ECON 2112. UNGRADED PROBLEM SET 2

Exercise 1. Finding Nash equilibria

Find all pure and mixed strategy Nash equilibria of the following games.

(i)

(ii)

(iii)

(iv)

Exercise 2. Voting

Voters in Suburbia are choosing between two candidates, L and R. L is proposing new infras-tructure funded with new taxes. Candidate R is in favour of preserving the status quo, i.e., no new taxes and no new spending. Consider three voters - Alison, Beth, and Chandra - whose preferences and payoff are as follows.

• Alison’s payoff is 2 if L wins but 0 otherwise.

• Beth’s payoff is 2 if R wins but 0 otherwise.

• Chandra is not interested in electoral politics. Nonetheless voting is compulsory in Suburbia so he always votes. He gets 0 if the candidate he votes for wins. Otherwise, his payoff is -1.

All three individuals - Alison, Beth, and Chandra - vote. Candidates receiving majority of the votes wins.

(i) Write down the 3-person normal form. game where Alison, Beth, and Chandra are three players and each of them has two strategies - vote for L or vote for R. As described above, their payoffs depend on the outcome of the election.

(ii) Find out all Nash equilibria in pure strategies.

(iii) Suppose we are interested in Nash equilibria where neither strictly nor weakly domi-nated strategies are played. A Nash equilibrium satisfying this property is called ad-missible. Are all Nash equilibria in (ii) admissible?

Exercise 3. Entry into new markets

Hunter and Melissa are two competing coffee growers who are exploring new markets to sell their product. There are two potential markets - Large and Small - referred to as L and S hereafter. Entry costs are large so each of them can enter only one of the two markets.

• If one of them enters L while the other one enters S, then one who enters L gets 36, while the other one gets 9.

• If both enter L, each gets 36α.

• If both enter S, each gets 9α.

where 0 ≤ α ≤ 2/1.

(i) The game described above has at least one Nash equilibrium in pure strategies irre-spective of the value of α - Do you agree?

(ii) The game described above has at least one Nash equilibrium in strictly mixed strategies irrespective of the value of α - Do you agree? [Note: By strictly mixed strategies we refer to strategies of the form. (xL + (1 − x)S, yL + (1 − y)S) where 0 < x < 1 and 0 < y < 1.]

(iii) Suppose you do not know any of the actual values except the following. If both enter L, each gets X. If both enter S, each gets Y. If one enters L and the other enters S, one who enters L gets W and the other one gets Z. You only know that X ≠ Y ≠ W ≠ Z. Can you answer (i) and (ii) with that information.

Exercise 4. Price competition

There are 100 consumers in the suburb Little Italy all of whom buy either one medium pizza or nothing. Each consumer has a reservation price of $15. That is, each consumer is willing to pay at most $15 for a medium pizza. There are two pizza stores - A and B - located next to each other. It costs $7 to make a medium pizza. The price of pizza is in whole dollars (e.g., $8, $7, but not $6.99 or $8.50)

Let pa and pb denote the price charged by stores A and B respectively. For simplicity, assume pi ∈ {1, 2, ..., 15} for both i = A, B. If pi < pj , all consumers buy pizza from store i as long as pi ≤ $15. If pi = pj = p, and p ≤ 15, then 50 consumers buy from store A while 50 buy from store B.

(i) Write down A’s and B’s payoff/profit in terms of pa and pb. Note, we are not asking for payoff matrix

(ii) Find out all Nash equilibria in pure strategies.

(iii) Are all equilibria in (ii) admissible?

(iv) Suppose store B plays a mixed strategy where it chooses 7, 8, and 9 with equal (prob-ability 1/3). What is the best response for player A?