Two airlines (TAP and Ryanair) compete on the Lisbon–London route. Each can set a High or Low fare.

Payoffs: (TAP’s profit , Ryanair’s profit). Hypothetical illustrative example.

How to read: Each cell shows what happens for a specific combination of choices.

• If both set High prices: each earns €500k (they share the market at high margins)
• If TAP goes Low, Ryanair stays High: TAP steals customers → TAP earns €600k, Ryanair earns only €200k
• If both go Low: price war → each earns only €350k

👉 Each player picks a row (TAP) or column (Ryanair). The cell at the intersection is the outcome.

TAP’s reasoning (looking at its own payoffs — the first number):

. . .

👉 By the same logic, Ryanair also finds Low is dominant (check the second numbers!).

Two suspects are arrested and interrogated separately. Each can Confess (betray) or Stay Silent (cooperate).

Payoffs are years in prison (lower is better).

The dilemma:

• If both stay Silent: only 1 year each (best collective outcome!)
• But Confess is a dominant strategy for each: whatever the other does, confessing gives a shorter sentence
• Result: both confess → 5 years each — worse than if they’d cooperated!

👉 Individual rationality leads to a collectively worse outcome. This is the Prisoner’s Dilemma.

Look at our airline game again:

⭐ = best collective outcome. 🔒 = dominant strategy outcome.

Both airlines want (High, High) = €500k each. But each has an incentive to deviate to Low and steal customers.

Since both follow their dominant strategy (Low), they end up at (€350k, €350k) — €150k worse for each than if they’d cooperated!

. . .

💡 This explains price wars in tourism: airlines, hotels, and tour operators often end up competing on price even though they’d all be better off with higher prices. The structure of the game traps them.

In the airline game: (Low, Low) is a Nash Equilibrium because:

• TAP plays Low. Could TAP do better by switching to High? No — TAP would go from €350k to €200k. ❌
• Ryanair plays Low. Could Ryanair do better by switching to High? No — same reasoning. ❌

👉 Neither player wants to deviate → it’s a Nash Equilibrium!

💡 A dominant strategy equilibrium is always a Nash Equilibrium, but not all Nash Equilibria involve dominant strategies (as we’ll see).

Step-by-step method to find Nash Equilibria in any 2×2 game:

1️⃣ For each of Ryanair’s strategies (each column), find TAP’s best response — underline TAP’s highest payoff in that column.

2️⃣ For each of TAP’s strategies (each row), find Ryanair’s best response — underline Ryanair’s highest payoff in that row.

3️⃣ Any cell where both payoffs are underlined is a Nash Equilibrium.

👉 Only (Low, Low) has both payoffs underlined → it’s the unique Nash Equilibrium.

Two hotels in Porto decide their marketing strategy: focus on Luxury positioning or Budget positioning.

Hypothetical illustrative example.

Does Hotel A have a dominant strategy? Let’s check:

• If B chooses Luxury: A prefers Budget (€400k > €300k)
• If B chooses Budget: A prefers Luxury (€500k > €200k)

❌ No dominant strategy! A’s best choice depends on what B does.

Nash Equilibria (underline method): (Luxury, Budget) and (Budget, Luxury) are both NE!

👉 This game has two Nash Equilibria — differentiation is better than being identical. The market “wants” the hotels to be different from each other.

What NE tells us:

• ✅ It’s a stable outcome — no one wants to deviate alone
• ✅ It’s a prediction of what rational players will do
• ✅ Every game has at least one Nash Equilibrium (though it might involve randomization — “mixed strategies,” which we won’t cover)

What NE does NOT tell us:

• ❌ NE is not necessarily efficient (Prisoner’s Dilemma: NE is worse for both!)
• ❌ NE is not necessarily unique (the hotel game has two)
• ❌ NE doesn’t require dominant strategies

Today’s Key Takeaways:

1. Game theory studies strategic interaction — your outcome depends on others’ choices
2. Three ingredients: players, strategies, payoffs
3. Payoff matrix: shows outcomes for all strategy combinations (read the row player’s payoff first)
4. Dominant strategy: best regardless of what others do
5. Prisoner’s Dilemma: individual rationality → collectively bad outcome. Both players have a dominant strategy, but the equilibrium is not the best outcome for the group
6. Nash Equilibrium: no player can improve by unilaterally deviating. Found via the underline method
7. NE can be unique or multiple, efficient or inefficient
8. Tourism: price wars (airlines, hotels), entry decisions (Airbnb), overbooking — all have game-theoretic structure

Next (Lecture 19, April 23): Sequential Games & Backward Induction — what happens when one player moves first?

Question: Two tour operators in Lisbon decide simultaneously whether to offer Free cancellation or No cancellation policies. The payoff matrix (profits in €k) is:

What is the Nash Equilibrium?

A. (Free, Free)
B (No, No)
C. (Free, No)
D. (No, Free)

Question: In a Nash Equilibrium:

A. Both players are maximizing their joint profit

B. Neither player can improve their payoff by changing their strategy alone

C. Both players must have dominant strategies

D. The outcome is always the best possible for society

Two beach bars in Albufeira (Bar A and Bar B) must decide their pricing strategy for the summer: High prices (€8 per cocktail) or Low prices (€5 per cocktail). The payoff matrix (daily profits in €) is:

Hypothetical illustrative example.

a) Does Bar A have a dominant strategy? Does Bar B? Explain.

b) Find the Nash Equilibrium using the underline method. Show your work.

c) Is the Nash Equilibrium efficient? That is, is there another outcome that makes both bars better off?

d) This is a Prisoner’s Dilemma. Explain why, using the definition from today’s lecture.

e) Suppose the two bars are owned by the same company. What pricing strategy would the company choose? What does this tell us about the value of coordination (or merger)?

f) Now suppose the bars compete every day for the entire summer (90 days), not just once. How might this change the outcome? Think about reputation and retaliation.

a) Bar A’s dominant strategy:

• If B plays High: A prefers Low (€450 > €400)
• If B plays Low: A prefers Low (€250 > €150)

Low is dominant for A. By symmetry (the game is symmetric), Low is also dominant for B.

b) Underline method:

Column “B: High”: A’s best = 450 (Low) → underline. Column “B: Low”: A’s best = 250 (Low) → underline.

Row “A: High”: B’s best = 450 (Low) → underline. Row “A: Low”: B’s best = 250 (Low) → underline.

Both underlined in (Low, Low) → Nash Equilibrium = (Low, Low) with payoffs (€250, €250).

c) Is (Low, Low) = (250, 250) efficient?

No! The outcome (High, High) = (400, 400) gives both bars higher profits. Each bar earns €150 more per day if they both keep prices high. The NE is Pareto inefficient — there exists another outcome that is better for everyone.

d) This is a Prisoner’s Dilemma because:

• Both players have a dominant strategy (Low)
• The dominant strategy equilibrium (Low, Low) = (250, 250) is worse for both players than the cooperative outcome (High, High) = (400, 400)
• Each player has an individual incentive to deviate from cooperation (if B plays High, A gains €50 by switching to Low: 450 > 400)
• But when both follow this incentive, they end up worse off

👉 Individual rationality → collective irrationality. This is the essence of the Prisoner’s Dilemma.

e) If both bars are owned by the same company:

The company maximizes total profit = A’s profit + B’s profit.

• (High, High): total = €400 + €400 = €800 ⬅️ Best!
• (Low, High) or (High, Low): total = €450 + €150 = €600
• (Low, Low): total = €250 + €250 = €500

The company chooses (High, High) → total profit = €800.

👉 This shows the value of coordination (or merger): by internalizing the competition, the company avoids the Prisoner’s Dilemma and achieves €300/day more in total profit. This is also why cartels are tempting (and illegal) — they replicate the merger outcome.

f) With 90 days of repeated interaction:

Bars can adopt strategies like tit-for-tat: start with High, and match whatever the other bar did yesterday. If Bar B cuts to Low, Bar A retaliates the next day. The threat of future punishment can sustain cooperation at (High, High) — as long as both bars value future profits enough. Repeated games can solve the Prisoner’s Dilemma through reputation!