Game Theory questions

[jkw1]

I'm not sure if this is an appropriate venue for this sort of post, so if anyone knows of more appropriate forums for this, please let me know.

I'm working on a practice exam for an undergrad micro exam and this question popped up. I'm having a terrible time with 1 and 2 and 5. The sample exam is at the end of the post.

So far the symmetrical equilibria i've found are Da(v)=Db(v)=K/2, K. One of these is given! I reason that the contribution functions (as in, I give 0 if my valuation is less than what I plan to give, otherwise what I give what I announce) cannot be constantly 0 (if my v is greater than K, I have incentive to pay for all because I would be better off). It does not make sense that I contribute between 0 and K/2 because then my utility would always be 0 as the strategies are symmetric and my partner and I would never reach K. By the same reasoning, it is wasteful to give between K/2 and K because I could get away with spending less.

For the asymmetric equilibria, I think one person contributing 0 can be an equilibrium because the possible payoff would be all the value of k times the probability that my partner's valuation of the good is greater than its cost. The problem is that because both players go at the same time, and they are indistinguishable, this strategy becomes symmetric and both players adopt the 0, tragedy of the commons style. (maybe this is a symmetric equilibrium for question 1, though it is not a step-function).

The maximum revenue for C will be K, in the case that both players pay for all of the good. It will never be greater than K.

The minimum revenue of C will be 0, and this is embodied by any equilibrium in which the two donors do not sucessfully complete the project (are there any? perhaps 0,0) and when they perfectly complete the project (aka when they both spend K/2)

When k=1/2, Da(v)=Db(v)=hv+j.... The situation is always symmetric. I suspect this will be much easier if I find more examples of equilibria.

Same when k=1/4... I suspect the variables will be different

I think I can explain the 1/2k+q and 1/2k-q using trembling hand analysis.

For the greedy collector I run into a fundamental problem. Won't the equilibria change in all of these cases for small values of K? The chance of player evaluations greater than K is much greater if K is closer to 0 than to 1. The only restriction placed on this K is K<1. The players know what K is, though. Certainly that would figure into their calculations! This is analyzed in numbers 5 and 6 and thinking about it really makes me wonder about the other parts of the exam.

As for his strategy backfiring... if there are equilibria from K/2 to K in the first instance I suspect that if there are no greedy collector equilibria for K<1 then he loses all of those possible outcomes.

my biggest problem is the change of probabilities for different, unspecified values of K<1.

I would appreciate any ideas or pointers toward appropriate literature. MWG's section on public economics helps my intuition but offers no good examples. I would say my general knowledge about public economics is decent, but applications and case analysis with uniform distribution is terribly weak.

thanks a lot! here is the exam:


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).