Difficulty: Intermediate-Advanced | Prerequisites: Monopoly, perfect competition, basic algebra
Oligopoly sits between perfect competition and monopoly: a small number of firms interact strategically. Cournot models simultaneous quantity competition; Stackelberg models sequential quantity competition with a leader and follower. Game theory provides the toolkit (Nash equilibrium, dominant strategies, backward induction, mixed strategies) for analysing strategic decisions more broadly. The exam tests you on solving reaction functions, finding equilibria, and computing welfare.
Reaction function (best response function)
Shows a firm's profit-maximising output as a function of the other firm's output. Derived by setting the partial derivative of profit with respect to own quantity equal to zero and solving for own quantity in terms of the rival's quantity.
Cournot-Nash equilibrium
Simultaneous quantity competition. Each firm chooses its output taking the other firm's output as given. The equilibrium is where the two reaction functions intersect: neither firm wants to change its output, given what the other is doing.
To solve: derive each firm's reaction function from its first-order condition, then solve the two equations simultaneously.
Cournot output per firm is less than the competitive output but more than half the monopoly output. Market price is between the competitive price and the monopoly price.
Stackelberg equilibrium
Sequential quantity competition. The leader moves first, the follower observes and then chooses. Solved by backward induction: first find the follower's reaction function, substitute it into the leader's profit function, then maximise.
The leader produces more than in Cournot, the follower produces less. Total output is higher and price is lower than in Cournot. The leader earns higher profit than the follower.
Dominant strategy
A strategy that is best regardless of what the other player does. If both players have a dominant strategy, the intersection is the equilibrium and is very easy to find.
Nash equilibrium
A set of strategies (one per player) where no player can improve their payoff by changing their strategy, given the other players' strategies. Every dominant strategy equilibrium is a Nash equilibrium, but not every Nash equilibrium involves dominant strategies.
To find in a payoff matrix: for each cell, check whether either player would want to deviate. If neither does, it is a Nash equilibrium.
Maximin strategy
The strategy that maximises the worst-case payoff. The player looks at each of their strategies, finds the minimum payoff from each, then picks the strategy whose minimum is highest. This is a conservative, risk-averse approach.
Backward induction
Used in sequential (extensive-form) games represented as game trees. Start at the end of the tree: each player at the final decision node picks their best option. Then work backwards, with each earlier player anticipating the choices that follow. This yields the subgame-perfect Nash equilibrium.
Mixed strategy equilibrium
When no pure strategy Nash equilibrium exists (or in addition to one), players randomise. Each player chooses a probability distribution over their strategies such that the other player is indifferent between their own strategies.
To find: set up the condition that the expected payoff from each of one player's strategies is equal, given the other player's mixing probabilities, and solve.
For Cournot and Stackelberg, once you have the equilibrium quantities, you can find the market price from the demand curve, then compute:
Consumer surplus: the triangle under demand and above price
Producer surplus: (price - MC) * quantity for each firm (or revenue minus variable cost)
Deadweight loss: the triangle between demand and MC, from market quantity to competitive quantity
Students often confuse Cournot (simultaneous) with Stackelberg (sequential). In Cournot, both firms choose at the same time. In Stackelberg, the leader commits first. The solution method is different: simultaneous equations versus backward induction.
A Nash equilibrium does not mean both players are happy with the outcome. It means neither can do better by changing their strategy alone. The Prisoner's Dilemma is a Nash equilibrium, even though both players would prefer a different outcome.
Students sometimes think the maximin strategy and the Nash equilibrium strategy are always the same. They can differ. Maximin is about minimising downside risk; Nash is about mutual best responses.
In mixed strategy problems, students often try to find the probability that makes their own payoffs equal. The correct approach is to find the probability that makes the other player indifferent.
⚠️ The Cournot/Stackelberg problem is one of the seven large final exam problems. You need to derive reaction functions, graph them, find both Cournot-Nash and Stackelberg equilibria in quantities, then compute market price, CS, PS, and DWL.
⚠️ The game theory problem is another large final exam problem. Be prepared for dominant strategies, Nash equilibria in normal-form games, maximin strategies, backward induction in game trees, and mixed strategy equilibria.
⚠️ For Stackelberg, always solve the follower's problem first (backward induction). Solving the leader's problem first is a common error.
True or false: In a Cournot duopoly, firms choose prices simultaneously. (False, they choose quantities.)
Fill in the blank: In Stackelberg, solve the ______'s problem first. (Follower's)
True or false: A dominant strategy equilibrium is always a Nash equilibrium. (True)
True or false: In a mixed strategy equilibrium, each player randomises to make themselves indifferent. (False, each player randomises to make the other player indifferent.)
Q: Two firms face demand P = 100 - Q, where Q = q1 + q2, and both have MC = 10. Find the Cournot-Nash equilibrium.
A: Firm 1's profit = (100 - q1 - q2) * q1 - 10 * q1. FOC: 100 - 2q1 - q2 - 10 = 0, so q1 = (90 - q2) / 2. By symmetry, q2 = (90 - q1) / 2. Solving simultaneously: q1 = q2 = 30. Total Q = 60, P = 40.
Q: Using the same setup, find the Stackelberg equilibrium with Firm 1 as leader.
A: Firm 2's reaction function: q2 = (90 - q1) / 2. Substitute into Firm 1's profit: (100 - q1 - (90 - q1)/2) * q1 - 10 * q1 = (55 - q1/2) * q1 - 10 * q1. FOC: 55 - q1 - 10 = 0, q1 = 45. Then q2 = (90 - 45)/2 = 22.5. Total Q = 67.5, P = 32.5. The leader produces more and earns more profit.
Q: In a 2x2 game, Player 1 chooses Up or Down, Player 2 chooses Left or Right. Payoffs (P1, P2): (Up, Left) = (3,2), (Up, Right) = (1,3), (Down, Left) = (2,1), (Down, Right) = (4,2). Find all Nash equilibria.
A: Check each cell. (Up, Left): P1 could switch to Down for 2 (worse), P2 could switch to Right for 3 (better). Not NE. (Up, Right): P1 could switch to Down for 4 (better). Not NE. (Down, Left): P1 could switch to Up for 3 (better). Not NE. (Down, Right): P1 could switch to Up for 1 (worse), P2 could switch to Left for 1 (worse). Neither wants to deviate. Nash equilibrium: (Down, Right) with payoffs (4, 2).
Cournot and Stackelberg outcomes fall between perfect competition and monopoly. As the number of Cournot firms increases, the outcome converges to the competitive equilibrium. With just one firm, it collapses to monopoly.
Game theory also applies to the antitrust/HHI topic: merger analysis is about strategic interaction between firms.
The welfare calculations (CS, PS, DWL) use the same geometry as in the monopoly and supply/demand topics.
Related Terms / Search Tags
Cournot duopoly, Stackelberg duopoly, reaction function, best response, Nash equilibrium, dominant strategy, maximin, minimax, mixed strategy, game tree, extensive form, normal form, payoff matrix, backward induction, subgame perfect equilibrium, oligopoly, strategic interaction, first-mover advantage, quantity competition