Lecture 6: Nash Equilibrium in Dating and Cournot Duopoly

What this lecture covers

The lecture opens by finishing the discussion of coordination games from the previous class, introducing the idea of "strategic complements" and showing where leadership and communication can help groups coordinate. It then plays out a new example, the "Going to the Movies" game, to show that not every coordination game is conflict-free: players can each prefer a different equilibrium, which is why this class of game is called the Battle of the Sexes.

The second half is a full derivation of the Cournot duopoly model, a classic economics application of Nash equilibrium where two firms choose production quantities. You'll see how to set up profit functions, use calculus to find each firm's best-response function, plot them to find the Nash equilibrium, and compare the resulting quantity, price, and profit to the monopoly and perfectly competitive outcomes.

Key ideas

• Strategic complements: in games like the partnership or investment game, the more one player does, the more the other player wants to do; best-response lines slope upward.
• Strategic substitutes: in games like Cournot duopoly, the more one player produces, the less the other wants to produce; best-response lines slope downward.
• Scope for leadership: pure coordination games (no conflict of interest) are especially receptive to a small nudge or announcement that helps everyone settle on the same equilibrium.
• Battle of the Sexes: a coordination game where multiple Nash equilibria exist, but different players prefer different ones, so communication may still fail to reach agreement.
• Best-response function via calculus: to find a firm's best output, differentiate its profit function with respect to its own quantity, set the result to zero, and check the second-order condition.
• Cournot quantity: the equilibrium output per firm is (a - c) / 3b, found where the two firms' best-response lines cross.
• Collusion is unstable: firms producing half the monopoly quantity each have an individual incentive to cheat and overproduce, and best responses to anticipated cheating pull the outcome back toward the Cournot equilibrium.
• Ordering of outcomes: total Cournot output lies between the monopoly quantity and the perfectly competitive quantity, so Cournot prices and profits sit between those two benchmarks.

Walkthrough

Leadership and strategic complements (0:01)

Recapping last class's Investment Game, the lecture notes that communication helped resolve a coordination problem in a way it never could in a Prisoner's Dilemma, because a Nash equilibrium is a self-enforcing agreement. This motivates the idea of "scope for leadership": in coordination games, a small suggestion about which equilibrium to play can shift everyone's behavior, unlike in games with a dominant strategy. Both the Investment Game and the earlier partnership effort game are labeled games of strategic complements, since each player's best response rises with the amount the other player contributes.

The Battle of the Sexes (4:59)

A couple must independently choose which of three movies to see, with going to Snow White strictly dominated for both, leaving a choice between two remaining films that each player ranks differently. Two students play the game live, first without communication (they happen to coordinate) and then with a brief phone-style negotiation. The lecture identifies two Nash equilibria - both attending the same movie - but points out the key difference from prior examples: each player prefers a different equilibrium, so this is not a pure coordination problem but a conflict wrapped inside one, named the Battle of the Sexes.

Setting up the Cournot duopoly model (18:37)

The lecture introduces Cournot duopoly as a classic Nash equilibrium application sitting between the extremes of monopoly and perfect competition, and as a case with few players but a continuum of strategies. Two firms choose quantities q1 and q2 of an identical good, face constant marginal cost c, and sell at a price determined by a downward-sloping demand curve p = a - b(q1 + q2). Each firm's profit is revenue minus cost, expressed as a function of both quantities.

Deriving best responses with calculus (roughly 29:00)

To find Firm 1's best response, the lecture differentiates its profit function with respect to q1, sets the derivative to zero, and confirms a maximum with the negative second derivative. Solving gives the best-response function q1 = (a - c)/(2b) - q2/2, and by symmetry q2 = (a - c)/(2b) - q1/2. The lecture connects the two endpoints of this best-response line to familiar economics concepts: when the rival produces nothing, the best response is the monopoly quantity (a - c)/(2b); when the rival already produces the competitive quantity (a - c)/b, the best response drops to zero because further output would push price below marginal cost.

Finding and interpreting the Nash equilibrium (roughly 49:00)

Plotting both best-response lines, their intersection gives the Cournot Nash equilibrium, solved algebraically as q1* = q2* = (a - c)/(3b). This is contrasted with the partnership and investment games: here the best-response lines slope downward, making the strategies substitutes rather than complements. The lecture then asks whether this equilibrium maximizes total industry profit and shows it does not - industry profit is maximized at the monopoly quantity, achievable only through collusion.

Why collusion breaks down (roughly 58:00)

Two students role-play managers of rival firms agreeing informally to each produce half the monopoly quantity. Because such agreements cannot be legally enforced, each firm has an incentive to secretly overproduce, and anticipating the other's cheating pulls both firms' best responses back toward the Cournot equilibrium rather than the collusive one. The lecture also notes historical examples - restrictions on paper, rubber, steel, and the OPEC oil cartel - where high collusive prices attracted new entrants who undercut the agreement, reinforcing why sustaining monopoly-level output is difficult without enforcement.

Comparing quantities, prices, and profits (roughly 1:09:00)

The lecture closes by comparing total Cournot output, 2(a - c)/(3b), to the monopoly and competitive benchmarks, showing it falls strictly between them. Consequently, Cournot prices and producer profits sit between the high monopoly price and the low competitive price, while consumers are better off than under monopoly but worse off than under perfect competition.

Before you watch

• Review the definition and method for finding Nash equilibrium from the previous lecture, including best-response comparison in payoff matrices.
• Be comfortable with basic single-variable calculus (differentiation, first- and second-order conditions), used here to derive best-response functions.
• Recall monopoly, perfect competition, and demand-curve concepts from an introductory economics course, since the lecture assumes familiarity with marginal revenue and marginal cost.

Check your understanding

1. What distinguishes strategic complements from strategic substitutes, and which category does Cournot duopoly fall into?
2. Why is the Battle of the Sexes different from earlier coordination games in the course, even though it also has multiple Nash equilibria?
3. Starting from the profit function p*q1 - c*q1, how do you derive Firm 1's best-response function?
4. Why does an informal agreement to each produce half the monopoly quantity tend to unravel back toward the Cournot equilibrium?
5. How does total output, and therefore price, under Cournot competition compare to monopoly and perfect competition, and why?