Open Rule Legislative Bargaining

This paper revisits the open rule legislative bargaining model of Baron and Ferejohn (1989) — the dominant workhorse model in political economy for analyzing how legislatures divide a surplus — and provides a more complete characterization of its stationary equilibria. The core research question is whether the equilibrium typically cited in the literature as the “open rule equilibrium” is actually the unique equilibrium, or whether it rests on implicit and unstated assumptions that, once relaxed, reveal a much richer equilibrium set.

The model features n=3 negotiators dividing a surplus normalized to one, operating under simple majority rule (2 of 3 votes required). The common discount factor is Delta in (0,1). In each period, a proposer is selected uniformly at random; under the open rule, an amender is then selected uniformly at random from the two non-proposers and may either accept or counter-propose. Sincere voting determines the outcome. The authors analyze stationary subgame perfect equilibria (SSPE), in which strategies depend only on current role, not history.

The existing literature implicitly adopted what the authors call the “standard assumption”: when given the opportunity to amend, the amender proposes the same allocation she would propose as a proposer in a closed rule game. Under this assumption, the unique SSPE has the proposer receiving share 1-Delta and each of the other two negotiators receiving Delta/2 (in the Pareto-efficient equilibrium). The literature treated this as the definitive open rule solution.

The paper’s first main result is that this standard-assumption equilibrium is indeed a valid SSPE, but it is not the only one. The key mechanism generating multiplicity is the treatment of off-path behavior: what the amender does when the proposer deviates to a non-equilibrium proposal. With n=3, a deviating proposer can exploit the structure so that the amender becomes a “free” coalition member — the proposer does not need to buy the amender’s vote separately, because the amender is already included in the majority once she counter-proposes. This expands the set of credible threats and supports a continuum of additional Pareto-undominated SSPEs.

The paper’s second main result characterizes the broader equilibrium set: all Pareto-undominated SSPEs belong to a class in which the proposer offers (1-Delta) to herself and equal shares to both other negotiators. In the non-standard equilibria, the amender always amends, generating equilibrium delay — agreements are not reached immediately, and payoffs are discounted by Delta^(t-1) for each period of delay.

The third main result is that among all Pareto-undominated SSPEs, the unique Pareto-efficient one is the standard-assumption equilibrium (no delay). All other equilibria involve delay and are therefore Pareto-inferior in expectation.

The institutional design implication reverses a widely held view: the open rule was thought to promote more egalitarian allocations relative to the closed rule. The authors show this is not the case for Pareto-efficient equilibria. The Pareto-efficient open rule equilibrium is actually a special case of the closed rule equilibrium — the proposer captures 1-Delta and offers Delta to the coalition. More broadly, open rule bargaining tends to generate longer equilibrium delays and less egalitarian surplus allocations than previously predicted by Baron and Ferejohn. Scope conditions: the formal analysis is restricted to n=3 negotiators; generalization to larger legislatures is noted as an open direction.

Q: What is the “standard assumption” and why does the existing literature rely on it?

A: The standard assumption holds that when an amender gets the opportunity to counter-propose, she proposes the same allocation she would choose if she were the proposer in a closed rule game. The existing open rule literature — including Baron and Ferejohn (1989), Jackson and Morelli (2004), Baron (2012), van Weelden (2013), and Austen-Smith and Banks (1999) — accepted this assumption implicitly, treating the resulting equilibrium as the unique open rule equilibrium. The assumption sidesteps the question of off-path behavior: what happens when the proposer deviates to a non-equilibrium proposal that the amender would want to amend. Because deviations are resolved within the same bargaining session under the open rule, off-path specifications are consequential.

Q: What is the unique SSPE under the standard assumption, and what are its payoff implications?

A: Under the standard assumption with n=3 and discount factor Delta, the unique SSPE has the proposer receiving a share of 1-Delta of the surplus and each of the other two negotiators receiving Delta/2. There is no delay: the proposal passes immediately in the period it is made. This equilibrium is Pareto-efficient relative to all other stationary equilibria identified in the paper.

Q: What is the mechanism by which the equilibrium set is larger than the standard assumption predicts?

A: With n=3, when a proposer deviates to a non-equilibrium proposal, the amender — who responds by counter-proposing — automatically becomes part of the passing coalition without the proposer needing to separately compensate her. This makes the amender a “free” coalition member in the deviation subgame, which changes the cost structure of deviations and expands the range of proposals the proposer can credibly make. Consequently, a wider set of strategies by the amender can be sustained as equilibrium responses, yielding a continuum of additional Pareto-undominated SSPEs beyond the standard-assumption equilibrium.

Q: What do the non-standard equilibria look like in terms of proposals, delay, and payoffs?

A: In the non-standard Pareto-undominated SSPEs, the proposer offers (1-Delta) to herself and equal shares (Delta/2 each) to the other two negotiators — note the proposer’s own share is the same as in the standard equilibrium, but the off-path behavior differs — and the amender always chooses to amend rather than accept. The amendment triggers a vote in which the amendment fails (or the process repeats), pushing resolution to the next period. This generates equilibrium delay: agreements take multiple periods to reach, and all payoffs are discounted by Delta^(t-1) per period of delay, making these equilibria Pareto-inferior to the no-delay equilibrium.

Q: Which equilibrium is Pareto-efficient among all Pareto-undominated SSPEs, and why?

A: The unique Pareto-efficient SSPE is the standard-assumption equilibrium, because it is the only one that involves no delay. All other Pareto-undominated SSPEs involve at least one period of delay, which destroys surplus through discounting (payoffs shrink by a factor of Delta per period). Since delay is costly for all negotiators and generates no compensating redistribution, any equilibrium with delay is Pareto-dominated by the no-delay equilibrium.

Q: What are the implications for the classic efficiency comparison between open and closed rules?

A: The closed rule always generates an efficient outcome (no delay in SSPE). The open rule can also generate an efficient outcome — under the standard-assumption equilibrium — but uniquely admits a continuum of inefficient equilibria involving delay. Therefore the open rule is weakly dominated by the closed rule from an efficiency standpoint: at best it matches the closed rule (one efficient equilibrium), and at worst it generates costly delay. This reverses the common inference that open rule unambiguously improves outcomes.

Q: What are the implications for the classic fairness comparison between open and closed rules?

A: The open rule was commonly believed to promote more egalitarian surplus divisions relative to the closed rule, which allows the proposer to extract a large share. The paper shows this view is misleading. In the Pareto-efficient open rule equilibrium, the proposer still captures 1-Delta — the same as under the closed rule — and the result is no more egalitarian. In the delay equilibria, the proposer does offer equal shares to both other negotiators, but this comes at the cost of inefficiency (delay). There is no Pareto-undominated open rule equilibrium that is both efficient and more egalitarian than the closed rule.

Q: What is the class of “Pareto-undominated stationary strategies” and why does the paper focus on it?

A: A stationary strategy profile is Pareto-undominated if no other stationary strategy profile gives every negotiator at least as high an expected payoff with at least one strictly better off. The paper focuses on this class to provide a tractable but principled selection criterion within the large set of SSPEs: it eliminates equilibria that are dominated from every player’s perspective, retaining only those that could plausibly arise if players coordinate on mutually beneficial outcomes. The characterization of this class reveals that equilibrium multiplicity is already substantial even after imposing this selection.

Q: What is the scope of the formal results, and what is left open?

A: The formal analysis is restricted to n=3 negotiators with simple majority rule (2 of 3 votes). The authors acknowledge that generalization to larger n is an important open question. The three-legislator case is the simplest non-trivial instance of the majority-rule bargaining problem, and the authors use it to isolate the mechanism cleanly. The model assumes sincere voting, a common discount factor Delta in (0,1), and stationary strategies.

Q: How does this paper relate to Baron and Ferejohn (1989)?

A: Baron and Ferejohn (1989) originated both the closed rule and open rule bargaining frameworks and derived the standard-assumption equilibrium for the open rule. Subsequent literature (Eraslan 2002, Cho and Duggan 2003, 2009, Banks and Duggan 2000) extended various aspects of the B&F framework. The present paper takes the B&F open rule model as given but demonstrates that B&F’s open rule analysis was incomplete: it did not systematically address off-path behavior, and as a result the equilibrium it identified is not unique. The paper’s main contribution is to show that the B&F open rule predictions — more egalitarian allocations and prompt agreement — do not hold generally across the full equilibrium set.

Open Rule: A bargaining protocol in which, after an initial proposal is made, a nominated amender may make a counter-proposal before a vote is taken; contrasted with the closed rule, under which the initial proposal is voted on without amendment.

Closed Rule: A bargaining protocol in which a vote is taken directly on the first proposal, with no opportunity for amendment.

Standard Assumption: The implicit assumption, used by Baron and Ferejohn (1989) and subsequent literature, that when the amender counter-proposes under the open rule, she proposes the same allocation she would choose as a proposer in a closed rule game; the paper shows this assumption is consequential for equilibrium uniqueness.

Stationary Subgame Perfect Equilibrium (SSPE): An equilibrium concept in which each player’s strategy depends only on her current role (proposer, amender, or voter) and not on the history of play; the paper characterizes SSPEs of the open rule model.

Pareto-Undominated Stationary Strategy Profile: A stationary strategy profile for which no other stationary strategy profile gives every negotiator weakly higher expected payoff with at least one strictly higher; used as a selection criterion to prune the large equilibrium set.

Equilibrium Delay: The phenomenon in which agreement is not reached in the current period because the amender always counter-proposes and the counter-proposal also fails, pushing resolution to a future period and discounting payoffs; all non-standard-assumption Pareto-undominated SSPEs involve delay.

Off-Path Behavior: The specification of what strategies players use following a deviation from equilibrium play; the paper shows that different specifications of off-path behavior by the amender support different equilibria, and that the existing literature was not systematic about this.