Economic and Game Theory
|"Inside every small problem is a large problem struggling to get out."|
Thread and Full Text View
I am now facing a challenging Buyer-Seller-Game which i cannot solve.
Do you know the official Title of that Game, or the Authors?
Especially (1b) finding 3 NE seems to be not possible for me and i cannot figure out the USPNE.
Also 2c) and 3 seem to be very hard.
I hope you are a little interested now. If not - I am very sorry for disturbing you.
Consider the following game. There are two players, the seller and the buyer.
In stage 1 of the game the seller has to make an investment i which can either be zero or eight. In case of i = 0 the buyer’s value of the traded product amounts to v(0) = 10. If i = 8, the buyer’s value is v(8) = 22. Finally note that the seller’s investment decision is observed by the buyer. In stage 2 the buyer proposes a price 0 ≤ p ≤ 22 for the product. Finally, in stage 3, the seller may either accept or reject the buyer’s proposal. If the seller rejects, his profit amounts to s = −i and the buyers profit is b = 0. Yet if the seller accepts the buyer’s proposal trade takes place and players’ profits are given by s = p − i and b = v(i) − p.
a) Describe both players strategy space.
b) Find three Nash equilibria of this game and show that they are indeed Nash equilibria. [Hint: Some of these equilibria will have incredible threats. If you still have problems in finding more than one equilibrium, analyze simplified versions of some subgames with only two or three feasible prices.]
c) Find the unique subgame perfect Nash equilibrium of the game. Explain whether this equilibrium is Pareto-optimal.
Part 2 Modify the above game by assuming that both players had conducted an option contract in advance. This contract fixes the price at p = 15. As before, in stage 1 the seller has to decide how much money to invest (zero or eight). The impact of this investment is the same as in Part 1. In stage 2, however, the buyer cannot propose a price anymore. Yet he can choose whether the trade takes place at all. If he decides not to exercise his call option, i.e., he does not want trade to take place, then the players’ payoffs are given by s = −i and b = 0.
If, in contrast, he exercises his call option, then payoffs are given by s- = 15 − i and b = v(i) − 15.
a) Determine the game tree of the extensive form of this game.
b) Determine the strategic form of this game.
c) Determine all pure Nash equilibria of this game. Which of these is subgame perfect and why?
Both games have been tested in economic experiments.
Briefly discuss the explanatory power of Nash equilibria and subgame perfect Nash equilibrium in these games.