Non-cooperative Nash Equilibrium Research Articles

Public Input Competition under Stackelberg Equilibrium: A Note

This paper examines the Stackelberg equilibrium for public input competition and compares it with the non-cooperative Nash equilibrium. Given two asymmetric regions, we show that under the Nash equilibrium, the more productive region tends to spend more on public input, which results in this region attracting more capital than the less productive region. The comparison of the two equilibria reveals that the leader region obtains a _rst-mover advantage under the stackelberg setting. This suggests that if regions interact with each other sequentially as in the Stackelberg equilibrium, then the regional disparity that is due to the heterogeneity of productivity is likely to be mitigated or enlarged, depending on which region performs the leadership role in the competition process.

Decomposition algorithm of searching equilibria in a dynamic game

This paper considers a noncooperative game of several players (governments of neighboring countries) participating in emission reduction trading. A special emphasis is placed on the case of two players, one representing Eastern European countries and the other being the countries of the former Soviet Union. We perform statistical analysis of the model parameters based on real data under quadratic cost functions and logarithmic utility functions. The concepts of a noncooperative Nash equilibrium and cooperative Pareto maxima are introduced and connections between them are established. A new concept, i.e., a market equilibrium, which combines the properties of Nash and Pareto equilibria, is rigorously defined. An analytical solution of the market equilibrium problem is given. This analytical solution can serve for verification of numerical search algorithms. In addition, we propose a computational algorithm of market equilibrium search, which shifts a competitive Nash equilibrium to a cooperative Pareto maximum. The algorithm is interpreted as a repeated auction, where the auctioneer possesses no information about the cost functions and the functions of environmental effect from emission reduction of the participating countries. An auctioneer strategy leading to market equilibrium attainment is considered. From the game-theoretic viewpoint, a repeated auction describes a learning process in a noncooperative repeated game under uncertainty. We compare the results gained by the proposed computational algorithms with their analytical counterparts. And finally, numerical calculations of equilibrium and algorithm trajectories converging to the equilibrium are demonstrated.

On the Complementarity between Law and Social Norms

AbstractThis article aims to clarify the relationship between the law and social norms and examine how they interact – whether the law completely replaces, or coexists with, the pre-existing social norms. We model a close-knit society consisting of injurers and victims, assuming that both the law and social norms maximize social welfare and that the costs of enforcing legal penalties are greater than those of enforcing social norms. We find that social norms completely replace the law, even in a non-cooperative Nash equilibrium; we then develop exceptions to this result. In particular, when community sanctions for violating social norms are calibrated without consideration of its marginal social benefit due to a reduction in law enforcement cost, our model can have multiple equilibria, including cases in which legal sanctions persist even in the long run, in spite of their inefficiently high enforcement costs. However, we also show that this possibility of an inefficient non-cooperative equilibrium can be eliminated if the government behaves as a Stackelberg leader for the formation of social norms.

Generalized Nash Equilibrium Problems in Banach Spaces: Theory, Nikaido--Isoda-Based Path-Following Methods, and Applications

Building upon the results in [M. Hintermuller and T. Surowiec, Pac. J. Optim., 9 (2013), pp. 251--273], a class of noncooperative Nash equilibrium problems is presented, in which the feasible set of each player is perturbed by the decisions of their competitors via a convex constraint. In addition, for every vector of decisions, a common “state” variable is given by the solution of an affine linear equation. The resulting problem is therefore a generalized Nash equilibrium problem (GNEP). The existence of an equilibrium for this problem is demonstrated, and first-order optimality conditions are derived under a constraint qualification. An approximation scheme is proposed, which involves the solution of a parameter-dependent sequence of standard Nash equilibrium problems. An associated path-following strategy based on the Nikaido--Isoda function is then proposed. Function-space-based numerics for parabolic GNEPs and a spot-market model are developed.

Non-Cooperative Decentralized Charging of Homogeneous Households' Batteries in a Smart Grid

For the efficient operation of the smart grid, it is important that there is an instant by instant matching between the electricity supply and the power consumption. Electrical power storage provides a viable solution to managing power supply and electrical loads as well as unexpected imbalances. Electricity suppliers could deploy electricity storage facilities at various levels of the smart grid system: generation, transmission, substations and residential level. Storage would significantly address the power quality and reliability problems through peak shaving and frequency control. It also reduces the need for huge infrastructural expenditures by making them more efficient. At the residential level, smart storage together with dynamic pricing in the deregulated electricity markets presents the electricity suppliers with a strategy to achieve grid stability. In this paper, we consider a smart grid environment with a high penetration of households' storage batteries. By using an appropriate electricity price structure, the electricity supplier influences households' electricity consumption. On the other hand, the households aim to minimize their electricity bills by capitalizing on price fluctuation to schedule their electrical appliances and coordinate the charging and discharging of their batteries. The electricity supplier has a dynamic power limit for each hour that must not be exceeded by the hourly aggregate load of the households. Further, we assume that in supplying electrical power, the households' electrical devices are given priority over their storage devices. The policy is such that batteries will be charged by the residual power after the appliances loads have been satisfied. The households have to compete for the residual electricity so as to maximize the state of charge of their batteries. We have therefore modeled this system as a non-cooperative Nash equilibrium game where the households are considered as selfish but rational players whose objectives are to optimize their individual utilities.

R&D INVESTMENTS AS PREBARGAINING STRATEGIES

This paper analyzes a semicollusive, differentiated duopoly. Firms first compete in cost reducing R&D and then cooperate on the output market. The sharing of the joint profit on the output market is modeled as a Nash bargaining game. We study an asymmetric setting in which one firm has a lower unit cost of production than the other firm, before any R&D expenditures. If firms do not agree on how to share their joint profit, they play a noncooperative Nash equilibrium. Assuming linear demand functions, we show that the Nash bargaining outcome is independent of whether firms play a Cournot or a Bertrand Nash equilibrium, as long as both firms supply positive outputs in these equilibria. If the two products are sufficiently differentiated, there is a unique equilibrium in which both firms supply a positive output, and in which the low cost firm always invests more in R&D than the high cost firm. If the two products are not very differentiated, and if the difference in unit costs between the two firms is not too large, there exist two equilibria. In each of these equilibria only one firm supplies a positive output. This can be the low cost or the high cost firm. In the latter case, the initially high cost firm invests so much in R&D that its unit cost after R&D is lower than that of the other firm. This firm then leapfrogs the other firm. If the two products are very similar and if firms apply Bertrand strategies when disagreeing, there exist equilibria in which only one firm supplies a positive output, while in the noncooperative Nash equilibrium that same firm can prevent the other firm from entering the market. We show that, in the context of the Nash bargaining model, this latter firm still has the power to claim a share of the joint profit.

Nash equilibrium solution in a vendor–buyer supply chain model with permissible delay in payments

In practice, vendors (or sellers) often offer their buyers a fixed credit period to settle the account. The benefits of trade credit are not only to attract new buyers but also to avoid lasting price competition. On the other hand, the policy of granting a permissible delay adds not only an additional cost but also an additional dimension of default risk to vendors. In this paper, we will incorporate the fact that granting a permissible delay has a positive impact on demand but negative impacts on both costs and default risks to establish vendor–buyer supply chain models. Then we will derive the necessary and sufficient conditions to obtain the optimal solution for both the vendor and the buyer under non-cooperative Nash equilibrium. Finally, we will use two numerical examples to show that (1) granting a permissible delay may significantly improve profits for both the vendor and the buyer, and (2) the sensitivity analysis on the optimal solution with respect to each parameter.

Distributed Regret Matching Algorithm for Dynamic Congestion Games with Information Provision

The advances in adaptive learning dynamics to pure Nash equilibria in game theory provide promising results for the modeling of selfish agents with limited information in congestion games. In this study, a distributed game-theoretical learning algorithm with real-time information provision for dynamic congestion games is proposed. The learning algorithm is based on the regret matching process by considering a user's previously realised payoffs and real-time information. The numerical studies show that the proposed algorithm can converge to a non-cooperative Nash equilibrium in both static and dynamic congestion networks. Moreover, the proposed algorithm leads to a plausible real-time route choice modeling framework based on a user's perception being updated by incorporating the user's past experience, real-time information and behaviour inertia.

When Samuelson Met Veblen Abroad: National and Global Public Good Provision when Social Comparisons Matter

This paper derives Pareto efficient policy rules for the provision of national as well as global public goods in a two-country world, where each individual cares about relative consumption within as well as between countries. Furthermore, we compare these policy rules with those that follow from a non-cooperative Nash equilibrium. The results show that both global and national public goods are systematically under-provided in Nash equilibrium under such relative consumption concerns.

Heterogeneous-Oriented Resource Allocation Method in Internet of Things

For various features of heterogeneous network access, combined with multi-master multiple slave Stackelberg game model, network pricing and resource allocation scheme of a heterogeneous IOT is proposed. A mobile end-user utility function based on income and spending is designed. After the operator's price is determined, the utility function satisfies the concave conditions to ensure the existence of mobile user's non-cooperative Nash equilibrium point, resulting in multiple resource participants in the joint optimal objective.

Mixed Taxation and Transboundary Externalities

This article concerns income taxation and commodity taxation in a multijurisdiction framework with transboundary environmental damage. The decision problem facing the government in each such jurisdiction is represented by a two-type model (with asymmetric information between the government and the private sector). We show how the possibility to influence the world market producer price adds mechanisms of relevance for redistribution and externality correction which, in turn, affect the domestic use of taxation. Finally, with the noncooperative Nash equilibrium as a reference case, we consider the welfare effects of policy coordination.

Game theoretic approach for fertilizer application: looking for the propensity to cooperate

This paper continues the research implemented in previous work of (Schreider et al. in Environ. Model. Assess. 15(4):223–238, 2010) where a game theoretic model for optimal fertilizer application in the Hopkins River catchment was formulated, implemented and solved for its optimal strategies. In that work, the authors considered farmers from this catchment as individual players whose objective is to maximize their objective functions which are constituted from two components: economic gain associated with the application of fertilizers which contain phosphorus to the soil and environmental harms associated with this application. The environmental losses are associated with the blue-green algae blooming of the coastal waterways due to phosphorus exported from upstream areas of the catchment. In the previous paper, all agents are considered as rational players and two types of equilibria were considered: fully non-cooperative Nash equilibrium and cooperative Pareto optimum solutions. Among the plethora of Pareto optima, the solution corresponding to the equally weighted individual objective functions were selected. In this paper, the cooperative game approach involving the formation of coalitions and modeling of characteristic value function will be applied and Shapley values for the players obtained. A significant contribution of this approach is the construction of a characteristic function which incorporates both the Nash and Pareto equilibria, showing that it is superadditive. It will be shown that this approach will allow each players to obtain payoffs which strictly dominate their payoffs obtained from their Nash equilibria.

On the Efficient Provision of Public Goods by Means of Lotteries

We provide a solution to the free-rider problem in the provision of a public good. To this end we define a biased indirect contribution game which provides the efficient amount of the public good in non-cooperative Nash equilibrium. No confiscatory taxes or other means of coercion are used. We rather extend the model of Morgan (2000), who used fair raffles as voluntary contribution schemes, to unfair or biased raffles, which we show to be equivalent to fair raffles whose tickets are sold to consumers at different individual prices. We give a detailed account of the solution for the case of two different consumers and discuss its implications for the general case of many consumers. JEL-Code: C720, D720, H410.

On the Efficient Provision of Public Goods by Means of Lotteries

We provide a solution to the free-rider problem in the provision of a public good. To this end we define a biased indirect contribution game which provides the efficient amount of the public good in non-cooperative Nash equilibrium. No confiscatory taxes or other means of coercion are used. We rather extend the model of Morgan (2000), who used fair raffles as voluntary contribution schemes, to unfair or biased raffles, which we show to be equivalent to fair raffles whose tickets are sold to consumers at different individual prices. We give a detailed account of the solution for the case of two different consumers and discuss its implications for the general case of many consumers.

Endogenous Participation in a Partial Climate Agreement with Open Entry: A Numerical Assessment

Our purpose is to analyse the effectiveness and efficiency of a Partial Climate Agreement with open entry under a non-cooperative Nash-Equilibrium framework. We evaluate a partial agreement policy in which non-signatory countries can decide to join or to leave a coalition of the willing at any point in time. By means of a simple analytical model and of a numerical integrated assessment model, we assess different coalition structures, and different minimum admission requirements. Our results indicate that a Partial Climate Agreement with open entry can be effective, achieving climate stabilization between 2C and 3C depending on the composition of the coalition of the willing. The policy turns out to be also rather efficient, with only minor losses with respect to a full cooperation agreement. Finally, we quantify the optimal admission requirement in about 40-50% of cumulative abatement.

Informational Benefits of International Treaties

International treaties allow a minister in one country to learn about the beliefs held by a minister in another country, so allowing each to make better decisions. When the net benefits of environmental policy are uncertain, a country which participates in negotiating an environmental treaty, or which supports strong environmental action, thereby provides information to another country that such action can be worthwhile. This dimension of international environmental agreements helps explain why countries negotiate treaties, why these treaties often implement the Nash non-cooperative equilibrium, why agreement to a treaty by a country deemed not particularly favorable to it can strengthen environmental action, and why treaties may invoke a minimum participation clause. The informational mechanism we discuss holds even if physical spillovers are absent, and even if monitoring and enforcement are infeasible. Our approach can explain additional phenomena: how requirements for approval by several bodies within a country can increase support for a policy, and why simultaneous action can lead to better outcomes than sequential action.

Competition and market power in physician private practices

This article presents a theoretical and empirical model to examine competition in physician private practices implementing a conjectural variation framework. Our study uses the 1998 American Medical Association Physician Socioeconomic Monitoring Survey and tests for collusion and market power in physician private practices. The year 1998 is of particular interest due to charges filed in Federal court by The United States Department of Justice (DOJ) against a number of large physician practices, ruling that physicians could no longer engage in joint negotiations. The indictments by the DOJ were based on anecdotal economic and legal observations rather than the result of empirical evidence from accepted econometric modeling. Our model indicates that the behavior of physicians in medical subspecialties and surgical subspecialties is consistent with a non-cooperative Nash equilibrium.

A Numerical Toolbox to Solve N-Player Affine LQ Open-Loop Differential Games

We present an algorithm and a corresponding MATLAB numerical toolbox to solve any form of infinite-planning horizon affine linear quadratic open-loop differential games. By rewriting a specific application into the standard framework one can use the toolbox to calculate and verify the existence of both the open-loop non-cooperative Nash equilibrium (equilibria) and cooperative Pareto equilibrium (equilibria). In case there is more than one equilibrium for the non-cooperative case, the toolbox determines all solutions that can be implemented as a feedback strategy. Alternatively, the toolbox can apply a number of choice methods in order to discriminate between multiple equilibria. The user can predefine a set of coalition structures for which they would like to calculate the non-cooperative Nash solution(s). It is also possible to specify the relative importance of each player in any coalition structure. Furthermore, the toolbox offers plotting facilities as well as other options to analyse the outcome of the game. For instance, it is possible to disaggregate each player’s total loss into its contributing elements. The toolbox is available as a freeware from the authors of this paper.

A Macroeconomic Policy Game for a Monetary Union with Adaptive Expectations

We consider a dynamic game model of a two-country monetary union. Governments (fiscal policies) pursue national goals while the common central bank’s monetary policy aims at union-wide objectives. For a symmetric demand shock, we derive numerical solutions of the dynamic game between the governments and the central bank. We consider conflicting (non-cooperative Nash equilibrium) and coordinated policy-making (cooperative Pareto solutions). We show that there is a trade-off between the deviations of instruments and targets from desired paths; the volatility of output and inflation increases when private agents react more strongly to changes in actual inflation.

Research on the dynamic advertising strategies of the duopoly manufacturers—the application of advertising differential game model

In tradition the duopoly manufacturer theories, which all assumed that the time was static, deduced the solutions of Cournot-Nash, Leader-Follower or Collusion under the consideration of the reaction function. But in reality, the time should be dynamic, and the targets of different decision-makers maybe conflict with or depend on each other. So, not only should the decision-makers foresee the strategies of the other decision-makers before hand, more over they should also consider the effect that their own strategies have on the strategies of the other decision-makers in the future. And in this paper this argument is used to build up the dynamic game model to research on the dynamic advertising strategies of the duopoly manufacturers. We deduce the non-cooperative Nash equilibrium of the open-loop and feedback strategies, then after through the analysis of the equilibrium solution, we obtain 5 propositions to provide theoretic basis for the duopoly manufacturers to make advertising strategies in the planning period.