Source: Velez & Guo lecture notes, Pindyck & Rubinfeld Ch. 10-13, 18 | Texas A&M University
Tags: game theory, normal form game, Nash equilibrium, dominant strategy, dominated strategy, iterated elimination, monopoly, marginal revenue, deadweight loss, price discrimination, oligopoly, Cournot duopoly, Bertrand duopoly, Stackelberg duopoly, externalities, Pigouvian tax, intermediate microeconomics
When markets are not perfectly competitive, firms have strategic power. Game theory provides the tools to predict behaviour in these settings. Monopolies restrict output and raise prices, creating deadweight loss. Oligopolies (Cournot, Bertrand, Stackelberg) produce outcomes between the monopoly and competitive extremes. Externalities cause further market failures, which targeted taxes or subsidies can correct.
Normal form game
A representation of a strategic situation consisting of: (1) players, (2) actions available to each player, (3) payoffs for every combination of actions. Typically shown as a matrix for two-player games.
Action profile
A combination of one action per player, e.g. (Confess, Confess) in the Prisoner's Dilemma.
Strictly dominant action
An action that gives a player a higher payoff than any other action, regardless of what the other players do.
Equilibrium in strictly dominant actions
An action profile where every player is playing a strictly dominant action. Strongest solution concept; not every game has one.
Strictly dominated action
An action for which some other action always gives a higher payoff, regardless of what others do. A rational player will never choose a strictly dominated action.
Iterated elimination of strictly dominated actions (IESDA)
Repeatedly remove strictly dominated actions from the game. After each round, check again for new strictly dominated actions in the reduced game. May or may not narrow down to a unique prediction.
Best response
An action that maximises a player's payoff given a specific action by the other player.
Nash equilibrium (NE)
An action profile where each player is playing a best response to the other player's action. No player has a profitable unilateral deviation. A game may have zero, one, or multiple Nash equilibria in pure strategies. Every finite game has at least one NE in mixed strategies (Nash, 1950).
Monopoly
A firm that controls 100% of the production of a good. The monopolist is a price-maker: it chooses output and the market demand determines the price.
Marginal revenue (MR)
MR(q) = dR/dq, where R(q) = p(q) * q. For a monopolist facing a downward-sloping demand, MR < p because selling one more unit requires lowering the price on all units. When demand is linear Q = a - bp, MR = a/b - (2/b)q (same y-intercept as the inverse demand, double the slope).
Price discrimination
Charging different prices to different consumers (or for different quantities) for the same good.
First-degree: charge each consumer their exact willingness to pay
Second-degree: charge different unit prices for different quantities (e.g. quantity discounts)
Third-degree: divide consumers into groups and charge each group a different price
Oligopoly
A market controlled by a small number of firms.
Cartel (cooperative oligopoly)
Firms cooperate on prices and quantities, behaving collectively like a monopolist.
Cournot duopoly
Two firms simultaneously choose quantities. The market price is determined by the aggregate quantity via the demand function.
Bertrand duopoly
Two firms simultaneously choose prices. Consumers buy from the firm with the lower price (split equally on ties).
Stackelberg duopoly
One firm (the leader) chooses its quantity first; the other firm (the follower) observes and then chooses. Modelled as an extensive form (sequential) game.
Externality
A cost or benefit that affects a third party who is not part of the economic transaction, and is not reflected in market prices.
Negative externality: a cost borne by third parties (e.g. pollution)
Positive externality: a benefit enjoyed by third parties (e.g. well-maintained gardens)
Social cost / social benefit
Private cost or benefit plus external cost or benefit. Social cost = private cost + external cost.
Pigouvian tax / subsidy
A tax set equal to the marginal external cost (or a subsidy equal to the marginal external benefit) at the socially optimal quantity. Restores efficiency by making private agents internalise the externality.
Why game theory matters for markets
Competitive markets assume no agent affects prices and no agent considers others' behaviour
Real markets often have few dominant firms (AT&T/Verizon wireless example, automobile manufacturers, OPEC)
Game theory provides a framework for predicting outcomes when agents are strategic
Constructing a game
Every normal form game has three elements:
Players: the decision-makers
Actions: what each player can choose
Payoffs: the outcome (utility or profit) for each player under each action profile
Payoffs are written as ordered pairs in each cell: (Row player's payoff, Column player's payoff).
Solution concepts, from strongest to weakest
Equilibrium in strictly dominant actions: each player has a single best action regardless of others. Example: (Confess, Confess) in the Prisoner's Dilemma.
Iterated elimination of strictly dominated actions (IESDA): remove actions no rational player would choose, then repeat. Example: the three-quantity oligopoly game reduces from 3x3 to (Q=64, Q=64).
Nash equilibrium: each player best-responds to the other. Applies even when neither dominance nor IESDA narrows things down. Example: Battle of the Sexes has two NE, (Movies, Movies) and (Football, Football).
The Prisoner's Dilemma
Both players have "Confess" as a strictly dominant action
The equilibrium (Confess, Confess) gives each player a payoff of 1
But (Don't Confess, Don't Confess) gives each player 3
Individual rationality leads to a collectively worse outcome
This tension is central to understanding why cartels are unstable and why cooperation is hard
Battle of the Sexes
No player has a dominant action; no action is strictly dominated
Two pure-strategy Nash equilibria exist
Illustrates that games can have multiple equilibria and that coordination matters
The monopolist's problem
Maximise pi(q) = R(q) - C(q), where R(q) = p(q) * q and p(q) is the inverse demand function (the maximum price at which q units can be sold).
Two conditions for the optimum:
MR(q*) = MC(q*)
p(q*) >= AVC(q*)
Finding p(q) from demand
Given Q(p) = a - bp, solve for p: p(q) = a/b - q/b.
The MR shortcut for linear demand
If inverse demand is p(q) = A - Bq, then MR(q) = A - 2Bq.
MR and demand share the same vertical intercept. The horizontal intercept of MR is half that of demand.
Monopoly vs. competition
Compared to a competitive market with the same cost structure, a monopolist:
Produces less
Charges a higher price
Sells at a price above marginal cost
Earns higher profits
Creates deadweight loss (the triangle between demand and MC, from q-monopoly to q-competitive)
Regulating a monopoly with a price cap
Set a maximum price equal to the competitive price p*
The regulated monopolist's effective demand becomes a horizontal line at p* until it meets the original demand curve, then follows the original demand
MR becomes flat at p* up to q*, then follows the original MR
Result: the monopolist produces at the competitive quantity q*
Price discrimination
Requires three conditions: market power, consumer heterogeneity, and limited resale.
First-degree price discrimination
Charge each consumer their exact willingness to pay
Captures all consumer surplus; CS = 0, PS = total welfare, DWL = 0
Efficient (no deadweight loss) but maximally redistributive towards the firm
Rarely achievable in practice; requires knowing each buyer's valuation
Second-degree price discrimination
Different unit prices for different quantities (e.g. "buy 2 get 1 half price")
Exploits diminishing marginal willingness to pay
Third-degree price discrimination
Segment the market into groups with different demand curves
Set the monopoly price independently in each segment: MR1(q1*) = MC and MR2(q2*) = MC
Always at least as profitable as a single-price strategy (the firm can always choose to set the same price in both segments)
Worked example: women's demand Q1 = 10 - p, men's demand Q2 = 9 - 1.5p, MC = 1. Discriminating yields profit of 29.625 vs. 27.225 under a single price.
Cournot duopoly (simultaneous quantity competition)
Two firms simultaneously choose quantities qA and qB
Market price is p(qA + qB), found by inverting the demand function
Each firm's profit depends on both quantities: pi_A = p(qA + qB) * qA - CA(qA)
Nash equilibrium: each firm's quantity is a best response to the other's
Solve by setting dpi_A/dqA = 0 and dpi_B/dqB = 0 simultaneously (two equations, two unknowns)
Cournot with equal costs, worked example
Q(p) = 1000 - 2p, CA = CB = 200q.
Inverse demand: p(q) = 500 - q/2
Best response for A: qA = 300 - qB/2
Best response for B: qB = 300 - qA/2
Equilibrium: qA* = qB* = 200, total output = 400, p = 300
Profit per firm = $20,000
Comparison:
Monopoly/cartel: q = 300, p = 350, total profit = $45,000
Competitive market: q = 600, p = 200, profit = $0
Cournot sits between monopoly and competition in output, price, and profit
Cournot with different costs
Q(p) = 250 - p, CA = 50q, CB = 45q.
Equilibrium: qA* = 65, qB* = 70, p = 115
Firm B (lower cost) produces more and earns higher profit ($4,900 vs. $4,225)
Part of the DWL comes from the high-cost firm producing a positive amount, something that would not happen under perfect competition or a monopoly with access to both technologies
Bertrand duopoly (simultaneous price competition)
Two firms simultaneously choose prices
Consumers buy from the cheaper firm; equal split on ties
With equal marginal costs (MC = c), the unique Nash equilibrium is pA = pB = c
Result: competitive outcome despite only two firms
Both firms earn zero profit
Logic: any price above MC invites the rival to undercut by epsilon
Stackelberg duopoly (sequential quantity competition)
The leader moves first; the follower observes and then chooses
Modelled as an extensive form game, solved by backward induction (subgame perfect NE)
The leader typically earns more than in Cournot (first-mover advantage), because it commits to a quantity that shapes the follower's best response
Not solved algebraically in this course, but the ranking is: leader profit (Stackelberg) > Cournot profit > follower profit (Stackelberg) >= Bertrand profit = 0
Summary ranking of market structures (equal costs, linear demand)
From least to most output (and from highest to lowest price):
Monopoly / Cartel
Stackelberg
Cournot
Bertrand / Perfect competition
What externalities are
An externality is a direct effect on a third party's welfare (or a firm's production) that is not mediated through prices. The key word is "direct": a price change caused by new supply is not an externality.
Negative externality example
A taxi ride: the direct effect is the transport service and payment. The indirect effect (externality) is pollution and congestion that affect everyone else. These costs are not reflected in the fare.
Positive externality example
Professional landscaping: the homeowner pays, neighbours enjoy the view for free. Fertiliser runoff into water sources is a simultaneous negative externality.
Market failure with negative externalities
The firm's private marginal cost (MC_p) is below the social marginal cost (MC_s = MC_p + MC_external)
The firm produces where p = MC_p, which exceeds the socially optimal quantity (where p = MC_s)
Result: overproduction relative to the social optimum, and DWL
Market failure with positive externalities
The marginal social benefit (MSB) exceeds the marginal private benefit (MPB = demand)
The market produces where p = MPB, which falls short of the socially optimal quantity (where p = MSB)
Result: underproduction relative to the social optimum, and DWL
Corrective taxes and subsidies (Pigouvian)
Negative externality: impose a tax tau = MC_external evaluated at the socially optimal quantity. This shifts the firm's effective cost up to match social cost, restoring efficiency. Justifies taxes on petrol, tobacco, alcohol, sugary drinks.
Positive externality: provide a subsidy equal to the marginal external benefit at the socially optimal quantity. This shifts effective cost down (or effective benefit up), restoring efficiency. Justifies subsidies for scientific research, historic preservation, education.
Berkeley soda tax example
One cent per ounce on distributors of sugary drinks (took effect January 2015)
Rationale: each sugary drink imposes roughly 10 cents of health costs on others (Medicare, Medicaid, private insurance)
Result: consumption dropped by over 20%
Estimated nationwide welfare gain from a well-designed soda tax: $2.4 to $7 billion per year
Monopolist's MR for linear demand
If Q(p) = a - bp, then p(q) = a/b - q/b, and MR(q) = a/b - 2q/b.
Monopoly optimality conditions
MR(q*) = MC(q*), and p(q*) >= AVC(q*)
Cournot best response (linear demand, equal costs)
If p(q) = A - Bq and MC = c:
Best response for firm i: qi = (A - c)/(2B) - qj/2
Bertrand equilibrium (equal costs)
pA = pB = MC. Zero economic profit.
Pigouvian tax
tau = marginal external cost at the socially optimal output
Social cost
C_social(q) = C_private(q) + C_external(q)
Social benefit
MSB(q) = MPB(q) + MEB(q)
⚠️ A Nash equilibrium requires mutual best responses. Check both players, not just one.
⚠️ Strictly dominant ≠ strictly dominated. Dominant means "always best"; dominated means "always beaten by something."
⚠️ IESDA depends on the order being valid. You can only eliminate an action if it is dominated in the current reduced game, not just the original game.
⚠️ For the monopolist, MR ≠ p. The monopolist faces a downward-sloping demand, so selling one more unit requires lowering the price on all units. Students frequently forget this and set p = MC instead of MR = MC.
⚠️ The MR shortcut (same intercept, double slope) works only for linear demand.
⚠️ In Cournot, each firm's best response is a function of the other firm's quantity. Solve the system of best-response equations simultaneously.
⚠️ Bertrand competition with equal marginal costs yields the competitive outcome (p = MC, zero profit), even with just two firms. This dramatically differs from Cournot.
⚠️ A price change caused by new market entry is not an externality. Externalities are direct effects on third parties, not effects transmitted through prices.
⚠️ Third-degree price discrimination requires separate demand curves and the ability to prevent resale between groups.
⚠️ DWL from monopoly comes from the gap between the quantity the monopolist produces and the competitive quantity. It is the triangle between demand and MC over that range.
Q: In the Prisoner's Dilemma, why don't the players cooperate even though mutual cooperation gives both a higher payoff?
A: Each player has "Confess" as a strictly dominant action, meaning confessing is individually optimal regardless of what the other does. Cooperation (Don't Confess) is not individually rational because each player can gain by deviating to Confess. Without a binding agreement, the dominant-strategy equilibrium prevails.
Q: A monopolist faces demand Q(p) = 100 - 2p and has MC = 10. What quantity does the monopolist produce, and at what price?
A: Inverse demand: p(q) = 50 - q/2. MR = 50 - q. Set MR = MC: 50 - q = 10, so q* = 40. Price: p(40) = 50 - 20 = 30.
Q: Two Cournot firms face Q(p) = 200 - p and both have C(q) = 20q. Find the equilibrium quantities, price, and profits.
A: p(q) = 200 - q. pi_A = (200 - qA - qB)qA - 20qA = (180 - qA - qB)qA. FOC: 180 - 2qA - qB = 0, so qA = 90 - qB/2. By symmetry, qB = 90 - qA/2. Substituting: qA = 90 - (90 - qA/2)/2 = 90 - 45 + qA/4, so 3qA/4 = 45, qA = 60. By symmetry qB = 60. Price = 200 - 120 = 80. Profit per firm = (80 - 20)(60) = $3,600.
Q: Why does Bertrand competition with equal costs lead to the competitive outcome?
A: If either firm sets a price above MC, the other can undercut by a tiny amount, steal the entire market, and earn positive profit. This undercutting continues until both prices equal MC. At p = MC, neither firm can profitably deviate: lowering the price means selling at a loss, and raising it means losing all customers.
Q: A factory's production creates pollution costing society $5 per unit. The factory's private MC is 10 + q, and demand is Q = 50 - p. What is the socially optimal output, and what Pigouvian tax restores it?
A: Social MC = private MC + 5 = 15 + q. Set p = social MC: using inverse demand p = 50 - q, we get 50 - q = 15 + q, so 2q = 35, q = 17.5. The private market produces where 50 - q = 10 + q, giving q = 20 (too much). The Pigouvian tax is $5 per unit, which equals the marginal external cost.
Q: A monopolist can segment the market into two groups. Group 1 has demand Q1 = 20 - p, group 2 has demand Q2 = 16 - 2p. MC = 2. Find the profit-maximising price in each segment.
A: Group 1: p1(q1) = 20 - q1, MR1 = 20 - 2q1. Set MR1 = 2: q1 = 9, p1 = 11. Group 2: p2(q2) = 8 - q2/2, MR2 = 8 - q2. Set MR2 = 2: q2 = 6, p2 = 5. The firm charges $11 to group 1 and $5 to group 2.
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