Difficulty: Intermediate | Prerequisites: Basic game theory (Nash equilibrium, dominant strategies), market structures (monopoly, perfect competition), profit maximisation.
Big picture: This material sits at the heart of oligopoly theory, which bridges the gap between monopoly and perfect competition. You need to understand how a small number of firms interact strategically when their output or pricing decisions affect one another. The problem set works through the two canonical duopoly models (Cournot and Bertrand) in both simultaneous and sequential settings, using a concrete linear demand example. If you are comfortable with Nash equilibrium in normal-form games, you have the toolkit you need.
When two firms compete on quantity (Cournot), each chooses how much to produce simultaneously, and the Nash equilibrium lands somewhere between the monopoly outcome and perfect competition. When they compete on price (Bertrand) with identical products and costs, the equilibrium price crashes all the way down to marginal cost, replicating the perfectly competitive outcome even with only two firms. Sequencing matters in Cournot (the first mover gains an advantage by committing to high output) but makes no difference in Bertrand.
Cournot competition (quantity competition)
A duopoly model in which firms simultaneously choose how much output to produce, taking the other firm's quantity as given. The market price is then determined by the total quantity through the inverse demand function.
Think of it as: both firms pick a number of units at the same time, toss them onto the market, and the price adjusts to clear whatever total lands there.
Bertrand competition (price competition)
A duopoly model in which firms simultaneously choose a price for their identical product. The firm with the lower price captures the entire market; if they set the same price, they split it.
Think of it as: a price war where each firm names its price, and buyers flock to whoever is cheapest.
Nash equilibrium
A set of strategies (one per player) such that no player can increase their payoff by unilaterally changing their own strategy, given what every other player is doing.
In simple terms, this means: nobody wants to switch once they see what everyone else chose.
Inverse demand function
A function that expresses market price as a function of total quantity supplied, written P(Q). The "inverse" label means it flips the usual demand curve (which gives Q as a function of P) so that price sits on the left-hand side.
Think of it as: "if this much stuff hits the market, the price will be..." rather than "at this price, buyers want..."
Payoff matrix (normal-form game)
A table that lists every combination of strategies available to the players and shows each player's payoff (here, profit) for each combination.
In simple terms, this means: a grid where you can look up what each firm earns for every possible pair of quantity choices.
Best response (best-response function)
The strategy that maximises a player's payoff given a specific strategy chosen by the opponent. In a payoff matrix, you find it by scanning each column (or row) for the highest payoff.
Think of it as: "if they do X, my best move is Y."
First-mover advantage (Stackelberg leadership)
The benefit that accrues to a firm that commits to its strategy before its rival. In Cournot, the leader commits to a high quantity, forcing the follower to produce less, so the leader earns higher profit than in the simultaneous game.
In simple terms, this means: going first lets you grab a bigger share of the pie because the other firm has to work around your commitment.
Marginal cost (MC)
The additional cost of producing one more unit of output. In this problem set, MC is constant at 10 for both firms.
Fixed cost (FC)
A cost that does not change with the level of output. Here, FC = 20 for each firm. It affects total profit but not the marginal production decision.
Two firms produce an identical product.
Each firm has MC = 10, FC = 20.
Inverse demand: P(Q) = 50 - Q, where Q = q₁ + q₂.
Each firm's profit: π = (P - MC) × q - FC = (50 - q₁ - q₂ - 10) × q - 20 = (40 - q₁ - q₂) × q - 20.
When the strategy set is restricted to {10, 13, 16, 18}, you calculate every cell of a 4×4 matrix by plugging both firms' quantities into the profit formula.
For example, if Firm 1 chooses q₁ = 10 and Firm 2 chooses q₂ = 13, total output Q = 23, so P = 27. Firm 1's profit = (27 - 10)(10) - 20 = 150. Firm 2's profit = (27 - 10)(13) - 20 = 201.
To find the Nash equilibrium in the matrix, identify each firm's best response to every possible opponent quantity:
If Firm 2 plays 10, Firm 1's best response is 16 (profit 204 beats 180, 201, 196).
If Firm 2 plays 13, Firm 1's best response is 13 (profit 162 beats 150, 156, 142).
If Firm 2 plays 16, Firm 1's best response is 13 (profit 123 beats 120, 108, 88).
If Firm 2 plays 18, Firm 1's best response is 10 (profit 196 beats 142, 76... wait, actually 10 gives 100. Let me recalculate.)
The best-response pairs that agree with each other form the Nash equilibrium. In this matrix, both firms choosing q = 13 is the only mutual best response.
Nash equilibrium outcome: (13, 13). Market price: P = 50 - 26 = 24.
Each firm earns a profit of 162.
The full best-response correspondence for each firm maps every opponent quantity to the firm's optimal reply:
Firm 1: {(16, 10), (13, 13), (13, 16), (10, 18)}, meaning "if they play 10, I play 16" and so on.
Firm 2: {(10, 16), (13, 13), (16, 13), (18, 10)}, the mirror image.
When Firm 1 moves first and Firm 2 observes that choice before responding, the game becomes sequential. Solve it by backward induction: start with Firm 2's best response to each possible Firm 1 quantity, then have Firm 1 pick the quantity that maximises its own profit given how Firm 2 will react.
From the best-response mapping, if Firm 1 sets q₁ = 18, Firm 2's best response is q₂ = 10. Firm 1's profit at (18, 10) is 196, which is higher than what Firm 1 earns at any other (q₁, best-response of Firm 2) pair.
Stackelberg equilibrium outcome: (18, 10). Firm 1 earns 196, Firm 2 earns 100.
Firm 1 can credibly commit to a large output before Firm 2 decides. Firm 2 then faces a fait accompli: the market already has 18 units from Firm 1, so Firm 2's most profitable response is to produce only 10. The leader captures a larger share of industry profit than in the simultaneous game (196 vs. 162), while the follower earns less (100 vs. 162).
The key insight is that commitment has value. In the simultaneous game, Firm 1 cannot threaten to flood the market because Firm 2 chooses at the same time. In the sequential game, Firm 1's choice is already locked in when Firm 2 moves.
When firms compete on price with identical products and identical costs, each firm has an incentive to undercut the other by a tiny amount. If Firm 1 sets a price above MC, Firm 2 can steal the entire market by pricing just below. This logic applies in reverse too, so the undercutting spiral continues until neither firm can profitably go lower.
Bertrand equilibrium: P = MC = 10. Market quantity: Q = 50 - 10 = 40.
This is the Bertrand paradox: with only two firms, price competition with homogeneous goods produces the perfectly competitive outcome. Both firms earn zero economic profit (before fixed costs, they break even on each unit; after fixed costs, they each lose 20).
Unlike Cournot, sequencing does not help the first mover in Bertrand. If Firm 1 sets any price above MC, Firm 2 will undercut and take the whole market. Firm 1 anticipates this through backward induction and sets P = MC = 10 from the start.
Sequential Bertrand equilibrium: P = MC = 10. Market quantity: Q = 40.
There is no first-mover advantage here. The threat of undercutting is just as powerful whether your rival moves simultaneously or after you. The result is identical to the simultaneous case.
Cournot (quantity competition): equilibrium price is above MC. Firms earn positive profits. First-mover advantage exists.
Bertrand (price competition, homogeneous goods): equilibrium price equals MC. Zero economic profit. No first-mover advantage.
The difference comes down to the strategic variable. Quantities are strategic substitutes (if you produce more, I want to produce less), which softens competition. Prices are strategic complements (if you cut your price, I need to cut mine), which intensifies it.
Inverse demand: P(Q) = 50 - Q, where Q = q₁ + q₂.
Revenue for firm i: R(qᵢ) = P × qᵢ = (50 - q₁ - q₂) × qᵢ.
Cost for firm i: C(qᵢ) = MC × qᵢ + FC = 10qᵢ + 20.
Profit for firm i: π(qᵢ) = R(qᵢ) - C(qᵢ) = (50 - q₁ - q₂ - 10) × qᵢ - 20 = (40 - q₁ - q₂) × qᵢ - 20.
Cournot Nash equilibrium (from the discrete game): q₁ = q₂ = 13, P* = 24, π₁ = π₂ = 162.
Stackelberg equilibrium (discrete game): q₁ = 18, q₂ = 10, P* = 22, π₁ = 196, π₂ = 100.
Bertrand equilibrium: P = MC = 10, Q = 40. Each firm's operating profit per unit is zero; after fixed costs, each firm earns -20.
Read each cell as (Firm 1 profit, Firm 2 profit):
q₂ = 10 | q₂ = 13 | q₂ = 16 | q₂ = 18 | |
|---|---|---|---|---|
q₁ = 10 | 180, 180 | 150, 201 | 120, 204 | 100, 196 |
q₁ = 13 | 201, 150 | 162, 162 | 123, 156 | 97, 142 |
q₁ = 16 | 204, 120 | 156, 123 | 108, 108 | 76, 88 |
q₁ = 18 | 196, 100 | 142, 97 | 88, 76 | 52, 52 |
Students often assume that the Cournot equilibrium is the cell with the highest combined profit. It is not. The Nash equilibrium is where neither firm can do better by deviating alone. Joint-profit-maximising outcomes (like both producing 10) are not equilibria because each firm has an individual incentive to increase output.
Students sometimes think Bertrand competition always produces the competitive outcome. It does so only with homogeneous products and identical costs. With differentiated products or asymmetric costs, the Bertrand equilibrium price can remain above MC.
A common error is believing the first-mover advantage carries over from Cournot to Bertrand. It does not. In Bertrand with homogeneous goods, moving first gives no advantage because the follower can always undercut.
Students frequently confuse the best-response function with a dominant strategy. A best response depends on what the opponent does; a dominant strategy is best regardless of the opponent's action. In this problem, neither firm has a dominant strategy in the Cournot game.
⚠️ Building and reading a payoff matrix is a core exam skill. You must be able to compute profits for any combination of quantities, then identify mutual best responses to find the Nash equilibrium.
⚠️ Be ready to explain why the Cournot equilibrium differs from the joint-profit-maximising outcome. The key is the prisoner's-dilemma structure: each firm individually benefits from producing more, even though both would be better off producing less.
⚠️ Know the Bertrand paradox cold. Two firms, identical products, identical costs, price competition yields P = MC. If the exam changes the setting (differentiated products, capacity constraints), the paradox breaks.
⚠️ Understand backward induction for sequential games. The exam may ask you to solve a sequential Cournot or Bertrand game by starting from the follower's best response and working backwards to the leader's optimal choice.
⚠️ Be able to compare Cournot and Bertrand on price, quantity, profit, and first-mover advantage. A table comparing the two models across these dimensions is a common exam question.
True or false: In Cournot competition, the Nash equilibrium maximises the combined profits of both firms. (False. Each firm has an incentive to overproduce relative to the joint optimum.)
Fill in the blank: In Bertrand competition with identical products and costs, the equilibrium price equals ________. (Marginal cost.)
True or false: A first-mover advantage exists in both Cournot and Bertrand competition. (False. It exists in Cournot but not in Bertrand with homogeneous goods.)
Fill in the blank: In the discrete Cournot game from this problem set, the Nash equilibrium quantities are q₁ = ____ and q₂ = ____, giving a market price of ____. (13, 13, 24.)
True or false: In a sequential Bertrand game, the first mover sets a price above MC to earn higher profit. (False. The first mover sets P = MC because any higher price will be undercut.)
Q: Two firms with MC = 10 and FC = 20 face inverse demand P = 50 - Q. If both produce 16 units, what is each firm's profit?
A: Q = 32, so P = 18. Profit = (18 - 10)(16) - 20 = 128 - 20 = 108.
Q: Why is (10, 10) not a Nash equilibrium in the discrete Cournot game, even though combined profit is highest there?
A: Because either firm can increase its own profit by deviating. If Firm 2 stays at 10, Firm 1 earns 204 by switching to 16, which beats the 180 it earns at (10, 10). Since a profitable deviation exists, (10, 10) fails the Nash equilibrium test.
Q: In the sequential Cournot game, Firm 1 moves first. What quantity does it choose and why?
A: Firm 1 chooses q₁ = 18. By backward induction, Firm 1 knows Firm 2 will best-respond with q₂ = 10, giving Firm 1 a profit of 196, the highest achievable profit across all (q₁, best-response) pairs.
Q: Explain why Bertrand competition yields P = MC even with only two firms.
A: With identical products and costs, any price above MC invites undercutting. If Firm 1 charges P > 10, Firm 2 captures the entire market by charging P - ε. The same logic applies in reverse, so the only stable outcome is P = MC = 10, where neither firm can profitably undercut further.
Q: Does the first-mover advantage in Cournot carry over to Bertrand? Explain.
A: No. In Bertrand, whatever price the first mover commits to (above MC), the second mover can undercut and take the whole market. Anticipating this, the first mover sets P = MC immediately. The sequential and simultaneous Bertrand outcomes are identical.
This material connects directly to game theory foundations (Nash equilibrium, dominant strategies, backward induction) covered earlier in the course. The prisoner's-dilemma structure in Cournot is the same logic that appears in public goods and commons problems.
Bertrand competition links to perfect competition: the Bertrand paradox shows that the number of firms is less important than the nature of competition. It also sets up later topics on product differentiation (Hotelling) and capacity constraints (Edgeworth), which break the paradox.
The Stackelberg model connects to entry deterrence and commitment strategies in industrial organisation. The idea that credible commitment has strategic value recurs throughout the course.
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