Game Theory Assignment 2

Question 1: Ordered voting
Three people are voting on a policy, which can be a, b, or c.  The three players’ preferences for different policies succeeding are as follows:
v_1 (a)=3; v_1 (b)=2; v_1 (c)=1
v_2 (b)=3; v_2 (c)=2; v_2 (a)=1
v_3 (c)=3; v_3 (a)=2; v_3 (b)=1
The three vote for one of these policies in order of player 1, player 2, and then player 3, and their votes are observed by all parties.  The winner has the most votes, and tiebreakers are determined by the following: a wins all ties, and b wins a tie over c.
        Who wins the vote, in any SPNE?  Who voted for the winner?  Explain, informally, why this happened.
        Find a Nash Equilibrium where policy a wins the vote.
        If Player 1 could choose to change the voting order from 1,2,3 to 3,2,1, would he do so?  Explain in terms of the resultant SPNE.

Question 2: Entry deterrence
Consider potential duopolists with cost functions c(q_i )=10q_i and a mutual demand of p=100-q, where q represents total quantity supplied, a la Cournot.  Now suppose player 1 decides to enter (E) or not (N), and then, upon observing player 1’s action, player 2 decides to enter or not.  Not entering grants 0 payoff, while entering itself incurs a fixed cost of F, but the player can produce and sell in the market.  If no one enters, the game ends with payoffs of 0.  If one player enters, he becomes a monopolist, obtaining monopoly profits minus F.  If both players enter, they play a classic normal form Cournot game, enjoying duopolist profits minus F.
        Suppose F=1000.  Describe the outcome of the unique SPNE.
For parts (b) through (d), assume F=800.
        Construct a Nash Equilibrium which is not subgame perfect where Player 2 becomes a monopolist.
        Suppose Player 1 can invest in a technology to reduce his own marginal cost from 10 to 1 at cost K.  What is the maximum cost of this investment K^* where Player 1 would choose to invest?
        Now suppose that Player 1 has the choice of the technological investment from part (c) after both players have made their entry decisions but before any production, and K=K^*.  Describe the actions and outcomes of the unique SPNE.

Question 3: Debt and inconsistency
A lender can offer a loan of value c with a particular payment schedule to a debtor in period 1.  The debtor can then accept or reject the loan; if he rejects, the game ends, and payoffs are 0 apiece.  If the debtor accepts, he enjoys the loan in period 1 and repays it in periods 2 and 3.  The lender offers either a structured debt contract (S) or a flexible debt contract (F).  The structured contract makes the debtor pay 5 in period 2 and 6 in period 3.  The flexible contract gives the debtor a choice in period 2: pay back 10 in period 2 and 0 in period 3, or pay 0 in period 2 and 20 in period 3.  For both the lender and the debtor, the standard exponential time discount factor δ=1.  But the debtor also has a hyperbolic discount factor β≤1.  The courts favor the lender, so the debtor must repay the loan according to the contract (i.e., the debtor only has a choice to accept or reject the contract, and then when to pay in a flexible loan).
        Suppose β=1.  Solve for the unique SPNE.
For parts (b) through (d) suppose β=2/5.
        Suppose the debtor is sophisticated, i.e. he understands β and performs backward induction on his own behavior.  Find the unique SPNE, making sure to specify complete plans of play.
        Suppose the debtor is naïve and assumes his actions are time-consistent, not performing backward induction.  What contract does the lender offer?  What happens?
        Suppose the debtor performs backward induction but has mis-estimated β, such that he believes β=β ̂=3/5 in period 1, while the lender understands that the true β=2/5.  What contract does the lender offer?  What happens?

Question 4: Sufficient punishment
Consider a two-stage game, where stage 1 is a prisoner’s dilemma and stage 2 is a dangerous coordination game, as described below:
Prisoner’s Dilemma
        Player 2


m       


f

M       
0,0       
-6,2

Player 1
F       
2,-6       
-4,-4

Dangerous
Coordination
        Player 2


D       


s

D       
1,1       
-12,-12

Player 1
S       
-12,-12       
0,0

The players have a common discount factor of δ.  There is no public signal that may allow for coordination of equilibria (don’t worry about it; that’s a quibble from chapter 10).
        Find the lowest δ where a play of (M,m) in the first round can be supported by a SPNE.  Explain your answer.
        Suppose there are three rounds; the first round is the prisoner’s dilemma, and then there are two rounds of dangerous coordination.  Find the lowest δ where playing (M,m) can possibly be supported via SPNE.