* T is always distributed equally among n districts so <math>t_{i}=T/n</math>.
* Proposals are fully characterized by <math>b\in B</math> and net benefits are <math>z_{i}=b_{i}-T/n</math>.
* Payoffs are discounted: <math>\delta^{\tau}z_{i}=U_{i}(z,\tau)</math>. Extensive form is the same as before for closed rule.
Structure of game:
* P is drawn (which implies a ratio of B/T).
* A random legislator is chosen to distribute B. Note that per the above, all T are distributed equally no matter what.
* Legislators vote against the status quo, in which everyone gets nothing and is taxed nothing.
Stationarity implies members are paid their continuation value in equilibrium in exchange for their votes. <math>\delta v(g,t), \forall t\in\Tau</math>
Proposition 1: With closed rule the stationary EQM has the following properties:
* (i) Inefficient pork barrel programs will be adopted. Inefficiency is increasing in <math>n</math>
* (ii) Possible set of programs is increasing in <math>\delta</math>.
* Proposal will be accepted if <math>(n-1)/2</math> members vote yes, therefore proposals will be of the form of: Keep <math>B-\frac{n-1}{2}(\frac{T}{n+\delta\bar{V}}</math>. Give <math>T/n +\delta\bar{V}</math> to <math>(n-1)/2</math> others, and the rest zero.
* <math>\bar{V}=P(selected)E[Value of being selected|p^{\ast})+P(not selected)(value of not being selected)</math>.
* <math>\bar{V}=\frac{1}{n}(B-\frac{n-1}{2}(T/n+\delta\bar{V}))+\frac{n-1}{n}(\frac{1}{2}(T/n+\delta\bar{V}) +\frac{1}{2}(-T/n))</math>. Solve for <math>\bar{V}=\frac{B-T}{n}</math>.
* Offer is <math>T/n+\frac{\delta(B-T)}{n}=\frac{\delta B-(1-\delta)T}{n}</math>.
... unfinished. Sorry.
Open rule:
* Never get universalism w/ inefficient program.
* Inefficent program minimum winning coalition (MWC).
* Amendments shift power to voters with inefficiency.