Non-symmetric Nash Bargaining 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.

Multi-Objective Surface Water Resource Management Considering Conflict Resolution and Utility Function Optimization

In the present research, a multi-objective model is developed for surface water resource management in the river basin area which is connected to the lake. This model considers different components of sustainable water resource management including economic, social and environmental aspects, and simultaneously tries to resolve conflicts between different stakeholders by means of non-symmetric Nash bargaining, which is linked to the multi-objective optimization method. This study proposes a new methodology to improve Nash Conflict Resolution through finding the optimum degree of the utility function. The proposed model is examined in the Zarrineh River basin in Iran. The results show that the amount of available resources or volume of reservoirs play a significant role in determining the optimal degree of the utility function and efficiency of the proposed method in such a way that the higher amount of resources or the larger reservoirs will result in the higher optimal degree of the utility function. In the proposed multi-objective model, two different amounts of surface water inflow are considered. The first assumed amount is the long-term average flow rate and the second one is equal to 80% of the first mode, which is reduced based on the estimated impacts of climate changes. This multi-objective allocation model could supply 100 and 97.5% of the environmental demand of Lake Urmia in the first and second situations, respectively.

Multi-Objective Reservoir Operation with Sediment Flushing; Case Study of Sefidrud Reservoir

In this study, the non-dominated sorting genetic algorithm (NSGA-II) is used for the multi-objective optimization of the Sefidrud reservoir in Northern Iran. The main objectives include water supply, hydropower generation and sediment flushing. In addition to some physical constraints such as the reservoir storage and the outlet flow, maximum flushing outflow and non-flushing in irrigation seasons, the environmental constraints as fish migration and spawning are taken into an account. After obtaining the Pareto optimal solutions by means of the weighted objective functions and non-symmetric Nash bargaining, various scenarios are defined. Then these scenarios are analyzed by introducing a new sustainability index. Furthermore, the different percentages of downstream water demand are considered in order to achieve a better evaluation of the scenarios. The results of study indicate that the optimal solutions are more sustainable than those of the current operation of Sefidrud reservoir, which increase the sediment flushing by 37 million tons compared to the current operation, with the same hydroelectric energy and downstream water supply. The proposed methodology could be used successfully in other reservoir operations including the sediment flushing.

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.

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.