Game Theory: Homework 4
Problem 1: You manage an airline that operates on the route from Moscow to Almaty.
You have one competitor. You compete in quantities (Cournot duopoly). Assume that
neither you nor your competitor has any fixed cost. Both your and your competitor’s
marginal cost (MC) can end up being either 2 or 6, depending on circumstances such as
personnel turnover and oil price that you cannot foresee in advance. You need to decide
how many planes to fly on the route, i.e., you need to decide on your quantity for the
next year before you know the analogous decision of your competitor. Before you decide,
you will be aware of your MC, but not of the MC of your competitor. You believe that
your competitor’s MC is going to be 2 with probability 1/2 and it is going to be 6 with
probability 1/2. Assume that your competitor is in a symmetric position. The overall
market demand is given by Q = 20 − P .
(a) Suppose that the realization of your competitor’s MC is independent of your own
realization (for example, due to independent personnel turnover issues). Suppose
your competitor produces Q1 if his MC is low and Q2 if his MC is high. What is
your best response as a function of Q1 and Q2 if your own MC is low and what is
it if your own MC is high?
(b) Assuming that your competitor thinks alike, what are the Bayesian Nash Equilibrium
values of output of each firm if its own MC is low and if its own MC is high? (Hint:
assume the equilibrium is symmetric and hence the optimal values of output given
a particular level of the MC is the same for both firms.)
(c) What is the resulting statistical distribution of quantity and price?
(d) Now suppose that the realization of your competitor’s MC is the same as your own
realization (for example, due to oil price shocks that are common to both firms).
Repeat the analysis from parts (a)-(c).
Problem 2: All the following questions consider single-unit auctions.
(a) What is the best strategy to bid in the second-price auction?
(b) What sealed-bid auction is equivalent to the English auction?
(c) What sealed-bid auction is equivalent to the Dutch auction?
(d) What is the difference between a private-values auction and a common-value auction?
Do both of these types of auctions have the problem of the winner’s curse?
(e) If bidders were hyperintelligent, would the winner’s curse be a problem?
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Problem 3: Imagine that 3 bidders want to buy a good in a sealed-bid auction. The
3 bidders come secretly together (they form a so-called “bidding ring”) and discuss their
positions. They find out each other’s values. Suppose the A’s value is 9, B’s value is 8 and
C’s value is 3. For simplicity, assume that only integer bids are allowed in the auction.
(a) First, suppose the auctioneer uses a second-price sealed-bid auction. What kind of
collusive arrangement would you imagine the bidders agree on? Would it be possible
or likely for one of the bidders to cheat on the collusive arrangement?
(b) Now suppose the auctioneer uses a first-price sealed-bid auction. What kind of collusive arrangement would you imagine the bidders agree on in this case? Would it
be possible or likely for one of the bidders to cheat on the collusive arrangement?
(c) Suppose you are an auctioneer and you are worried about bidder collusion. Which
of the two auction types would you prefer to use so as to minimize the likelihood of
collusion or in order to maximize the sale price?
Problem 4: Consider a private-value first-price sealed-bid auction with two bidders.
Each bidder’s value is drawn from the uniform distribution on [0, 1]. Unlike in the case
we studied in the class, assume that rather than being risk-neutral, bidders may also be
risk-averse. or risk-loving. In particular, the utility function of each bidder is given by
u(x) = xγ with γ > 0. Note that γ = 1 corresponds to risk neutrality, γ < 1 corresponds
to risk aversion and γ > 1 corresponds to risk-loving.
(a) Solve for the symmetric Bayesian Nash Equilibrium bidding functions. What proportion of their value do bidders bid? How does it depend on their risk aversion?
Explain.
(b) How do your answers change if there are n > 2 bidders? Provide an intuition for the
change.
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