# BPP Field Exam 2006 Answers

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**Answer A.1: The Theory of Partnerships**

**A.1: The Theory of Partnerships**

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) - Close Rule, Finite Session (e.g. 2 sessions). 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 2 total 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).**

We now have an application of Baron & Ferejohn (1989) - Closed Rule, Infinite Session

From Proposition 2 in the paper, if: [math]1 \gt \delta \gt \frac{(n+2)}{2(n-1)} \mbox{ and } n \ge 5\,[/math] then: Any distribution of benefits ([math]x\,[/math]) may be supported.

This is accomplished through use of punishment strategies for any voter who attempts to deviate from the allocation ([math]x\,[/math]), as discussed at length on pp1189-1191. Such punishment strategies suffer from being at times only weakly credible, in that punishers may be indifferent to carrying out their threats, leading voters to anticipate that enforcement may occur with less than probability 1, and thus unraveling the equilibrium. Baron and Ferejohn propose a refinement called Stationary Equilibrium, where members take the same actions in structurally equivalent subgames. Note that two sub-games are structurally equivalent iff: (i) the agenda is identical, (ii) set members who may be recognized (at the next node) are identical, (iii) the strategy sets of the members are identical.

In the case of equal probabilities, majority rule and infinite session, Proposition 3 in the paper states that for all [math]\delta \in [0,1]\,[/math] a stationary SPNE in pure strategies exists iff:

- A recognized member proposes to give [math]\frac{\delta}{n}\,[/math] to [math]\frac{(n-1)}{2}\,[/math] randomly chosen other members, and to keep [math]1-\frac{\delta (n-1)}{2n}\,[/math] for himself.
- Each member votes for any proposal that gives him at least [math]\frac{\delta}{n}\,[/math].
- 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. 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%. Also, there is no longer a "cheap" strategy for buying votes, as the marginal cost of each vote is roughly equal (up to a factor of [math]\delta[/math]) to its marginal benefit, in the sense that each partner invited to join the marginal winning coalition must be compensated only [math]\delta p_i\,[/math], where [math]p_i[/math] is their share of the total votes (and also thus their probability of being called upon in the next period to make a proposal). In eqm, we thus expect that the first partner called up on will choose a coalition of any k other partners such that the sum of the k partners' shares plus the first partner's shares will just exceed 50%, and each of the k partners will be compensated [math]\delta p_k\,[/math], with the original proposer keeping the remainder of the total surplus.