The Tempered Aspirations solution for bargaining problems with a reference point

Gupta and Livne (1988) modified Nash’s (1950) original bargaining problem through the introduction of a reference point restricted to lie in the bargaining set. Additionally, they characterized a solution concept for this augmented bargaining problem. We propose and axiomatically characterize a new solution concept for bargaining problems with a reference point: the Tempered Aspirations solution. In Kalai and Smorodinsky (1975), aspirations are given by the so called ideal or utopia point. In our setting, however, the salience of the reference point mutes or tempers the negotiators’ aspirations. Thus, our solution is defined to be the maximal feasible point on the line segment joining the modified aspirations and disagreement vectors. The Tempered Aspirations solution can be understood as a “dual” version of the Gupta–Livne solution or, alternatively, as a version of Chun and Thomson’s (1992) Proportional solution in which the claims point is endogenous. We also conduct an extensive axiomatic analysis comparing the Gupta–Livne to our Tempered Aspirations solution.     

The Kalai-Smorodinsky bargaining solution with loss aversion

We consider bargaining problems under the assumption that players are loss averse, i.e., experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on an outcome: an outcome z in a bargaining problem is self-supporting under a given bargaining solution, whenever transforming the problem using outcome z as a reference point, yields a transformed problem in which the solution is z.We show that n-player bargaining problems have a unique self-supporting outcome under the Kalai-Smorodinsky solution. For all possible loss aversion coefficients we determine the bargaining solutions that give exactly these outcomes, and characterize them by the standard axioms of Scale Invariance, Individual Monotonicity, and Strong Individual Rationality, and a new axiom called Proportional Concession Invariance (PCI). A bargaining solution satisfies PCI if moving the utopia point in the direction of the solution outcome does not change this outcome.     

Stochastic evolution of 2D crystals

We study a crystalline version of the modified Stefan problem in the plane. The feature of our approach is that, we consider a model with stochastic perturbations and assume the interfacial curve to be a polygon. The existence of solution to our stochastic system is established. Galerkin’s method is one of the main tools used in the proof of our assertion.     

g-dominance: Reference point based dominance for multiobjective metaheuristics

One of the main tools for including decision maker (DM) preferences in the multiobjective optimization (MO) literature is the use of reference points and achievement scalarizing functions [A.P. Wierzbicki, The use of reference objectives in multiobjective optimization, in: G. Fandel, T. Gal (Eds.), Multiple-Criteria Decision Making Theory and Application, Springer-Verlag, New York, 1980, pp. 469–486.]. The core idea in these approaches is converting the original MO problem into a single-objective optimization problem through the use of a scalarizing function based on a reference point. As a result, a single efficient point adapted to the DM’s preferences is obtained. However, a single solution can be less interesting than an approximation of the efficient set around this area, as stated for example by Deb in [K. Deb, J. Sundar, N. Udaya Bhaskara Rao, S. Chaudhuri, Reference point based multiobjective optimization using evolutionary algorithms, International Journal of Computational Intelligence Research, 2(3) (2006) 273–286]. In this paper, we propose a variation of the concept of Pareto dominance, called g-dominance, which is based on the information included in a reference point and designed to be used with any MO evolutionary method or any MO metaheuristic. This concept will let us approximate the efficient set around the area of the most preferred point without using any scalarizing function. On the other hand, we will show how it can be easily used with any MO evolutionary method or any MO metaheuristic (just changing the dominance concept) and, to exemplify its use, we will show some results with some state-of-the-art-methods and some test problems.     

Duality,area-considerations,and the Kalai–Smorodinsky solution

We introduce a new solution concept for $2$-person bargaining problems, which can be considered as the dual of the Equal-Area solution (EA) (see Anbarc? and Bigelow (1994)). Hence, we call it the Dual Equal-Area solution (DEA). We show that the point selected by the Kalai–Smorodinsky solution (see Kalai and Smorodinsky (1975)) lies in between those that are selected by EA and DEA. We formulate an axiom–area-based fairness–and offer three characterizations of the Kalai–Smorodinsky solution in which this axiom plays a central role.     

The ordered field property and a finite algorithm for the Nash bargaining solution

This note proves that the two person Nash bargaining theory with polyhedral bargaining regions needs only an ordered field (which always includes the rational number field) as its scalar field. The existence of the Nash bargaining solution is the main part of this result and the axiomatic characterization can be proved in the standard way with slight modifications. We prove the existence by giving a finite algorithm to calculate the Nash solution for a polyhedral bargaining problem, whose speed is of orderBm(m-1) (m is the number of extreme points andB is determined by the extreme points).     

On the time discretization for the globally modified three dimensional Navier-Stokes equations

In this work, we analyze the discrete in time 3D system for the globally modified Navier-Stokes equations introduced by Caraballo (2006) [1]. More precisely, we consider the backward implicit Euler scheme, and prove the existence of a sequence of solutions of the resulting equations by implementing the Galerkin method combined with Brouwer’s fixed point approach. Moreover, with the aid of discrete Gronwall’s lemmas we prove that for the time step small enough, and the initial velocity in the domain of the Stokes operator, the solution is H² uniformly stable in time, depends continuously on initial data, and is unique. Finally, we obtain the limiting behavior of the system as the parameter N is big enough.     

Kalai-Smorodinsky Bargaining Solution Equilibria

Multicriteria games describe strategic interactions in which players, having more than one criterion to take into account, don’t have an a-priori opinion on the relative importance of all these criteria. Roemer (Econ. Bull. 3:1–13, 2005) introduces an organizational interpretation of the concept of equilibrium: each player can be viewed as running a bargaining game among criteria. In this paper, we analyze the bargaining problem within each player by considering the Kalai-Smorodinsky bargaining solution (see Kalai and Smorodinsky in Econometrica 43:513–518, 1975). We provide existence results for the so called Kalai-Smorodinsky bargaining solution equilibria for a general class of disagreement points which properly includes the one considered by Roemer (Econ. Bull. 3:1–13, 2005). Moreover we look at the refinement power of this equilibrium concept and show that it is an effective selection device even when combined with classical refinement concepts based on stability with respect to perturbations; in particular, we consider the extension to multicriteria games of the Selten’s trembling hand perfect equilibrium concept (see Selten in Int. J. Game Theory 4:25–55, 1975) and prove that perfect Kalai-Smorodinsky bargaining solution equilibria exist and properly refine both the perfect equilibria and the Kalai-Smorodinsky bargaining solution equilibria.     

Independence of irrelevant expansions

Nash characterized the only bargaining solution to satisfy a well-known list of axioms. Independence of Irrelevant Alternatives states invariance of the solution outcome under certain contractions of the bargaining problem. A dual of this axiom is proposed here, stating invariance under certainexpansions of the bargaining problem andNash's solution is characerized by substituting this axiom for IIA in Nash's original list. After a transposition from the domain of bargaining solutions to the domain of choice rules, and a weakening of Invariance with respect to Positive Affine Transformations toTranslation Invariance, this new list of axioms is shown to characterizeUtilitarian rules.     

A Twofold Spline Chebyshev Linearization Approach for a Class of Singular Second-Order Nonlinear Differential Equations

The aim of this paper is to present a twofold approach for the numerical solution of a class of singular second-order nonlinear differential equations. The first is based on a modified version of an adaptive spline collocation method (ASCM). The second is a patching approach (PASCM) that splits the problem domain into two subintervals: Chebyshev economization procedure is implemented in the vicinity of the singular point and outside this domain the resulting initial or boundary value problem is handled by the (ASCM). The second strategy is based on the linearization of the nonlinear term about the given initial condition at the singular point. The choice of either technique relies on the specified boundary or initial conditions. Performance of the approach is investigated numerically through a number of application examples that demonstrate the efficiency of the approach and that it has O(h ⁴) rate of convergence. Results confirm that the scheme yields highly accurate results when compared with the exact and/or numerical solutions that exist in the literature.     

Endogenous reference points in bargaining

We allow the reference point in (cooperative) bargaining problems with a reference point to be endogenously determined. Two loss averse agents simultaneously and strategically choose their reference points, taking into consideration that with a certain probability they will not be able to reach an agreement and will receive their disagreement point outcomes, whereas with the remaining probability an arbitrator will distribute the resource by using (an extended) Gupta–Livne bargaining solution (Gupta and Livne in Manag Sci 34:1303–1314, 1988). The model delivers intuitive equilibrium comparative statics on the breakdown probability, the loss aversion coefficients, and the disagreement point outcomes.     

On convergence of the modified Newton’s method under Hölder continuous Fréchet derivative

In this paper, the upper and lower estimates of the radius of the convergence ball of the modified Newton’s method in Banach space are provided under the hypotheses that the Fréchet derivative of the nonlinear operator are center Hölder continuous for the initial point and the solution of the operator. The error analysis is given which matches the convergence order of the modified Newton’s method. The uniqueness ball of solution is also established. Numerical examples for validating the results are also provided, including a two point boundary value problem.     

Non-existence of super-additive solutions for 3-person games

The super-additive solution for 2-person Nash bargaining games (with constant threat) was defined axiomatically inPerles/Maschler [1981]. That paper contains also a study of its basic properties. In this paper we show that the axioms are incompatible even for 3-person unanimity games. This raises the problem of finding a satisfactory generalization of this solution concept to multi-person games.     

A modified Newton’s method for best rank-one approximation to tensors

In this paper, a modified Newton’s method for the best rank-one approximation problem to tensor is proposed. We combine the iterative matrix of Jacobi-Gauss-Newton (JGN) algorithm or Alternating Least Squares (ALS) algorithm with the iterative matrix of GRQ-Newton method, and present a modified version of GRQ-Newton algorithm. A line search along the projective direction is employed to obtain the global convergence. Preliminary numerical experiments and numerical comparison show that our algorithm is efficient.     

Solvability of the 3D rotating Navier–Stokes equations coupled with a 2D biharmonic problem with obstacles and gradient restriction

A coupled system by the 3D rotating Navier–Stokes equations with a mixed boundary condition and a 2D biharmonic problem with two obstacles and the gradient restriction is investigated in this paper. Using the Schauder’s fixed point theorem, we show the existence of a strong solution for a sufficiently large viscosity ν and sufficiently small data.     

Equilibrium selection in a bargaining problem with transaction costs

It is the purpose of the paper to analyse a bargeining situation with the help of the equilibrium selection theory of John C. Harsanyi and Reinhard Selten. This theory selects one equilibrium point in every finite non-cooperative game. The bargaining problem is the following one: the two bargainers — player 1 and player 2 — simultaneously and independently propose a payoffx of player 1 in the interval 〈0, 1〉. If agreement is reached player 2's payoffs is 1?x. Otherwise both receive zero. Each playeri has a further alternativeW _i , namely not to bargain at all (i=1, 2). Thereby he avoids transaction costsc andd of bargaining which arise whether an agreement is reached or not. One may think of an illegal deal where bargaining involves a risk of being punished — independently whether the deal is made or not. The model has the form of a (K+1)×(K+1)-bimatrix game. It is assumed that there is an indivisable smallest money unit. The game hasK+1 pure strategy equilibrium points.K of them correspond to an agreement and the last one is the strategy pair where both players refuse to bargain. Each of theK+1 equilibrium points can be the solution of the game. The aim of the Harsanyi-Selten-theory is to select in a unique way one of these equilibrium points by an iterative process of elimination (by payoff dominance and risk dominance relationships) and substitution. For each parameter combination (c, d) a sequence of candidate sets arises which becomes smaller and smaller until finally a candidate set with exactly one equilibrium point — the solution of the game — is found. For the sake of shortness the paper will report results without detailed proofs, which can be found elsewhere [Leopold-Wildburger].     

Cooperative games with imcomplete information

A bargaining solution concept which generalizes the Nash bargaining solution and the Shapley NTU value is defined for cooperative games with incomplete information. These bargaining solutions are efficient and equitable when interpersonal comparisons are made in terms of certainvirtual utility scales. A player's virtual utility differs from his real utility by exaggerating the difference from the preferences of false types that jeopardize his true type. In any incentive-efficient mechanism, the players always maximize their total virtual utility ex post. Conditionally-transferable virtual utility is the strongest possible transferability assumption for games with incomplete information.     

Wave front set at infinity for tempered ultradistributions and hyperbolic equations

Abstract We study the propagation of singularities and the microlocal behaviour at infinity for the solution of the Cauchy problem associated to an SG-hyperbolic operator with one characteristic of constant multiplicity. We perform our analysis in the framework of tempered ultradistributions, cf. Introduction, using an appropriate notion of wave front set. Keywords: Wave front set at infinity, Tempered ultradistributions, Hyperbolic equations     

A characterization of the Kalai–Smorodinsky bargaining solution by disagreement point monotonicity

We provide a new axiomatization of the Kalai–Smorodinsky bargaining solution, which replaces the axiom of individual monotonicity by disagreement point monotonicity and a restricted version of Nash’s IIA.     

Remarks on Caristi’s fixed point theorem and Kirk’s problem

Without assumptions on the continuity and the subadditivity of η, by means of Caristi’s fixed point theorem, we investigated the existence of fixed points for a Caristi type mapping which partially answered Kirk’s problem and improved Caristi’s fixed point theorem, Jachymski’s fixed point theorem and Khamsi’s fixed point theorem since φ is not necessarily assumed to be bounded below on X.