Changes

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=2sessions)]. 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 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.

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 Also, there is no longer a "cheap" strategy of achieving this outcome would be for buying votes, as the proposer marginal cost of each vote is roughly equal (up to build a coalition starting with factor of <math>\delta</math>) to its marginal benefit, in the least likely sense that each partner invited to join the marginal winning coalition must be recognized compensated only <math>\delta p_i\,</math>, where <math>p_i</math> is their share of the total votes (and also thus cheapest) voter and continuing their probability of being called upon in this fashion until the total vote share exceeded 50%next period to make a proposal). It is then possible In eqm, we thus expect that the first partner called up on will choose a coalition of any k other partners such that the results in part (c) may hold, in sum of the sense that low probability voters would achieve over-sized gains from k partners' shares plus the gamefirst partner's shares will just exceed 50%, but this effect would likely and each of the k partners will be counter-balanced by compensated <math>\delta p_k\,</math>, with the fact that high probability voters are more likely to be recognized first and are a more necessary component original proposer keeping the remainder of any winning coalitionthe total surplus.