Abstract Bounded rationality, asymmetric information, and R&D spillovers are widely existed in monopoly markets, and they have been researched separately by a large number of literatures; however, there are few works that discussed both R&D spillovers and asymmetric information in oligopolistic games with bounded rational firms. Considering that R&D spillovers only flow from the R&D leader to the R&D follower, a duopoly Cournot game with heterogeneous expectations and asymmetric information is presented. In our model, a firm with private information of his marginal cost is designed, and the coefficient of R&D spillovers is introduced. Interesting findings show the following: (i) In a static duopoly Cournot game with perfect rationality, the equilibrium output of firm 1 with private information is negatively related to R&D spillovers and the probability of high marginal cost, while firm 2’s equilibrium output is positively correlated with them. (ii) In a dynamic duopoly Cournot game with asymmetric information and heterogeneous expectations, if firms adopt adaptive expectation and naïve expectation respectively, the Nash equilibrium is always globally asymptotically stable; if they use adaptive expectation and gradient dynamical expectation respectively, the Nash equilibrium tends to be locally asymptotically stable under certain conditions. Furthermore, the bigger the probability of high marginal cost or R&D spillovers are, the more volatile the monopoly market is, while higher technology innovation efficiency (TIE) of firm 1 is conducive to the stability of the product market. Our study would have theoretical and practical significance to the technological innovation activities of homogeneous products in oligopoly markets.

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.

have the best response based on the other firms output. Within the game, firms reach the Nash equilibrium when the Cournot equilibrium is achieved. The Bertrand Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed two-person zero-sum games, in which a participant's gains or losses are exactly balanced by the losses and gains of the other participant. In the 1950s, it Some… Unlike those in economics, the payoffs for games in biology are often interpreted as corresponding to fitness. In addition, the focus has been less on equilibria that correspond to a notion of rationality and more on ones that would be maintained by evolutionary forces. The best-known equilibrium in biology is known as the evolutionarily stable strategy (ESS), first introduced in (Maynard Smith & Price 1973). Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium. In biology, game theory has been used as a model to understand many different phenomena. It was first used to explain the evolution (and stability) of the approximate 1:1 sex ratios. (Fisher 1930) suggested that the 1:1 sex ratios are a result of evolutionary forces acting on individuals who could be seen as trying to maximize their number of grandchildren. Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of… The Cournot competition model involves players choosing quantity of a homogenous product to produce independently and simultaneously, where marginal cost can be different for each firm and the firm's payoff is profit. The production costs are public information and the firm aims to find their profit-maximizing quantity based on what they believe the other firm will produce and behave like monopolies. In this game firms want to produce at the monopoly quantity but there is a high incentive to deviate and produce more, which decreases the market-clearing price. For example, firms may be tempted to deviate from the monopoly quantity if there is a low monopoly quantity and high price, with the aim of increasing production to maximize profit. However this option does not provide the highest payoff, as a firm's ability to maximize profits depends on its market share and the elasticity of the market demand. The Cournot equilibrium is reached when each firm operates on their reaction function with no incentive to deviate, as they have the best response based on the other firms output. Within the game, firms reach the Nash equilibrium when the Cournot equilibrium is achieved.

In this paper, we focus on a dynamic Cournot game in the market with a nonlinear (isoelastic) demand function. In our model, there are N competing firms featured by nonlinear cost functions, which enhances our model’s resemblance to real-world scenarios. Firstly, we focus on the homogeneous case where firms’ marginal costs change at equal rates. We establish analytical expressions of the market supply at equilibrium and perform comparative static analysis. In addition, we investigate the local stability under different economies of scale and show that there could be multiple stable equilibria

Cournot and Bertrand oligopolies constitute the two most prevalent models of firm competition. The analysis of Nash equilibria in each model reveals a unique prediction about the stable state of the system. Quite alarmingly, despite the similarities of the two models, their projections expose a stark dichotomy. Under the Cournot model, where firms compete by strategically managing their output quantity, firms enjoy positive profits as the resulting market prices exceed that of the marginal costs. On the contrary, the Bertrand model, in which firms compete on price, predicts that a duopoly is enough to push prices down to the marginal cost level. This suggestion that duopoly will result in perfect competition, is commonly referred to in the economics literature as the "Bertrand paradox".In this paper, we move away from the safe haven of Nash equilibria as we analyze these models in disequilibrium under minimal behavioral hypotheses. Specifically, we assume that firms adapt their strategies over time, so that in hindsight their average payoffs are not exceeded by any single deviating strategy. Given this no-regret guarantee, we show that in the case of Cournot oligopolies, the unique Nash equilibrium fully captures the emergent behavior. Notably, we prove that under natural assumptions the daily market characteristics converge to the unique Nash. In contrast, in the case of Bertrand oligopolies, a wide range of positive average payoff profiles can be sustained. Hence, under the

An oligopoly is a market in which the price of goods is controlled by a few firms. Cournot introduced the simplest game-theoretic model of oligopoly, where profit-maximizing behavior of each firm results in market failure. Furthermore, when the Cournot oligopoly game is infinitely repeated, firms can tacitly collude to monopolize the market. Such tacit collusion is realized by the same mechanism as direct reciprocity in the repeated prisoner’s dilemma game, where mutual cooperation can be realized whereas defection is favorable for both prisoners in a one-shot game. Recently, in the repeated p