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Game Theory: Theory Field Exam on Asset Trading and Bargaining Games, Exams of Economics

A theory field exam focusing on game theory, specifically on asset trading and bargaining games. It covers topics such as seller-buyer interactions, investment strategies, subgame perfect equilibrium, and bargaining solutions. The exam includes questions on proving efficiency of exchange, analyzing take-it-or-leave-it offers, and finding equilibrium strategies.

Typology: Exams

2011/2012

Uploaded on 12/04/2012

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Theory Field Exam
August 
Consider a risk-neutral seller and a risk-neutral buyer. The seller owns an asset
(an invention, small firm, etc.) in which she can invest. Let I[0,) denote
her investment. After investing, an opportunity arises in which the seller can
sell the asset to a buyer. To motivate trade, assume the buyer, if he acquires
the asset, can take a subsequent action, b[0,¯
b], that affects the asset’s return
(to him at least). Finally, the asset yields a return, r, to its then owner, where
the return depends on investments made in it. Note the realization of roccurs
after the point at which exchange can occur. The payoffs to the buyer and
seller—ignoring transfers—are, respectively,
UB=0,if no exchange
rb , if exchange and US=rI , if no exchange
I , if exchange ,
where “exchange” means ownership of the asset was passed to the buyer. Not
surprisingly, the buyer takes no action if there isn’t exchange.
Given investment Iand action b, the return rhas an expected value R(I, b).
Assume the expected return function, R:R+×[0,¯
b]R, has the following
properties:
For all b[0,¯
b], R(·, b) : R+Ris a twice continuously differentiable,
strictly increasing, and strictly concave function;
For any b[0,¯
b], there exists an ¯
I(b)<such that ∂R(I , b)/∂I < 1 if
I > ¯
I(b);
∂R(0,0)/∂ I > 1; and
For any IR+, argmaxb[0,¯
b]R(I, b)bexists and, if I > 0, it is a subset
of (0,¯
b].
Define V(I) = maxb∈B R(I, b)b. Finally, assume:
There exists an I>0 such that I= argmaxIV(I)I.
The expected return to the seller if she retains ownership is R(I , 0).
The timing of the game is first the seller decides her investment. At this time,
no contract exists between buyer and seller. Then there is possible exchange.
Then, if he took possession, the buyer decides on his investment. Finally ris
realized. Assume that the buyer never observes the seller’s investment nor any
signal of it prior to exchange. Assume the seller never observes the buyer’s
investment (if any). Assume only the owner of the asset at the time ris realized
observes r.
(a) Prove that, if I > 0, then efficiency dictates that exchange take place (i.e.,
the buyer end up with the asset).
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Theory Field Exam

August 

Consider a risk-neutral seller and a risk-neutral buyer. The seller owns an asset (an invention, small firm, etc.) in which she can invest. Let I ∈ [0, ∞) denote her investment. After investing, an opportunity arises in which the seller can sell the asset to a buyer. To motivate trade, assume the buyer, if he acquires the asset, can take a subsequent action, b ∈ [0, ¯b], that affects the asset’s return (to him at least). Finally, the asset yields a return, r, to its then owner, where the return depends on investments made in it. Note the realization of r occurs after the point at which exchange can occur. The payoffs to the buyer and seller—ignoring transfers—are, respectively,

UB =

0 , if no exchange r − b , if exchange and US =

r − I , if no exchange −I , if exchange

where “exchange” means ownership of the asset was passed to the buyer. Not surprisingly, the buyer takes no action if there isn’t exchange. Given investment I and action b, the return r has an expected value R(I, b).

Assume the expected return function, R : R+ ×[0, ¯b] → R, has the following

properties:

  • For all b ∈ [0, ¯b], R(·, b) : R+ → R is a twice continuously differentiable, strictly increasing, and strictly concave function;
  • For any b ∈ [0, ¯b], there exists an I¯(b) < ∞ such that ∂R(I, b)/∂I < 1 if I > I¯(b);
  • ∂R(0, 0)/∂I > 1; and
  • For any I ∈ R+, argmaxb∈[0,¯b] R(I, b) − b exists and, if I > 0, it is a subset of (0, ¯b]. Define V (I) = maxb∈B R(I, b) − b. Finally, assume:
  • There exists an I∗^ > 0 such that I∗^ = argmaxI V (I) − I. The expected return to the seller if she retains ownership is R(I, 0). The timing of the game is first the seller decides her investment. At this time, no contract exists between buyer and seller. Then there is possible exchange. Then, if he took possession, the buyer decides on his investment. Finally r is realized. Assume that the buyer never observes the seller’s investment nor any signal of it prior to exchange. Assume the seller never observes the buyer’s investment (if any). Assume only the owner of the asset at the time r is realized observes r.

(a) Prove that, if I > 0, then efficiency dictates that exchange take place (i.e., the buyer end up with the asset).

(b) Prove the following: If the buyer has the ability to make a take-it-or-leave- it offer to the seller, then no pure-strategy equilibrium exists. The same is true if the seller can make a take-it-or-leave-it offer unless welfare given exchange and no investment exceeds maximum possible welfare given no exchange (i.e., unless V (0) ≥ maxI R(I, 0) − I).

Consider the game in which, at time of exchange, the seller makes the buyer a take-it-or-leave-it offer. Assume R(I, 0) − I is strictly concave in I.

(c) Find an equilibrium in which the seller chooses a particular Iˆ as a pure strategy. What condition must Iˆ satisfy? Prove I < Iˆ ∗.

Now consider the game in which the seller mixes over investments according to the differentiable strategy F (·). Take F (·) as given.

(d) What mechanism would the buyer offer the seller if the buyer could make the seller a take-it-or-leave-it offer? Hint: A mechanism maps the seller’s announcement into R^2.

In calculating the full equilibrium of the game in which the buyer has the ability to make the take-it-or-leave-it offer, one must solve for the equilibrium strategy F (·) (i.e., F (·) is endogenous).

(e) Limiting attention to everywhere differentiable strategies, what is the seller’s equilibrium strategy for this game?