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1. Two buyers are trying to buy a horse from a seller.  Both buyers are willing to pay at most v  > 0 for the horse.   Each buyer must simultaneously bid a real number bi  > 0. The horse is sold to the person making the higher bid, but both buyers must pay their bids (if the bids are tied, then each buyer gets the horse with probability one-half).

(a)  Formulate this situation as a strategic game.

Answer:  There are two players.  The set of strategies is the positive half of the real line for both players. The player who chooses the higher number wins. The payoff is 1 - bi  for the winner and -bi  fo the loser. If the bids tie,

the payoff is 1/2 - bi  for each player.                                                        口

(b) Show that this game does not have a pure strategy Nash equilibrium.

Answer:  Suppose b1  < b2 . This cannot be a NE since bidder 2 can reduce his bid and still win.  If b1  = b2 , then bidder 2 can increase his bid by ε and win with probability one, thereby raising his payoff by  - ε > 0 for ε small.

口

(c) Solve for a Nash equilibrium.

Answer:  Let bidder j bid according to the density function f (b) on [0, 1] and let F (b) be the associated cdf. If bidder i bids b, then his payoff is

U (b) = F (b) x v - b.

(We have generalized the problem somewhat by assuming that the value of the object is v rather than 1.) Differentiating this wrt b, we get

U (b) = f (b)v\ - 1.

Suppose that we want player i to be indifferent between all his bids in the interval [0, v]. Then at bid b in this interval, U\ (b) must equal zero, i.e.

U (b) = f (b)v\ - 1 = 0.                                   (1)

Thus, this yields

f (b) = 1

So bidder i will be indifferent between all his bids on  [0, v] if U\ (b)  = 0 on  [0, v]. So if bidder j\ s bid function is uniform on  [0, v], then bidder i

will be indifferent between all bids on [0, v]. Thus, there exists a symmetric equilibrium where each bidder bids according to the uniform distribution on [0, v], i.e. with the density function  on [0, v], with f (b) = 0. elsewhere.    Notice that the first order condition,  (1) is satisfied for all values of b in [0, v], not just at a single value of b. This implies that the player finds all these values of b optimal, and is indifferent between them. This is different from what you are used to, e.g. in the Cournot duopoly, where the first order condition yields a unique best response for any strategy of the opponent. 口

2.  General Spartacus is defending a territory which is accessible by two mountain passes against general  Crassus.   Spartacus  has 3 divisions at  his disposal and Crassus has two divisions.  Each general allocates his divisions between the two passes.  Spartacus wins the battle at a pass if and only if he assigns at least as many divisions to the pass as does Crassus.  Spartacus successfully defends his territory if and only if he wins the battle at both passes. Assume that each general only cares whether or not Spartacus defends his territory (i.e.  they do not care additionally about the number of battles that they win).

(a)  Formulate this situation as a strategic game.

Answer: The game modeling this scenario includes as players the two gen- erals.  The strategies of general Spartacus are {0, 1, 2, 3} where the index stands for the armies allocated to the first pass, and the strategies of general Crassus are {0, 1, 2} where the index stands for the armies allocated to the first pass. Assigning payoff 1 to winning the war and 0 to losing it, the payoff matrix is:

口

(b) What are the weakly dominated strategies for each player?

Answer:

i.  Note that for Spartacus, 0 is weakly dominated by 1 and 3 is weakly dominated by 2.

ii. This implies that if Crassus plays strategy  1 with positive probability, then strategies 0 and 3 cannot be best responses for player A.

iii.   For Crassus, there are no weakly dominated strategies.

iv.  Clearly, there is no pure strategy Nash equilibrium.

口

(c)  Find a mixed strategy Nash equilibrium. Do players assign positive probabili- ties to weakly dominated strategies in this equilibrium? Explain why they do or do not.

Answer: Let us denote a strategy profile by p =(p0 , p1 , p2 , p3 ) for Spartacus and q =(q0 , q1 , q2 ) for Crassus.

Let us first construct an equilibrium where Spartacus does not play his weakly dominated strategies, so that p0  = p3  = 0.For Spartacus to randomize be- tween 1 and 2, it must be that player 2 plays 0 and 2 with equal probability. For Crassus to randomize between 0 and 2, it must be that Spartacus plays 1 and 2 with equal probability. When Spartacus does not play 0 nor 3, Crassus never plays 1. In sum, a mixed strategy equilibrium is such that p1  = p2  = 1/2 and q0  = q2  = 1/2.

This is not the only equilibrium.  Let us now consider equilibria where Spar- tacus plays his weakly dominated strategies with positive probability.  From (2) above, this implies that Crassus cannot play 1 with positive probability. For Crassus to be indifferent between 0 and 2, we must have

p2 + p3  = p0 + p1 .

So for any values of x, y  e  [0, 0.5], the  profile p  =  (x, 0.5 - x, y, 0.5 - y) satisfies this equation.  Since Spartacus must be indifferent between his strategies, this implies q0  = q2  = 1/2, as before. So there is a continuum of equilibria (i.e. infinitely many), corresponding to these values of x and y .    Is there an equilibrium where q1   >  0?  NO! To show this,  let  us assume q1  > 0. If q1  > 0, then p0  = 0, since for Spartacus, 1 is strictly better than 0. Similarly, p3  = 0, since if q1  > 0. But (2) establishes that 0 and 3 cannot be played with positive probablity.  But in this case, if Spartacus only plays strategies 1 and 2, then strategy 1 is strictly inferior for player 2. So q1  = 0. In other words,  if we assume q1   > 0 we can deduce q1   = 0, which is a contradiction, and so the intial assumption must be false.  Hence q1  cannot be strictly positive in any equilibrium.

To summarize, in any equilibrium q0  = q2  = 0.5, and we have infinitely many values of pi  satisfying the conditions set out above.                                  口

3. A musician is considering buying a unique instrument from a seller. The musician can be of two types:  a professional or an amateur.  The seller cannot observe the type of the musician, but she knows that the musician is a professional with probability α .  If the musician is a professional, he is willing to pay at most v for the instrument, and if he is an amateur, he values the instrument at w < v . The seller values the instrument at zero.

Suppose each  professional  musician  has a  mustache  (think of it as a  publicly observable attribute) and all amateur musicians are clean-shaven.

In order to sell the instrument, the seller makes a take-it-or-leave-it offer quoting a price of her choice to the musician.

(a) Suppose the seller observes the musician’s attribute (mustache) or the ab-

sence of thereof before making an offer. Solve for an equilibrium of this game and provide economic intuition for your findings. Is equilibrium unique?       Answer:     This  is an extensive form game with  perfect  info, so we will look at SPNE. The equilibrium is unique. The seller sets the price v for the professional musician—the largest price at which he is willing to purchase the instrument—and w for the amateur.  The musician buys the instrument as long as the price is weakly below the value of the instrument.

In this case, since the seller has all the bargaining power, the buyer’s surplus is fully extracted.                                                                                      口

(b) Suppose the buyer must wear a face mask.   In this case the seller cannot

observe the musician’s attribute (mustache) or the absence of thereof before making an offer. Solve for an equilibrium of this game and provide economic intuition for your findings. Is equilibrium unique?

Answer: There is generically unique weak sequential equilibrium. There are two prices that can potentially be set in equilibrium:  v and w .  If the seller sets price w, the musician buys the instrument with probability 1, hence the seller’s payoff is w .  If the price is v, the musician will buy the instrument only if he is a professional. Therefore, the expected payoff of the seller is αv . If α < w/v, the equilibrium price is w . Otherwise the price is v . The musician buys as long as the price is weakly below the value of the instrument.          In this case, the seller has the full bargaining power, but the buyer has private information. In order to elicit this info, the buyer has to leave the buyer that has high willingness to pay with some surplus.                                          口

(c) Suppose the seller observes the musician’s attribute (mustache) or the ab- sence of thereof before making an offer. However, prior to visiting the seller, the musician can secretly remove his attribute (i.e., shave the mustache off) at a cost c  < v - w .   Solve for an equilibrium of this game and provide economic intuition for your findings.

Answer: There are two cases:

i. α  <  w/v .   In this case, since c  <  v - w, the  musician shaves the mustache off with probability 1. The rest of the equilibrium is the same as in question 3b.

ii. α  > w/v .   In this case, the equilibrium is in mixed strategies.   The musician shaves the mustache off with probability σ .  If the seller sees the buyer with the mustache, the price is set at v .  If the seller sees the musician without the mustache, the price is set at v with probability β

and at w with probability 1 - β .

The musician’s indifference condition:

c    

β(v - w) - c = 0  =÷  β =

The seller’s belief about the type of the clean-shaven buyer is

σα       Pr{professional I clean - shaven} =

The seller is indifferent between setting price w and price v if

w            σα      

v      σα + 1 - α ,

hence

(1 - α)w

α(v - w) .

Note that σ < 1 since α > w/v .

Ability to hide the attribute provides the buyer with a chance to generate some private information.   However, the value of this private information is endogenous.  In particular, if the buyers with high willingness to pay are numerous, they cannot hide among the buyers with low willingness to pay, so, in equilibrium, the value of the private information is fully offset by the cost of generating it.                                                                                 口

4. An eastern merchant is shipping her produce along the silk road to sell it in the west. Suppose the price of the produce is normalized to 1 and the cost of producing q units of good for the merchant is q2 /2. The merchant is maximizing the profit. The silk road passes through some states i = 1, ..., k (the merchant cannot sell her goods in those states). Each state i imposes a per-unit tax τi  on all merchandise that travels through the state. This tax is imposed without coordinating with the other states.  The objective of the state is to maximize tax revenues.  When the merchant plans her journey (i.e., when she decides how much merchandise to bring with her), she observes the taxes that are imposed along the route. Assume that the merchant has  “deep pockets”— i.e., she can pay a tax amount that exceeds her revenue from selling the good.

(a) Suppose k = 1. Solve for an equilibrium of this game.

Answer:  When choosing how much merchandise to bring along, the mer- chant solves

max(1 - τ )q - q2 /2

q>0

hence her equilibrium strategy is q(τ ) = (1 - τ ).  The state maximizes tax revenue:

max τ q(τ ).

T >0

Thus, if k = 1, the tax is τ = 1/2 and the tax revenue is 1/4.                  口

(b) Suppose k = 2. Solve for an equilibrium of this game. Compare your answer

to the one you obtained in question 4a and provide intuitive explanation for the difference between the answers.

Answer:  Let τ = τ1 + τ2  be an aggregate tax rate payed by the merchant. The merchant will bring q(τ ) = (1 - τ ) of the good with her.  The state 1 maximizes tax revenue:

max τ1 q(τ1 + τ2 ).

T1>0

Thus, state 1’s best reply is τ1 (τ2 ) = (1 - τ2 )/2.  There are two equilibria: a trivial one  in which τ  >  1 and the  merchant does  not travel,  and an interesting one:  τ1  = τ2  = 1/3.  In the latter, each state collects the tax revenue of 1/9. This is less than what they could achieve if the coordinated on their tax rates. That’s because they impose negative externality on each other: tax rate of state 1 reduces a taxable base for state 2 (via merchant’s best reply) and vice versa.                                                                         口

(c) Solve for an equilibrium for any k .  What happens to the equilibrium tax revenues and merchant’s trade if k becomes arbitrarily large?

Answer:  Similarly to the previous question, τi  = 1/(k + 1).  The total tax revenues are  and the merchant’s volume of trade is 1/(k + 1). All of these quantities become arbitrarily small as k becomes arbitrarily large. The externality imposed by the tax setters on each other completely destroys the tax revenues and merchant’s trade.                                                           口