Bargaining Solution Research Articles

A note on indeterminacy of unemployment equilibrium

According to Mortensen (1989), increasing returns to scale in matching may cause multiple equilibria if collective bargaining or efficiency wage is applied. I show that a unique equilibrium exists, also if the Nash bargaining solution is used, if productivity is constant.

Gately values of cooperative games

AbstractWe investigate Gately’s solution concept for cooperative games with transferable utilities. Gately’s conception introduced a bargaining solution that minimises the maximal quantified “propensity to disrupt” the negotiation process of the players over the allocation of the generated collective payoffs. We show that Gately’s solution concept is well-defined for a broad class of games and that it can be interpreted as a compromise solution. We also consider a generalisation based on a parameter-based quantification of the propensity to disrupt. We provide an axiomatic characterisation of the original Gately value as well as these generalised Gately values. Furthermore, we investigate the relationship of these Gately values with the Core and the Nucleolus and show that Gately’s solution is in the Core for all regular 3-player games, but is fundamentally different from the Nucleolus. We identify exact conditions under which these Gately values are Core imputations for arbitrary regular cooperative games. Finally, we investigate the relationship of the Gately value with the Shapley value.

The Nash bargaining solution and utilitarian–egalitarian social welfare functions

The 2-person Nash bargaining solution is the only scale covariant one that maximizes, on every problem, some mixture of the utilitarian and egalitarian social welfare functions. With three or more players, no solution satisfies these two axioms.

Step-by-step negotiations and the Nash bargaining solution: Efficiency-free characterizations

Real-world bargainings often proceed step-by-step: the agents make intermediate agreements today and continue to negotiate tomorrow to reach a final agreement. We consider two natural properties of the bargaining solutions in such step-by-step negotiations and show that the Nash solution is the only solution that satisfies either of them, a very weak axiom of individual rationality, and Nash’s axioms except for Pareto optimality.

Capacity and operation joint optimization for integrated energy system based on Nash bargaining game

Integrated energy systems (IESs) can uniformly coordinate different locations and diversified forms of energy systems, which is critical for ensuring low-carbon and sustainable energy usage. This paper proposed a capacity and operation joint optimization method for an IES with multiple interconnected energy hubs (EHs) based on the Nash bargaining game. Firstly, considering the uncertainty of the EH, robust optimization was carried out for both long-term capacity planning and short-term operational strategies at the same stage. Furthermore, the optimization of the single EH is embedded in the cooperative framework of the IES, and it is formulated as an asymmetrical Nash bargaining game model, in which the players bargain over the transaction energy. Finally, this game model is solved by the alternating direction method of multipliers algorithm. Case studies with a typical IES including three EHs are conducted to verify the effectiveness of the proposed methods. It was found that the proposed joint optimization method has better computational ability than traditional two-stage method. The asymmetrical bargaining solution can not only effectively reduce the cost of the IES, but also help achieve a fair benefit balance among EHs.

Reputational Bargaining with External Resolution Opportunities

Abstract Two parties negotiate in the presence of external resolution opportunities (e.g. court, arbitration, or war). The outcome of external resolution depends on the privately held justifiability/strength of their claims. A justified party issues an ultimatum for resolution whenever possible, but an unjustified party strategically bluffs with an ultimatum to establish a reputation for being justified. We show that the availability of external resolution opportunities can benefit or hurt an unjustified party in equilibrium. When the chances of being justified become negligible, agreement is immediate and efficient; and if the set of justifiable demands is rich, our solution modifies the Nash–Rubinstein bargaining solution of Abreu and Gul ((2000), Econometrica, 68, 85–117) in a simple way.

Sum-rate maximization for a distributed space-time block code-aided cooperative NOMA with energy harvesting

AbstractIn this paper, we exploit the spatial and transmission diversities in cooperative non-orthogonal multiple access (C-NOMA) networks to improve the system sum-rate. To achieve this, we propose a user-pairing scheme where near-field user pairs serve as relays for user pairs in the far-field region. Based on this pairing scheme, we incorporate a space-time block code transmission technique at the near-field user pairs to maximize the transmission diversity in the cooperative phase. Moreover, we consider a non-linear energy harvesting model at the near-field user pair to alleviate the problem of energy consumption during the cooperative transmission phase. Further to this, we formulate a sum-rate maximization problem that is addressed from the viewpoint of joint power allocation factor and power splitting ratio optimization. We develop a low-computational iterative algorithm based on the concepts of the Stackelberg game and the Nash bargaining solution. We benchmark our findings with different scenarios, such as energy harvesting C-NOMA with a fixed power allocation factor and power splitting ratio, energy harvesting C-NOMA without STBC, non-cooperative NOMA, and orthogonal multiple access. The results obtained via simulations outperform the benchmark schemes.

Target setting for airlines: A DEA bargaining approach with network structure

AbstractThe objective of this study is to find out how airlines can reduce their inputs and provide appropriate passenger and cargo capacity, thereby increasing passenger and cargo traffic and reducing CO2 emissions. A network data envelopment analysis (DEA)–based Nash bargaining model is developed incorporating intermediate products and weakly disposable undesirable outputs and is applied to a dataset of 33 airline companies to obtain the bargaining solution. According to the results, airlines that are weak efficient or inefficient could establish their targets to attain efficiency through a fair and logical negotiation among the inputs, intermediate products, good outputs and bad output. Appropriate policies need to be implemented that would improve the airlines’ performance in terms of improvement in not only the efficiency of inputs and desirable outputs along with intermediate products but also with regard to environmental performance. This would ensure sustainable growth in airlines’ performance.

Fair Exploration via Axiomatic Bargaining

Exploration is often necessary in online learning to maximize long-term rewards, but it comes at the cost of short-term “regret.” We study how this cost of exploration is shared across multiple groups. For example, in a clinical trial setting, patients who are assigned a suboptimal treatment effectively incur the cost of exploration. When patients are associated with natural groups on the basis of, say, race or age, it is natural to ask whether the cost of exploration borne by any single group is “fair.” So motivated, we introduce the “grouped” bandit model. We leverage the theory of axiomatic bargaining, and the Nash bargaining solution in particular, to formalize what might constitute a fair division of the cost of exploration across groups. On one hand, we show that any regret-optimal policy strikingly results in the least fair outcome: such policies will perversely leverage the most “disadvantaged” groups when they can. More constructively, we derive policies that are optimally fair and simultaneously enjoy a small “price of fairness.” We illustrate the relative merits of our algorithmic framework with a case study on contextual bandits for warfarin dosing where we are concerned with the cost of exploration across multiple races and age groups. This paper was accepted by David Simchi-Levi, data science. Funding: This work was supported by the National Science Foundation, Division of Civil, Mechanical and Manufacturing Innovation [Grant 1727239]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.01985 .

Collaborative game-based task offloading scheme in the UAV-TB-assisted battlefield network platform

In the sixth-generation (6G) wireless networks, the use of unmanned aerial vehicles (UAVs) and tethered balloons (TBs) to assist cellular networks has attracted considerable attentions due to their dynamic and quick deployment with their relative low cost. In this article, we propose a new task offloading scheme for smart devices in the modern battlefield area. By the integrative platform of TBs, UAVs and smart devices, the main challenges are (i) providing a task splitting algorithm for the partial offloading service, and (ii) develop a TB resource sharing algorithm to handle the offloading requests. For convenient wireless communications, UAVs work as relay nodes between TBs and individual devices. To achieve a mutually desirable solution, our proposed scheme is formulated as cooperative game models. First, the sequential Raiffa bargaining solution is applied to split the computation-intensive task of each smart device in the battlefield area. Second, the average-surplus value is adopted to effectively share the TB computing resource. Based on the reciprocal combination of two cooperative game solutions, we explore the sequential interaction of TBs, UAVs and battlefield devices, and jointly design our integrated control scheme for offloading services. According to the synergy effect, our hybrid approach can provide a fair-efficient solution in the UAV-TB-assisted battlefield network infrastructure. Finally, extensive simulations are conducted, and the results demonstrate the superiority of our proposed scheme over the existing baseline protocols.

Collective or individual rationality in the Nash bargaining solution: efficiency-free characterizations

AbstractIn the classical bargaining problem, we propose a very mild axiom of individual rationality, which we call possibility of utility gain. This requires that for at least one bargaining problem, there exists at least one player who reaches a higher utility level than their disagreement utility. This paper shows that the Nash solution (Nash in Econometrica 18(2):155–162, 1950) is characterized by possibility of utility gain and continuity with respect to feasible sets together with Nash’s axioms except weak Pareto optimality. We also show that in Nash’s theorem, weak Pareto optimality can be replaced by conflict-freeness (introduced by Rachmilevitch in Math Soc Sci 76(C):107–109, 2015). This demands that when the agreement most preferred by all players is feasible, this should be chosen. Furthermore, we provide alternative and unified proofs for other efficiency-free characterizations of the Nash solution. This clarifies the role of each axiom in the related results.

Strengthening Cooperative Consensus in Multi-Robot Confrontation

Multi-agent reinforcement learning (MARL) has proven effective in training multi-robot confrontation, such as StarCraft and robot soccer games. However, the current joint action policies utilized in MARL have been unsuccessful in recognizing and preventing actions that often lead to failures on our side. This exacerbates the cooperation dilemma, ultimately resulting in our agents acting independently and being defeated individually by their opponents. To tackle this challenge, we propose a novel joint action policy, referred to as the consensus action policy (CAP). Specifically, CAP records the number of times each joint action has caused our side to fail in the past and computes a cooperation tendency, which is integrated with each agent’s Q -value and Nash bargaining solution to determine a joint action. The cooperation tendency promotes team cooperation by selecting joint actions that have a high tendency of cooperation and avoiding actions that may lead to team failure. Moreover, the proposed CAP policy can be extended to partially observable scenarios by combining it with Deep Q network or actor-critic–based methods. We conducted extensive experiments to compare the proposed method with seven existing joint action policies, including four commonly used methods and three state-of-the-art methods, in terms of episode rewards, winning rates, and other metrics. Our results demonstrate that this approach holds great promise for multi-robot confrontation scenarios.

On bargaining norms as solutions to cost–minimization problems

This paper studies bargaining outcomes in economies in which agents may be able to impose outcomes that deviate from the relevant social norms but incur costs when they do so. It characterizes bargaining outcomes that are easiest for a society to sustain as part of a social norm that everybody will want to follow. Depending on technological assumptions, the approach yields the Nash bargaining solution, the Kalai–Smorodinsky solution, the equal monetary split, and other bargaining solutions. Set‐valued solution concepts are derived that are relevant if one is unable or unwilling to make specific technological assumptions.

A Fair Multi-Partner Profit Allocation for Islanded Micro-grid

A micro-grid (MG) is regarded as a localized small-scale power supply system designed for providing electrical power for a remote community, which may be connected to the grid. The paper considers six participants’ sites in a remote community. The peak energy consumption patterns for these participants must be different to ensure coalition for mutual benefits and this would facilitate efficient utilization of energy resources. This community concerned may include rural communities having participants such as schools, housing estates, industrial sites, etc. The propose approach would ensure that many participants and their needs are captured and optimized profit of the participants would ensure mutual benefits. In islanded MG, most of the cost/profit distributions in energy trading problems are done based on the self-interest of the participants. Some consider a fair (acceptable or reasonable) cost/profit distribution using the cooperative game theory (CGT) based on the Nash bargaining solution (NBS). However, this method leads to a high degree of dissatisfaction. To improve the fairness and mutual benefits for all MG participants, a new flexible approach is developed. Specifically, CGT which uses a generalized NBS (GNBS) has been proposed, which relaxes certain axiom of the NBS to ensure that fairness of game theory is achieved. The approach allows the profit of the MG participants to be shared using a mechanism based on the negotiation power (i.e. weighted fairness) so that the MG would attain an improved economic outcome. Empirical evaluation of the approach indicates that overall profits using CGT is higher when compared with an independent approach. Moreover, different in negotiation power is achieved in which higher profit is allocated to any participant based on participant's mutual agreement in achieving fair profit distribution. A robust optimization algorithm called Teaching-Learning-Based-Optimization (TLBO) is developed to obtain optimal results. The heuristics algorithms are used in testing the effectiveness of TLBO, which confirm that the TLBO is indeed powerful and robust enough to give quality solutions in solving energy-trading problems.

Cooperative advertising and coordination in a supply chain: the role of Nash bargaining fairness concerns

This study constructs a game-theoretic model of cooperative advertising in a supply chain with Nash bargaining solutions as the fairness reference point. We use a square root response function to describe the saturation effect of advertising. We find that the retailer’s Nash bargaining fairness concerns (NBFC) improve the local advertising investment even more than the level of the centralized case. The effect of NBFC on the retailer’s profit is inverted U-shaped, rising first and then falling, and vice versa for the manufacturer. As the fairness-concerned coefficient increases from small to large, the efficiency of the supply chain changes from improvement to decline. Moreover, we find that a two-way subsidy mechanism in cooperative advertising still works on coordination although the retailer possesses NBFC. The study offers practical management insights into the operational strategies of supply chain members.

Fixed cost allocation using asymmetrical core-Nash bargaining DEA game

This paper introduces cooperative game theory into the data envelopment analysis (DEA) method for building a novel fixed cost allocation (FCA) approach from group and individual consciousness. We start by acknowledging the divergences of decision-making units (DMUs) and propose an asymmetrical Nash bargaining DEA game, where DMUs’ bargaining powers are defined through the functions of cross-efficiency and operation scale. However, the asymmetrical Nash bargaining solution is proposed from an individual perspective, without providing the theoretical support of full cooperation. To narrow this gap, we further introduce the core concept of a coalitional cooperative game into the built asymmetrical Nash bargaining DEA game, resulting in the asymmetrical core-Nash bargaining DEA game. The asymmetrical core-Nash bargaining solution is based on the common weights, with the properties of stability, fairness, Pareto-efficiency, and invariance to affine transformation. Furthermore, our proposed approach inherently ensures the uniqueness of the fixed cost scheme. Finally, numerical analysis and case study are provided to show the concrete application of the new method.

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.

Dominant Smart Contracts Based on Major Bargaining Solutions

We consider a situation in which two parties have concluded an efficient contract corresponding to one major bargaining solution. After the parties have agreed on one particular contract, an unanticipated shock may change the contract outcomes in a way that benefits one party but harms the other party. If this happens, they have the option to either stay with the original exchange contract or adjust some contract parameters such as the price. We propose a model to perform such adjustments automatically, to obtain the same bargaining solution as in the initial contract under the restriction that the new contract dominates the outcomes of the original contract. We study several bargaining solutions within this general framework. These bargaining solutions offer various sharing rules to distribute the benefit between the parties. To reflect practical considerations, we only consider adjustments made via one contract parameter (the price), while all other parameters result from the original contract and the random shock. To evaluate the efficiency of the proposed approach, we also compare it to a full re-negotiation scenario, in which all parameters can be modified within the boundaries resulting after the random shock. However, waiting and re-negotiation might be costly compared to the situation when the smart contract executes the adjustment automatically. Therefore, the automatic adjustment might be more efficient compared to the other types of contracts. We present several numerical examples and run large random simulations, which we also check statistically.

Cooperation maintenance in dynamic discrete-time multicriteria games with application to bioresource management problem

We consider a dynamic, discrete-time, game model where the players use a common resource and seek to optimize different criteria. The center (referee) shares the resource available for exploitation between the competitors. To construct a multicriteria Nash equilibrium the bargaining solution is adopted. To obtain a multicriteria cooperative equilibrium, a modified bargaining scheme that guarantees the fulfillment of individual rationality conditions is applied. Two approaches to maintain cooperative behavior, namely, the cooperative incentive equilibrium and the time-consistent payoff distribution procedure, are considered. Moreover, the cooperative incentive equilibrium is constructed under the assumption that the center punishes players for deviations. A dynamic bi-criteria bioresource management problem is investigated to illustrate the solution concepts proposed.

Overwhelmed by routine tasks: A multitasking principal agent perspective

Employees often report being overloaded with routine tasks, which distracts them from more demanding and rewarding work. We analyze this in a multitasking model involving a verifiable routine task and a skill-dependent task characterized by moral hazard. Multitasking is costly so that high- and low-skilled employees specialize, and only employees with intermediate productivity handle both tasks. In the Nash bargaining solution of contract negotiations, work overload in the routine task is used to lower the informational rent from the non-verifiable task. Compared to the first-best solution, more employees perform both tasks and effort in the routine task is inefficiently large. Moreover, the payment structure is such that the employee would prefer to allocate more effort to the demanding task, which we interpret as a loss of control over work autonomy. Reducing employees' bargaining power exacerbates the problem, while the effects of improved monitoring are mixed.