Prisoner's Dilemma, Collusion and Tit for Tat

Two ferry companies are the only way onto an island. One night, their owners secretly shake hands: both will keep fares high. Within weeks, one of them quietly cuts its fare. Why do deals like this keep breaking, when keeping them would make both richer? This is Module 4.5B, oligopoly and game theory: collusion, changing payoffs and the prisoner's dilemma.

Meet Puffin Ferries and Otter Ferries, the island's only two ferries. Docks and boats cost a fortune, so no rival gets in.

Quick refresher from Module 4.5A: an oligopoly is a few interdependent firms behind high barriers to entry; two firms make a duopoly. Each firm's profit depends on the other's choices.

Their best move together is collusion: cooperating to raise combined profit. Firms that formally agree to raise prices and cut output form a cartel, which tries to act like one monopoly.

Quick refresher from Module 4.2A: a monopoly is the only seller, so it sells less, at a higher price, than a competitive market would.

Cartels are illegal in the United States. And each member is tempted to cheat: cut your fare a little, steal your rival's riders, keep every extra sale.

Each ferry picks High fare or Low fare, and the payoff matrix shows weekly profit for every combination.

Quick refresher from Module 4.5A: Puffin picks a row, Otter picks a column, and in each cell the first number is Puffin's profit, the second Otter's.

Both High: Puffin $70,000, Otter $55,000. Puffin High, Otter Low: Puffin $5,000, Otter $85,000. Puffin Low, Otter High: Puffin $80,000, Otter $25,000. Both Low: Puffin $45,000, Otter $35,000.

Solve it, starting with Puffin. Quick refresher from Module 4.5A: a best response fixes the other player's choice, then compares only your own two payoffs.

If Otter goes High, Puffin's Low earns $80,000, more than High's $70,000. If Otter goes Low, Puffin's Low earns $45,000, more than $5,000.

Quick refresher from Module 4.5A: a dominant strategy pays more whatever the other player does, so it takes two comparisons, each with your own payoffs. Low won both, so it's Puffin's dominant strategy.

Now Otter, using only the second numbers. If Puffin goes High, Low earns $85,000 against $55,000; if Puffin goes Low, $35,000 against $25,000. Low is Otter's dominant strategy too.

Quick refresher from Module 4.5A: at a Nash equilibrium, neither player gains by switching alone, so check both. From Low, Low, Puffin would fall from $45,000 to $5,000, and Otter from $35,000 to $25,000.

So the Nash equilibrium is Low, Low: Puffin earns $45,000 a week and Otter $35,000. Yet both High pays Puffin $70,000 and Otter $55,000. That's a prisoner's dilemma: each player's dominant strategy leaves both worse off than cooperating. That's our answer. From the handshake, cheating lifts Puffin from $70,000 to $80,000, or Otter from $55,000 to $85,000. Each gains by cheating, whatever the other does, so the deal breaks.

So what just happened? One: Low fare is each ferry's dominant strategy. Two: Low, Low is the Nash equilibrium, $45,000 and $35,000. Three: both High pays each more, but cheating tempts: a prisoner's dilemma.

Now the payoffs change. A higher harbor fee hits night trips, which only Otter's low-fare plan uses: Otter's profit from Low fare falls $20,000, whatever Puffin does. Redraw the matrix. Change only Otter's numbers in its Low column: $85,000 becomes $65,000, and $35,000 becomes $15,000. Everything else stays. Exam trap: cutting Puffin's numbers in that column too. The fee hits one player's one strategy, so only Otter's two payoffs move. Then redo the best responses. Puffin's numbers didn't change, so Low is still its dominant strategy. For Otter, if Puffin goes High, Low still wins, $65,000 to $55,000; if Puffin goes Low, High now wins, $25,000 to $15,000. Otter has no dominant strategy.

Puffin plays Low, so Otter plays High. Check both: Puffin switching falls from $80,000 to $70,000, Otter from $25,000 to $15,000. New Nash equilibrium: Puffin Low, Otter High. Report each profit with its firm: Puffin $80,000 a week, Otter $25,000. On a recent AP exam, the old equilibrium's profits, or no firm names, cost students this point.

So what just happened? One: the fee cut only Otter's two Low payoffs. Two: Puffin still plays Low, so Otter's best response is High. Three: the new Nash equilibrium pays Puffin $80,000 and Otter $25,000.

On the redrawn matrix, how big a boost to Otter's High-fare profit, the same whatever Puffin does, would make High its dominant strategy? Find the comparison High still loses. Against Puffin's Low, High already wins, $25,000 to $15,000. Against Puffin's High, it loses, $55,000 to $65,000: short by $10,000. So the boost must be more than $10,000. Exactly $10,000 only ties, and a dominant strategy must pay more, not the same. Exam trap: from a list like $5,000, $10,000, $12,000 or $20,000, pick $12,000, the smallest amount above the shortfall. $20,000 works, but it isn't the minimum.

So what just happened? One: High fare already won when Puffin went Low. Two: it lost by $10,000 when Puffin went High. Three: so the boost must top $10,000.

Back to the original game, before the fee. So far each ferry chooses once: a one-shot game, where cheating pays. But real rivals meet week after week: a repeated game. That allows tit for tat: cooperate first, then copy your rival's last move. Say Puffin plays tit for tat for three rounds. If Otter cheats every round, it grabs $85,000 in round one, then Puffin copies the low fare: $85,000 + $35,000 + $35,000 = $155,000. If Otter keeps its fare high, so does Puffin, and Otter earns $55,000 three times: $165,000. That beats cheating every round.

So what just happened? One: played once, cheating pays. Two: tit for tat punishes cheating with low fares. Three: over three rounds, cooperating's $165,000 beats cheating's $155,000.

Keeping prices high like this, with no formal agreement, is called tacit collusion. But it often slips, as in the prisoner's dilemma, so oligopolists struggle to reach the monopoly outcome. Still, their prices are generally higher, and quantities lower, than in perfect competition.

One more tool. If Puffin and Otter merge and keep both ferries, the new monopoly picks the cell with the most combined profit. Add the two payoffs in every cell. Both High: $70,000 + $55,000 = $125,000. The others: $90,000, $105,000 and $80,000. The largest is $125,000, at both High: exactly what the cartel wanted. Still, check every sum; the biggest isn't always where both cooperate. Exam trap: merged profit is the largest cell sum, not the Nash total, $80,000, and not all four cells added, $400,000. Calculate means showing the addition.

So what just happened? One: add both payoffs in every cell. Two: the largest sum, $125,000 at both High, is the merged profit. Three: it beats the Nash total, $80,000.

Exam moment: two questions the way the AP exam asks them, then one free-response part.

Question 1. Rival internet providers Orbit and Nova each choose Upgrade or Wait. The first entry in each cell is Orbit's profit; the second is Nova's. A fee cuts Nova's profit from Upgrade by $12,000, whatever Orbit does. At the new Nash equilibrium, what does each firm earn? A) Orbit $14,000, Nova $33,000 B) Orbit $17,000, Nova $27,000 C) Orbit $24,000, Nova $6,000 D) Orbit $24,000, Nova $18,000 E) Orbit $33,000, Nova $14,000

The answer is E. Nova's Upgrade payoffs fall to $6,000 and $15,000, so Wait now wins for Nova either way, and Orbit still upgrades: Orbit $33,000, Nova $14,000. D is the old equilibrium. C applies the fee but skips the new best responses. B cuts Orbit's payoffs instead. Option A: firms swapped.

Question 2. Lark and Wren, a town's only two arcades, each choose Expand or Hold. The first entry in each cell is Lark's profit; the second is Wren's. If they merge, keeping both arcades and the same payoffs, what is the maximum combined profit? A) $30,000 B) $34,000 C) $40,000 D) $45,000 E) $153,000

The answer is D. The cell sums are $30,000, $45,000, $38,000 and $40,000; the largest is $34,000 + $11,000 = $45,000, Lark Expand with Wren Hold. Option A: the Nash total. B is one payoff, not a sum. C picks both Hold without checking every sum. E adds all four cells.

Free response. Inkwell and Pressly, a city's only two print shops, each choose Discount or Regular pricing. The first entry in each cell is Inkwell's weekly profit; the second is Pressly's. If the firms merge, keeping both shops and the same payoffs, calculate the maximum combined profit. Show your work.

To earn the point: maximum combined profit = $23,000 + $18,000 = $41,000, at both Regular, the largest of the four cell sums. The costly mistake: adding all four cells, $133,000, or writing $41,000 with no addition. On a recent AP exam, only about half of students earned this point.

So what just earned the point? One: add both payoffs in every cell. Two: pick the largest sum. Three: show the work, $23,000 + $18,000 = $41,000.

Recap: dominant strategies can trap rivals in a prisoner's dilemma, so cartels often break. A payoff change moves only one player's payoffs for one strategy. Merged profit is the largest cell sum. Next: the Unit 4 review, worked.