View source for Baron Ferejohn (1989) - Bargaining In Legislatures

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*This page is referenced in [[BPP Field Exam Papers]]

[http://www.edegan.com/repository/Lucy%20Hu%20Writeup%20Baron%20and%20Ferejohn.pdf Handwritten Notes from Class]. Also: [http://www.edegan.com/wiki/images/1/1c/Legislative_bargaining.pdf Summary] downloaded from [http://www.stanford.edu/~niederle Muriel Niederle's] homepage [http://www.stanford.edu/~niederle/legislative_bargaining.pdf here]. 

==Reference(s)==
Baron, D. and J. Ferejohn (1989), Bargaining in Legislatures, American Political Science Review 83 (December), 1181. [http://www.edegan.com/pdfs/Baron%20Ferejohn%20(1989)%20-%20Bargaining%20in%20Legislatures.pdf pdf]


==Abstract==
Bargaining in legislatures is conducted according to formal rules specifying who may make proposals and how they will be decided. Legislative oucomes depend on those rules and on the structure of the legislature. Althrought the social choice literature provides theories about voting equilibria, it does not endogenize the formation of the agenda on which the voting is based and rarely takes into account the institutional structure found in legislatures. In our theory members of the legislature act noncooperatively in choosing strategies to serve their own districts, explicitly taking into account the strategies that members adopt in response to the sequential nature of proposal making and voting. The model permits the characterization of a legislative equilibrium reflecting the structure of the legislature and also allows consideration of the choice of elements of that structure in a context in which the standard, institution-free model of social choice yields no equilibrium.

==Key Concepts==

Amendment rules:
*Open: An amendment may be added and voted on (essentially instead of the motion), or the motion can be moved (to an immediate vote). 
*Closed: Motion voted on immediately against the status quo. There are no amendments.


==The Model==

The legislature consists of:
*<math>n\,</math> members - each represents a district. <math>n\,</math> is assumed odd.
*a recognition rule that determines who may make a proposal: this is random and exogenous
*an amendment rule: Open or Closed
*a voting rule: simple majority voting is used


There is one (non-negative) unit of benefits to be split among the districts. The members are risk neutral and their utility depends only on the benefits to their district. The game is one of perfect information. Members can not make binding commitments - their strategies must be self-enforcing at all points. There for the solution concept is SPNE. 


The model assumes:
*<math>p_i\,</math> is the probability that a member is recognized. 
*<math>x^i = (x_1^i, \ldots, x_n^i)\,</math> is a proposal for the distribution such that <math>\sum_j x_j^i \le 1\,</math>
*The status quo is no allocation and under a closed rule members vote against the status quo.
*For open rules an amendment is a new proposal from a different member (<math>j \ne i\,</math>) who is recognised with probability <math>\frac{p_j}{\sum_{k \ne i} p_k}\,</math>. Only one amendment can be made, or the motion can be passed forward. If an amendment is made the vote is between the original motion and the amended motion. If the vote is for the amendment it becomes the motion on the floor, otherwise the motion on the floor persists into the next period. The game ends when a vote on the motion on the floor is passed.
*The discount factor is common: <math>\delta \le 1\,</math>.
*Members have utility: <math>u^j(x^k,t) = \delta^t x_j^k\,</math>
*A pure strategy at time <math>\tau\,</math> is: <math>s_{\tau}^i: H_{\tau} \to [yes,no]\,</math>, where <math>H_{\tau}\,</math> is the history
*Mixed strategies <math>\sigma_{\tau}^i\,</math> is a probability distribution over <math>s_{\tau}^i\,</math>
*<math>v_i(t,g)\,</math> is the value of sub-game <math>g\,</math>, and <math>\delta v_i(t,g)\,</math> is the continuation value if the legislature moves to sub-game <math>g\,</math>. <math>v_i\,</math> is the ex-ante value at the beginning of the game.
*Voting occurs sequentially and openly, allowing the elimination of weakly dominated strategies.


===Closed Rule - Two Sessions===

Suppose equal probabilities of recognition, and that members get zero if nothing is past at the end of the final session. The tie-break rules are: 
#A member votes for a bill if indifferent between the its distribution and the continuation value.
#A member whose vote will not be decisive votes for a bill iff its distribution is at least as great as the continuation value.


The game is solved by backwards induction.


An strategy is a SPNE iff:
*If recognized in period 1 propose:
**Give <math>\frac{\delta}{n}\,</math> to any <math>\frac{(n-1)}{2}\,</math> other members
**Keep <math>1-\frac{\delta (n-1)}{2n}\,</math> for himself
*If recognized in period 2 propose:
**Keep everything
*Vote for:
**Any first period proposal that gives at least <math>\frac{\delta}{n}\,</math>
**Vote for any second period proposal


The proof is straight forward:

:<math>v_i(2,g) = 0\,</math> 


As the game ends after period two, the continuation value is zero.

:<math>v_i(1,g) = \frac{1}{n}\,</math>


As each member has equal probability of being recognized in the next period, is risk neutral, and can assign all benefits to themselves.


Therefore vote yes iff offered at least <math>\frac{\delta}{n}\,</math>. And the minimal majority needed to pass the vote is:

:<math>\frac{(n-1)}{2}\,</math>.


Therefore keep:

:<math>1-\frac{\delta (n-1)}{2n}\,</math> 


In a three member legislature this is:

:<math>1-\frac{\delta (1)}{3}\,</math>


As <math>n \to \infty\,</math> this becomes:

:<math>1-\frac{\delta (1)}{2}\,</math> 


Therefore the payoffs to the first proposer are:

:<math>u \in \left[1-\frac{\delta (1)}{2}, 1-\frac{\delta (1)}{3} \right]\,</math>


Key notes:
#The distribution reflects the majority rule used
#Recognition in the first period, in conjuntion with the closed rule, gets the member the largest share. This is agenda power. The proposed gets at least half the benefits.
#The initial offer is accepted and the legislature adjourns after 1 period. This results from impatience.
#If the members have different probabilities of recognition their continuation value in period 1 is equal to their probability of recognition.
#A recognised member will choose to share with the members with the lowest continuation values, and the member with the highest probability will have the lowest ex-ante value for the game.
#The SPNE doesn't say who should be choosen to share with, and can randomize. Randomization provides a stationary symmetric solution.

===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'''.


To support an arbitrary distribution <math>x \in X\,</math> then:
#A member proposes <math>x\,</math> when recognized, everyone is to vote for <math>x\,</math>
#If a majority rejects <math>x\,</math>, then the next member proposes <math>x\,</math>
#If a member is recognized and proposes <math>y \ne x\,</math> then
##A majority <math>M(y)\,</math> is to reject <math>y\,</math>
##The next member proposes <math>z(y)\,</math> such that for the deviator <math>z_j(y) = 0\,</math> and everyone in <math>M(y)\,</math> is to vote for <math>z(y)\,</math> over <math>y\,</math>
##If the next member doesn't propose <math>z(y)\,</math> repeat the above stage to punish that member.


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'''.

Two sub-games are '''structurally equivalent''' iff:
#The agenda is identical
#Set members who may be recognized (at the next node) are identical
#The strategy sets of the members are identical

An equilibrium is stationary if the continuation values for each structurally equivalent subgame are the same. This necessarily has that strategies are stationary - members take the same actions in structurally equivalent subgames. Not that stationary strategies are history independent.

In the case of equal probabilities, majority rule and infitite session, proposition 3 in the paper states that for all <math>\delta \in [0,1]\,</math> a stationary SPNE in pure strategies exists iff:
*A recognised member proposes to give <math>\frac{\delta}{n}\,</math> to <math>\frac{(n-1)}{2}\,</math> randomly choosen other members, and to keep <math>1-\frac{\delta (n-1)}{2n}\,</math> for himself
*Each member votes for any proposal that gives him at least <math>\frac{\delta}{n}\,</math>
*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 

A: <math>\frac{1-\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.