Graphs

The Coase Conjecture and Agreement Rules in Policy Bargaining
Updated version coming soonAn agendasetter proposes a spatial policy to voters and can revise the initial proposal if it gets rejected. Voters can communicate with each other and have distinct but correlated preferences, which the agendasetter is uncertain about. I investigate whether the ability to make a revised proposal is valuable to the agendasetter. When a single acceptance is required to pass a policy, the equilibrium outcome is unique and has a screening structure. Because the preferences of voters are singlepeaked, the Coase conjecture is violated and the ability to make a revised proposal is valuable. When two or more acceptances are required to pass a policy, there is an interval of the agendasetter's equilibrium expected payoffs. The endpoints have a screening structure, leading to the same conclusions as in the case of a single acceptance. Interestingly, an increase in the required quota $q$ may allow the agendasetter to extract more surplus from voters. An application to spending referenda suggests that the expected budget may increase in response to allowing the bureaucrat to make a revised proposal and/or an increase in the number of voters whose acceptance is required.A violation of the Coase conjecture: as players become perfectly patient (i.e., as the lower parabola converges to the upper one), the initial proposal (triangle) converges to the anticipated revised proposal (square) only when the latter is greater than the ideal policy (circle) of the target voter.
The anticipated revision (square) is a policy that sets the target voter in the low state to the statusquo level of utility. You can adjust it by dragging left and right. The ideal policy of the target voter in the high state (circle) is also adjustable. You can also adjust the patience parameter by dragging the lover parabola up and down.A decrease in equilibrium expected policy in required quota: when the limit sets of expected policies overlap, there exists an increasing policy selection . The passed policy may increase on average when the policymaximizing agendasetter must secure approval from a larger number of voters.
The shaded regions represent the limit sets of expected policies for different numbers of voters who must accept a policy proposal. The equilibrium policies for the cases when the state is known (the prior is either 0 or 1) are represented by circles (higher required quota) and squares (lower required quota) and can be adjusted by dragging up and down. The depicted case is when voters cannot communicate with each other. 
Equality in Legislative Bargaining
R&R at Journal of Economic TheoryI study a distributive model of legislative bargaining in which the surpluses generated by coalitions equal the sums of productivities of coalition members. The heterogeneous ability of players to generate surplus leads to asymmetric bargaining prospects in otherwise symmetric environments. More productive players are recruited more often by other players despite having higher expected payoffs; however, the players who are recruited in every coalition have equal expected payoffs despite having different productivity. I show that an increase in the required quota raises equality as measured by the Gini coefficient.The radii of blue and red bubbles depict the productivity and equilibrium expected payoffs of players as functions of recognition probabilities and discount factors. The vector of expected payoffs is unique by Proposition 1. The recognition probability and discount factor of each of five (in this example) players can be adjusted by dragging the bubbles.
The Lorenz curves of the distributions of productivity and equilibrium expected payoffs of players for the recognition probabilities and discount factors set above.
You can adjust the productivity of each player and the required quota for the above example. The productivity ranges between 1 and 20 and the required quota ranges between 1 and 5.
The expected payoffs are computed by solving a system of two nonlinear equations in two unknowns (see Footnote 17). The values above are obtained using a straightforward implementation of NewtonRaphson algorithm with the Jacobian approximated by finite differences. As shown in the paper, the system is only piecewise differentiable and therefore the vanilla NewtonRaphson is not guaranteed to converge. Nonetheless, it does surprisingly well in this context even for large numbers of players.