Strategy thinking

Dr. Vidya Atal Strategic Thinking & Game Theory

Module 8 Assignment

Answer all the questions. Remember that every part of the question has points assigned; by just submitting an answer to each part you can get 50% submission credit, but if you do not submit an answer to a part, you get 0. So play strategically! Please upload the images of your answers on Canvas before the due date.

1. Two players simultaneously and independently have to decide how much to contribute to a public good. If player 1 contributes x ≥ 0 and 2 contributes y ≥ 0, their payoffs are:

π1 = 2(x + y + xy)− x2, π2 = 2(x + y + xy)− ty2,

where t is a private information of player 2. Player 1 knows that t is 2 with probability 1 2 and 3 with probability

1 2 . Find out the best response functions. Show your work. What is the Bayes-Nash Equilibrium of this game? Show your work. (10+20 = 30 points)

2. You are considering an investment of $250. The investment could be bad with 20% probability, mediocre with 40% probability, and good with the remaining 40% probability. You seek advice from a financial adviser who examines the investment, finds its true quality, and gives you a report whether it is good, bad, or mediocre. Assume that he never underplays an investment. If you invest after seeing the report, you have to pay a fee of 2% of the $250 investment plus 25% of any gains from it. The return from the investment is (−$120) if it is a bad investment, (+$6) if it is a mediocre one and (+$160) if it is a good one. The adviser suffers a reputation cost of $S if he overplays the investment slightly, but $L if he overplays a bad investment to be a good one, where 0 ≤ S ≤ L.

(a) Give a set of values for (S,L) so that there exists a “partial revelation”Bayes Nash Equilibrium. Show your work. (10 points)

(b) Give a set of values for (S,L) so that there exists a “full revelation”Bayes Nash Equilibrium. Show your work. (10 points)

(c) Give a set of values for (S,L) so that the “babbling”Bayes Nash Equilibrium is the only equilibrium. Show your work. (10 points)

1