Skip to main content

Game Theory

tip

The following questions have been taken from the Midterm Exam of ECON 159 of Yale University.

Q1: State whether each of the following claims is true or false (or cannot be determined). For each, explain your answer in (at most) one short paragraph. Explaining an example or a counter-example is sufficient. In the absence of this, a clear and concise intuition will suffice; a formal proof is not required.

  1. A strictly dominated strategy can never be the best response.
  2. In the candidate-voter model, if two people are standing, one to the left of center and one to the right of the centre, and neither of them is 'too extreme', then it is an equilibrium.
  3. If (s, s) is a Nash equilibrium of a symmetric, two-player game, then s is evolutionarily stable.

Q2: Josh has invited Caleb to his party. Josh must choose whether or not to hire a clown. Simultaneously, Caleb must decide whether or not to go to the party. Caleb likes Roger but he hates clowns — he even hates other people seeing clowns! Caleb's payoff from going to the party is 4 if there is no clown, but 0 if there is a clown there. Caleb's payoff for not going to the party is 3 if there is no clown, but 1 if there is a clown at the party. Roger likes clowns —he especially likes Caleb's reaction to them — but does not like paying for them. Roger's payoff if Caleb comes to the party is 4 if there is no clown, but 8 - x if there is a clown (x is the cost of a clown). Roger's payoff if Caleb does not come to the party is 2 if there is no clown, but 3-x if there is a clown there.

  1. Write down the payoff matrix of this game.
  2. Suppose x = 0. Identify any dominated strategies. Explain and find the Nash equilibrium. What are the equilibrium payoffs?
  3. Suppose x = 2. Identify any dominated strategies. Explain and find the Nash equilibrium. What are the equilibrium payoffs?
  4. Suppose x = 3. Identify any dominated strategies. Explain and find the Nash equilibrium. What are the equilibrium payoffs?
  5. Suppose x = 5. Identify any dominated strategies. Explain and find the Nash equilibrium. What are the equilibrium payoffs?

Q3: Six validators in a network must decide whether to invest in improve the security measure. The network is vulnerable to attacks if too many validators are insecure. Validators aim to maximize their expected rewards.

  • Security Cost: Validator i incurs cost i. (E.g. Validator 1: cost 1, validator 6: cost 6).
  • Reward: A secure validator earns 6 units. An insecure validator earns 0 if compromised.
  • Attack probability: If k validators are insecure, each one of them faces a k/6 chance of being slashed.
  1. Is it a Nash equilibrium for validators 1-4 to secure their nodes and validators 5-6 to remain insecure?
  2. Is it a Nash equilibrium for Validators 1-3 to secure and Validators 4-6 to remain insecure?
  3. Which validators have strictly/weakly dominated strategies?
  4. After removing dominated strategies, do others become dominated?
  5. Find all pure/mixed Nash equilibriums.

How to Submit Your Work​

  • All work for your chosen challenge must be committed to the GitHub repository assigned to you during onboarding.
  • Include diagrams/charts to illustrate key points.
  • Maintain a technical focus throughout.
  • No need to delve extremely deep into the underlying systems—focus on tokenomics, and wisely manage your time.
  • Structure your commits clearly, with meaningful messages that outline the progress of your work, see Git Practices for reference.
  • Ensure your final submission is well-organized, with supporting files, diagrams, or models included as needed.

🍀 Good luck!