AP® Microeconomics review sheet from Aim for Five (aimforfive.com/micro/units/4/4-5)
Unit 4 · Topic 4.5
4.5 Oligopoly and Game Theory
An oligopoly is a market with a few large firms that watch each other closely. Because each firm's best move depends on what its rivals do, economists use game theory and payoff matrices to predict the outcome. The prisoner's dilemma explains why firms that would gain from cooperating often don't.
Key terms
- oligopoly
- collusion
- payoff matrix
- dominant strategy
- Nash equilibrium
- prisoner's dilemma
What makes an oligopoly
In an oligopoly, a few firms produce most of the market's output. Examples include cell phone carriers, airlines on a route, and commercial jet makers. The product can be identical (like steel) or differentiated (like cars). High barriers to entry, often economies of scale, keep the number of firms small.
The key feature is interdependence: each firm's profit depends on its rivals' choices. If one airline cuts fares, the others lose passengers unless they respond. So every decision about price, output or advertising involves guessing what rivals will do.
Collusion and cartels
Collusion is when rival firms agree to act together, for example by limiting output or fixing prices. A group of firms that formally agrees to do this is a cartel. If the firms act like one monopolist, they can share monopoly profit, which is more than they'd earn competing.
Collusion is hard to keep going. Each member has an incentive to cheat: if everyone else keeps the price high, a firm that secretly cuts its price a little or sells extra output can steal customers and earn more. If everyone cheats, the price falls and the agreement falls apart. Collusion is also illegal in the United States under antitrust law (6.4).
Reading a payoff matrix
A payoff matrix is a table showing what each player earns for every combination of choices. On the AP exam, it's two players with two choices each. One player picks a row, the other picks a column. Each cell lists two payoffs: by convention, the row player's payoff comes first and the column player's second. Always check the question's key, though.
A dominant strategy is a choice that gives a player a higher payoff no matter what the other player does. To test for one, fix the other player's choice, compare your two payoffs, then repeat for the other player's other choice. If the same choice wins both times, it's dominant.
A Nash equilibrium is a cell where neither player can do better by changing only its own choice, given what the other player is doing. To find one, check each of the four cells: would the row player switch? Would the column player switch? If neither would, it's a Nash equilibrium. A game can have one, more than one, or none in pure strategies, and you don't need a dominant strategy to have one.
The prisoner's dilemma
In a prisoner's dilemma, each player has a dominant strategy to not cooperate (for example, to charge a low price). So the Nash equilibrium is both firms not cooperating. But if both had cooperated (charged a high price), both would have earned more.
That's the logic of why cartels break down: each firm's self-interest leads to an outcome that's worse for both. Repeated play can help: if firms meet again and again, the threat of a price war next round can keep them cooperating.
What's in and out of scope
The exam uses only simple games: two players, two choices each, chosen at the same time. You won't be tested on bigger games, mixed strategies (choosing randomly) or game trees for moves made in sequence.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
A pricing prisoner's dilemma
Two gas stations, Alpha (rows) and Beta (columns), each choose a High or Low price. Payoffs are daily profits, with Alpha's listed first. High/High: ($400, $400). Alpha High, Beta Low: ($100, $500). Alpha Low, Beta High: ($500, $100). Low/Low: ($200, $200). (a) Does each firm have a dominant strategy? (b) What is the Nash equilibrium? (c) Is it the best outcome for the two firms together?
Show the solutionHide the solution
- Step 1: (a) Alpha: if Beta picks High, Alpha gets $400 with High or $500 with Low, so Low is better. If Beta picks Low, Alpha gets $100 with High or $200 with Low, so Low is better again. Low is Alpha's dominant strategy.
- Step 2: The game is symmetric, so the same check shows Low is Beta's dominant strategy too.
- Step 3: (b) Both choose Low. Check the Low/Low cell: if Alpha switched to High, it would get $100 instead of $200; if Beta switched, it would also drop to $100. Neither wants to switch, so Low/Low ($200, $200) is the Nash equilibrium.
- Step 4: (c) No. At High/High each earns $400, which is better for both. But High/High isn't stable: each station could earn $500 by switching to Low. This is a prisoner's dilemma.
Answer: (a) Yes, both have Low as a dominant strategy. (b) Low/Low, with $200 each. (c) No; High/High ($400 each) is better for both, but each has an incentive to cheat.
- Example 2
A Nash equilibrium when only one player has a dominant strategy
Firm A (rows) and Firm B (columns) each decide whether to Advertise or Not advertise. Payoffs are in thousands of dollars, A's first. Both Advertise: (30, 20). A Advertises, B Not: (50, 15). A Not, B Advertises: (25, 35). Neither advertises: (40, 40). Find each firm's dominant strategy, if any, and the Nash equilibrium.
Show the solutionHide the solution
- Step 1: Firm A: if B Advertises, A gets 30 (Advertise) or 25 (Not), so Advertise. If B doesn't, A gets 50 (Advertise) or 40 (Not), so Advertise. A's dominant strategy is to Advertise.
- Step 2: Firm B: if A Advertises, B gets 20 (Advertise) or 15 (Not), so Advertise. If A doesn't, B gets 35 (Advertise) or 40 (Not), so Not. B's best choice depends on A, so B has no dominant strategy.
- Step 3: B can predict that A will Advertise, since that's A's dominant strategy. B's best response is to Advertise.
- Step 4: Check Advertise/Advertise: A would drop from 30 to 25 by switching; B would drop from 20 to 15. Neither switches, so it's a Nash equilibrium. Check Not/Not: A would switch to Advertise (50 > 40), so it isn't one.
Answer: A's dominant strategy is to Advertise; B has none. The Nash equilibrium is both Advertise, with payoffs ($30,000, $20,000).
- Example 3
How big a fee changes a dominant strategy
Use the game above. Suppose a new rule makes Firm A pay a fixed fee every time it advertises, taking that amount off both of A's Advertise payoffs. What is the smallest fee that makes Not advertising A's dominant strategy?
Show the solutionHide the solution
- Step 1: For Not to be dominant, A must prefer Not whichever choice B makes.
- Step 2: If B Advertises: A needs 30 − fee < 25, so the fee must be more than 5.
- Step 3: If B doesn't advertise: A needs 50 − fee < 40, so the fee must be more than 10.
- Step 4: Both conditions must hold, so the fee must be more than 10 (thousand dollars). A fee between 5 and 10 would only remove A's dominant strategy, not make Not dominant.
Answer: Any fee greater than $10,000 makes Not advertising A's dominant strategy.
Common mistakes
- Picking the cell with the biggest combined payoff as the Nash equilibrium. A Nash equilibrium is where neither player wants to switch alone, which is often not the best joint outcome.
- Comparing a player's payoffs across the wrong cells. Hold the other player's choice fixed and compare only that player's own payoffs.
- Reading the wrong number in a cell. The first payoff usually belongs to the row player; check the key.
- Thinking a game has no Nash equilibrium unless both players have dominant strategies. Work cell by cell.
On the exam
- A short free-response question often gives a payoff matrix and asks for dominant strategies, the Nash equilibrium, whether a player's choice depends on the other's, and how a change in payoffs (a fee, a subsidy, a new deal) changes the outcome. Show the comparisons you made, with numbers.
- When asked to explain why collusion is unstable, say that each firm can raise its own profit by cheating, assuming the others stick to the deal.
Connected topics
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Check yourself
5 questions on 4.5 Oligopoly and Game Theory. Pick an answer to see if you got it, and why.
| Alpha's strategy | Beta charges a high price | Beta charges a low price |
|---|---|---|
| Alpha charges a high price | Alpha $40, Beta $40 | Alpha $10, Beta $50 |
| Alpha charges a low price | Alpha $50, Beta $10 | Alpha $20, Beta $20 |
Hypothetical payoff matrix: daily profits in thousands of dollars for two firms, Alpha and Beta, which each choose a price once, at the same time, and know all the payoffs
Which of the following describes Alpha's dominant strategy?
What is the Nash equilibrium of this game?
Suppose the firms secretly agree to both charge a high price. If Alpha breaks the agreement and charges a low price while Beta keeps its price high, by how much does Alpha's daily profit change?
Which statement best describes the outcome of this game?
Suppose Alpha's payoff when it charges a high price and Beta charges a low price rises from $10 thousand to $25 thousand. All other payoffs stay the same. Which of the following would then be true?
0 of 5 answered