A bargaining model of endogenous procedures

Questions of procedure often seem more contentious than policy substance. As Rep. Robert H. Michel (R-IL), U.S. House minority leader from 1981–1995, once stated: “Procedure hasn’t simply become more important than substance—it has, through a strange alchemy, become the substance of our deliberations.” Veteran lawmaker Rep. John Dingell (D-MI) made a similar remark: “If you let me write the procedure, and I let you write the substance, I’ll [beat] you every time.” (both quoted in Oleszek 2007).

For example, in the U.S. every 2 years when a new Congress convenes, Representatives and Senators first propose and vote over candidates for presiding officer, as well as committee chairs and members. The proposals and votes are public information. Similar protocols are followed by non-majoritarian bodies like the IMF Executive Board which uses weighted voting.

See Banks and Duggan (2006a) for details. LSWP is similar to, but weaker than, single-peakedness.

Cho and Duggan (2003) in fact use a stronger condition than order-restriction, namely a quadratic functional form which they show produces order-restriction of preference profiles. Cardona and Ponsati (2007, 2011) provide existence and uniqueness results for one-dimensional policy bargaining with a zero default payoff. As we assume a default from the choice set, their results do not translate to our environment.

This is in contrast to McCarty’s (2000) model of exogenous procedures where greater procedural prerogatives in distributive bargaining do not necessarily lead to higher expected payoffs for those holding them. One reason is that more powerful members are more expensive to include in a minimum-winning coalition.

In an extension of their dynamic model, Duggan and Kalandrakis (2012) allow a proposer as well as a voting rule to be chosen through a more general sequential voting procedure. They provide existence and continuity results, but no direct characterization results, for this extension of their model.

In Diermeier et al. (2015) equal recognition, full preference symmetry, and majority voting provide simple sufficient conditions for existence of (strictly or weakly presistent) no-delay procedural equilibria, as they ensure that all players prefer the mean-preserving median monopoly power to the symmetric continuation lottery resulting from delay. By comparison, here in the absence of symmetry the argument for existence of no-delay equilibria involves theoretically identifying a stochastically dominant procedure preferred to delay. The LPS property of the dominant procedure plays a key role in this argument.

Examples of voting rules sharing these properties are: majority rule, \( \mathcal {D}= \{ C\subseteq N\mid \vert C \vert >\frac{n}{2} \},\) dictatorship, \(\mathcal {D}= \{ C\subseteq N\mid i\in C \}\), for some fixed player \(i\in N,\) and weighted voting, \( \mathcal {D}= \{ C\subseteq N\mid \sum _{i\in C}\omega _{i}>\frac{1}{2} \},\) where the \(\omega _{i}\)’s are nonegative voting weights that sum up to one.

The policy discount factor \(\gamma \) can be interpreted as the conditional probability that bargaining continues for another round; then \(\gamma <1\) implies that policy bargaining ends in finite time with probability one. Note that the extensive form avoids the complexities of history dependence as policy bargaining histories do not affect the timing of the next procedural period.

Quadratic loss is an example utility function that makes preferences order-restricted with respect to ideal points (Cho and Duggan 2003; Banks and Duggan 2006b).

Austen-Smith and Banks (2006, Sec. 6.2) show why a bargaining equilibrium in the space of stationary strategy profiles is also a bargaining equilibrium in the space of all strategy profiles. Ruling out weakly dominated voting strategies is standard and serves to eliminate voting outcomes disconnected from preferences.

The no-delay claim in Proposition 1 is stronger than its counterpart in Banks and Duggan (2006a) since it does not rely on interior recognition probabilities, i.e., that all players have positive proposal power; see their Corollary 2. The reason is that we require players to resolve indifference between continuing and ending bargaining by ending it. Otherwise, there may be delay equilibria, but if so, they are all outcome-equivalent to the unique no-delay equilibrium, i.e., the outcome in every period is the default policy \(p^{D}\). Our tie-breaking assumption eliminates this outcome-equivalent equilibrium multiplicity.

In a symmetric majoritarian environment the procedure giving the median monopoly proposal power would always belong to the procedural collective acceptance set. See Diermeier et al. (2015).

Denote the bounds of the upper-contour set of \(u_{k}\) at \(p^{D}\) by \( \overline{p}_{k}^{D}=\max \{ p\in [ 0,1] \mid u_{k}(p)\ge u_{k}(p^{D}) \} \) and \(\underline{p}_{k}^{D}=\min \{ p\in [ 0,1] \mid u_{k}(p)\ge u_{k}(p^{D}) \} .\) By Proposition 1 and Lemma 1 in the Appendix, policy bargaining cannot yield policy outcomes outside \(Z^{v}=[ \underline{p}_{k}^{D},\overline{p}_{k}^{D}] \cap [ {\tilde{p}}_{1},{\tilde{p}}_{n}].\)

Moreover, as this hypothetical discrete policy space becomes less coarse, the fractional proposals would converge to the single-policy proposals of the bargaining model with a continuous policy space. We thank an anonymous reviewer for suggesting this analogy.

With equal recognition, full preference symmetry, and majority voting, as assumed in Diermeier et al. (2015), existence of a procedural equilibrium would be more immediate. The core player, i.e., the median, would be indifferent whether a given policy deviation occurs on its left or on its right. Thus, the policy majority acceptance set (the median’s policy acceptance set) is symmetric with respect to the median’s ideal point. This, together with symmetric ideal points and equal recognition, ensures that the continuation policy lottery induced by procedural bargaining is symmetric, so the median, and all other players, prefer median monopoly power to the symmetric continuation lottery resulting from delay. Thus, not only is the procedural collective acceptance set non-empty, the equilibrium also has to be no-delay.

A detailed solution to this example is presented in Appendix B.

The only (trivial) exception is the case of three players, when two of them may share proposal power, thus forming a procedural majority.

In this respect, the equilibrium intuition is similar to the minimum winning coalition logic of sequential distributive models, e.g., Diermeier and Vlaicu (2011b), and different from the tyranny/oligarchy logic of dynamic distributive models, e.g., Jeon (2015), where one/two players monopolize society’s economic and political resources.