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ECON3073 Games and Economic Decisions

23-24 Semester 1

Assignment 1

Instruction: Please answer all 5 questions and submitted your written answers in hard copy to Ms. Eko Chen at T1-301-R5-H17 by 5p.m., Monday, November 6. Using AI tools is strictly prohibited.

Question 1 (20 marks)

Consider the following game, where a two-player game in which Player 1 and Player 2 must simultaneously choose one of two strategies: "Cooperate" or "Defect."

The payoff matrix is shown below.

However, due to some mistakes the players may have got the wrong payoff matrix shown instead as below.

a. Find all NE in the correct payoff matrix. (5 marks)

b. Find all NE in the wrong payoff matrix. (5 marks)

c. Assume Player 1 got the correct payoff matrix and Player 2 got the wrong payoff matrix. Player 1 knows about the wrong matrix and Player 1 knows that Player 2 got the wrong matrix but Player 2 does not know that. What strategy will both players play?  (5 marks)

d. Assume now Player 2 got the correct payoff matrix and Player 1 got the wrong payoff matrix. Player 2 knows about the wrong matrix and Player 2 knows that Player 1 got the wrong matrix but Player 1does not know that. What strategy will both players play? (5 marks)

Question 2 (20 marks)

Two friends, Alice and Bob, are standing at opposite ends of a long corridor. They each have a water balloon, and they can choose to either throw the water balloon or hold onto it. They are trying to soak each other with water and stay dry themselves.

The payoffs are as follows:

If both Alice and Bob throw their water balloons at each other, they both get wet, and each receives a payoff of -10 points (they both get wet and no one wins).

If one of them throws a water balloon, and the other holds onto theirs, the person who throws it wins and gets a payoff of +10 points, while the person who holds onto it loses and receives -5 points.

If both Alice and Bob hold onto their water balloons, they both receive a payoff of 0 points.

a. Represent the game in a suitable format. (4 marks)

b. Draw the best response functions for both players. (6 marks)

c. Fina all NE in this game. (4 marks)

d. Suppose Bob has a secret crush on Alice and he does not really want to get Alice soaked with water. The actual payoff that Bob gets is this payoff function (a, b) =3a+b, where a is Alice’s payoff and b is Bob’s original payoff. Represent Bob’s actual payoff in a suitable format. Discuss what outcome will be reached. (6 marks)

Question 3 (20 marks)

Consider the following two player game.

a. Is there a strategy of player 1 that is strictly dominated? Is there a strategy of player 2 that is a never best response? Is there a strategy of player 2 that is strictly dominated? (4 marks)

b. What outcome is reached if we carry out iterated elimination of strictly dominated strategies in this game? (4 marks)

c. Describe the rationality assumptions needed to carry out iterated elimination of strictly dominated strategies in this game. (2 marks)

d. Now, we allow for mixed strategies. Is there a pure strategy of player 2 which is not strictly dominated in the original game but becomes strictly dominated by a mixed strategy? (6 marks)

e. What outcome is reached if we allow mixed strategies and carry out iterated elimination of strictly dominated strategies in this game? (4 marks)

Question 4 (20 marks)

Consider the following single object allocation problem. A single object needs to be allocated to one of  agents. Each agent has a value  for the object – this is the payoff he will derive from the object if given for free.

The game is as follows:

· Each agent  announces a non-negative number  ∈ R+ as her bid.

· Highest bidder wins the object – in case of a tie, the bidder with the lowest index wins (for instance, if agents 2, 3, 5 have the highest bid, then 2 wins the object).

· The winner gets the object for free, i.e., does not pay anything and does not receive any payment.

· All other agents receive a payment equal to the highest bid amount. Payoffs are as follows: winner’s payoff is her value; each loser’s payoff is the amount she receives (highest bid amount).

Prove that bidding at  is a dominant strategy (playing  earns player i a payoff at least as high as any other strategies regardless of what any other players do) by discussing the following two cases.

a. Discuss the case when agent  wins the object by bidding at . (10 marks)

b. Discuss the case when agent  does not win the object by bidding at . (10 marks)

Question 5 (20 marks)

Three players live in a town, and each can choose to contribute to fund a streetlamp. The value of having the streetlamp is 3 for each player, and the value of not having it is 0. The mayor asks each player to contribute either 1 or nothing. If at least two players contribute, then the lamp will be erected. If one player or no player contribute then the lamp will no be erected, in which case any person who contributed will not get her money back.

a. Write out each player’s best response. (10 marks)

b. Find all pure-strategy Nash equilibria. (10 marks)