Business Economics · Profit maximisation under imperfect competition

Oligopoly and Game Theory: Models and Nash Equilibrium

Updated 11 October 2026 · Fact-checked

An oligopoly is a market with a few large firms whose decisions depend on each other. To solve questions, identify the model: kinked demand, Cournot (firms choose quantities), Bertrand (firms choose prices) or a payoff matrix. Then find each firm's best response and the point where no firm wants to change: the Nash equilibrium.

Understand Oligopoly and Game Theory

An oligopoly is a market dominated by a small number of firms, with high barriers to entry. Examples are Indian telecom, airlines and cement. The key feature is interdependence: one firm's price or output changes the profit of its rivals, so each firm must guess how rivals will react before it acts.

Because of this, there is no single oligopoly model. Firms can collude, agreeing on price or output. A formal agreement is a cartel, which acts like a joint monopoly and restricts output to raise price. Cartels are usually illegal and unstable, because each member gains by secretly cheating on its quota.

The kinked demand curve explains why prices can be sticky. The firm assumes that rivals will match a price cut but ignore a price rise. So demand is elastic above the current price (a rise loses many customers) and inelastic below it (a cut gains few customers). The marginal revenue curve then has a vertical gap at the kink. If marginal cost moves within that gap, the profit-maximising price and output do not change. The model describes price stability but does not explain how the price was set in the first place.

Two classic models cover rivalry. In Cournot, each firm chooses its quantity, taking the rival's quantity as given. In Bertrand, firms with identical products choose prices, and each undercuts the other until price equals marginal cost. So Cournot gives a price above marginal cost, while basic Bertrand gives a competitive outcome even with two firms.

Game theory studies strategic choice using a payoff matrix. A dominant strategy is best whatever the rival does. A Nash equilibrium is a set of strategies where each player's choice is the best response to the others' choices, so nobody gains by changing alone. In the prisoner's dilemma, both firms are better off cooperating, but each has an incentive to cheat, so both end up in a worse outcome. This is why cartels break down.

Key rules to remember

Profit maximisation: MR = MC; Every firm still uses this rule. Under kinked demand, check whether MC lies in the gap of the MR curve.
Linear inverse demand (two firms): P = a − b(q₁ + q₂); Standard set-up for Cournot questions. Market quantity Q = q₁ + q₂.
Cournot best response (zero or constant MC = c): q₁ = (a − c − b·q₂) ÷ (2b); Comes from MR₁ = a − 2b·q₁ − b·q₂ set equal to c. Firm 2's response is the mirror image.
Cournot equilibrium, n identical firms: qᵢ = (a − c) ÷ (b(n + 1)); P = (a + n·c) ÷ (n + 1); For two firms, each produces one third of (a − c) ÷ b. Valid for linear demand and identical constant MC.
Bertrand equilibrium (identical products, same constant MC): P = MC; Each firm earns zero economic profit. The result needs identical goods and no capacity limits.
Nash equilibrium test: Each player's strategy is a best response to the other's strategy; In a matrix, mark each player's best payoff in response to each rival choice. A cell with both marks is a Nash equilibrium.
Cartel (joint monopoly) output: Set MR of total market = MC, then share output; Total profit is maximised, but each member is tempted to produce more than its quota.

How to solve Oligopoly and Game Theory questions

Use this order for any oligopoly or game theory question. Decide the model first, because it fixes what the firms choose.

1Read what the firms choose: quantity (Cournot), price (Bertrand), whether they move together (collusion), or a list of strategies with payoffs (matrix).
2Write down the given data: demand function, marginal cost, number of firms, and whether products are identical.
3For Cournot, derive each firm's marginal revenue holding the rival's output fixed. Set MR = MC to get the best response function.
4Solve the best response functions together. For identical firms you can set q₁ = q₂. Then find total Q, price P and profit.
5For a payoff matrix, find each player's best response to each rival strategy. Mark them. Cells with both marks are Nash equilibria. Check for dominant strategies first.
6For kinked demand, locate the kink at the current price, draw the MR gap, and check whether MC cuts MR in the gap. If yes, price stays the same.
7Compare with the cooperative outcome (cartel or both cooperate). State why firms are tempted to cheat and why the cartel is unstable.
8State your answer with units, and write any assumption you used, such as identical products and constant MC.

Quickest way: Best-response marking in a payoff matrix

When to use it: Use this for any MCQ or short question with a 2 by 2 or 3 by 3 payoff matrix asking for Nash equilibrium or dominant strategy.

For each column the rival could pick, circle the row player's highest payoff in that column.
For each row the rival could pick, circle the column player's highest payoff in that row.
Any cell with both payoffs circled is a Nash equilibrium.
If one row is best against every column, that is a dominant strategy for the row player. Same logic for columns.
Check whether the equilibrium is worse for both than another cell. If so, it is a prisoner's dilemma.

Common mistakes in Oligopoly and Game Theory

Saying Bertrand and Cournot give the same result.
Both are 'two-firm competition' models, so students blur them.
Fix: Remember the choice variable. Cournot: quantity, price above MC. Bertrand with identical goods: price, and price equals MC.
Treating the kinked demand curve as an explanation of how the price is set.
The diagram looks like a full theory of pricing.
Fix: State that it only explains why price stays stable once it has been reached. Also mention it assumes rivals match cuts but not rises.
Setting MR = MC with the market demand slope instead of the firm's own slope in Cournot.
Students forget that the firm's MR is a − 2b·q₁ − b·q₂, not a − 2b(q₁ + q₂).
Fix: Differentiate the firm's own revenue q₁·P with q₂ held constant. The own-output term doubles; the rival's term does not.
Picking the best joint outcome in a matrix as the Nash equilibrium.
The highest total payoff looks like the 'best' answer.
Fix: Nash equilibrium is about no one gaining by changing alone, not about total payoff. Test each cell using best responses.
Assuming a cartel is stable because it maximises joint profit.
Students focus on the joint outcome and ignore individual incentives.
Fix: Show that each member earns more by raising output while others keep to quota. That is the prisoner's dilemma structure.
Missing that a matrix can have more than one Nash equilibrium, or none in pure strategies.
Students stop after finding the first cell.
Fix: Check every cell. Report all equilibria, or state that none exists in pure strategies.

Worked examples

Example 1

Two firms sell identical goods. Market inverse demand is P = 100 − Q, where Q = q₁ + q₂. Each firm has constant marginal cost ₹20 and no fixed cost. Find the Cournot equilibrium output of each firm, the market price and each firm's profit.

Show the solution

Firm 1 revenue = q₁(100 − q₁ − q₂). So MR₁ = 100 − 2q₁ − q₂.
Set MR₁ = MC: 100 − 2q₁ − q₂ = 20, so q₁ = (80 − q₂) ÷ 2 = 40 − 0.5q₂.
By symmetry, q₂ = 40 − 0.5q₁.
Set q₁ = q₂ = q: q = 40 − 0.5q, so 1.5q = 40, q = 26.67 (exactly 80 ÷ 3).
Total Q = 160 ÷ 3 = 53.33. Price P = 100 − 160 ÷ 3 = 140 ÷ 3 = ₹46.67.
Profit per firm = (P − MC) × q = (140 ÷ 3 − 20) × (80 ÷ 3) = (80 ÷ 3) × (80 ÷ 3) = 6,400 ÷ 9 = ₹711.11.

Answer: Each firm produces 80 ÷ 3 ≈ 26.67 units. Price is about ₹46.67. Each firm earns profit of about ₹711.11.

Example 2

Two firms, A and B, choose High price or Low price. Payoffs (profit in ₹ crore) are listed as (A, B). Both High: (10, 10). A High, B Low: (2, 14). A Low, B High: (14, 2). Both Low: (6, 6). Find the Nash equilibrium and explain why it is a prisoner's dilemma.

Show the solution

Firm A's best response: if B plays High, A gets 10 from High or 14 from Low, so Low is best. If B plays Low, A gets 2 from High or 6 from Low, so Low is best.
So Low is a dominant strategy for A.
Firm B is symmetric: Low is best whether A plays High (14 versus 10) or Low (6 versus 2). Low is dominant for B.
The only cell where both play best responses is (Low, Low), with payoffs (6, 6). This is the unique Nash equilibrium.
Both High gives (10, 10), which is better for both firms than (6, 6).
Cooperation is not stable, because each firm gains by cutting its price if the other stays High.

Answer: The Nash equilibrium is both firms choosing Low price, with payoffs of ₹6 crore each. It is a prisoner's dilemma because both High would give ₹10 crore each, but each firm has an incentive to deviate.

Exam tips

For MCQs, identify the model from one phrase: 'chooses quantity' means Cournot, 'chooses price with identical goods' means Bertrand, 'price rigidity' means kinked demand.
In written answers, show the best response derivation, not just the final numbers. Marks are often given for the method and for stating assumptions.
Always give a short verbal conclusion: compare the Nash outcome with the cartel outcome and say which is better for firms and for consumers.
Draw a clean kinked demand diagram with the kink, the broken MR curve and the MC lines passing through the gap. Label elastic and inelastic sections.
In a matrix, show your best-response marks so the examiner can follow your reasoning even if you slip on one number.

Practice questions from Profit maximisation under imperfect competition

Oligopoly and Game Theory in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Oligopoly and Game Theory: frequently asked questions

What is the difference between Cournot and Bertrand models?

In Cournot, firms choose quantities and the market price follows from total output, so price stays above marginal cost. In basic Bertrand, firms with identical goods choose prices and undercut each other until price equals marginal cost. The Bertrand result needs identical products and no capacity limits.

How do I find the Nash equilibrium in a payoff matrix?

For each strategy of the rival, mark your best payoff. Do the same for the rival. Any cell where both players' payoffs are marked is a Nash equilibrium. Always check every cell, as there may be more than one.

Why does the kinked demand curve show price stickiness?

The firm believes rivals match a price cut but not a price rise. This makes demand elastic above the kink and inelastic below it. Marginal revenue has a gap at the current output, so costs can change within the gap without changing price.

Why do cartels tend to collapse?

Each member can raise its profit by producing more than its quota while others keep to theirs. This is the prisoner's dilemma. Because everyone has the same incentive, the agreement breaks down unless it can be monitored and enforced.