Multiple Nash Equilibria Research Articles

Service provision decisions of competitive manufacturers: From service spillover perspective

This paper investigates competing manufacturers’ strategic choice of service provision in a two-level supply chain composed two manufacturers and one common retailer. Our results indicate that the strategy that two competing manufacturers provide services is always the dominant choice, and two competing manufacturers could have the highest profit in that case. But when only one of them provides services, the result is related to the potential market size, the unit service cost coefficient and the service spillover intensity. Finally, through the cost sharing and quantity competition extensions, we find the Pareto regions of cost sharing under the equilibrium strategy, and there will be multiple Nash equilibrium regions under the quantity competition.

Berge equilibrium, altruism and social welfare

AbstractWelfare and other properties of Berge equilibria are investigated. In particular, we address the questions to what extent Berge equilibrium can select from multiple Nash equilibria; can serve as a substitute for Nash equilibria; can Pareto improve upon Nash equilibrium. Furthermore, some of the recent results on the relation between Berge equilibria and Kantian equilibria are summarized.

A new bargaining solution for finite offer spaces

AbstractThe bargaining problem deals with the question of how far a negotiating agent should concede to its opponent. Classical solutions to this problem, such as the Nash bargaining solution (NBS), are based on the assumption that the set of possible negotiation outcomes forms a continuous space. Recently, however, we proposed a new solution to this problem for scenarios with finite offer spaces de Jonge and Zhang (Auton Agents Multi-Agent Syst 34(1):1–41, 2020). Our idea was to model the bargaining problem as a normal-form game, which we called the concession game, and then pick one of its Nash equilibria as the solution. Unfortunately, however, this game in general has multiple Nash equilibria and it was not clear which of them should be picked. In this paper we fill this gap by defining a new solution to the general problem of how to choose between multiple Nash equilibria, for arbitrary 2-player normal-form games. This solution is based on the assumption that the agent will play either ‘side’ of the game (i.e. as row-player or as column-player) equally often, or with equal probability. We then apply this to the concession game, which ties up the loose ends of our previous work and results in a proper, well-defined, solution to the bargaining problem. The striking conclusion, is that for rational and purely self-interested agents, in most cases the optimal strategy is to agree to the deal that maximizes the sum of the agents’ utilities and not the product of their utilities as the NBS prescribes.

Environmental delegation versus sales delegation: a game-theoretic analysis

AbstractRecently, in their 2019 paper, Poyago-Theotoky and Yong consider a managerial Cournot duopoly with pollution externalities and emission taxes and propose an explicit environmental incentive in a managerial compensation contract. The authors compare several exogenous equilibria emerging in the symmetric sub-games in which the owner offers either the environmental delegation contract or the standard sales delegation contract: abatement and social welfare (resp. emission taxes) under environmental delegation are higher (resp. lower) than under sales delegation. The present work extends their model using a game-theoretic approach to analyse the asymmetric sub-games, in which only one firm adopts the environmental contract, and adds the contract decision stage. Results show that the environmental contract never emerges as the unique sub-game perfect Nash equilibrium of this non-cooperative managerial decision game. Indeed, if the green R&D technology is efficient, the sales contract emerges as the unique Pareto-inefficient Nash equilibrium. Otherwise, if the green R&D technology is inefficient, multiple Nash equilibria in pure strategies exist (coordination game). Our findings offer direct policy implications.

A two-population game in observable double-ended queuing systems

In this study, the existing game theoretical framework is extended to strategic queuing in search of solutions for a two-population game in observable double-ended queuing systems with zero matching times. We show that multiple Nash equilibria and one unique subgame perfect Nash equilibrium exist in this game.

The Frequency of Convergent Games under Best-Response Dynamics

We calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of n-player, m-strategy normal-form games. To obtain the ensemble, we generate payoff matrices at random. Games with a unique pure strategy Nash equilibrium converge to the Nash equilibrium. We then consider a wider class of games that converge under a best-response dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games with a given number of pure Nash equilibria goes to zero as the number of players or the number of strategies goes to infinity. In the 2-player case, we show that for large games with at least 10 strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an n-partite graph to describe games.

A Multivariate Load Trading Optimization Method for Energy Internet Based on LSTM and Gaming Theory

Energy Internet is a complex nonlinear system. There are many stakeholders in the load trading market, which is usually regarded as a multi-player gaming. Although gaming theory has been introduced to solve Multivariate Load trading problems, different conditions should be considered to accurately optimize the multivariate load trading problem. For example, the selling side needs to reduce the reserve capacity and improve profits, but the consumer side needs to reduce costs and minimize the impact on its own electricity consumption. These contradictory conditions require multiple Nash equilibrium to achieve obviously. To address this issue, a unified architecture of the power system cloud trading is constructed in this paper, which is combined with the multiple load classification of the power system. In addition, according to the power market operation mechanism, a price-guided multivariate load trading game strategy is designed. More importantly, a multivariate load trading optimization method based on LSTM (Long Short-Term Memory) and gaming theory is proposed in this work. LSTM is introduced for real time prediction, which can be combined with the game theory for strategy searching. The global stability and optimal solution theory prove the feasibility of the proposed neural network, and finally the effectiveness of the proposed method is verified by using numerical simulation.

To abate, or not to abate? A strategic approach on green production in Cournot and Bertrand duopolies

This research analyses firms' strategic choice of adopting an abatement technology in an environment with pollution externalities when the government levies an emission tax to incentivise firms to undertake emission-reducing actions. A set of different Nash equilibria – ranging from dirty to green production – arises in both quantity-setting and price-setting duopolies. Results show that if societal awareness towards a clean environment is relatively low (resp. high) and the index measuring the relative cost of abatement is relatively high (resp. low), the strategic interaction between two independent, competing, and selfish firms playing the abatement game leads them not to abate (resp. to abate) as the Pareto efficient outcome: no conflict exists between self-interest and mutual benefit to not undertake (resp. to undertake) emissions-reducing actions. Multiple Nash equilibria or a “green” prisoner's dilemma can also emerge in pure strategies. When the choice of adopting a green technology is an anti-prisoner's dilemma (deadlock), the society is better off, as social welfare under abatement is always larger than under no abatement. These findings suggest that living in a sustainable environment encourages the improvement of public education systems for the achievement of an eco-responsible attitude and the development of clean technologies through ad hoc R&D. • The article analyses the firms' abatement game in a duopoly. • There are pollution externalities and a welfare-maximising emission tax. • A set of different Nash equilibria, ranging from dirty to green production, arises. • Results hold both in quantity-setting and price-setting strategic contexts. • Sustainability requires ad hoc R&D and an eco-responsible attitude.

The value of communication and cooperation when servers are strategic

In 2015, Guglielmi and Badia discussed optimal strategies in a particular type of service system with two strategic servers. In their setup, each server can be either active or inactive and an active server can be requested to transmit a sequence of packets. The servers have varying probabilities of successfully transmitting when they are active, and both servers receive a unit reward if the sequence of packets is transmitted successfully. Guglielmi and Badia provided an analysis of optimal strategies in four scenarios: where each server does not know the other’s successful transmission probability; one of the two servers is always inactive; each server knows the other’s successful transmission probability and they are willing to cooperate.
 
 Unfortunately, the analysis by Guglielmi and Badia contained some errors. In this paper we correct these errors. We discuss three cases where both servers (I) communicate and cooperate; (II) neither communicate nor cooperate; (III) communicate but do not cooperate. In particular, we obtain the unique Nash equilibrium strategy in Case II through a Bayesian game formulation, and demonstrate that there is a region in the parameter space where there are multiple Nash equilibria in Case III. We also quantify the value of communication or cooperation by comparing the social welfare in the three cases, and propose possible regulations to make the Nash equilibrium strategy the socially optimal strategy for both Cases II and III.
 
 doi:10.1017/S1446181120000048

An Improvement of Quantum Prisoners’ Dilemma Protocol of Eisert-Wilkens-Lewenstein

In this paper, an improvement for quantum Prisoners’ Dilemma protocol of Eisert-Wilkens-Lewenstein (EWL) is presented. This improvement differs from EWL protocol in that the initial quantum states of Alice and Bob are changeable by themselves. The research results show that when the social optimal initial quantum states are chosen, the payoffs of Alice and Bob from Mixed Strategies at multiple Nash equilibrium are higher than those of EWL protocol. Moreover, with certain payoff matrix and entanglement degree, this improvement can find pure strategy Nash equilibrium where EWL protocol has none.

Strategic Capacity Planning Problems in Revenue‐Sharing Joint Ventures

We study strategic capacity investment problems in joint ventures (JVs) with fixed‐rate revenue‐sharing contracts. We adopt a game‐theoretical approach to study two types of JVs depending on how individual resources determine the effective capacity of a JV. With complementary resources, the effective capacity of a JV is constrained by the most scarce resource. We show that multiple Nash equilibria could exist. Nevertheless, there exists a unique Strong Nash equilibrium. We show that there is an efficient and fair fixed‐rate revenue‐sharing contract which induces the system optimal outcome in the Strong Nash equilibrium. On the other hand, with substitutable a resource, the effective capacity of a JV is measured by aggregating individual contributions. We show that there does not exist a fixed‐rate revenue‐sharing contract that induces the system optimum. We quantify that the efficiency of a JV which decreases with the number of participants, the cost asymmetry and the cost margin of the JV. We propose provably‐good fixed‐rate revenue‐sharing contracts with performance guarantees. We also propose a simple modified contract to achieve the channel coordination. Finally, we fit our model with historical data to shed some insights on two JV examples in the motion picture industry.

The neural correlates of reaching focal points

In pure-coordination games where there are multiple Nash equilibria, the selection of coordinated responses is inexplicable by rational-choice theory – yet coordination is ubiquitous in daily interactions. The psychological game theory therefore evokes the idea of focal-points: some equilibria being chosen due to its salience, as well as through predicting (i.e., mentalizing) other's response. Previous work has been limited to investigating how structural atrophy relates to deficits in coordination, or how brain activations differ between intuitive and deliberated coordination. In this study, we investigated how the strategy of coordination is reflected in the brain, compared to when no coordination is required. Using functional near-infrared spectroscopy (fNIRS), we examined the neural correlates of deriving a response to a category where participants had to either answer freely (i.e., a survey) or try to match their response with another participant (i.e., coordinate). We found that the coordination trials elicited significantly larger changes in oxyhemoglobin [HbO] concentration than survey trials in frontopolar and lateral prefrontal cortex (PFC). Individual differences in behavioral focal index was significantly correlated with [HbO] concentration in lateral PFC. Granger Causality (GC) analysis revealed greater effective connectivity from frontopolar to lateral PFC, and less GC from lateral PFC to frontopolar in the coordination condition. Our findings highlight the crucial role of frontopolar and lateral PFC in human coordination.

Quantum repeated pricing game

We study a twice-repeated pricing game in the quantum domain. Two firms play this game, and both firms have three pricing strategies. The classical version of the repeated game has multiple Nash equilibria in every stage. We prove that the classical version of this twice-repeated game is a subset of the quantum game. We find that if the initial state is prepared to be entangled, then under certain conditions, there are three pure-strategy Nash equilibria in every stage of the game, and both firms can obtain higher payoffs than they can in the corresponding classical game.

THE VALUE OF COMMUNICATION AND COOPERATION WHEN SERVERS ARE STRATEGIC

In 2015, Guglielmi and Badia discussed optimal strategies in a particular type of service system with two strategic servers. In their setup, each server can be either active or inactive and an active server can be requested to transmit a sequence of packets. The servers have varying probabilities of successfully transmitting when they are active, and both servers receive a unit reward if the sequence of packets is transmitted successfully. Guglielmi and Badia provided an analysis of optimal strategies in four scenarios: where each server does not know the other’s successful transmission probability; one of the two servers is always inactive; each server knows the other’s successful transmission probability and they are willing to cooperate.Unfortunately, the analysis by Guglielmi and Badia contained some errors. In this paper we correct these errors. We discuss three cases where both servers (I) communicate and cooperate; (II) neither communicate nor cooperate; (III) communicate but do not cooperate. In particular, we obtain the unique Nash equilibrium strategy in Case II through a Bayesian game formulation, and demonstrate that there is a region in the parameter space where there are multiple Nash equilibria in Case III. We also quantify the value of communication or cooperation by comparing the social welfare in the three cases, and propose possible regulations to make the Nash equilibrium strategy the socially optimal strategy for both Cases II and III.

Multiple Nash Equilibria in Electricity Markets With Price-Making Hydrothermal Producers

This paper proposes a novel approach incorporating scheduling decisions into a multinodal multiperiod Cournot game. Through applying the Nikaido–Isoda function, market clearing is conducted without dual values being required. Maps of Nash equilibria are obtained through a branch-and-cut algorithm, based on tailored cutting rules. A case study of inertial response requirements shows that these maps and the resulting range of potential player profits can be used to analyze the impacts of policy decisions influenced by discontinuous variables. The case study also shows the financial impact on neighboring producers to the node with applied inertial response requirements.

A decision approach with multiple interactive qualitative objectives for product conceptual schemes based on noncooperative-cooperative game theory

Product development based on a morphological matrix involves the process of decision-based design. Although the decision process can generate conceptual schemes under the guidance of qualitative decision objectives, analysis of the interactions among the qualitative objectives is seldom considered, which can lead to unreliable optimal solutions by combining conflicting principle solutions. In addition, due to the ambiguity of the constraints among the qualitative objectives, multiple feasible schemes with equilibrium states are not considered in the concept decision stage. To solve these problems, a decision approach with multiple interactive qualitative objectives is developed for conceptual schemes based on noncooperative-cooperative game theory to consider the tradeoffs among objectives (e.g., cost, quality and operability) using discrete principle solution evaluation data. First, the morphological analysis method can obtain feasible schemes and determine the principle solutions for each subfunction. Second, the principle solutions are quantified using linguistic terms. Then, the subfunctions are categorized through cluster analysis to determine the suitable principle solution. Third, based on the clustering results, a noncooperative game decision model is constructed to identify multiple Nash equilibrium solutions that satisfy the constraints among the objectives. Fourth, a cooperative game decision model is constructed to obtain the optimal scheme as screened by the noncooperative game model. The case study proves that this approach can choose a relatively superior scheme under the existing technical conditions, thereby preventing inconsistency with the actual design expectations.

On the impact of information disclosure on advance reservations: A game-theoretic view

In many branches of the economy, customers can reserve resources in advance. Yet, service providers often differ in the information they disclose to customers. In this paper, we evaluate how information about server availability impacts the strategic behavior of customers in a loss system with N servers, where each customer can either reserve a server at a certain cost, or take the risk of finding no server available. We formulate the problem as a non-cooperative game with a random number of players. Our main contributions are to establish the existence, structure, and uniqueness (or lack thereof) of pure Nash equilibria, depending on the information disclosed to customers. Specifically, we first prove that if the number of available servers is always disclosed, then there exists exactly one pure Nash equilibrium. High reservation costs lead to an equilibrium in which all servers remain unreserved, while low reservation costs lead to an equilibrium that consists of N “time-thresholds.” A customer that observes n available servers, makes a reservation only if she makes her inquiry before the nth time-threshold. Next, we consider the case where the number of available servers is disclosed only when that number falls below a certain threshold. We show that, in this game, the same types of equilibria prevail. However, multiple Nash equilibria may exist. Finally, we numerically compare the performance of the different policies and we formulate the conjecture that it is preferable for a provider to hide information about the number of available servers.

Efficiency of Revenue Sharing Joint Ventures with Capacity Investment Decisions and Spillovers

We study strategic capacity investment problems in joint ventures (JVs) with fixed-rate revenue sharing contracts. We distinguish two types of JVs, depending on how individual resources determine the effective capacity of a JV. Despite its popularity across many industries, JVs tend to suffer from high failure rates and studies suggested misaligned incentives among JV participants as one of the leading causes. To investigate this phenomenon, we propose a game-theoretical approach that allows asymmetric players and nonlinear convex costs to model the strategic behavior of JV participants. With complementary resources, the effective capacity of a JV is constrained by the most scarce resource. We show that multiple Nash equilibria could exist. Nevertheless, there exists a unique Strong Nash equilibrium. On the other hand, with substitutable resources, the effective capacity of a JV is measured by aggregating individual contributions. We show that there does not exist a fixed-rate revenue sharing contract that induces the system optimum. We quantify that the efficiency of a JV which decreases with the number of participants, the cost asymmetry and the cost margin of the JV. We also consider an extension with spillover effect which results from the knowledge and technology transfer among the JV participants. From a practical perspective, we show that for complementary resource sharing, there is an efficient and fair fixed-rate revenue sharing contract which induces the system optimal outcome in the Strong Nash equilibrium. With substitutable resources, we propose provably-good revenue sharing contracts with performance guarantees. Lastly, we also fit our model with historical data to shed some insights on two JV examples in the motion picture industry.

Social Interaction, Stochastic Volatility, and Momentum

Social interaction contributes to stochastic volatility and momentum in financial markets. By developing a simple evolutionary model of asset pricing and population game, we incorporate social interaction among investors with information uncertainty and show that social interaction leads to the existence of multiple Nash equilibria that characterize two volatility regimes in asset returns and bi-modal distribution in population dynamics. Consequently, the stochastic switching between the two volatility regimes and the persistent dominance of the population with different beliefs generates mis-pricing (bubbles), stochastic volatility, volatility clustering, and time series momentum in the short-run and reversal in the long-run.

Strategic customer behavior facing possible stockout: An experimental study

This paper studies strategic customer behavior in a monopolistic seller system. Given the stock quantity and the selling prices, a customer chooses either to buy at the full price to certainly obtain a product or to wait for a discount with the risk of stockout. We model the behavior of strategic customers as a customer-game. It is shown that the customer-game may have multiple Nash equilibria. Specifically, in the condition of weak incentive compatibility, the trembling hand perfect equilibrium is “all-wait”, which is inconsistent with the “all-buy” assumption in the literature. We conduct laboratory experiments to investigate behavior of strategic customers. The results show that customers' behaviors exhibit a deviation from Nash equilibria. Using a framework of quantal response equilibrium, we develop behavioral models to effectively characterize the behavior of strategic customers.