Non-symmetric Solution Research Articles

A two-stage stochastic game model for elective surgical capacity planning and investment

Waiting for elective procedures has become a major health concern in both rich and poor countries. The inadequate balance between the demand for and the supply of health services negatively affects the quality of life, mortality, and government appraisal. This study presents the first mathematical framework shedding light on how much, when, and where to invest in health capacity to end waiting lists for elective surgeries. We model the healthcare system as a two-stage stochastic capacity expansion problem where government investment decisions are represented as a non-symmetric Nash bargaining solution. In particular, the model assesses the capacity requirements, optimal allocation, and corresponding financial investment per hospital, region, specialty, and year. We use the proposed approach to target Chile’s elective surgical waiting lists (2021–2031), considering patients’ priorities, 10 regional health services, 24 hospitals, and 10 surgical specialties. We generate uncertain future demand scenarios using historical data (2012–2021) and 100 autoregressive integrated moving average prediction models. The results indicate that USD 3,331.677 million is necessary to end the waiting lists by 2031 and that the Nash approach provides a fair resource distribution with a 6% efficiency loss. Additionally, a smaller budget (USD 2,000 million) was identified as necessary to end the waiting lists in a longer planning horizon. Further analysis revealed the impact of investment in patient transfer and a decline in investment yield.

Two-party Bargaining Processes Based on Subjective Expectations: A Model and a Simulation Study

This work presents a model for a two-party bargaining process in which multiple offers are exchanged as the negotiation goes on, under a risk of breakdown. Typical game theoretical analyses of such settings assume the breakdown risk is known and the parties are able to calculate an initial offer that is immediately accepted by the other party, ending the negotiation. Aiming to develop a model that is closer to real-life situations, in which parties do exchange many offers in a bargaining process, we consider the parties are unable to compute the far-reaching consequences of their offers, and are guided by their subjective expectations of the outcome of the negotiation. This introduces a new perspective to the analysis of two-party bargaining processes: the confidence of the bargainers in terms of what they hope to achieve by bargaining with each other. The resulting model can be seen as an extension of the Zeuthen-Hicks bargaining model. We show analytically that under the assumption of concave utilities of both parties, the bargaining process converges to the nonsymmetric Nash bargaining solution, where the asymmetry is caused by differences in expectations. This result provides a new interpretation of the parameters of the nonsymmetric Nash bargaining solution, linking them to behavior in the bargaining process. As an additional contribution, we perform a simulation study to examine the role of confidence and to analyze the outcomes for non-concave utility functions.

Asymmetric Propellant Positioning in Symmetric Tanks and Propellant Management Devices

Dangers of assuming symmetric liquid and gas positioning in symmetric tanks with symmetric propellant management device vanes are identified. Examples from spaceflight history plus modeling of proposed designs illustrate the issue in numerous geometries. Spaceflight hardware and research geometries present the same behavior: that the minimum-energy state for static positioning of liquid and gas in a partially filled vessel with axial symmetry or with, for example, fourfold symmetry of propellant management device vanes need not be symmetric. Because it is simple to assume a symmetric solution, the reality of the nonsymmetric solution may surface as a potentially crippling anomaly in spaceflight or in 0 research operations. Altered mass properties and thermal response of tanks to heaters are two important changes that nonsymmetric liquid positioning can produce. The surprises are avoidable by ground-based analysis and modeling that consider the possibility of such solutions.

Radial confinement of deeply trapped particles in a non-symmetric magnetohydrodynamic equilibrium

Quasisymmetry and omnigeneity of an equilibrium magnetic field are two distinct properties proposed to ensure radial localization of collisionless trapped particles in any stellarator. These constraints are incompletely explored, but have stringent restrictions on a magnetic geometry. This work employs an analytical approach to understand the implications of the constraints. The particles move in an intrinsically three dimensional equilibrium whose representation is given by the earlier work of Weitzner and its extension here. For deeply trapped particles, a local equilibrium expansion around a minimum of the magnetic field strength along a magnetic line suffices. This analytical non-symmetric equilibrium solution enables explicit representation of the constraints. The results show that it is far easier to satisfy the omnigeneity condition than the quasisymmetry requirement. Correspondingly, there exists a large class of equilibrium close to quasisymmetry that remains omnigeneous while allowing inclusion of error fields, which may destroy quasisymmetry.

حل تعارضات برای مدیریت پایدار منابع آب بر اساس تئوری بازی؛ مطالعه موردی

About 90 percent of water consumption is allocated to the agricultural sector in Iran. Obviously, the amount of agricultural benefit depends on the amount of water resource using. The farmer imagines that increasing the amount of cultivated area and irrigation will also increase amount of the crop and consequently his benefit. The assumptions of farmers lead to the destruction of the environment which finally will result in the tragedy of common pool resources. Preventing water resource overdraft in order to support sustainable development as the environmental object and maximizing the farmers' net benefits as the economic object, are in conflict in the short-term. It is necessary to find a balance between them. For this purpose, the optimum consumption of a water resource and farmers' net benefits are determined for the Urmia study region during the 2014-2015 crop year. This is done by using conflict resolution methods including a non-symmetric Nash, non-symmetric Kalai-Smorodinsky, non-symmetric area monotonic and non-symmetric equal loss solution in four crop pattern scenarios. The results revealed that increasing environmental weight decreases cultivated area and consequently agricultural net benefit in all crop scenarios. The fourth scenario is more reasonable and admissible due to its feasibility and acceptability by farmers. Also a comparison of the fourth scenario with the current situation in equal weights of environmental and economic objects and by assuming cultivation of major crops in the region showed that in this scenario the water consumption rate reduces 17% and the net benefit of farmers increases 10%.

Intergovernmental fiscal relationship in China: a simple model based on the nonsymmetric Nash solution

We propose a new empirical approach to analyze fiscal decentralization and apply it to Chinese intergovernmental fiscal relationships between the central government and provincial governments, using nonsymmetric Nash solution. In calculating budgetary revenue and expenditure shares, we include extra budgetary revenue and expenditure. We find that although an increase in either income inequality or real per capita GDP lowers local governments’ bargaining power within the budgetary system, local governments can offset this by obtaining more bargaining power over extra budgetary expenditures. Another finding is that although urbanization increases provincial governments’ budgetary revenues, it also restricts the scope for further budgetary expenditure.

Competitive Equilibrium in Concave Linearly Homogeneous Exchange Economies and the Non-Symmetric Nash Bargaining Solution: A Note on a Theorem Due to Eisenberg

In this paper we show that for concave linearly homogeneous exchange economies every competitive equilibrium satisfies the property that the competitive allocation is a non-symmetric Nash bargaining solution with weights being the initial income of individual agents evaluated at the equilibrium price vector. This follows as a simple consequence of the same result for linear exchange economies which we also prove here. We also show that a competitive equilibrium exists for such economies. On the way we provide a new proof of a result due to Eisenberg (1961) that says an allocation which is a non-symmetric Nash bargaining solution with weights being the initial income of individual agents is a market equilibrium allocation corresponding to the given initial incomes. All our proofs are very simple in comparison to alternative routes to the results that are currently in existence.

Existence of Competitive Equilibrium in Linear Exchange Economies: A Simple Proof Using Brouwer’s Fixed Point Theorem

In this paper we show that for concave piecewise linear exchange economies every competitive equilibrium satisfies the property that the competitive allocation is a non-symmetric Nash bargaining solution with weights being the initial income of individual agents evaluated at the equilibrium price vector. We prove the existence of competitive equilibrium for concave piecewise linear exchange economies by obtaining a non-symmetric Nash bargaining solution with weights being defined appropriately. In a later section we provide a simpler proof of the same result using the Brouwer’s fixed point theorem, when all utility functions are linear. In both cases the proofs pivotal step is the concave maximization problem due to Eisenberg and Gale and minor variations of it. We also provide a proof of the same results for economies where agents’ utility functions are concave, continuously differentiable and homogeneous. In this case the main argument revolves around the concave maximization problem due to Eisenberg. Unlike previous results, we do not require all initial endowments to lie on a fixed ray through the origin. 1. Introduction : Eisenberg and Gale (1959) showed that for linear exchange economies with individual initial endowment bundles lying on a fixed ray through the origin, the proof of existence of a competitive equilibrium reduces to finding a solution for a concave programming problem. In particular they showed that the corresponding competitive allocation is a nonsymmetric Nash bargaining solution (see Kalai (1977)) with weights being given by the share of each individual in the aggregate initial endowment. Moore (2007) refers to the non-symmetric Nash bargaining solution as a solution to the Cobb-Douglas-Eisenberg (CDE) aggregator function. In this paper we first show that for concave piecewise linear exchange economies every competitive equilibrium satisfies the property that a competitive allocation is a non-symmetric

Game Theory Applications in a Water Distribution Problem

A water distribution problem in the Mexican Valley is modeled first as a three-person noncooperative game. Each player has a five-dimensional strategy vector, the strategy sets are defined by 15 linear constraints, and the three payoff functions are also linear. A nonlinear optimization problem is first formulated to obtain the Nash equilibrium based on the Kuhn-Tucker conditions, and then, duality theorem is used to develop a computational procedure. The problem can also be considered as a conflict between the three players. The non-symmetric Nash bargaining solution is suggested to find the solution. Multiobjective programming is an alternative solution concept, when the water supply of the three players are the objectives, and the water authority is considered to be the decision maker. The optimal water distribution strategies are determined by using these solution concepts and methods.

Discussion on symmetry of optimum topology design

Recent study by Rozvany discussed an important topic in structural topology optimization: non-uniqueness and symmetry. This paper demonstrates that for the frame topology optimization with frequency objective or frequency constraints, the topology optimum is not symmetric even when the geometry, material distribution and ground structure are symmetric. Two benchmark examples with fundamental frequency constraints support our view. Nonlinear and non-convex constraints are contributors to non-symmetric optimum solution for the class of topology optimization problems considered here.

Hollow sphere models, conic programming and third stress invariant

In this paper the hollow sphere model is investigated within the framework of limit analysis (LA), using the classical two-part velocity field, i.e., the exact solution for hydrostatic loading plus a linear solution, in both cases of von Mises and Drucker–Prager matrices. We use the kinematic LA approach in a quasi-analytical approach by imposing the plastic admissibility (PA) condition (Drucker–Prager matrix) or upper bounding the dissipated power (von Mises matrix) in a sufficiently high number of distributed points, thanks to conic programming formulations. Then we analyze the “porous Drucker–Prager” case to confirm that the so-called UBM (Upper Bound Model) approach of Guo et al. (2008) is only an estimate, although a good one in fact. Moreover, it is shown that the mean stress axis should not be a strict axis of symmetry for the macroscopic criterion. Then, considering the “Porous von Mises” case, we obtain that the real criterion is not only lower than the Gurson criterion, but probably non-symmetric with respect to the mean stress axis, more than in the Drucker–Prager case. We finally use ad hoc updated 3D-FEM LA codes to confirm the previous results and to evaluate the entire influence of the third stress invariant: the classical (Σm, Σeqv) formulation of the Gurson criterion clearly overestimates the real, non-symmetric solution of the hollow sphere model, at least for porosities of the same order of magnitude as the value used in this work.

Positive eigenvector of nonlinear perturbations of nonsymmetric M-matrix and Newton iterative solution

Positive eigenvector of nonlinear perturbations of nonsymmetric M-matrix and its Newton iterative solution are studied. It is shown that any number greater than the smallest positive eigenvalue of the M-matrix is an eigenvalue of the nonlinear problem and that the corresponding positive eigenvector is unique and the Newton iteration of the positive eigenvector is convergent. Moreover, such positive eigenvectors form a monotone increasing and continuous function of the corresponding eigenvalues.

A Two-Person Conflict Model in Train Scheduling

This paper presents a two-person bargaining model in which railway company and its passenger community are the players in the train-scheduling problem. The fuel consumption cost and the total passenger-time constitute the payoff functions. First, the Pareto frontier is determined using the ∊-constraint method and then conflict resolution is performed using five particular resolution methodologies, namely the non-symmetric Nash solution, the non-symmetric Kalai-Smorodinsky solution, the non-symmetric area monotonic solution, the non-symmetric equal loss solution, and a one-shot solution concept. A numerical example is presented to illustrate the methodology.

Energy adaptation of non-associated plasticity tangent matrices for symmetric solvers

The behaviour of most civil engineering materials is considered by many to be best characterised by non-associated plasticity models which generate unsymmetric stiffness matrices. Various non-symmetric solution algorithms have been developed for solving such problems but these require large core storage, (approximately twice the size required for symmetric solution algorithms), and the computational process is consequently slowed down especially for very large problems. Two groups of eight symmetric solution techniques are proposed in this paper for non-associated plasticity problems using the total and plastic work done as scalar measures to adapt their tangent matrices for symmetric solvers. Five numerical examples are used to illustrate the performance and limitations of the techniques in terms of numerical stability, solution accuracy and computation times.

A new Characterization of the non-symmetric Nash solution

A one-step stochastic bargaining procedure is examined which is based on assuming offer-dependent break-down probabilities and that at each step, the players optimize their expected payoffs. It is shown that there is a unique solution of the bargaining process and it reduces to the non-symmetric Nash solution if each player initially offers the disagreement payoff value to the other player. Therefore the one-step bargaining solution is a further generalization of the non-symmetric Nash solution with arbitrary initial offers.

A STOCHASTIC BARGAINING PROCESS AND CORRESPONDING ONE-SHOT SOLUTION CONCEPT

A stochastic bargaining process is first introduced to solve two-person conflicts. The outcome of the process depends on the initial offers of the two players, as well as on the offer-dependent breakdown probabilities. After the convergence (in most cases the finiteness of the process) is verified, a one-shot solution is introduced. The existence of a unique solution is then proven, and its relation to the non-symmetric Nash solution is discussed.

A bifurcation study of natural convection in porous media with internal heat sources: the non-Darcy effects

Multiplicity features of natural convection flow in porous media, generated and sustained by a uniform internal heat source are investigated. The flow, in a two-dimensional enclosure, is described by the Brinkman's extension of the Darcy equation. No-slip boundary conditions are used. The focus is on the role of the Brinkman viscous term in influencing the location of singular points. The behavior of the system is regulated by two control parameters, the Rayleigh number (the dynamic parameter) and the Darcy number. The singular solutions are constructed using algorithms from bifurcation theory. Multiple solutions consisting of symmetric and nonsymmetric solution branches, are revealed as the control parameters change. The range of the Rayleigh number for which a unique solution exists is enlarged when the Darcy number is increased.

Some properties of rotating coherent structures in a non-neutral plasma column

Coherent structures rotating at the angular velocity ω in a two-dimensional pure electron plasma confined inside a conducting grounded cylinder are considered. These structures are described by a streamfunction ψ, obeying a nonlinear boundary value problem. It is shown, in particular, that (i) this problem has a minimal cylindrically symmetric solution if it has a solution at all. (ii) For a nonsymmetric solution, 2ω is necessarily comprised between the minimum and maximum values taken by the local diocotron frequency in the cylinder. (iii) Bifurcation of a symmetric radially decreasing solution to a nonsymmetric one can occur only when 2ω is equal to the mean diocotron frequency.

On bargaining position descriptions in non-transferable utility games

We present a generalization to the Harsanyi solution for non-transferable utility (NTU) games based on non-symmetry among the players. Our notion of non-symmetry is presented by a configuration of weights which correspond to players' relative bargaining power in various coalitions. We show not only that our solution (i.e., the bargaining position solution) generalizes the Harsanyi solution, (and thus also the Shapley value), but also that almost all the non-symmetric generalizations of the Shapley value for transferable utility games known in the literature are in fact bargaining position solutions. We also show that the non-symmetric Nash solution for the bargaining problem is also a special case of our general solution. We use our general representation of non-symmetry to make a detailed comparison of all the recent extensions of the Shapley value using both a direct and an axiomatic approach.

An axiomatization of the weighted NTU value

Non-symmetric generalizations of the non-transferable utility (NTU) are defined and characterized axiomatically. The first of these is a weighted NTU value that is identical to the (symmetric) NTU value when players have the same weights. On the class of transferable utility games, this weighted NTU value coincides with the weighted Shapley value and on the pure bargaining games it coincides with the non-symmetric Nash bargaining solution. A further extension, the random order NTU value, is also defined and axiomatized and its relationship to the core is discussed.