Lecture 7: Bertrand Competition and the Candidate-Voter Model

What this lecture covers

The lecture reworks last class's duopoly setting but has the two firms compete on price instead of quantity. This "Bertrand" version produces a strikingly different result: with identical products, the only Nash equilibrium has both firms pricing at marginal cost, mimicking perfect competition even though there are only two firms. The lecture flags this as troubling for how much modeling choices can matter, then shows how relaxing the assumption of identical products (previewing a homework assignment on the "linear city" model) restores an outcome that sits between monopoly and perfect competition, closer to intuition.

The second half returns to the elections model introduced weeks earlier, but changes two assumptions: the number of candidates is no longer fixed, and candidates cannot choose their position - each voter is a potential candidate at their own fixed location. Through a live in-class game with students standing up to "run," the lecture builds toward characterizing which entry patterns are Nash equilibria in this candidate-voter model.

Key ideas

• Bertrand competition: firms set prices instead of quantities; demand goes entirely to the lower-priced firm, split evenly on a tie.
• Bertrand equilibrium: with identical products and constant marginal cost, the unique Nash equilibrium has both firms pricing at marginal cost, yielding zero profit.
• Discontinuous payoffs: Bertrand best responses cannot be found with calculus because a firm's profit jumps discontinuously as it crosses the rival's price.
• Model sensitivity: changing only the strategy variable (quantity versus price) in an otherwise identical market produces dramatically different predictions, which is a caution about taking any single model's result too literally.
• Differentiated products / linear city model: representing products (or firms) as points on a line lets a transport-cost term stand in for consumers' preference for products closer to their ideal, restoring realistic in-between outcomes.
• Candidate-voter model: unlike earlier election models, the number of candidates is endogenous and each candidate's political position is fixed at their own location rather than freely chosen.
• Payoffs in the candidate-voter game: entering costs C, winning pays B, and every voter suffers a distance-based disutility cost from whichever candidate wins.
• Equilibrium with the median candidate: having exactly one candidate, the median voter, run unopposed is a Nash equilibrium, since any other entrant would simply lose.

Walkthrough

Reviewing Cournot and framing Bertrand (0:00)

The lecture briefly recaps the previous class's Cournot result - quantity, price, and profit all sitting between the monopoly and competitive benchmarks - and introduces Bertrand competition as an alternative model of the same underlying market where firms choose prices rather than quantities. It sets up the demand structure: total demand is 1 - P, where P is the lower of the two firms' prices, and a firm's own quantity sold depends on whether it is strictly cheaper, strictly more expensive, or tied with its rival.

Finding Bertrand best responses (roughly 11:00)

Because the payoff function is discontinuous, the lecture reasons through cases rather than using calculus. If the rival prices below cost, the best response is to price above them and sell nothing. If the rival prices above cost but below the monopoly price, the best response is to undercut them by a small amount ε. If the rival prices above the monopoly price, the best response is to price at the monopoly price rather than keep undercutting. If the rival prices exactly at cost, any price at or above cost is a best response, since further sales earn no profit.

The Bertrand Nash equilibrium (roughly 18:15)

Despite the complicated best-response structure, the lecture shows both firms pricing at marginal cost is the unique Nash equilibrium: at that point neither firm can profitably deviate, and any other candidate outcome unravels because one firm always wants to slightly undercut the other. The result is that price, profit, and consumer surplus all look identical to what a perfectly competitive market with many firms would produce, even though there are only two firms - a result the lecture flags as an uncomfortably strong conclusion for real-world antitrust policy.

Restoring realism with differentiated products (roughly 28:24)

To address the discomfort with the identical-products result, the lecture sketches the linear city model that the homework will ask students to solve: firms are located at opposite ends of a line representing a market of evenly distributed consumers, and each consumer pays the firm's price plus a transport cost that rises with the square of the distance to that firm. This transport cost can represent literal travel distance or more abstractly a mismatch between a consumer's ideal product characteristics (illustrated with a beer-flavor example) and what is actually on offer. The lecture notes that solving this model with price competition should produce an equilibrium closer to the Cournot-style outcome, between monopoly and perfect competition.

Setting up the candidate-voter model (roughly 41:43)

Returning to the earlier Downs-Hotelling election setup, the lecture changes two assumptions: the number of candidates is endogenous (anyone can choose to run or not), and candidates cannot choose a position - each voter occupies a fixed point on the left-right line and may become a candidate at that same point. Voters always vote for the nearest running candidate, and the winner is decided by plurality. Payoffs combine a prize B for winning, a cost C for running (with B = 2C for simplicity), and a distance-based disutility from whichever candidate ultimately wins.

Playing the entry game in class (roughly 53:10)

A row of students, arranged along the political spectrum, repeatedly decides simultaneously whether to "run." Several outcomes are tested: when a non-central candidate wins alongside a losing rival, the losing candidate should not have entered (their best response was not to run); when two candidates run right on top of each other in the middle, a third candidate can profitably enter and split votes to win, showing that outcome is not an equilibrium either.

Characterizing equilibria (roughly 1:01:00)

The class works out several equilibrium properties: no candidate standing at all cannot be an equilibrium, since anyone could profitably enter and win unopposed. Exactly one candidate, the median voter, running alone is an equilibrium, because any other entrant would simply lose. Two identical candidates both located exactly at the median is not an equilibrium, because an outside candidate could win by picking up one full side of the spectrum plus half the middle. The lecture then checks a more complex configuration with candidates spread on both sides of a central winner and confirms it is an equilibrium, reasoning through three types of possible deviations: entering further out, entering at the center, and dropping out.

Before you watch

• Review the Cournot duopoly derivation from the previous lecture, since Bertrand is presented as a direct point of comparison.
• Recall the median voter or Downs-Hotelling model from earlier in the course, which set up candidates on a left-right line with fixed voter preferences.
• Be comfortable identifying Nash equilibria by checking that no player has a profitable deviation, the method used throughout.

Check your understanding

1. Why does the Bertrand model predict zero profit and marginal-cost pricing with just two firms, unlike the Cournot model?
2. Why can't calculus be used directly to find best responses in the Bertrand pricing game?
3. In the linear city model, what role does the transport cost term play, and why does it help produce a more realistic outcome than the basic Bertrand model?
4. In the candidate-voter model, why is it a Nash equilibrium for only the median voter to run, but not for two identical centrist candidates to both run?
5. In the final equilibrium checked in class, why is it not a profitable deviation for a near-median candidate to drop out of the race?