In 1838 — a century before Nash — Cournot set two firms choosing quantities against a linear demand and solved where their reaction curves cross. That crossing is a Nash equilibrium: each firm's output is a best response to the others'. The primitives go in, the engine solves the fixed point, and the equilibrium comes out — for n symmetric firms, exactly (a−c)/((n+1)·b) each. The blue team proves it; the red team breaks it.
source A. A. Cournot, Recherches sur les principes mathématiques de la théorie des richesses (1838), Bacon tr. Researches into the Mathematical Principles of the Theory of Wealth — archive.org/details/researchesintom00fishgoog. Rendered, not quoted.
Each firm maximizes profit (p − c)·qi with p = a − b·Q, taking rivals' output as given. Setting ∂πi/∂qi = 0 gives the best response to the others' total Q₋ᵢ:
qi = (a − c − b·Q₋ᵢ) / (2b)
The equilibrium is the fixed point: every firm simultaneously on its own reaction curve. Live check of the current solution — how far each firm sits from its best response:
| firm | qi | best resp. | |gap| |
|---|
This is the very first instance of the Nash equilibrium. Cournot solved a two-firm fixed point in 1838 — mutual best response — 112 years before Nash (1950) proved that every finite game has such a point.
Cournot found one equilibrium by hand for one game; Nash proved the existence Cournot had quietly assumed. Oligopoly-as-fixed-point is the premise; the existence theorem is the next sphere's conclusion.
The blue team's live check: re-solve the fixed point for many n, confirm the closed form (a−c)/((n+1)b) and mutual best-response to 1e-9, and that price falls toward c while Q rises. If red tampers, this badge is where it shows.
A Cournot market needs only four numbers. Demand is linear inverse: the more total quantity Q reaches the market, the lower the price.
| symbol | meaning | role |
|---|---|---|
| a | demand intercept | price at Q=0 |
| b | demand slope | price drop per unit |
| c | marginal cost | cost per unit made |
| n | number of firms | who competes |
Price is set by the market: p = a − b·Q, Q = Σ qi. Each firm chooses its own qi simultaneously, then feeds the panel below.
The engine solves the fixed point where every firm is on its reaction curve — computed live, never looked up.
The two reaction lines above are the duopoly Cournot solved in 1838; they cross at the equilibrium — each firm best-responding to the other.
What the machine proves, live: for n symmetric firms each produces exactly (a−c)/((n+1)·b); the price strictly between the monopoly price (a+c)/2 and marginal cost c; total quantity rising with n toward the competitive limit p→c. The current market's numbers are above; the closed form is the output.
The blue team's witness (left) re-solves these live; the red team (right) tries to make firms overshoot.
It also assumes a homogeneous good, simultaneous moves (Stackelberg's leader breaks symmetry), constant marginal cost with no capacity limits, one shot (repetition enables collusion), and complete information. Linear demand keeps the fixed point closed-form; general demand may have no equilibrium, or several.
"Competition means firms are price-takers." Cut. Here each firm has market power — it moves the price by (b·qi) — yet still reaches an equilibrium. Price-taking is the n→∞ limit, not the setup.
"Two firms is basically a monopoly." Cut. Duopoly output (2/3 of competitive) already beats monopoly (1/2); the engine shows Q rising and p falling at every added firm.
"Each firm should just produce the monopoly quantity." Kept, corrected. That is exactly the tamper (window 6) — best-responding as if rivals make nothing. It is not a mutual best response, and the fixed point rejects it.
The red team's move: make every firm assume the others produce zero and best-respond as a lone monopolist — so all n firms overshoot. The blue team's witness (window 7) is watching.
When all firms best-respond to zero, total quantity is too high, price too low, and no firm is actually on its reaction curve — the residual jumps off zero. The witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.