Continuity properties of bargaining solutions

Three different concepts of continuity of bargaining solutions are examined: Pareto continuity, Hausdorff continuity, and Kuratowski continuity. A new axiomatic characterization of the Nash solution is proposed. In this axiomatization, Kuratowski continuity plays a major role.     

Uncovered bargaining solutions

An uncovered bargaining solution is a bargaining solution for which there exists a complete and asymmetric relation (tournament) such that, for each feasible set, the bargaining solution set coincides with the uncovered set of the tournament. We provide a characterization of a class of uncovered bargaining solutions.     

Replication invariance of bargaining solutions

This note is concerned with the behavior of bargaining solutions under replication of bargaining problems. A notion of replication, alternative to that studied by Kalai, is proposed, and it is shown that for both the Nash and Kalai-Smorodinsky solutions, the sum of what the agents of each type obtain in the replicated problem is equal to what the agent they are representing would obtain in the original problem under an application of the asymmetric generalization of the solution with weights proportional to the orders of replication of the different types. This similarity in behavior of the two solutions is in contrast with the conclusions reached by Kalai.     

Multiple-issue bargaining and axiomatic solutions

We study two-person, multiple-issue bargaining problems and identify four procedures by which the bargaining may take place. Drawing on some logic from non-cooperative game theory, we propose axioms which relate the outcomes of the procedures. We also promote a weak monotonicity axiom on solutions, called issue-by-issue monotonicity, which is geared toward multiple-issue bargaining. Our main result concerns the relationship between a sequential bargaining procedure — with the rule that agreements are implemented only after all issues are resolved — and global bargaining (in which all issues are negotiated simultaneously). If a bargaining solution predicts the same outcome with these two procedures, then we say that it satisfiesagenda independence. We prove that a solution satisfies axioms of efficiency, symmetry, scale invariance, issue-by-issue monotonicity, and agenda independence if and only if it is the Nash solution. This result provides new intuition for Nash's independence of irrelevant alternatives axiom. Among other results, we show that a solution is invariant to all four of the procedures and satisfies efficiency and symmetry if and only if it is the utilitarian solution with equal weights. We comment on the results of other authors who address multiple-issue bargaining.     

Inequality averse multi-utilitarian bargaining solutions

This paper introduces and analyzes the class of inequality averse multi-utilitarian solutions for cooperative bargaining problems. We show that generalized Gini solutions and inequality averse Choquet solutions are particular cases of this new multi-valued solution concept and provide a complete characterization in which an invariance property, consisting of a weakening of both the linear invariance axiom in Blackorby et al. (Econometrica 62:1161–1178, 1994) and the restricted invariance axiom in Ok and Zhou (Games Econ Behav 33:249–264, 2000), plays an important role. Moreover, by relaxing the assumptions involved in the characterization, the class is extended to include inequality loving multi-utilitarian solutions which are also studied in the paper.     

Nonzero-sum differential games with bargaining solutions

Nash's bargaining solution for finite games is extended to differential games with nonzero-sum integral payoffs. Sufficient conditions for the optimality of a strategy pair are established. An example is given.     

Continuity of solutions to n-harmonic equations

In this paper, we study the nonhomogeneous n-harmonic equation $$-{\rm div}\,(|{\nabla} u|^{n-2}{\nabla} u)=f$$ in domains ${\Omega\subset {\mathbb {R}^n}}$ (n?≥?2), where ${f\in W^{-1,\frac{n}{n-1}}(\Omega)}$ . We derive a sharp condition to guarantee the continuity of solutions u. In particular, we show that when n?≥ 3, the condition that, for some ${\epsilon >0 ,}$ f belongs to $${\mathfrak{L}}({\rm log}\,{\mathfrak{L}})^{n-1}({\rm log}\,{\rm log}\,{\mathfrak{L}})^{n-2}\cdots({\rm log}\cdots{\rm log}\,{\mathfrak{L}})^{n-2}({\rm log}\cdots{\rm log}\,{\mathfrak{L}})^{n-2+\epsilon}(\Omega)$$ is sufficient for continuity of u, but not for ${\epsilon=0}$ .     

Nonsymmetric Nash solutions and replications of 2-person bargaining

A 2-person fixed threat bargaining problem is considered. A full characterization of the solutions which satisfy all of Nash's axioms except for the axiom of symmetry is given. It is also shown that these nonsymmetric Nash solutions are precisely the solutions that arise from symmetric Nash solutions through replications.     

Continuity of weak solutions of elliptic partial differential equations

The continuity of weak solutions of elliptic partial differential equations

Continuity properties of solutions to some degenerate elliptic equations

We consider a nonlinear (possibly) degenerate elliptic operator where the field a and the function b are (unnecessarily strictly) monotonic and a satisfies a very mild ellipticity assumption. For a given boundary datum ? we prove the existence of the maximum and the minimum of the solutions and formulate a Haar-Radò type result, namely a continuity property for these solutions that may follow from the continuity of ?. In the homogeneous case we formulate some generalizations of the Bounded Slope Condition and use them to obtain the Lipschitz or local Lipschitz regularity of solutions to Lu=0. We prove the global Hölder regularity of the solutions in the case where ? is Lipschitz.     

Continuity of weak solutions to certain singular parabolic equations

Sunto Le equazioni paraboliche singolari (1.1)nella introduzione, si presentano come modelli di una classe generale di fenomeni di diffusione con cambio di fase. Le soluzioni deboli sono trovate in senso globale come classi di equivalenza in certi spazi di Sobolev. In questo lavoro si dimostra che le soluzioni deboli ammettono delle rappresentanti continue nell'interno del dominio di definizione. Si danno anche delle condizioni di continuità fino alla frontiera.

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Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based upon work supported by the National Science Foundation under Grant No. MCS78-09525 A01.     

Continuity of solutions of a quasilinear hyperbolic equation with hysteresis

This paper is devoted to the investigation of quasilinear hyperbolic equations of first order with convex and nonconvex hysteresis operator. It is shown that in the nonconvex case the equation, whose nonlinearity is caused by the hysteresis term, has properties analogous to the quasilinear hyperbolic equation of first order. Hysteresis is represented by a functional describing adsorption and desorption on the particles of the substance. An existence result is achieved by using an approximation of implicit time discretization scheme, a priori estimates and passage to the limit; in the convex case it implies the existence of a continuous solution.     

Continuity of solutions for parametric variational inequalities in Banach space

We consider here a type of pseudo-monotone parametric variational inequalities on a class of Banach spaces and show that such problems admit continuous (with respect to the parameter) solutions, as long as generic existence and uniqueness conditions for these solutions are satisfied. In particular, we show that such results are valid on a class of Banach spaces whenever we deal with strong pseudo-monotonicity, while others are valid in Hilbert spaces, whenever strict monotonicity is present. We also provide examples to illustrate the new results.