Download Theory Field Exam

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, b̄], 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,
0 , if no exchange
r − I , if no exchange
UB =
and US =
,
r − b , if 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, b̄] → R, has the following
properties:
• For all b ∈ [0, b̄], R(·, b) : R+ → R is a twice continuously differentiable,
strictly increasing, and strictly concave function;
¯ < ∞ such that ∂R(I, b)/∂I < 1 if
• For any b ∈ [0, b̄], there exists an I(b)
¯
I > I(b);
• ∂R(0, 0)/∂I > 1; and
• For any I ∈ R+ , argmaxb∈[0,b̄] R(I, b) − b exists and, if I > 0, it is a subset
of (0, b̄].
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).
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(b) Prove the following: If the buyer has the ability to make a take-it-or-leaveit 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 Iˆ < I ∗ .
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 R2 .
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?
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Theory Field Examination
Game Theory (209A)
Aug 2010
Good luck!!!
Question 1
Consider the variant of the bargaining game of alternating offers in which each player  incurs the cost
  0 for every period in which agreement is not reached. That is, player ́’s payoff if the agreement
 is concluded in period  is  −  .
1. By using the one-deviation property of subgame perfect equilibrium, show that if 1  2 then
the game has a unique subgame perfect equilibrium in which
— player 1 always proposes ∗ = (1 0) and accepts a proposal  if and only if 1 ≥ 1∗
— player 2 always proposes  ∗ = (1 − 1  1 ) and accepts a proposal  if and only if 2 ≥ ∗1 .
2. Show that if 1 = 2 =   1 then the game has many subgame perfect equilibrium outcome
including, if  = 13 , equilibria in which agreement is delayed.
Question 2
Let B be the set of all convex, compact and comprehensive sets in R2+ with nonempty intersection with
R2++ (a set  in R2+ is comprehensive if  ∈  and 1 ≤ 1 and 2 ≤ 2 then  ∈ ).
1. Show that Kalai bargaining solution
  ( ) = {1 = 2 :  ∈ } ∩   ()
does not satisfy   .
2. Show that Kalai-Smorodinsky bargaining solution
  ( ) = {
1
2
=
:  ∈ } ∩   ()
̄1
̄2
where ̄ is player ’s the maximum utility does not satisfy .
Question 3
1. Give an example of a normal-form game with a Nash equilibrium that is not trembling-hand
perfect equilibrium.
2. Give an example of an extensive-form game of perfect information with a subgame perfect equilibrium that is not trembling-hand perfect equilibrium.
3. Give an example of an extensive-form game of perfect information whose strategic form has
trembling-hand perfect equilibrium that is not a subgame perfect equilibrium
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