Malice and patience in Rubinstein bargaining

If two players playing a Rubinstein alternating offers game are highly malicious (getting a high utility from “malice” in every period when the other player does not obtain a share in a fixed pie), and highly patient, no equilibrium with an agreement exists and players choose perpetual disagreement. This does not change if the players are subjected to a known deadline after which the pie will be appropriated by outside agencies or disappear: perpetual disagreement is still the only outcome. If in addition players are required to pay endogenously determined fines if they fail to reach agreement, players with discount factors in a certain range do reach agreement, but only at the deadline. However, infinitely patient players would never reach agreement for any feasible level of one-time fines. The result contrasts with spiteful or envious preferences. Our results highlight a novel reason for failure to resolve property disputes.     

PRIZES VERSUS WAGES WITH ENVY AND PRIDE*

We show that if agents are risk‐neutral, prizes outperform wages if and only if there is sufficient pride and envy relative to the noisiness of performance. If agents are risk‐averse, prizes are a necessary supplement to wages (as bonuses).     

ON THE NEUTRALITY OF COALITION FORMATION IN A PURE BARGAINING PROBLEM*

We consider the sequential bargaining game à la Stahl–Binmore–Rubinstein with random proposers, juxtaposing an ex ante coalition formation stage to their bargaining game. On the basis of the expected outcomes in the negotiation over how to split a dollar, players can form coalitions in a sequential manner, within each of which they can redistribute their payoffs. It turns out that the grand coalition does form, and that each player receives his discounted expected payoff, which is obtained by playing as a single player in the negotiation, although there could be many equilibria in the bargaining stage.     

Blindfolded vs. Informed Ultimatum Bargaining – A Theoretical and Experimental Analysis

This paper analyzes blindfolded vs. informed ultimatum bargaining where proposer and responder are both either uninformed or informed about the size of the pie. Considering the transition from one information setting to another suggests that more information induces lower (higher) price offers and acceptance thresholds when the pie is small (large). While our experimental data confirm this transition effect, risk aversion leads to diverging results in blindfolded ultimatum bargaining where task‐independent strategies such as ‘equal sharing’ or the ‘golden mean’ are implemented more frequently.     

Justifying social discounting: The rank-discounted utilitarian approach

The discounted utilitarian criterion for infinite horizon social choice has been criticized for treating generations unequally. We propose an extended rank-discounted utilitarian (ERDU) criterion instead. The criterion amounts to discounted utilitarianism on non-decreasing streams, but it treats all generations impartially: discounting becomes the mere expression of intergenerational inequality aversion. We show that more inequality averse ERDU societies have higher social discount rates when future generations are better off. We apply the ERDU approach in two benchmark economic growth models and prove that it promotes sustainable policies that maximize discounted utilitarian welfare.     

Bilateral bargaining in networks

Each connected pair of nodes in a network can jointly produce one unit of surplus. A maximum number of linked nodes is selected in every period to bargain bilaterally over the division of the surplus, according to the protocol proposed by Rubinstein and Wollinsky [Equilibrium in a market with sequential bargaining, Econometrica 53 (1985) 1133-1150]. All pairs, which reach an agreement, obtain the (discounted) payoffs and are removed from the network. This bargaining game has a unique subgame perfect equilibrium that induces the Dulmage-Mendelsohn decomposition (partition) of the bipartite network (of the set of nodes in this network).     

Bargaining with history-dependent preferences

Two impatient players bargain over a pie of size one according to the infinite alternating-offers procedure. Players’ payoffs depend not only on the outcome but also on the process of the bargaining. Specifically, they prefer impasse to any agreement that gives them lower discounted utility than would have been derived from accepting earlier offers. We characterize the essentially unique subgame perfect equilibrium path, which consists of gradual concessions. The more patient players are, the longer it takes them to reach an agreement. When players become infinitely patient, the efficiency loss is substantial, yet the equilibrium division converges to the Nash solution.     

The effect of decision weights in bargaining problems

Bargaining problems are considered where the preferences of the bargainers deviate from expected utility but can be modelled according to rank-dependent utility theory. Under rank-dependent utility both the utility function and the probability weighting function influence the risk attitude of a decision maker. The same definition of risk aversion leads to two forms of risk aversion: utility risk aversion and probabilistic risk aversion. The main finding is that these two forms can have surprisingly opposite consequences for bargaining solutions that exhibit a weak monotonicity property. In particular, in a large class of bargaining problems both increased utility risk aversion and decreased probabilistic risk aversion of the opponent are advantagous for a player. This is demonstrated for the Kalai-Smorodinsky bargaining solution. The Nash bargaining solution does not behave regularly in this respect.     

Bargaining and Boldness

We study a multiperson bargaining problem with general risk preferences through the use of Shaked's game of cycling offers with exogenous breakdown. If preferences are “smooth,” then as the risk of breakdown vanishes, the limiting outcome is one in which bargainers are equally marginally bold; where a bargainer's marginal boldness measures his willingness to risk disagreement in return for a marginal improvement in his position. Under smoothness, any (ordinal-)Nash solution is an equally marginally bold outcome. However, unlike the concept of the (ordinal-)Nash solution, a unique equally marginally bold outcome exists in natural cases—in particular, if all bargainers have risk-averse preferences of the rank-dependent expected utility type. For these preferences, the equally marginally bold outcome maximizes a “bargaining power”-adjusted (asymmetric) Nash product where the degree of asymmetry is determined by the disparity in the marginal valuation of certainty among bargainers. Journal of Economic Literature Classification Numbers: C72, C78, D81.     

Search theory,competitive equilibrium,and the Nash bargaining solution

We investigate a canonical search-theoretic model without entry. Two agents are randomly matched with a long side being rationed. The matched agents face a pair of randomly drawn non-transferable payoffs, and then choose whether or not to form a partnership subject to a small probability of exogenous break down. As this probability and friction vanish, the Nash bargaining solution emerges as the unique undominated strategy equilibrium outcome if the mass of each party is the same. If the size of one party is larger than the other, the short side extracts the entire surplus, a sharp contrast to Rubinstein and Wolinsky (1985) [16].     

Bargaining and inequity aversion: On the efficiency of the double auction

In a bargaining setting with asymmetrically informed, inequity-averse parties, a fully efficient mechanism (i.e., the double auction) exists if and only if compassion is strong. Less compassionate parties do not trade in the double auction in the limit of strong envy.     

Inequity Aversion and Team Incentives

We study optimal contracts in a simple model where employees are averse to inequity, as modeled by Fehr and Schmidt (1999) . A "selfish" employer can profitably exploit envy or guilt by offering contracts which create inequity off-equilibrium, i.e., when employees do not meet his demands. Such contracts resemble team and relative performance contracts. We derive conditions for inequity aversion to be in itself a reason to form work teams of distributionally concerned employees, even in situations in which effort is contractible.     

A Note on Franchising and Wage Bargaining

A franchising contract relocates distributable rent between franchisor and franchisee. With decentralized wage bargaining relocation modifies the position of the union in the wage bargaining. If the rent is relocated to the franchisor completely, then even a strong union is not able to raise wage above reservation level in the franchisee's firm. If franchisor and franchisee negotiate on rent division, there is an incentive to increase the franchise fee at the expense of the union. Therefore the overall rent assigned to labor depends on the differences of labor intensity in the franchisor's and franchisee's firms. Firm owners may be able to transfer distributable rents from a firm with a strong union to one with a weak union. Furthermore, a franchising contract shows a first mover advantage. A franchising contract is placed before wage bargaining, benefiting the franchisor.     

Knightian decision theory and econometric inferences

An uncertainty averse Knightian decision maker has a set of probability distributions over outcomes and chooses something other than the status quo only if the change increases the expected payoff according to all the distributions. It is possible to define a standardized degree of uncertainty aversion. To each such degree, there corresponds a set of prior distributions over the parameters of a Gaussian linear regression model, these priors being centered on a uniform prior. The set of posterior means corresponding to this set of priors has the same properties as a standard confidence region.     

The Formateurs’ Role in Government Formation

Following an election, Heads of State typically choose a formateur to report back a prospective government by some deadline. If this formateur fails then another party leader is chosen on the same terms. We model this mechanism as a bargaining game, in which a single party leader has exclusive access to the Head of State, but does not have a monopoly on making proposals. We characterize the unique equilibrium of the game in which all parties are symmetric and parties split a fixed-size pie. We also show that supra-minimal governments may form when parties are asymmetrically sized: a result consistent with the evidence on government formation in countries which use a formateur mechanism. We are grateful to Maria Montero, Rick Thomas and Paul Warwick for valuable conversations, and to seminar participants in UBC and Warwick.     

Inequality aversion and risk aversion

This note shows that for two social welfare functions which are inequality averse with respect to certainty equivalents, if one is more inequality averse for certainty equivalents than the other, the household preference induced by optimally allocating aggregate bundles according to this social welfare function is more risk averse than the other. We present examples showing that this comparative static can be reversed if absolute inequality aversion is dropped. We show that the utilitarian rule always induces the least risk averse household preference among all social welfare functions (this corresponds to the sum of certainty equivalents).     

Folk Theorem For Bilateral Bargaining with Vector Endowments

We analyse a bargaining game of two players on the division of the sum of their vector endowments, with alternating proposals and discounting of single period utilities. The pair of endowments is not weakly Pareto‐efficient. Until they reach an agreement, each of the parties can withdraw from bargaining and keep their endowments. Any strictly individually rational division of the sum of endowments can emerge in some subgame perfect equilibrium if discount factors are close enough to one. Each subgame perfect equilibrium, in which the parties’ decisions do not depend on past rejected proposals, leads to a weakly Pareto‐efficient agreement in the first period.     

An overlapping-generations model with search

Search is embedded in an overlapping-generations model. The young participate in a centralized market, and then are matched in pairs in a decentralized market. The old only participate in the centralized market. If the buyer's bargaining power in pairwise trade is close to unity and if the old are risk averse, then the golden-rule rate of money transfer is positive. Such risk aversion, the pairwise meetings, and dependence of the young's saving on the rate of return are necessary for this result.     

Bargaining and markets: complexity and the competitive outcome

Rubinstein and Wolinsky (Rev. Econ. Stud. 57 (1990) 63-78) consider a simple decentralised market game in which agents meet randomly or voluntarily and bargain over the terms of trade. They show that any individually rational price can be sustained as a sequential equilibrium even though the model has a unique competitive outcome. Here, I consider Rubinstein and Wolinsky's model and show that if complexity costs of implementing strategies enter players’ preferences, together with the standard payoff in the game, then every equilibrium is stationary/Markov and induces the unique competitive price. Thus, I demonstrate that aversion to complexity may provide a justification for the competitive outcome.     

Employment Protection and Unemployment Benefits: On Technology Adoption and Job Creation in a Matching Model

We analyse the effects of different labour‐market policies (employment protection, unemployment benefits, and payroll taxes) on job creation and technology choices in a model where firms are matched with workers of different productivity and wages are determined by ex post bargaining. The model is characterized by two intertwined sources of inefficiency, namely a matching externality and a hold‐up externality associated with the bargaining strength of workers. The results depend on the relative importance of the two externalities and on worker risk aversion. “Flexicurity”, meaning low employment protection and generous unemployment insurance, can be optimal if workers are sufficiently risk‐averse and the hold‐up problem is relatively important.