Changes

Note that we have minimum winning coalitions, a value of the game <math>v_{i}=1/n</math> proposal power equal to <math>x_{i}^{i}\geq 1/2 \geq \delta/n</math>. We have fairness because the value of the game is equal across all players.

Comparitive statics around proposal power: <math>x_{i}^{i}=1-\frac{\delta(n-1)}{2n}</math>. Note that patience reduces proposal power: <math>\frac{\partial x_{i}^{i}}{\partial\delta}=-\frac{n-1}{2n}<0</math>. Larger legislatures reduce proposal power:<math>\frac{\partial x_{i}^{i}}{\partial n}=-\frac{\delta}{2}+\frac{\delta(n-1)}{2n^{2}}=\frac{-\delta(n^{2}-n-1}{2n^{2}}<0</math>.

Difference in equality between proposer and coalition members=<math>\Delta=1-\frac{\delta(n-1)}{2n}-\frac{\delta}{n}</math>. <math>\frac{\partial \Delta}{\partial n}=\frac{\delta}{n}>0</math>. Coalition members also worse off as legislature size gets bigger.

===In the PaperOpen Rule: Infinite session===<i>Editor: What about open rule finite session? Doesn't seem to be in the paper anywhere</i>.

Open rule: Member is recognized and makes a proposal. Another member is recognized and can either move the previous question to a vote agains the status quo (which is zero for everyone), or offer an alterative proposal to be voted against the previous proposal. If first proposal ends: Game over. If amendment wins -- another recognition round to offer proposal.  The role equilibrium strategy is as follows: * If recognized, keep <math>\hat{y}^{a}</math> for yourself and distribute the remainder to <math>m(\delta,n) </math> other members, where <math>1-n\geq m \geq (n-1)</math>. <math>1>m/2\geq 1/2</math>. * If recognized as an amender who is part of the aforementioned group of <math>m</math>: Move to a vote. * If recognized as an amender who is NOT part of the aforementioned group of <math>m</math>: Make a proposal to keep <math>\hat{y}^{a}</math> for yourself and distribute the majority rule remainder to <math>m(rather than say unaminity\delta,n) </math> other members -- including all those who are not in the first proposer's majority. Recognizer is never included because he is too expensive to pay off. * If you're a voter: Same rules as above. Vote for whichever pays you higher, and for the amendement if you're indifferent.  In equilibrium, the game procedes sequentially until nature randomly selects an amender who is covered in the papercoalition of the original proposer, as when the proposal is approved. ====Proof ==== Let <math>y^{a}</math> be what proposer (j) keeps for self m members have continuation values offered to <math>V_{j}^{m}(y^{j})</math> be the case value of the stationary equilibrium for an opengame to <math>j</math> when <math>y^{j}</math> is on the floor.  Note: Stationarity implies that <math>y^{j\ast}</math> will be the same in all recognized rounds. Note: N members, member 1 is recognized, member j is excluded member recognized, member  A: <math>\frac{1-rule\hat{y}^{a}}{m}\geq \delta V_{i}(y^{i})</math>. This means that the amendment <math>\max V_{i}^{m\ast}(y_{i}). </math> ==== Substantive conclusions ====* Possibility of delay. * Size of winning coalition can be larger than the minimum winning coalition. * <math>\frac{\partial m}{\partial \delta}<0</math>, and (separately) <math>\frac{\partial m}{\partial n}<0</math> (Rui says this latter one needs to be checked). * More equal because proposer spreads wealth more broadly.