### Elevator Installation Dilemma: An Economic Game Theory Problem**Scenario:**An apartment building requires a new elevator, costing $10,000. There are 10 residents who need to decide independently whether or not to contribute to this cost. The rules of contribution are as follows:- If no resident is willing to contribute, the elevator will not be installed, and each resident will receive a payoff of $0.- If at least one resident is willing to contribute, the elevator will be installed. All residents, regardless of their contribution, will then receive a benefit valued at $2,000.**Cost Sharing:**- A resident who decides to contribute will share the installation cost of the elevator, which is calculated as $\frac{\$10,000}{n}$, where $ n $ is the number of contributing residents.- For example, if 5 residents decide to contribute, each of these residents will receive a net payoff of $2,000 - \frac{\$10,000}{5} = $0. The remaining residents, who do not contribute, will each receive the payoff of $2,000.**Questions for Analysis:**1. **Is every resident chipping in a Nash Equilibrium of the game? Explain.** 2. **Is no resident chipping in a Nash Equilibrium of the game? Explain.**3. **Are there other Nash equilibria? If so, specify them and verify that they are indeed equilibria. If no other Nash equilibria exist, explain why other action profiles are not Nash equilibria.**This problem is designed to explore concepts of Nash Equilibrium within game theory, presenting a practical scenario of decision-making and cost-benefit analysis. The discussion encourages a deeper understanding of strategic interactions among individuals in a shared environment.

A Nash equilibrium is a concept in game theory that represents a situation in which each player in a…

An apartment building needs a new elevator, which costs $10,000. 10 residents need to decide independently whether or not to chip in. If no resident is willing to bear the cost, the elevator will not be installed and each resident will receive zero payoff. However, if n > 0 residents are willing to chip in, the elevator will be installed. If the elevator is installed, each resident will receive a benefit equal to $2,000 regardless of whether they have chosen to chip in or not. However, a resident who has chosen to chip in will also bear their share of the For example, if 5 residents choose to chip in, they will receive a net $10,000 installation cost n payoff equal to $2,000 $10,000 5 or $0, while other residents will each receive payoff $2,000. (1) Is every resident chipping in a Nash equilibrium of the game? Explain. (2) Is no resident chipping in a Nash equilibrium of the game? Explain. (3) Are there any other Nash equilibria? If there are, specify such a Nash equilibrium and verify that it is indeed an equilibrium. If no other Nash equilibria exits, explain why the other action profiles are not Nash equilibria. .

An apartment building needs a new elevator, which costs $10,000. 10 residents need to decide independently whether or not to chip in. If no resident is willing to bear the cost, the elevator will not be installed and each resident will receive zero payoff. However, if n > 0 residents are willing to chip in, the elevator will be installed. If the elevator is installed, each resident will receive a benefit equal to $2,000 regardless of whether they have chosen to chip in or not. However, a resident who has chosen to chip in will also bear their share of the For example, if 5 residents choose to chip in, they will receive a net $10,000 installation cost n payoff equal to $2,000 $10,000 5 or $0, while other residents will each receive payoff $2,000. (1) Is every resident chipping in a Nash equilibrium of the game? Explain. (2) Is no resident chipping in a Nash equilibrium of the game? Explain. (3) Are there any other Nash equilibria? If there are, specify such a Nash equilibrium and verify that it is indeed an equilibrium. If no other Nash equilibria exits, explain why the other action profiles are not Nash equilibria. .

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

### Elevator Installation Dilemma: An Economic Game Theory Problem

**Scenario:**

An apartment building requires a new elevator, costing $10,000. There are 10 residents who need to decide independently whether or not to contribute to this cost. The rules of contribution are as follows:

- If no resident is willing to contribute, the elevator will not be installed, and each resident will receive a payoff of $0.
- If at least one resident is willing to contribute, the elevator will be installed. All residents, regardless of their contribution, will then receive a benefit valued at $2,000.

**Cost Sharing:**

- A resident who decides to contribute will share the installation cost of the elevator, which is calculated as $\frac{\$10,000}{n}$, where $ n $ is the number of contributing residents.
- For example, if 5 residents decide to contribute, each of these residents will receive a net payoff of $2,000 - \frac{\$10,000}{5} = $0. The remaining residents, who do not contribute, will each receive the payoff of $2,000.

**Questions for Analysis:**

1. **Is every resident chipping in a Nash Equilibrium of the game? Explain.**

2. **Is no resident chipping in a Nash Equilibrium of the game? Explain.**

3. **Are there other Nash equilibria? If so, specify them and verify that they are indeed equilibria. If no other Nash equilibria exist, explain why other action profiles are not Nash equilibria.**

This problem is designed to explore concepts of Nash Equilibrium within game theory, presenting a practical scenario of decision-making and cost-benefit analysis. The discussion encourages a deeper understanding of strategic interactions among individuals in a shared environment.

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