understand market competition today. The Cournot and Bertrand duopoly models offer contrasting perspectives on strategic competition, revealing how firms make decisions in markets dominated by just two players.

The foundations of quantity competition: Cournot’s insight 🔗

Named after French mathematician Antoine Augustin Cournot, who developed this framework in 1838, the Cournot model focuses on firms competing through output decisions. In this scenario, two firms simultaneously decide how much of a homogeneous product to produce, each assuming the rival’s output will remain fixed. The market then determines the price based on total supply, creating a fascinating interdependence between the competitors’ profit functions.

Picture two mineral water companies supplying the same market. Each company must commit to production levels before knowing what the other will do. If one company floods the market with output, the price drops for both producers. This creates a strategic dilemma: produce too much and depress prices, or produce too little and leave profits on the table.

How firms respond: reaction functions explained 🔗

The heart of Cournot competition lies in what economists call reaction functions. Each firm’s reaction function shows its profit-maximizing output given the rival’s production level. These functions typically slope downward, indicating that quantities are strategic substitutes. In practical terms, if your competitor produces more, your best response is to produce less.

Think of it like sharing a fixed-size pie. If one firm takes a larger slice by increasing production, the other firm finds it more profitable to scale back rather than flood an already saturated market. This negative relationship between rival outputs characterizes quantity competition and distinguishes it fundamentally from price competition.

Finding equilibrium: where strategy meets stability 🔗

The Cournot-Nash equilibrium occurs where the two reaction functions intersect. At this point, neither firm wants to change its output because each is producing the profit-maximizing quantity given the rival’s choice. For symmetric firms with identical costs, both companies produce the same amount, resulting in industry output that falls between monopoly levels and perfectly competitive levels.

Consider a numerical example: if market demand follows the equation P = 100 – Q, where Q is total output, and both firms have marginal costs of 10, the Cournot equilibrium has each firm producing 30 units. Total industry output reaches 60 units, with a market price of 40. Compare this to a monopolist who would produce just 45 units at a price of 55, or perfect competition where output would reach 90 units at a price equal to marginal cost.

The temptation of collusion and the prisoner’s dilemma 🔗

While the Cournot equilibrium is stable, it’s not the most profitable outcome for the firms. If they could collude and jointly restrict output to monopoly levels, both would earn higher profits by splitting the monopoly profit equally. However, collusive agreements are inherently unstable in the Cournot framework because each firm has a private incentive to cheat by producing slightly more than agreed.

This creates a classic prisoner’s dilemma scenario. Suppose both firms agree to produce 22.5 units each, mimicking monopoly output. Each firm can increase its profit by secretly producing more, say 25 units, while hoping the rival sticks to 22.5. But when both firms think this way, they end up back at the Cournot equilibrium of 30 units each, earning less than they would have under successful collusion but more than if only one cheated. The self-interested pursuit of individual profit undermines joint profit maximization.

Switching gears: Bertrand’s price competition model 🔗

In 1883, French mathematician Joseph Bertrand challenged Cournot’s assumptions by proposing that firms compete on price rather than quantity. In the Bertrand model, two firms simultaneously set prices for a homogeneous product, and customers naturally flock to the lower-priced option. This seemingly minor change in strategic variable leads to dramatically different outcomes.

The Bertrand paradox refers to the startling prediction that with just two competitors, prices get driven all the way down to marginal cost, resulting in zero economic profit. This outcome mirrors perfect competition despite having only two firms in the market, which seems counterintuitive given the concentrated market structure.

Understanding the logic of undercutting 🔗

Why does Bertrand competition yield such extreme results? The logic is straightforward but powerful. Suppose both firms initially set prices above marginal cost, earning positive profits. Each firm has an incentive to slightly undercut the rival’s price, capturing the entire market demand and substantially boosting profits. But the rival thinks the same way, leading to a downward spiral of price cuts.

This undercutting continues until neither firm can profitably reduce prices further, which occurs when price equals marginal cost. At this point, undercutting would mean selling at a loss, so both firms reluctantly accept zero economic profit. The only stable equilibrium involves both firms pricing at marginal cost, even though this leaves no room for covering fixed costs.

The mathematics of best responses in price competition 🔗

The best response functions in Bertrand competition differ fundamentally from Cournot due to the discontinuous nature of demand. If a firm’s rival prices above marginal cost, the best response is to undercut by the smallest possible amount, capturing the entire market. If the rival prices at marginal cost, the firm is indifferent between matching or pricing slightly higher.

This creates what economists call a “weakly dominated strategy” equilibrium. Pricing at marginal cost is a Nash equilibrium not because it’s highly profitable, but because no firm can do better given the rival’s price. The competitive outcome emerges from the aggressive dynamics of price competition rather than from actual efficiency gains.

Contrasting the two models: strategic substitutes versus strategic complements 🔗

The fundamental difference between Cournot and Bertrand models lies in how strategic variables interact. In Cournot competition, quantities are strategic substitutes, meaning firms accommodate rivals’ increases by reducing their own output. In Bertrand competition, prices are strategic complements, with firms aggressively countering rivals’ price increases by cutting their own prices even further.

This distinction matters enormously for market outcomes. Cournot industries tend to be less competitive, with prices above marginal cost and positive economic profits. Bertrand industries are extremely competitive, with prices driven to marginal cost and zero profits. The choice of model depends on market characteristics: industries where capacity is fixed and production decisions are made in advance fit the Cournot framework, while industries with flexible capacity and instant price adjustments align better with Bertrand.

Why reality rarely matches the extremes 🔗

Neither model perfectly describes real-world markets because both rest on extreme assumptions. The Bertrand paradox rarely appears in practice because most products have some differentiation, firms face capacity constraints, consumers incur search costs, and repeated interactions allow for tacit collusion. Similarly, pure Cournot competition is uncommon because firms often have some flexibility to adjust production in response to rivals’ choices.

Consider the airline industry. Airlines choose capacity (seat availability) well in advance, resembling Cournot competition. Yet they also engage in frequent price adjustments, incorporating elements of Bertrand competition. Most real industries blend features of both models, with the appropriate framework depending on whether quantity commitments or price flexibility dominates strategic decision-making.

Applications in modern markets 🔗

These classical models remain surprisingly relevant for understanding contemporary markets. The Organization of the Petroleum Exporting Countries (OPEC) often exhibits behavior consistent with Cournot competition, with member countries setting production quotas that affect global oil prices. Technology platforms competing for network effects sometimes engage in aggressive price competition reminiscent of Bertrand dynamics, with firms initially pricing below cost to capture market share.

Understanding these models helps regulators evaluate merger proposals and assess market power. A Cournot merger analysis would focus on how combined production capacity affects market output and prices, while a Bertrand analysis would examine whether merged entities could coordinate on higher prices. The models also inform antitrust policy by revealing how market structure influences competitive behavior and consumer welfare.

What do you think? Can you identify real-world examples of companies that engage primarily in quantity competition versus price competition? How might digital platforms and modern supply chain flexibility be changing the relevance of these traditional models in today’s economy?

References

1. https://en.wikipedia.org/wiki/Cournot_competition
2. https://policonomics.com/lp-game-theory2-cournot-duopoly-model/
3. https://www.economics.utoronto.ca/osborne/2×3/tutorial/COURNOT.HTM
4. https://www.economics.utoronto.ca/jfloyd/modules/dcne.html
5. https://en.wikipedia.org/wiki/Bertrand_competition
6. https://inomics.com/terms/bertrand-competition-1504578
7. https://data88e.org/textbook/content/07-game-theory/bertrand.html
8. https://en.wikipedia.org/wiki/Bertrand_paradox_(economics)
9. https://inomics.com/terms/cournot-competition-1525473