An oligopoly is a market with a few firms, each large enough to affect the price. Unlike monopoly (one firm) or competition (many firms), oligopolists must consider rivals' reactions. This makes oligopoly inherently game-theoretic. Cournot modeled quantity competition. Bertrand modeled price competition. Both arrive at outcomes between monopoly and perfect competition.
Cournot duopoly: quantity competition
Two firms simultaneously choose how much to produce. Market price depends on total output. Each firm's best response depends on the other's quantity. The Nash equilibrium (Cournot equilibrium) is where both reaction curves intersect: neither firm wants to change its output given the other's choice.
Scheme
; Cournot duopoly; Demand: P = 100 - Q, where Q = q1 + q2; Each firm has MC = 10
(define a 100) ; demand intercept
(define c 10) ; marginal cost; Reaction function: q_i = (a - c - q_j) / 2
(define (reaction q-other)
(/ (- a c q-other) 2))
; Cournot equilibrium: both on their reaction curves; q1 = (a-c)/3, q2 = (a-c)/3
(define q-cournot (/ (- a c) 3))
(define p-cournot (- a (* 2 q-cournot)))
(define profit-cournot (* q-cournot (- p-cournot c)))
(display "Cournot equilibrium:") (newline)
(display " q1 = q2 = ") (display q-cournot) (newline)
(display " Total Q = ") (display (* 2 q-cournot)) (newline)
(display " Price = $") (display p-cournot) (newline)
(display " Each firm's profit = $") (display profit-cournot) (newline)
; Compare to monopoly (one firm, Q = (a-c)/2)
(define q-monopoly (/ (- a c) 2))
(define p-monopoly (- a q-monopoly))
(display "Monopoly: Q=") (display q-monopoly)
(display ", P=$") (display p-monopoly) (newline)
; Compare to competition (P = MC)
(define q-competitive (- a c))
(display "Competition: Q=") (display q-competitive)
(display ", P=$") (display c)
If firms compete on price instead of quantity, the logic changes dramatically. With identical products, the firm with the lower price captures the entire market. Each firm undercuts the other until price equals marginal cost. The Bertrand paradox: just two firms produce the perfectly competitive outcome.
Firms can collectively profit by acting as a monopoly: restrict output and raise prices. But each member has an incentive to cheat by producing more at the high cartel price. Cartels are unstable for the same reason the prisoner's dilemma is: individual rationality undermines collective benefit.
Scheme
; Cartel instability; Two firms agree to split monopoly output
(define a 100) (define c 10)
(define monopoly-q (/ (- a c) 2)) ; total monopoly quantity
(define cartel-share (/ monopoly-q 2)) ; each firm's agreed share
(define (price total-q) (- a total-q))
(define (profit q total-q) (* q (- (price total-q) c)))
; Both cooperate
(define coop-profit (profit cartel-share monopoly-q))
(display "Both cooperate (cartel):") (newline)
(display " Each produces: ") (display cartel-share) (newline)
(display " Each earns: $") (display coop-profit) (newline)
; One firm cheats (produces Cournot best response)
(define cheat-q (/ (- a c cartel-share) 2))
(define total-if-cheat (+ cheat-q cartel-share))
(define cheater-profit (profit cheat-q total-if-cheat))
(define sucker-profit (profit cartel-share total-if-cheat))
(display "One cheats:") (newline)
(display " Cheater produces: ") (display cheat-q)
(display ", earns: $") (display cheater-profit) (newline)
(display " Loyal firm earns: $") (display sucker-profit) (newline)
(display " Cheating pays: $")
(display (- cheater-profit coop-profit)) (display " more")
Wikipedia: Oligopoly