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BPP Field Exam 2006 Answers (view source)
Revision as of 21:33, 20 February 2011
, 21:33, 20 February 2011no edit summary
*The first vote receives a majority, so the legislature completes in one session.
So in the setup described above, we anticipate that should punishment strategies be fully credible, any allocation can be implemented by the partnership in equilibrium. However, with the refinement to stationary equilibrium, we collapse back to the equilibrium prediction from part (a) above, except that now the allocation decision is repeated each year and each . So every time an allocation decision is to be made, a (potentially) different partner will enjoy the agenda power and associated rents that comes from being randomly selected to propose an allocation to the partnership first.
'''c.) Returning to the one-shot/no-reputation case, consider what would happen if partnership shares are not distributed evenly, and members have probabilities of being recognized which are proportional to their shares. What would be the equilibrium strategies and outcomes you would expect in this case?'''
Here we would predict that partners with the lowest probability of recognition (based on their shares) may have the highest ex ante value of the game, as they are the least costly members of any voting coalitions. In equilibrium, the randomly selected partner will always form a coalition of (n-1)/2 other partners with the lowest continuation values, where the continuation value is now given by <math>\delta p_i\,</math>, where <math>p_i\,</math> is the probability of partner i of being recognized in a given period. We conjecture that whenever shares are distributed across partners with a small variance (as discussed in the example from the paper below), this result will generally hold. But where there is a less equal distribution of shares (e.g. one partner has 50% of the shares, and the remaining partners evenly split the 50% remainder evenly across them), then this result may be less robust. Instead, the high probability of selecting the first partner combined with the large number of potential coalitions than can be built by him may lead to the opposite result.
As noted on the top of page 1189 of the article: "in a two-session legislature, if the members have different probabilities <math>p_i\,</math> of being recognized, each has a continuation value <math>v_i(1, g) = p_i\,</math> for any second session subgame. Then, if any member k is recognized in the first stage, he or she can offer <math>\delta p_i\,</math> to the ith member and that member will vote for the proposal. Member k will thus choose the <math>(n-1)/2\,</math> members with the lowest <math>p_i\,</math>. Note that depending on the probabilities the member with the lowest probability of recognition may have the highest ex ante value of the game, and the member with the highest probability of recognition may have the lowest ex ante value of the game. For example, if <math>n=3, p_1=\frac{1}{3}+\epsilon, p_2=\frac{1}{3}, p_3=\frac{1}{3}-\epsilon \,</math>, the ex-ante values <math>v_i\,</math> of the game have limits <math>v_1=\frac{2}{9},</math> <math>v_2=\frac{1}{3},</math> <math>v_3=\frac{4}{9},</math> as <math>\epsilon \rarr 0\,</math>. The member with the lowest probability of recognition thus can do better than the other members because he or she is a less costly member of any majority."
'''d.) Finally, consider what would happen if both voting rights and recognition probabilities were proportional to the shares held, what would you expect in this case?'''
This issue is not addressed directly in the paper, but we can note that the results from part (c) above held in part because each partner's vote was equally valuable in achieving a majority. Now that voting rights are no longer uniformly distributed, it is not necessary to collect a voting coalition of (n-1)/2 other voters, only to ensure that the total share of yes votes exceeds 50%. The cheapest strategy of achieving this outcome would be for the proposer to build a coalition starting with the least likely to be recognized (and thus cheapest) voter and continuing in this fashion until the total vote share exceeded 50%. It is then possible that the results in part (c) may hold, in the sense that low probability voters would achieve over-sized gains from the game, but this effect would likely be counter-balanced by the fact that high probability voters are more likely to be recognized first and are a more necessary component of any winning coalition.
