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===In A: <math>\frac{1-\hat{y}^{a}}{m}\geq \delta V_{i}(y^{i})</math>. This means that the Paper===amendment <math>\max V_{i}^{m\ast}(y_{i}). </math>
The role ==== Substantive conclusions ====* Possibility of delay. * Size of winning coalition can be larger than the majority rule minimum winning coalition. * <math>\frac{\partial m}{\partial \delta}<0</math>, and (separately) <math>\frac{\partial m}{\partial n}<0</math> (rather than say unaminityRui says this latter one needs to be checked) is covered in the paper, as is the case of the stationary equilibrium for an open-rule. * More equal because proposer spreads wealth more broadly.
Baron Ferejohn (1989) - Bargaining In Legislatures (view source)
Revision as of 19:14, 29 September 2020
, 19:14, 29 September 2020no edit summary
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|Has article title=Bargaining In Legislatures
|Has author=Baron Ferejohn
|Has year=1989
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*This page is referenced in [[BPP Field Exam Papers]]
===Closed Rule - Infinite Sessions===
====No equilibrium restriction ====
From proposition 2 in the paper, if:
<math>1 > \delta > \frac{(n+2)}{2(n-1)} \mbox{ and } n \ge 5\,</math>
then: '''Any distribution of benefits (<math>x\,</math>) may be supported'''.
However, this is unsatisfactory as it requires a complete history at all points (which is unrealistic if <math>\delta\,</math> is a reelection probability and new members can't know the history), and if a member were indifferent between enforcing and not, it is only weakly credible.
====With equilibrium restriction: Exactly the same as Closed Rule +Finite Horizon ====
To restrict the equilibrium space the paper considers '''Stationary Equilibrium'''.
*The first vote recieves a majority, so the legislature completes in one session
<b>This is the same as closed-rule, finite session.</b>
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.
===Open 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 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 remainder to <math>m(\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 in the coalition of the original proposer, 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 value of the game 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
