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Baron Ferejohn (1989) - Bargaining In Legislatures (view source)
Revision as of 20:25, 18 May 2010
, 20:25, 18 May 2010no edit summary
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
#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(2,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 n \to infty this becomes:
:<math>1-\frac{\delta (1)}{2}\,</math>
Therefore the payoffs to the first proposer are:
:<math>u \in [1-\frac{\delta (1)}{2}, 1-\frac{\delta (1)}{3}]\,</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.
