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12/04/2006 11:39 AM by name withheld; game theory help

Hello Everyone,
I'm really struggling with this material. Any help would be greatly appreciated. Without further ado:

Consider the following game. Two doners, A and B, are contributing to a fund drive to finance a binary public good, organized by a collector C. The value that A and B attach to the public good, when it is provided, are Va and Vb respectively, while the collector recives no utility from the public good. Values are private information (i.e. A knows Va, but not Vb). Assume that values are independent draws from a uniform distribution from [0,1].
The timing of the game is as follows. First, A and B simultaneously and independently decide on the amount of donations (Da and Db) to give to C. C follows this rule: if the total amount of donations Da + Db is smaller than the cost of production k, the public good is not produced, and donations are returned. In this case, the utility A, B, and C is zero. On the other hand, if Da + Db is larger or equal to k, the collector produces the public good, and keeps the difference, i.e. the amount (Da+Db)-k for himself. In this case, the utility of C is (Da+Db)-k, the utility of B is Vb-Db, and the utility of A is Va-Da.

Assume that K<1

1) Exhibit at least 4 different symmetric step-function equilibria. (One 1-step function equilibrium looks like this: Da(v)=Db(v)=0, if v<v*, and Da(v)=Db(v)=k/2 if v is larger or equal to v*. The value of v* is determined so that there are no profitable deviations.)
2. Are there any asymmetric step-function equilibria?
3. Among the equilibriayou found in part 1), which one maximizes the expected revenues of C?
4. Which step-function equilibrium minimizes the revenue of C?
5. Assume now that k=1/2. Is Da(v)=Db(v)=hv+j a symmetric equilibrium, and for what values of h and j?
6. Repeat question 5 for k=1/4. If the linear strategy above does not work, modify it accordingly.

Consider now the case where K is either Kh=1/2+q, or Kl=1/2-q. Both values of K occur with equal probability, and neither donor nor the collector knows the true value. As q remains strictly positive, but becomes smaller and smaller and closer to zero...

7. Are there any step-function equilibria?
8. Are there any linear equilibria, similar to the one in part 5?

Consider now a "greedier" collector: C still keeps the difference between Da and Db and K, when it is positive, but now C keeps the money when Da and Db are smaller than k.

9. Are there any step-function equilibria, if k<1?
10. Can you find a pair of equilibria such that the strategy of the collector backfires: C's expected revenue is smaller in one of the equilibria you found in 1) than in one of teh equilibria you found in 9).

Again, thank you for looking at this. Any help you can give me at this point would be extremely appreciated. [Manage messages]