# Two competing hospitals in small city can choose whether to offer only basic health care, all-

purpose care, or specialized care. The following payoff table denotes profits for each
hospital:
Hospital B's Services

Hospital A's
Services

Basic
All-Purpose
Specialty
Basic 5, 7 9, 8
6, 10
All-Purpose
5, 4 4, 7
6, 12
Specialty
13, 6
7, 4 3, 3

a) Does either hospital have a dominant strategy (or any dominated strategy)? Assuming
that they determine their strategies independently of one another, what are the hospitals' respective Nash equilibrium strategies? Explain why.
b) Suppose instead that the hospitals coordinate. What actions should they each take?
c) One way for the two hospitals to coordinate is for them to merge. What general
economic reasons might there be for a hospital merger to generate an increase in
profit?

## a) To determine if either hospital has a dominant strategy, we need to compare the payoffs for each hospital's services.

For Hospital B:
- If Hospital A offers basic care, the payoffs are 5 and 7 for basic, 9 and 8 for all-purpose, and 6 and 10 for specialty.
- If Hospital A offers all-purpose care, the payoffs are 5 and 4 for basic, 4 and 7 for all-purpose, and 6 and 12 for specialty.
- If Hospital A offers specialty care, the payoffs are 13 and 6 for basic, 7 and 4 for all-purpose, and 3 and 3 for specialty.

From these payoffs, we can see that for each service choice by Hospital A, the payoffs for Hospital B vary. Therefore, there is no dominant strategy for Hospital B.

Similarly, for Hospital A:
- If Hospital B offers basic care, the payoffs are 5 and 7 for basic, 5 and 4 for all-purpose, and 13 and 6 for specialty.
- If Hospital B offers all-purpose care, the payoffs are 9 and 8 for basic, 4 and 7 for all-purpose, and 7 and 4 for specialty.
- If Hospital B offers specialty care, the payoffs are 6 and 10 for basic, 6 and 12 for all-purpose, and 3 and 3 for specialty.

Again, there is no dominant strategy for Hospital A.

To find the Nash equilibrium strategies, we need to look for situations where neither hospital has an incentive to deviate from their chosen strategy. In this case, the Nash equilibrium occurs when both hospitals choose their all-purpose services. This is because if Hospital A chooses all-purpose, Hospital B's best response is also all-purpose, and vice versa.

b) If the hospitals coordinate and communicate with each other, they can jointly choose their strategies to maximize their combined profits. In this case, they should both choose the strategy that gives them the highest joint profit. Looking at the payoffs, the strategy that maximizes the joint profit is for both hospitals to offer specialty care.

c) A hospital merger may generate an increase in profit for several reasons:
- Economies of scale: Merging hospitals can lead to lower costs per patient by consolidating resources, sharing administrative costs, and optimizing the utilization of equipment and facilities.
- Enhanced bargaining power: A merged hospital may have more leverage in negotiations with insurance companies and suppliers, allowing them to secure better reimbursement rates or lower prices for medical supplies.
- Improved specialization: Merging hospitals can result in the consolidation of specialized services, leading to improved expertise, wider range of services, and attracting more patients.
- Increased market share: A merger can expand the merged hospital's market presence, allowing it to capture a larger share of patients and potentially increase revenue.
- Improved financial stability: A merger may provide financial stability by combining resources and pooling risks, making it easier to invest in advanced medical technologies and infrastructure.

It's important to note that the outcomes of hospital mergers can vary depending on the specifics of the situation and the local market conditions.

## a) To determine if either hospital has a dominant strategy, we need to compare the payoffs for each strategy option for both hospitals.

For Hospital A:
- If Hospital B offers basic care, Hospital A's best choice is to offer specialty care (payoff of 10 is higher than 7 or 8).
- If Hospital B offers all-purpose care, Hospital A's best choice is to offer basic care (payoff of 9 is higher than 4 or 6).
- If Hospital B offers specialty care, Hospital A's best choice is to offer all-purpose care (payoff of 12 is higher than 4 or 7).

Therefore, Hospital A does not have a dominant strategy.

For Hospital B:
- If Hospital A offers basic care, Hospital B's best choice is to offer all-purpose care (payoff of 9 is higher than 8 or 10).
- If Hospital A offers all-purpose care, Hospital B's best choice is to offer basic care (payoff of 7 is higher than 4 or 6).
- If Hospital A offers specialty care, Hospital B's best choice is to also offer specialty care (payoff of 7 is higher than 4 or 3).

Therefore, Hospital B's dominant strategy is to offer specialty care.

The Nash equilibrium strategies are when both hospitals choose their best responses to each other's strategies. In this case, Hospital A's best response to Hospital B's dominant strategy is to also offer specialty care. Therefore, the Nash equilibrium strategy is for both hospitals to offer specialty care.

b) If the hospitals coordinate, they should choose the strategy option that gives them the highest combined payoff. Looking at the table, the highest combined payoff occurs when Hospital A offers all-purpose care and Hospital B offers specialty care, with a total payoff of 12 + 7 = 19. Therefore, if the hospitals coordinate, they should choose this strategy.

c) A hospital merger may generate an increase in profit due to several economic reasons:
- Cost synergies: By combining resources and operations, the merged hospital may achieve economies of scale, leading to cost savings in areas such as purchasing, administration, and infrastructure.
- Enhanced bargaining power: The merged hospital might have more negotiating power with insurance providers, suppliers, and other stakeholders, leading to better terms and lower costs.
- Increased market share: A merger can result in a larger customer base and market share, providing the merged hospital with more opportunities to attract patients and generate revenue.
- Specialization and expertise: The merged hospital may be able to consolidate specialized services and attract top talent, which can lead to improved quality of care and reputation, attracting more patients and generating higher profits.
- Improved financial stability: A merger can provide financial stability by pooling resources and reducing financial risks, allowing the merged hospital to invest in new technologies, facilities, and services.

These are general economic reasons, and the actual impact will depend on the specific circumstances and market conditions.