Zurich and Munich. Both are considering adding an additional daily train to this route. The payoff matrix shows the payoffs for each railway, where Swiss Rail values are in italics and EuroRail values are in bold. Assume that both companies have complete knowledge of the other’s payoff matrix. | | Add train | Add train | |———-|———–|———–| | Add train|
1,500 | | Do not add train |
4,000 | | Do not add train |
2,000 | | Do not add train |
3,000 | Select the answer that best describes the strategies in this game. A. Both companies’ dominant strategy is to add the train. B. EuroRail’s dominant strategy is to add the train, whereas Swiss Rail does not have a dominant strategy. C. Neither company has a dominant strategy. D. EuroRail’s dominant strategy is to not add the train, whereas Swiss Rail’s dominant strategy is to add the train. Does a Nash equilibrium exist in this game? A. Yes, it exists in the lower left quadrant. B. No, it does not exist. C. Yes, it exists in the upper right quadrant. D. Yes, it exists in the upper left quadrant.
EuroRail and Swiss Rail are hypothetical railways that have a duopoly on the route that connects the cities of Zurich and Munich. Both are considering adding an additional daily train to this route. The payoff matrix shows the payoffs for each railway, where Swiss Rail values are in italics and EuroRail values are in bold. Assume that both companies have complete knowledge of the other’s payoff matrix. | | Add train | Add train | |———-|———–|———–| | Add train|
1,500 | | Do not add train |
4,000 | | Do not add train |
2,000 | | Do not add train |
3,000 | Select the answer that best describes the strategies in this game. A. Both companies’ dominant strategy is to add the train. B. EuroRail’s dominant strategy is to add the train, whereas Swiss Rail does not have a dominant strategy. C. Neither company has a dominant strategy. D. EuroRail’s dominant strategy is to not add the train, whereas Swiss Rail’s dominant strategy is to add the train. Does a Nash equilibrium exist in this game? A. Yes, it exists in the lower left quadrant. B. No, it does not exist. C. Yes, it exists in the upper right quadrant. D. Yes, it exists in the upper left quadrant.
The Correct Answer and Explanation is:
To solve this game theory problem, we’ll first reconstruct the payoff matrix, identify dominant strategies, and analyze for Nash equilibrium. Then, we’ll provide a comprehensive explanation of the strategic interactions involved.
1. Payoff Matrix Setup
From your description, here is the assumed structure of the payoff matrix for Swiss Rail (SR) and EuroRail (ER), formatted as (Swiss Rail, EuroRail):
| EuroRail: Add Train | EuroRail: Do Not Add Train | |
|---|---|---|
| Swiss: Add | (1500, 1500) | (4000, 1000) |
| Swiss: Do Not Add | (2000, 4000) | (3000, 3000) |
Note: Italics are replaced with normal text, and we represent the matrix as (Swiss, EuroRail). This aligns with the description and creates a clear format for analysis.
2. Dominant Strategies
A dominant strategy is one that gives a player a better payoff no matter what the other player does.
Let’s examine each player:
Swiss Rail (SR) Strategy
Compare SR’s payoffs in both ER scenarios:
- If ER Adds a Train:
- SR Adds: 1500
- SR Doesn’t Add: 2000 → Better
- If ER Doesn’t Add:
- SR Adds: 4000
- SR Doesn’t Add: 3000 → Better
So:
- When ER adds → SR prefers not to add.
- When ER doesn’t add → SR prefers to add.
🡺 Swiss Rail has no dominant strategy, since its best choice depends on what EuroRail does.
EuroRail (ER) Strategy
Compare ER’s payoffs in both SR scenarios:
- If SR Adds a Train:
- ER Adds: 1500
- ER Doesn’t Add: 1000 → Better to Add
- If SR Doesn’t Add:
- ER Adds: 4000
- ER Doesn’t Add: 3000 → Better to Add
🡺 EuroRail always prefers to Add a Train, regardless of Swiss Rail’s choice.
🡺 So, EuroRail has a dominant strategy: Add a Train.
Answer to Strategy Question
Correct Answer: B. EuroRail’s dominant strategy is to add the train, whereas Swiss Rail does not have a dominant strategy.
3. Nash Equilibrium
A Nash Equilibrium is a strategy profile where no player has an incentive to deviate unilaterally.
Let’s check each combination for a Nash equilibrium.
(Add, Add) = (1500, 1500)
- SR: If switches to “Don’t Add” → gets 2000 → better off
- ER: Already choosing dominant strategy
🡺 Not a Nash Equilibrium, because SR would switch.
(Add, Don’t Add) = (4000, 1000)
- SR: Best choice if ER doesn’t add
- ER: But ER can get 3000 by switching to Add → would switch
🡺 Not a Nash Equilibrium
(Don’t Add, Add) = (2000, 4000)
- SR: Better off than 1500
- ER: Best possible response (dominant strategy)
- SR: Could switch to Add and get 1500 → worse off
- ER: Won’t deviate (4000 is best)
🡺 No one wants to deviate → this is a Nash Equilibrium
✅ Yes, Nash equilibrium exists in lower left quadrant
(Don’t Add, Don’t Add) = (3000, 3000)
- SR: Could get 4000 by switching → would deviate
- ER: Could get 4000 by switching → would deviate
🡺 Not a Nash Equilibrium
Answer to Nash Equilibrium Question
Correct Answer: A. Yes, it exists in the lower left quadrant.
Explanation
Let’s now elaborate with a detailed explanation, bringing in key economic and game theory concepts.
A. Background: Oligopoly and Strategic Decisions
The market described is a duopoly, meaning two firms (Swiss Rail and EuroRail) dominate the train route between Zurich and Munich. In such markets, strategic decisions by one firm directly affect the other.
Here, both firms are deciding whether or not to add a new train. Each firm’s profits depend not only on its own decision but also on the decision made by the other company. This interdependence makes the situation ripe for game theoretic analysis.
B. Understanding Payoffs
Each firm’s payoffs are affected by competition:
- When both add a train, competition intensifies. Result: both earn less (1500 each).
- When only one adds, the firm that adds gains a first-mover or competitive advantage (4000), and the other suffers (1000 or 2000).
- When neither adds, both maintain current profitability (3000 each) — moderate but stable outcome.
The payoffs reflect classic tension in oligopoly:
- Aggressive strategy (Add Train): Could lead to competitive war.
- Passive strategy (Don’t Add): Risk being undercut.
C. Dominant Strategy Explained
A dominant strategy offers the best payoff regardless of what the opponent does. We find that:
- EuroRail has a dominant strategy: Always do better by adding a train.
- This reflects an aggressive strategy typical of firms with lower opportunity costs or better market reach.
- Swiss Rail lacks a dominant strategy: Its best choice depends on EuroRail’s move.
- If EuroRail adds, Swiss is better off not adding.
- If EuroRail doesn’t add, Swiss should add.
This illustrates strategic dependence: Swiss Rail must anticipate EuroRail’s move.
D. Nash Equilibrium: Stability in Strategy
A Nash Equilibrium is a pair of strategies where neither player benefits by deviating unilaterally.
Let’s focus on the (Don’t Add, Add) quadrant:
- EuroRail adds the train — its dominant strategy.
- Swiss Rail, knowing ER adds, chooses not to add since 2000 > 1500.
- If either deviates:
- ER would still choose to Add.
- SR would lose by deviating.
🡺 Neither firm benefits by changing its decision → stable outcome.
Thus, (Don’t Add, Add) is the only Nash Equilibrium in this game. It’s also asymmetric — one firm (EuroRail) is aggressive; the other (Swiss Rail) is cautious.
E. Real-World Implications
This scenario mirrors real-world oligopolies like:
- Airline route decisions
- Tech product launches
- Gas station pricing
In each, firms must anticipate rivals’ behavior and act strategically. Often, firms with dominant strategies (e.g., lower costs, higher capacity) force others to accommodate.
In this case:
- EuroRail’s aggressive strategy forces Swiss Rail to be reactive.
- If both were aggressive (Add/Add), both suffer — the “prisoner’s dilemma” of market competition.
This explains why Nash Equilibrium does not always align with the socially optimal outcome (e.g., (Do Not Add, Do Not Add) where both get 3000). Firms act in self-interest, not collective best interest.
F. Final Summary
| Aspect | Answer |
|---|---|
| Dominant Strategies | EuroRail has a dominant strategy to Add; Swiss Rail does not. |
| Correct Multiple Choice | B |
| Nash Equilibrium | Exists in Lower Left Quadrant — Swiss: Don’t Add, EuroRail: Add |
| Correct Multiple Choice | A |
