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We study the effects of uniting two separated markets, each monopolized by a firm, into a single integrated duopoly market. When one of the firms is consumer-friendly, we examine certain conditions under which integration turns out to be beneficial or detrimental to the actors involved. We show that consumers in the local market which the consumer-friendly firm is from may have their surplus reduced unless the firm has a cost disadvantage. We also find conditions under which the welfare of one market or the other can be reduced, even that of both simultaneously. Finally, we show that (i) it would be better that the firm is friendly only with the consumers of its original market and not with those of the global market and (ii) more competition in the presence of a consumer-friendly firm might not be beneficial to the local society.

We extend the dynamic Cournot duopoly framework with emission charges on output by Mamada and Perrings (Econ Anal Policy 66:370–380, 2020), which encompassed homogeneous products in its original formulation, to the more general case of differentiated goods, in order to highlight the richness in its static and dynamic outcomes. Each firm is taxed proportionally to its own emission only and charge functions are quadratic. Moreover, due to an adjustment capacity constraint, firms partially modify their output level toward the best response. Like in Mamada and Perrings (Econ Anal Policy 66:370–380, 2020), the only steady state coincides with the Nash equilibrium, and it will be considered admissible when it guarantees the positivity of the marginal emission charge. We find that the full efficacy of the environmental policy, which applies to an equilibrium that is globally asymptotically stable anytime it is admissible, is achieved in the case of independent goods, as well as with a low good interdependence degree in absolute value, independently of being substitutes or complements. When goods are substitutes and their interdependence degree is high, the considered environmental policy is still able to reduce pollution at the equilibrium, but the latter is stable just when the policy intensity degree is large enough. When instead goods are complements and their interdependence degree is high in absolute value, the considered environmental policy produces detrimental effects on the pollution level and the unique equilibrium is always unstable, when admissible. This highlights that, from the static viewpoint, even in the absence of free riding possibilities, the choice of the mechanism to implement has to be carefully pondered, according to the features of the considered economy.

In this paper, the robust game model proposed by Aghassi and Bertsimas (Math Program Ser B 107:231–273, 2006) for matrix games is extended to games with a broader class of payoff functions. This is a distribution-free model of incomplete information for finite games where players adopt a robust-optimization approach to contend with payoff uncertainty. They are called robust players and seek the maximum guaranteed payoff given the strategy of the others. Consistently with this decision criterion, a set of strategies is an equilibrium, robust-optimization equilibrium, if each player’s strategy is a best response to the other player’s strategies, under the worst-case scenarios. The aim of the paper is twofold. In the first part, we provide robust-optimization equilibrium’s existence result for a quite general class of games and we prove that it exists a suitable value ϵ such that robust-optimization equilibria are a subset of ϵ-Nash equilibria of the nominal version, i.e., without uncertainty, of the robust game. This provides a theoretical motivation for the robust approach, as it provides new insight and a rational agent motivation for ϵ-Nash equilibrium. In the last part, we propose an application of the theory to a classical Cournot duopoly model which shows significant differences between the robust game and its nominal version.

In this paper we investigate the dynamics of a duopoly game with ambiguity aversion regarding uncertainty in demand and constant expectations concerning competitor production. The focus is on an asymmetric Cournot game where players engage in robust optimization and have different beliefs about the possible realizations of the random parameters of the price function. The players’ ambiguity aversion introduces multiple equilibria and instability that otherwise would not be present. The investigation of the global dynamics of the game reveals the emergence, through border-collision bifurcations, of periodic and chaotic dynamics.

Amid growing environmental apprehensions, an increasing consumer demographic actively pursues genuine and transparent information traceability, aiming to bolster their commitment to sustainable practices. This study aims to investigate how consumer preferences for traceability, sustainability, and sustainable brand reputation of products influence the optimal decisions and selling quantity strategies adopted by firms operating in heterogeneous smart sustainable supply chains (SSCs). Utilizing both centralized and decentralized scenarios, this research constructs differentiated duopoly models, classifying participating entities as either boundedly rational or naive. Applying a differential game approach for an infinite time horizon, the study assesses the entities’ steady optimal decisions as well as the performances of the sustainable product across two scenarios. A cournot duopoly game is employed to examine the selling quantity strategies of differentiated smart SSCs. We leverage various tools for numerical simulations, including 1-D bifurcation diagrams, the largest Lyapunov exponent (LLE), phase diagrams, and basins of attraction. The outcomes of our investigation underscore the substantial influence of consumer preferences for sustainability and traceability on the transparency of information related to sustainable products. This influence reverberates across manufacturers and suppliers, where brand reputation emerges as a pivotal factor in shaping decision-making processes. Our study meticulously examines the repercussions of the profit-sharing ratio and the adjustment speed of boundedly rational entities.

This paper has two parts. In the mathematical part, we present two inexact versions of the proximal point method for solving quasi-equilibrium problems (QEP) in Hilbert spaces. Under mild assumptions, we prove that the methods find a solution to the quasi-equilibrium problem with an approximated computation of each iteration or using a perturbation of the regularized bifunction. In the behavioral part, we justify the choice of the new perturbation, with the help of the main example that drives quasi-equilibrium problems: the Cournot duopoly model, which founded game theory. This requires to exhibit a new QEP reformulation of the Cournot model that will appear more intuitive and rigorous. It leads directly to the formulation of our perturbation function. Some numerical experiments show the performance of the proposed methods.

Recently, 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.

This article develops a non-cooperative game with managerial quantity-setting firms in which owners choose whether to delegate output and abatement decisions to managers through a contract based on emissions (conventionally denoted as ‘green’ delegation, GD) instead of sales (sales delegation, SD), and the government levies an emissions tax to incentivise firms’ emissions-reduction actions. First, it compares the Nash equilibrium outcomes between GD and SD and then contrasts them also with profit maximisation (PM). A plethora of Nash equilibria emerges, especially in the case GD versus PM (the ‘green delegation game’), depending on the public awareness toward environmental quality, ranging from the coordination game to the ‘green’ prisoner's dilemma. Second, though the contract under GD incentivises managers for emissions, the environmental damage is lower than under SD. This is because the optimal tax more than compensates the incentive for emissions. These findings suggest that designing GD contracts paradoxically favours environmental quality.

We consider the problem of patent licensing in a Cournot duopoly in which the innovator (patentee) is one of the firms and it is capacity constrained. We show that when the patentee can produce a relatively small (relatively large) quantity, it prefers licensing by means of a fixed fee (unit royalty). When the patentee can set two-part tariffs in the form of combinations of fixed fees and unit royalties, it charges a positive fixed fee if and only if it is limited to producing a relatively small quantity. We also show that with combinations of fixed fees and royalties, the royalty rate is lower than is true for the standard case.

In this paper, we introduce a Cournot duopoly game that can characterize fierce competition in digital economies and employ it to examine the effects of research and development (R&D) spillovers while considering various competition intensities. We obtain the analytical solution of the Nash equilibrium and the expression of commodity price, firm production, and variable profit under some key competition intensities. Furthermore, we analyze the local stability of the Nash equilibrium and derive that the equilibrium may lose its stability only through a 1:4 resonance bifurcation. Numerical simulations are conducted, through which we find that the Nash equilibrium transitions to complex dynamics through a cascade of period-doubling bifurcations. Phase portraits are also provided to illustrate more details of the dynamics, which confirm the previous theoretical finding that the Nash equilibrium loses its stability through a 1:4 resonance bifurcation.