Cournot-Nash Game Research Articles

On the existence of solutions to stochastic quasi-variational inequality and complementarity problems

Variational inequality problems allow for capturing an expansive class of problems, including convex optimization problems, convex Nash games and economic equilibrium problems, amongst others. Yet in most practical settings, such problems are complicated by uncertainty, motivating the examination of a stochastic generalization of the variational inequality problem and its extensions in which the components of the mapping contain expectations. When the associated sets are unbounded, ascertaining existence requires having access to analytical forms of the expectations. Naturally, in practical settings, such expressions are often difficult to derive, severely limiting the applicability of such an approach. Consequently, our goal lies in developing techniques that obviate the need for integration and our emphasis lies in developing tractable and verifiable sufficiency conditions for claiming existence. We begin by recapping almost-sure sufficiency conditions for stochastic variational inequality problems with single-valued maps provided in our prior work Ravat and Shanbhag (in: Proceedings of the American Control Conference (ACC), 2010), Ravat and Shanbhag (SIAM J Optim 21: 1168–1199, 2011) and provide extensions to multi-valued mappings. Next, we extend these statements to quasi-variational regimes where maps can be either single or set-valued. Finally, we refine the obtained results to accommodate stochastic complementarity problems where the maps are either general or co-coercive. The applicability of our results is demonstrated on practically occuring instances of stochastic quasi-variational inequality problems and stochastic complementarity problems, arising as nonsmooth generalized Nash-Cournot games and power markets, respectively.

Shared stocks, game theory and the zonal attachment principle

The outcomes of a Nash-Cournot game and a game of cooperation supported by a threat strategy are compared. The discussion is related to the ongoing conflict over the mackerel stock in the Northeast Atlantic. Despite the absence of a comprehensive management agreement, the outcome of the mackerel fishery is nowhere near what is predicted by the Nash-Cournot equilibrium. To the contrary, the countries involved seem to be engaged in an informal cooperation, supported by an implicit threat of mutually assured destruction should any single one compete too aggressively. The zonal attachment principle of dividing the total catch from shared stocks is also examined and found wanting in many cases.

On smoothing, regularization, and averaging in stochastic approximation methods for stochastic variational inequality problems

Traditionally, most stochastic approximation (SA) schemes for stochastic variational inequality (SVI) problems have required the underlying mapping to be either strongly monotone or monotone and Lipschitz continuous. In contrast, we consider SVIs with merely monotone and non-Lipschitzian maps. We develop a regularized smoothed SA (RSSA) scheme wherein the stepsize, smoothing, and regularization parameters are reduced after every iteration at a prescribed rate. Under suitable assumptions on the sequences, we show that the algorithm generates iterates that converge to the least norm solution in an almost sure sense, extending the results in Koshal et al. (IEEE Trans Autom Control 58(3):594–609, 2013) to the non-Lipschitzian regime. Additionally, we provide rate estimates that relate iterates to their counterparts derived from a smoothed Tikhonov trajectory associated with a deterministic problem. To derive non-asymptotic rate statements, we develop a variant of the RSSA scheme, denoted by aRSSA $$_r$$ , in which we employ a weighted iterate-averaging, parameterized by a scalar r where $$r = 1$$ provides us with the standard averaging scheme. The main contributions are threefold: (i) when $$r<1$$ and the parameter sequences are chosen appropriately, we show that the averaged sequence converges to the least norm solution almost surely and a suitably defined gap function diminishes at an approximate rate $$\mathcal{O}({1}\slash {\root 6 \of {k}})$$ after k steps; (ii) when $$r<1$$ , and smoothing and regularization are suppressed, the gap function admits the rate $$\mathcal{O}({1}\slash {\sqrt{k}})$$ , thus improving the rate $$\mathcal{O}(\ln (k)/\sqrt{k})$$ under standard averaging; and (iii) we develop a window-based variant of this scheme that also displays the optimal rate for $$r < 1$$ . Notably, we prove the superiority of the scheme with $$r < 1$$ with its counterpart with $$r=1$$ in terms of the constant factor of the error bound when the size of the averaging window is sufficiently large. We present the performance of the developed schemes on a stochastic Nash–Cournot game with merely monotone and non-Lipschitzian maps.

Market Power in Interactive Environmental and Energy Markets: The Case of Green Certificates

Markets for environmental externalities are typically closely related to the markets causing such externalities, whereupon strategic interactions may result. Along these lines, the market for Green Certificates is strongly interwoven in the electricity market as the producers of green electricity are also suppliers of Green Certificates. In this paper, we formulate an analytic equilibrium model for simultaneously functioning electricity and Green Certificates markets, and focus on the role of market power. We consider two versions of a Nash-Cournot game; a standard Nash-Cournot game where the players treat the market for Green Certificates and the electricity market as separate markets; and a Nash-Cournot game with endogenous treatment of the interaction between the electricity and Green Certificates markets with conjectured responses. One result is that a certificate system faced with market power may collapse into a system of per unit subsidies, as the producers involved start to game on the joint functioning of markets.

Self-Tuned Stochastic Approximation Schemes for Non-Lipschitzian Stochastic Multi-User Optimization and Nash Games

We consider multi-user optimization problems and Nash games with stochastic convex objectives, instances of which arise in decentralized control problems. The associated equilibrium conditions of both problems can be cast as Cartesian stochastic variational inequality problems with mappings that are strongly monotone but not necessarily Lipschitz continuous. Consequently, most of the currently available stochastic approximation schemes cannot address such problems. First, through a user-specific local smoothing, we derive an approximate map that is shown to be Lipschitz continuous with a prescribed constant. Second, motivated by the need for a robust scheme that can be implemented in a distributed fashion, we develop a distributed self-tuned stochastic approximation scheme (DSSA) that adapts to problem parameters. Importantly, this scheme is provably convergent in an almost-sure sense and displays the optimal rate of convergence in mean squared error, i.e., ${\mathcal O}(1/k)$ . A locally randomized variant is also provided to ensure that the scheme can contend with stochastic non-Lipschitzian multi-user problems. We conclude with numerics derived from a stochastic Nash-Cournot game.

Market Power in Interactive Environmental and Energy Markets: The Case of Green Certificates

Markets for environmental externalities are typically closely related to the markets causing such externalities, whereupon strategic interactions may result. Along these lines, the market for Green Certificates is strongly interwoven in the electricity market as the producers of green electricity are also suppliers of Green Certificates. In this paper, we formulate an analytic equilibrium model for simultaneously functioning electricity and Green Certificates markets, and focus on the role of market power. We consider two versions of a Nash-Cournot game; a standard Nash-Cournot game where the players treat the market for Green Certificates and the electricity market as separate markets; and a Nash-Cournot game with endogenous treatment of the interaction between the electricity and Green Certificates markets with conjectured responses. One result is that a certificate system faced with market power may collapse into a system of per unit subsidies, as the producers involved start to game on the joint functioning of markets.

A Bi-level Formulation for the Combined Dynamic Equilibrium based Traffic Signal Control

This paper formulates the combined dynamic user equilibrium and signal control problem (DUESC) as a bi-level optimization problem. The signal control operator in the upper level optimizes the signal setting to minimize the system travel time whereas the road users in the lower level minimize their own costs (by changing departure times, paths or both) leading to dynamic user equilibrium behavior. Three components of the bi-level formulation are discussed including network loading model, the dynamic user equilibrium model and the signal control model. Then the combined problems are formulated as a Nash-Cournot game and a Stackelberg game. A solution technique based on the iterative optimization and assignment (IOA) method is proposed to solve the DUESC problem. We use the projection algorithm to solve the lower level and the mixed integer programming solver to solve the upper level. Extensive numerical results demonstrate the benefits of using this model.

On Nash–Cournot games with price caps

In this paper, we study an N-person Nash–Cournot game with a price cap. Under certain mild conditions, we show that this game can be formulated as a complementarity problem. Based on this formulation, we study various properties of the game, including equilibrium existence, computability, and a description of the equilibrium set when there are multiple equilibria.

Solving Discretely-Constrained Nash–Cournot Games with an Application to Power Markets

This paper provides a methodology to solve Nash–Cournot energy production games allowing some variables to be discrete. Normally, these games can be stated as mixed complementarity problems but only permit continuous variables in order to make use of each producer’s Karush–Kuhn–Tucker conditions. The proposed approach allows for more realistic modeling and a compromise between integrality and complementarity to avoid infeasible situations.

Impact of wind integration on electricity markets: a chance-constrained Nash Cournot model

SUMMARY Wind-powered electricity generation is growing at a high rate around the world, mainly driven by the associated environmental benefits. However, because of the uncertain nature of wind energy, large-scale integration of wind-powered generators into power systems introduces uncertainty of supply in the operation and planning of electric energy systems. A chance-constrained model is presented to study the strategic behavior of power suppliers (firms) with respect to the wind generation uncertainty in an electricity market. The linear complementarity problem of a Nash-Cournot competition is extended to account for the wind generation uncertainty. An iterative solution approach is introduced to solve the resultant joint chance-constrained programming problem. Five studies were conducted to show the impact of the amount, location and standard deviation of the wind generation and transmission line limits on the suppliers’ profit in the Nash-Cournot game for different confidence levels. The value of stochastic solution index is used to evaluate the suppliers’ loss of profit. The numerical results for a three-bus test system are given to verify the formulation and the solution approach. Copyright © 2011 John Wiley & Sons, Ltd.

On the Characterization of Solution Sets of Smooth and Nonsmooth Convex Stochastic Nash Games

Variational analysis provides an avenue for characterizing solution sets of deterministic Nash games over continuous-strategy sets. We examine whether similar statements, particularly pertaining to existence and uniqueness, may be made when player objectives are given by expectations. For instance, in deterministic regimes, a suitable coercivity condition associated with the gradient map is sufficient for the existence of a Nash equilibrium; in stochastic regimes, the application of this condition requires being able to analytically evaluate the expectation and its gradients. Our interest is in developing a framework that relies on the analysis of merely the integrands of the expectations; in the context of existence statement, we consider whether the satisfaction of a suitable coercivity condition in an almost-sure sense may lead to statements about the original stochastic Nash game. Notably, this condition also guarantees the existence of an equilibrium of the scenario-based Nash game. We consider a range of such statements for claiming the existence of stochastic Nash equilibria when payoff functions are both smooth and nonsmooth and when strategy sets are possibly coupled through a shared convex constraint. Notably the sufficiency conditions are less stringent, when one either imposes appropriate monotonicity requirements or requires that strategy sets be decoupled. Uniqueness, however, can be claimed by requiring that a strong monotonicity condition holds over a set of positive measure, rather than in an almost-sure sense. When strategy sets are coupled by shared convex expected-value constraints, a suitable regularity condition allows for claiming existence and uniqueness in the primal-dual space. We illustrate our approach by examining two extensions of stochastic Nash–Cournot games, of which the first allows for nonsmooth payoffs through the introduction of risk-measures, while the second allows for shared stochastic constraints.

Market Integration and Competition in Environmental and Trade Policies

Recent empirics suggest the relevance of transport cost reductions for world trade growth along with eliminations in protectionist trade barriers. To address the welfare effects of trade cost reductions in a context of ‘trade and the environment,’ we develop a two-stage game model where governments choose environmental and trade policies and firms play a Cournot-Nash game. We show that reductions in transport costs lead to lower emission taxes and higher tariffs. And, we find that the degree of pollution damage plays a central role in whether market integration is welfare-improving relative to autarky.

Lobbying, Regulation in Goods and Labor Markets, and Poisson Technical Change

AbstractI consider an economy in which monopolistic firms produce differentiated products, using skilled and unskilled labor. There is monopolistic competition in the goods market, which determines the size of the rents. There is a self-interested government that determines the minimum wage for unskilled labor, either directly or through the support of labor unions, and the number of firms that can enter the market. Firms devote skilled labor to research and development (R&D) to escape production costs. Interest groups that represent workers and firms lobby the government paying political contributions that are conditional on the government's prospective policy. The labor market is regulated, if the minimum wage is high enough to cause unemployment, and deregulated otherwise.Technically, the model is solved as follows. Monopolistic competition is modeled as a Cournot-Nash game: each firm sets production given the production of the other firms. Because the outcome of R&D in each firm follows a Poisson process, a firm's investment problem is solved by stochastic dynamic programming. The interest groups that represent workers and firms maximize the present value of the workers’ and shareholders’ income, respectively, through their political contributions to the government, while the government maximizes the present value of the political contributions it receives from these. The interest groups act as leaders, but the government as a follower in a common agency game. The outcome of this game determines the government's policy, i.e. the number of firms and the unskilled workers’ minimum wage.The technical contribution of the paper is to show how the Cournot-Nash game, the Poisson technological change due to R&D and the common agency game can be considered within a unifying extended game. The economic prediction of the model is the following. The smaller (greater) the elasticity of substitution between skilled and unskilled labor in production, the more (less) likely there is regulation in the labor market. An increase in the elasticity of substitution between skilled and unskilled labor leads to deregulation in the labor market.

Regulating Environmental Externalities Through Public Firms: A Differential Game

We investigate the possibility of using public firms to regulate polluting emissions in a Cournot oligopoly where production takes place at constant returns to scale and entails a negative environmental externality. We model the problem as a differential game and investigate (i) the Cournot-Nash game among profit-seeking firms; (ii) the Markov Perfect Nash equilibrium under social planning, where the industry output is entirely controlled by a benevolent planner aiming at the maximisation of social welfare; and (iii) the Markov Perfect Nash equilibrium in a mixed setup where at least one firm is public, while the others remain profit-seeking agents. Our analysis identifies the conditions whereby having a mixed market as a regulatory instrument suffices to drive the industry to the same output, externality and social welfare as under planning, both along the optimal path and in steady state.

The value of information in electricity investment games

In this paper we look at the assumptions behind a Cournot model of investment in electricity markets. We analyze how information influences investment, looking at the way common knowledge of marginal costs, expectations on the competitors’ marginal costs, expectations on the level and duration of demand, and conjectures on the others’ behavior, influence the value of a project. We expose how the results are highly dependent on the assumptions used, and how the investment Nash–Cournot game with perfect and complete information implies such a degree of coordination between players that the outcome of the game would be classified by any regulation law as collusive behavior. Furthermore, we introduce the concept of Nash Value of Complete Information. As an example we use a stylized model of investment in liberalized electricity markets.

How Does ANWR Exploration Affect OPEC Behavior? —A Simulation Study of an Open-loop Cournot-Nash Game

Exploring petroleum reserves in the Alaskan Arctic National Wildlife Refuge (ANWR) has been proposed to reduce the dependence on foreign oil and to ease the energy shortage in the United States. To investigate the impacts of the ANWR exploration on strategic behavior of OPEC members, a calibrated dynamic model of oligopolistic competition and cartel collusion in the U.S. petroleum market is built in this paper. Numerical simulations on an open-loop game are used here to examine the scope and magnitude of strategic interactions between OPEC's decisions and ANWR exploration. The simulation results show that OPEC's strategic postures have much stronger effects on the U.S. petroleum market than the ANWR exploration. The simulations in this paper indicate that preventing cartel collusion by OPEC is more effective than the ANWR exploration in alleviating short petroleum supplies of the United States in the near future.

A dynamic Cournot–Nash game: a representation of a finitely repeated feedback game

This paper studies market outcome equivalence of two dynamic production-capital investment games under uncertainty. One is played under complete information, while the other, feedback (FB) game, is played under incomplete information about the opponents’ costs and market demand. The FB game structure may be observed in some newly initiated industries, in which a homogeneous good is exchanged via an auction mechanism. In that case, the FB game setting may predict the complete information equilibrium market outcomes.

Differential variational inequalities and shipper dynamic oligopolistic network competition

In this paper, we show that the differential Cournot–Nash game describing dynamic oligopolistic network competition may be articulated as a differential variational inequality involving both control and state variables. We exploit this formulation to establish necessary conditions, an existence theorem and a nonconvex functional mathematical programming formulation. We show by example that, despite its nonconvexity, this mathematical program may be solved by the multi-start global optimization scheme found in the off-the-shelf software package GAMS when used in conjunction with the commercial solver MINOS. We also present a detailed numerical example. For that example, freight trip tables, formed ex post from the shipment patterns that are solutions to the dynamic oligopolistic network competition model we present, exhibit substantial temporal fluctuations. These observations are significant for they suggest that shippers must be extremely astute and capable of dramatically and rapidly altering production and distribution schedules if they are to compete in the final goods market successfully. If distribution services are provided by separate freight carriers, those carriers must be able to survive in a feast-or-famine environment.

Distribution Systems and Operating Leverage

This paper presents a model in which firms compete in a Cournot-Nash game. Firms can choose an exclusive agency (EA) or an independent agency (IA) distribution system. Firms can enter and exit the market and switch distribution systems with no cost. The only difference between an EA system and an IA system is the operating leverage or cost structure. An EA firm incurs low (per unit) variable cost but high fixed cost. An IA firm incurs high variable cost but low fixed cost. Under the different operating leverages, we investigate the equilibrium outcome in which no entry, no exit, and no switching occur. We find that coexistence is possible when IA firms are less efficient than EA firms. IA firms can dominate in the market even if IA firms are inefficient. An EA firm produces more than an IA firm. IA firms tend to dominate more in high risk lines than in low risk lines. We provide numerical examples showing how an inefficient system can survive and even dominate in the market.

Production of Chlorofluorocarbons in Anticipation of the Montreal Protocol

Anticipation of an International Environmental Agreement provides an incentive for countries to change their production behavior prior to negotiations in order to gain a favorable bargaining position. Increased historical production figures at the time of negotiations may influence the magnitude of the baseline from which cutbacks will be specified. In this paper we empirically measure the magnitude of such strategic production behavior in the case of the Montreal Protocol on Substances that Deplete the Ozone Layer of 1987. Due to data limitations we specify a two player Nash–Cournot game between the United States and the rest of the world. We find evidence of asymmetric strategic behavior, which resulted in a net increase of aggregate world chlorofluorocarbon (CFC) production.