Baron Ferejohn (1989) - Bargaining In Legislatures

• This page is referenced in BPP Field Exam Papers

Reference(s)

Baron, D. and J. Ferejohn (1989), Bargaining in Legislatures, American Political Science Review 83 (December), 1181. 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:

1. A member votes for a bill if indifferent between the its distribution and the continuation value.
2. 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 [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:

1. The distribution reflects the majority rule used
2. 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.
3. The initial offer is accepted and the legislature adjourns after 1 period. This results from impatience.
4. If the members have different probabilities of recognition their continuation value in period 1 is equal to their probability of recognition.
5. 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.
6. 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

From proposition 2 in the paper, if:

[math]1 \gt \delta \gt \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:

1. A member proposes [math]x when recognized, everyone is to vote for \lt math\gt x\,[/math]
2. If a majority rejects [math]x\,[/math], then the next member proposes [math]x\,[/math]
3. If a member is recognized and proposes [math]y \ne x\,[/math] then
   1. A majority [math]M(y)\,[/math] is to reject [math]y\,[/math]
   2. 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]
   3. If the next member doesn't propose [math]z(y)\,[/math] repeat the above stage to punish that member.