Uniqueness of stationary equilibria in bargaining one-dimensional policies under (super) majority rules

We consider negotiations selecting one-dimensional policies. Individuals have instantaneous preferences represented by continuous, concave and single-peaked utility functions, and they are impatient. Decisions arise from a bargaining game with random proposers and (super) majority approval, ranging from the simple majority up to unanimity. We provide sufficient conditions that guarantee the existence of a unique stationary subgame perfect equilibrium, and we provide its explicit characterization. The uniqueness of the equilibrium permits an analysis of the set of Pareto optimal voting rules. For symmetric distributions of peaks and uniform recognition probabilities unanimity is the unanimously preferred majority rule.     

The characterization of decisive weighted majority rules

This note supplies a rationale for the adoption of decisive weighted majority rules by small decision-making bodies (n<6). It proves that such rules are the only decision rules that satisfy the two common and quite mild conditions of monotonicity and neutrality.     

Nominal bargaining power, selection protocol, and discounting in legislative bargaining

The comparative static predictions of the Baron and Ferejohn [Baron, D.P., and Ferejohn, J.A., (1989). Bargaining in legislatures, American Political Science Review 83 (4), 1181-1206] model better organize behavior in legislative bargaining experiments than Gamson's Law. Regressions similar to those employed in field data produce results seemingly in support of Gamson's Law (even when using data generated by simulating agents who behave according to the Baron-Ferejohn model), but this is determined by the selection protocol which recognizes voting blocks in proportion to the number of votes controlled. Proposer power is not nearly as strong as predicted in the closed rule Baron and Ferejohn model, as coalition partners refuse to take the small shares given by the continuation value of the game. Discounting pushes behavior in the direction predicted by Baron and Ferejohn but has a much smaller effect than predicted.     

Demand bargaining and proportional payoffs in majority games

We study a majoritarian bargaining model in which players make payoff demands in decreasing order of voting weight. The unique subgame perfect equilibrium outcome is such that the minimal winning coalition of the players that move first forms with payoffs proportional to the voting weights. This result advances previous analysis in terms of one or more of the following: a) the simplicity of the extensive form (finite horizon with a predetermined order of moves); b) the range of the majority games covered; c) the equilibrium concept (subgame perfect equilibrium is sufficient for a unique prediction).     

Simple Markovian equilibria in dynamic spatial legislative bargaining

The paper proves, by construction, the existence of Markovian equilibria in a dynamic spatial legislative bargaining model. Players bargain over policies in an infinite horizon. In each period, a sequential protocol of proposal-making and voting, with random proposer recognitions and a simple majority, produces a policy that becomes the next period's status-quo; the status-quo is endogenous. The construction relies on simple strategies determined by strategic bliss points computed by the algorithm we provide. A strategic bliss point, the dynamic utility ideal, is a moderate policy relative to a bliss point, the static utility ideal. Moderation is strategic and germane to the dynamic environment; players moderate in order to constrain the future proposals of opponents. Moderation is a strategic substitute; when a player's opponents do moderate, she does not, and when they do not moderate, she does. We provide conditions under which the simple strategies induced by the strategic bliss points computed by the algorithm deliver a Stationary Markov Perfect equilibrium, and we prove its existence in generic games with impatient players and in symmetric games. Because the algorithm constructs all equilibria in simple strategies, we provide their general characterization, and we show their generic uniqueness.     

The network of weighted majority rules and its geometric realizations

In a previous work the first author considered the network G(n,p) of weighted majority rules (WMR) for n decision makers whose competencies are given by their probabilities p=(p₁,…,p_n) of making a correct decision. On this paper we consider chains of decision profiles, which must occur in G(n,p) in a fixed order, and show that they can be mapped onto straight lines in a low-dimensional geometric realization. The minimal number of directions which must used to separate all edges is given as the chromatic number of a certain incidence graph. We also define degenerate networks in which several nodes coalesce.     

Proposal power and majority rule in multilateral bargaining with costly recognition

This paper studies a sequential bargaining model in which agents expend efforts to be the proposer. In equilibrium, agents’ effort choices are influenced by the prize and cost effects. The (endogenous) prize is the difference between the residual surplus an agent obtains when he is the proposer and the payment he expects to receive when he is not. Main results include: (1) under the unanimity voting rule, two agents with equal marginal costs propose with equal probabilities, regardless of their time preferences; (2) under a nonunanimity rule, however, the more patient agent proposes with a greater probability; (3) while, under the unanimity rule, the social cost decreases in group heterogeneity, it can increase under a nonunanimity rule; and (4) when agents are identical, the unanimity rule is socially optimal.     

Efficient delays in a stochastic model of bargaining

Summary. We consider a k-player sequential bargaining model in which both the cake size and the identity of the proposer are determined by a stochastic process. For the case where the cake is a simplex (of random size) and the players share a common discount factor, we establish the existence of a unique stationary subgame perfect payoff which is efficient and characterize the conditions under which agreement is delayed. We also investigate how the equilibrium payoffs depend on the order in which the players move and on the correlation between the identity of the proposer and the cake size.Received: November 5, 1996; revised version: December 31, 1996This revised version was published online in February 2005 with corrections to the cover date.     

A theory of ethnic diversity and income distribution: A legislative bargaining approach

In this paper, I examine how the two dimensions of income inequality and ethnic diversity affect the formation of political coalitions and the reallocation of income. I construct a legislative bargaining model to analyze when coalitions form along the income dimension and when it forms along the ethnic dimension. The results of the model suggest that the agenda setter selects a partner with a lower aggregate income: a smaller population size and a lower income level. Further, I extend the model from a one-round to a two-round game. The extended model shows that forming an oversized coalition is a possible equilibrium outcome and is consistent with the empirical findings.     

The ratchet effect and bargaining power in a two-stage model of competitive bidding

Summary. In the model presented, a buyer uses competitive bidding to facilitate her purchase of a good (the primary good of the exchange). Not included in the original purchase is the possible procurement of a good related to the original purchase: the supplementary good. The primary and supplementary goods are closely related; knowing a bidder's cost of producing the primary good implies that the buyer can infer the bidder's cost of producing the supplementary good. I show that a bidding mechanism for the primary good will fail to ensure an efficient allocation if the buyer learns the bid of the winner and the price of the supplementary good is determined through sequential bargaining. Received: August 22, 1996; revised version: June 23, 1997     

Uniqueness of stationary equilibria in a one-dimensional model of bargaining

We prove uniqueness of stationary equilibria in a one-dimensional model of bargaining with quadratic utilities, for an arbitrary common discount factor. For general concave utilities, we prove existence and uniqueness of a “minimal” stationary equilibrium and of a “maximal” stationary equilibrium. We provide an example of multiple stationary equilibria with concave (nonquadratic) utilities.     

Pork versus public goods: an experimental study of public good provision within a legislative bargaining framework

We experimentally investigate a legislative bargaining model with both public and particularistic goods. Consistent with the qualitative implications of the model: there is near exclusive public good provision in the pure public good region, in the pure private good region minimum winning coalitions sharing private goods predominate, and in the “mixed” region proposers generally take some particularistic goods for themselves, allocating the remainder to public goods. As in past experiments, proposer power is not nearly as strong as predicted, resulting in public good provision decreasing in the mixed region as its relative value increases, which is inconsistent with the theory.     

Union bargaining power, employment, and output in a model of monopolistic competition with wage bargaining

This paper examines the implications of a rise in the bargaining power of workers on the real wage, income distribution, and the levels of employment and output using a macroeconomic model with monopolistic competition and worker-owner Nash bargaining at the firm level. It thereby provides optimizing microfoundations to Kalecki's macroeconomic analysis of the positive effect on output of a rise in trade-union power, and contrasts it with the neoclassical view based on the diminishing marginal productivity of labor.     

An informational role of supermajority rules in monitoring the majority party's activities

There often exists a supermajority rule that enables the minority party to delay or prevent a vote on a bill. I construct a two‐period model consisting of a representative voter, self‐interested parties, and a media outlet. In the model, the majority party has an incentive to misrepresent the voter's optimal policy. I show that the minority party's attempt to block a vote (e.g., a filibuster) can signal this misrepresentation. Interestingly, the key is that the minority party and the mass media are complementary in creating the signal. Overall, the results suggest that supermajority rules could be beneficial even for the majority of voters.     

A bargaining model in general equilibrium

The goal of the paper is to present a simple model of rational endogenous household formation in a general equilibrium framework in which Pareto optimality at the economy level is not necessarily obtained. The simplest example of household formation is the case in which pairs of individuals engage themselves in a bargaining process on the division of some wealth: if an agreement on the distribution is (not) reached, we can say that the household is (not) formed. The vast majority of existing bargaining models predicts agreements on an efficient outcome. A seminal paper by Crawford (Econometrica 50:607–637, 1982) describes a very simple game with incomplete information in which, even with rational agents, disagreement causes welfare losses. We embed that model in a general equilibrium framework and present some results on equilibria both in the bargaining game and the associated exchange economy. Crawford’s results support Schelling’s intuition on the reasons of disagreement: it may arise if players’ commitments are reversible. Crawford shows that high probabilities of reversibility tend to favor the bargaining impasse, in fact with low probability. We prove that even if those probabilities are arbitrarily close to zero, disagreement is an equilibrium outcome, with high probability. That conclusion seems to be an even stronger support to Schelling’s original viewpoint. In the exchange economy model with that noncooperative bargaining game as a first stage, we present significant examples of economies for which equilibria exist. Because of disagreement, Pareto suboptimal exchange economy equilibria exist for all elements in the utility function and endowment spaces and they may coexist with Pareto optimal equilibria even at the same competitive prices.     

A bargaining model of tax competition

This paper develops a model in which competing governments offer financial incentives to induce individual firms to locate within their jurisdictions. Equilibrium is described under three specifications of the supplementary taxes. There is no misallocation of capital under two of these specifications, and there might or might not be capital misallocation under the third. This result contrasts strongly with that of the standard tax competition model, which does not allow governments to treat firms individually. That model finds that competition among governments almost always leads to capital misallocation.     

Nash bargaining model of HMO premiums

This paper estimates a model of commercial HMO premiums based on Nash's axiomatic approach to bargaining between HMOs and employers. The bargaining models incorporate variables that measure the?‘power’?of the parties to affect the division of potential gains from a contract. It is found that an increase in the number of competing HMOs increases the employer's bargaining power and leads to lower premiums, especially for for-profit HMOs. It is also found that employers’?bargaining power over non-profit HMOs is positively related to the ratio of the HMO's administrative expenses/total expenses.     

The determination of public tuition fees in a mixed education system: A majority voting model

We study the determination of public tuition fees through majority voting in a vertical differentiation model where agents' returns on educational investment differ and public and private universities coexist and compete in tuition fees. The private university offers higher educational quality than its competitor, incurring higher unit cost per trained student. The tuition fee for the state university is fixed by majority voting while that for the private follows from profit maximization. Then agents choose to train at the public university or the private one or to remain uneducated. The tax per head adjusts in order to balance the state budget. Because there is a private alternative, preferences for education are not single‐peaked and no single‐crossing condition holds. An equilibrium is shown to exist, which is one of three types: high tuition fee (the “ends” are a majority), low tuition fee (the “middle” is a majority), or mixed (votes tie). The cost structure determines which equilibrium obtains. The equilibrium tuition is either greater (majority at the ends) or smaller (majority at the middle) than the optimal one.     

The bargaining set of a large game

We study the equivalence between the Mas-Colell bargaining set and the core in the general context of TU games with a measurable space of players. In the first part of the paper, we study the problem without imposing any restriction on the class of games we consider. In the second part, we first introduce a new class of exact games, which we call thin games. For these games, we show not only that the Mas-Colell bargaining set is equal to the core, but also that it is the unique stable set in the sense of von Neumann and Morgenstern. We then study the relation between thin games, exact non-atomic market games and non-atomic convex games. Finally, by further developing “thinness” related ideas, we prove new equivalence results for a class of non-exact market games as well as a class of non-exact, non-market games.     

Weak stability and a bargaining set for the marriage model

In this note we introduce weak stability, a relaxation of the concept of stability for the marriage model by assuming that the agents are no longer myopic in choosing a blocking pair. The new concept is based on threats within blocking pairs: an individually rational matching is weakly stable if for every blocking pair one of the members can find a more attractive partner with whom he forms another blocking pair for the original matching. Our main result is that under the assumption of strict preferences, the set of weakly stable and weakly efficient matchings coincides with the bargaining set of Zhou (1994, Games Econ. Behav. 6, 512–526) for this context.