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Q1 – Finding the equilibrium (equilibria) in a multi-stage game (30 marks) Six students are working to complete their video for the First Year Challenge Project. Call the amount of effort of student i, E_i and the vector of efforts of all six students E. The maximum a student can dedicate to the work is E_i = 8. The payoffs of student i have the following form ?_i(E) = ?_{j=1}^6 E_j – 1.5E_i. That is, every unit of effort has a benefit of 1 for every person of the group but there is a cost of effort of 1.5 for every unit of effort that a student puts in. (a) Each student has a different task which she does independently, simultaneously and without communicating with others. i. Does this game have a strictly dominated strategy (or strategies)? Describe what it is (they are). ii. Does the game have a Nash equilibrium (or equilibria)? Prove what it is (they are) and how you found it (them). iii. If there is a Nash equilibrium, is the outcome of the equilibrium Pareto efficient? Explain your answer. (b) Suppose the game is as in question (a) but individuals play sequentially. That is, Student 2 sees what Student 1 did, and then she does her task. Then Student 3 sees what Students 1 and 2 did, and so on. i. Draw a game tree to illustrate this new game for the six players. There is no need to show the payoffs for all players. Focus on showing the sequence of play and the strategies. ii. What would the Nash equilibrium of this new game be? Explain your reasoning. (c) Suppose that the game is as described in the question (a), but rather than caring only about themselves, each Student’s payoff is her own plus that of one other person in the group. For example, Student 1 adds the payoff of Student 2 to her own and vice versa, and Student 3 adds the payoff of Student 4 to her own and vice versa, and so on. i. Does this game have a strictly dominated strategy (or strategies)? Describe what they are. ii. Does the game have a Nash equilibrium (or equilibria)? Describe what they are. iii. If there is a Nash equilibrium, is the outcome of the equilibrium Pareto efficient?

          Q1 – Finding the equilibrium (equilibria) in a multi-stage game (30 marks)

Six students are working to complete their video for the First Year Challenge Project.

Call the amount of effort of student i, E_i and the vector of efforts of all six students E. The maximum a student can dedicate to the work is E_i = 8.

The payoffs of student i have the following form ?_i(E) = ?_{j=1}^6 E_j – 1.5E_i. That is, every unit of effort has a benefit of 1 for every person of the group but there is a cost of effort of 1.5 for every unit of effort that a student puts in.

(a) Each student has a different task which she does independently, simultaneously and without communicating with others.
i. Does this game have a strictly dominated strategy (or strategies)? Describe what it is (they are).
ii. Does the game have a Nash equilibrium (or equilibria)? Prove what it is (they are) and how you found it (them).
iii. If there is a Nash equilibrium, is the outcome of the equilibrium Pareto efficient? Explain your answer.

(b) Suppose the game is as in question (a) but individuals play sequentially. That is, Student 2 sees what Student 1 did, and then she does her task. Then Student 3 sees what Students 1 and 2 did, and so on.
i. Draw a game tree to illustrate this new game for the six players. There is no need to show the payoffs for all players. Focus on showing the sequence of play and the strategies.
ii. What would the Nash equilibrium of this new game be? Explain your reasoning.

(c) Suppose that the game is as described in the question (a), but rather than caring only about themselves, each Student’s payoff is her own plus that of one other person in the group. For example, Student 1 adds the payoff of Student 2 to her own and vice versa, and Student 3 adds the payoff of Student 4 to her own and vice versa, and so on.
i. Does this game have a strictly dominated strategy (or strategies)? Describe what they are.
ii. Does the game have a Nash equilibrium (or equilibria)? Describe what they are.
iii. If there is a Nash equilibrium, is the outcome of the equilibrium Pareto efficient?

Q1 – Finding the equilibrium (equilibria) in a multi-stage game (30 marks)

Six students are working to complete their video for the First Year Challenge Project.

Call the amount of effort of student i, Ei and the vector of efforts of all six students E. The maximum a student can dedicate to the work is Ei = 8.

The payoffs of student i have the following form ?i(E) = ?j=1^6 Ej – 1.5Ei. That is, every unit of effort has a benefit of 1 for every person of the group but there is a cost of effort of 1.5 for every unit of effort that a student puts in.

(a) Each student has a different task which she does independently, simultaneously and without communicating with others.
i. Does this game have a strictly dominated strategy (or strategies)? Describe what it is (they are).
ii. Does the game have a Nash equilibrium (or equilibria)? Prove what it is (they are) and how you found it (them).
iii. If there is a Nash equilibrium, is the outcome of the equilibrium Pareto efficient? Explain your answer.

(b) Suppose the game is as in question (a) but individuals play sequentially. That is, Student 2 sees what Student 1 did, and then she does her task. Then Student 3 sees what Students 1 and 2 did, and so on.
i. Draw a game tree to illustrate this new game for the six players. There is no need to show the payoffs for all players. Focus on showing the sequence of play and the strategies.
ii. What would the Nash equilibrium of this new game be? Explain your reasoning.

(c) Suppose that the game is as described in the question (a), but rather than caring only about themselves, each Student’s payoff is her own plus that of one other person in the group. For example, Student 1 adds the payoff of Student 2 to her own and vice versa, and Student 3 adds the payoff of Student 4 to her own and vice versa, and so on.
i. Does this game have a strictly dominated strategy (or strategies)? Describe what they are.
ii. Does the game have a Nash equilibrium (or equilibria)? Describe what they are.
iii. If there is a Nash equilibrium, is the outcome of the equilibrium Pareto efficient?

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