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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
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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,
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).
properties:
(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?