Changes

A partnership group has a surplus it needs to allocate to the partners at the end of the year. The procedure it uses (as enshrined in its Operating Agreement) is as follows. Decisions are made by a committee-of-the whole (i.e. the entire partnership). At the annual meeting for the partnership, one of the partners is chosen randomly (with each having an equal likelihood of being selected) to propose an allocation to each of the members, including herself. If her proposal is accepted by a majority of the partnership, then that proposal is implemented. If it is not passed by a majority, then another partner is chosen randomly to make a proposal and the procedure repeats. All of the partners prefer higher allocations to lower allocations, and faster decisions to slower decisions.'''

'''a.) Outline a model of the above situation. Assuming the game is a single-shot and there are no punishments by the partnership (no reputation effects), discuss the equilibrium for your model. Note: you do not have to solve the model; simply discuss the proposition(s) you expect that could be derived and the intuition(s) behind it (them).'''

Assuming the game is single-shot (a finite number of sessions will be held to consider distribution proposals for the surplus), we make the additional assumption that if no allocation is agreed in the final period, then the surplus is not distributed and all partners end up with 0. If so, we have a simple application of Baron & Ferejohn (1989) - [http://www.edegan.com/wiki/index.php/Baron_Ferejohn_%281989%29_-_Bargaining_In_Legislatures#Closed_Rule_-_Two_Sessions Close Rule, Finite Session (e.g. n=2)]. Backwards induction thus yields a SPNE in which each member, if recognized, makes the same majoritarian proposal to distribute the benefits to a minimal winning majority (characterized in Proposition 1). For simplicity, normalize the total surplus to 1, and consider n=2 sessions. In equilibrium, the first partner (randomly selected in the first session), proposes an allocation of <math>\frac{\delta}{n}\,</math> to any <math>(n-1)/2\,</math> other selected partners (this is their continuation value for being selected with probability <math>\frac{1}{n}\,</math> and claiming the entire surplus of 1, discounted by <math>\delta\,</math>, in the next and final period) and proposes to keep the remaining <math> 1 - \frac{\delta(n-1)}{2n}\,</math> for himself. The proposal is approved by a majority (the proposer plus his <math>(n-1)/2\,</math> allies receiving positive shares), and the game ends in the first period. Note that the proposer receives the largest share (ranging between <math>(1-\frac{\delta}{3})</math> and <math>(1-\frac{\delta}{2})\,</math>, so at least one half of the total surplus) due to the agenda power from being recognized first, as well as the institutional setup of the closed rule, which excludes amendments from immediate consideration by the voting body.

'''b.) Suppose that the partnership (membership) is stable, infinitely-lived, and makes a surplus allocation decision every year. How would you account for this in your model? Discuss equilibrium behavior and strategies using these assumptions (again you do not need to explicitly solve the model, simply explain your reasoning).'''

'''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?'''

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?'''