Bargaining

1. Bargaining Advanced Political Economics Fall 2012 Riccardo Puglisi

2. Bargaining in Legislatures • Equilibrium induced by the institutional structure. • Sequential nature of proposal making (amendments) and voting • 3 Steps: Recognition / Proposal / Voting

3. Legislature Structure The legislature consists of: • n members/districts • Recognition rule determining who makes the proposal • Amendment Rule • Voting Rule • Task: Choose a non-negative distribution of one unit of benefits among members, on the basis of majority voting (pie splitting again) • Perfect information

4. Recognition Rule • Neutral random draw among the n members pi = Probability of that representative i is recognized • Proposal is a distribution s.t. • Status Quo (S.Q.) is the zero vector.

5. Amendment Rule: Closed rule • Closed rule: unchanged proposal vs Status Quo

6. Amendment Rule: Open rule

7. FRAMEWORK • Discount rate, d<1, is common to all members • History htincludes: 1) who was recognized 2) what was proposed 3) what was voted • Strategy maps from history into a motion or a vote • Equilibrium concept: Subgame-perfect equilibrium (SPE) • Discount rate, d<1, is common to all members • History htincludes: 1) who was recognized 2) what was proposed 3) what was voted

8. CLOSED RULE and FINITE NUMBER of SESSION • Two sessions • n (odd) members • Equal probability of recognition • Ties are broken in favor of the proposal Idea: Work by Backward Induction: think about what are the optimal proposal and voting choices in the second period. In the first period the recognized member would offer “a proposal that cannot be refused” (no more, no less) to the minimum number of members, in order to secure a majority of votes: minimum winning coalition. It is optimal to offer the discounted expected payoff each member would get in the second period.

9. Characterization of the SPE Proposition 1: A strategy configuration is a SPE for a two-sessions, n (odd) members legislature with a closed rule and equal probability of recognition if and only if: • if recognized in the first session, a member makes a proposal to distribute d/n to any (n-1)/2 other members and to keep to herself. If recognized in the second period, a member proposes to keep all the benefits; • each member votes for any first session proposal in which she receives at least d/n and votes for any second session proposal.

10. CLOSED RULE and INFINITE NUMBER of SESSIONS Idea: Any distribution of the benefits may be supported as a subgame perfect equilibrium if there is a sufficient number of members and they are patient enough [Folk theorem] Proposition 2: For a n-member, majority rule legislature with an infinite number of sessions and closed rule, if and n>5, any distribution of the benefits may be supported as a SPE. Proof: Constructing a strategy which supports an arbitrary distribution .

11. Proof of Proposition 2 (hints) The strategy supporting an arbitrary distribution x is: • whenever a member is recognized, she has to propose x, everyone has to vote for x; • if a majority rejects x, the next member recognized is to propose x; • if a member j is recognized and deviates to propose y≠x, then • a majority M(y) is to reject y; • the member k recognized next is to propose z(y) s. t. for the deviator zj(y)=0, and everyone in M(y) is to vote for z(y) over y. • If in strategy (3b) member k is recognized and proposes s ≠ z(y), repeat strategy (3) with s replacing y and k replacing j. Intuition: for any given distribution members who deviate from the equilibrium strategy are harshly punished. And the same happens to those that do not punish those who deviated, and so on and so forth…

12. STATIONARY EQUILIBRIUM The agents have to take the same action in structurally equivalent subgames. Definition: Structurally equivalent subgames are subgames such that: • The ex-ante agenda is the same; • The sets of members who may be recognized is the same; • The strategy set is identical Note that previous strategies are not stationary, they are history dependent.

13. CLOSED RULE, STATIONARY STRATEGIES, INFINITE NUMBER OF SESSIONS

14. Characterization of the stationary SPE Proposition 3: For every a configuration of pure strategies is a stationary subgame-perfect equilibrium in a infinite session majority rule, n-member legislature governed by a closed rule if and only if it has the following form: • a member recognized proposes to receive and to offer d/n to (n-1)/2 other members selected at random; • Each member votes for any proposal in which at least d/n is received The first proposal receives a majority vote, so the legislature complete its task in the first session. The ex-ante value of the game is v=1/n for each member.

15. OPEN RULE More complicated because the recognized voter has to take into account the possibility of future amendments to the proposal. Idea: The initially recognized voter could distribute the benefits among all n-1 voters to be sure of getting a majority in the next period, but that is very expensive. Therefore, she chooses m(d,n) voters among whom to divide the benefits. If in the next recognition one of these voters get recognized she does not amend the proposal, which gets voted against the Status Quo. Otherwise the new recognized voter selects other m(d,n) voters and so on.

16. Redistribution in a Divided Society • Austen-Smith and Wallerstein (JPubE, 2006) • Individuals different along two dimensions: • Economic element: employed-unemployed (but also low and high human capital types) • Social/racial cleavage: blacks and whites • Policy is multidimensional: • Redistribution in the labor market: unemployment benefits (from employed to unemployed) • Affirmative action (quota on the jobs to minorities)

17. Redistribution in a Divided Society • Political Decision through Legislative Bargaining (infinite horizon) among 3 parties: • White Workers with High Human Capital • White Workers with Low Human Capital • Blacks (both low and high Human Capital) • Preferences: • High Human Capital types are more likely to be employed, thus they like unemployment benefits less than low types. • Blacks like Affirmative action more than whites • The result holds even if in the presence of Affirmative action -and controlling for human capital-, blacks like UB more than whites(see page 1801 in the paper)

18. Redistribution in a Divided Society • Main Result (Proposition 1): the possibility of affirmative action reduces the expected redistribution through taxes and UB • IDEA: Potential alliance between high human capital whites and blacks to lower taxes and UB and increase the use of affirmative action. • APPLICATION: Nixon policies in the 70s.