Half of the producers in an industry produce with high quality and the rest with low quality. The producers can sell their product by themselves or else use a distributor.
A distributor who is not used gets payoff 0. A producer who chooses not to use a distributor gets a payoff of 3 if it has high product quality and a payoff of 2 otherwise.
If a producer chooses to use a distributor, the distributor chooses one of three possible actions: working, shirking, or else cheating the producer (not paying the share of revenueagreed upon).
With a high quality product, a distributorís payoff is 5 from working, 2 from shirking, and 6 from cheating.
With a low quality product, a distributorís payoff is 1 from working, 2 from shirking, and 0 from cheating.
A producer using a distributor gets payoff 0 if the distributor cheats and gets 3 otherwise.
A distributor cannot tell the quality of the product directly, though it might infer information from the producer ís behavior.
(a) Draw a possible game tree representing the interaction between a typical producer selected at random and a distributor in the industry above. In this game, the producerlearns the quality of its product and decides if it will use the distributor. If it does, then the distributor decides whether to work, shirk or cheat. The playersíactions and payoff are as described above, and all of this information is common knowledge.The remaining parts of the problem refer to the game in part a.
(b) How many pure strategies does the producer have? Give an example of one.
(c) Find every Nash equilibrium in which the producer and distributor choose pure strategies.
(d) Show that there is a sequential equilibrium in which the producer sells by itself no matter what happens. Compare the efficiency of the outcome in this equilibrium to that of the other Nash equilibrium outcomes in part c.
(e) Is there a sequential equilibrium of the game of part a in which the outcome is worse for the distributor than if it were impossible for the distributor to cheat?Explain.