Duopoly Model Research Articles

Three Duopoly Game-Theoretic Models for the Smart Grid Demand Response Management Problem

Demand response management (DRM) significantly influences the prospective advancement of electricity smart grids. This paper introduces three distinct game-theoretic duopoly models for the smart grid demand response management problem. It delineates several rational assumptions regarding the model variables, functions, and parameters. The first model adopts a Cournot duopoly form, offering a unique closed-form equilibrium solution. The second model adopts a Stackelberg duopoly structure, also providing a unique closed-form equilibrium solution. Following a comparison of the economic viability of the two model equilibria and an assessment of their sensitivity to parametric changes, the paper proposes a third model with a Cartel structure and discusses its advantages over the earlier models. Finally, the paper examines how demand forecasting affects the equilibrium quantities and pricing solutions of each model.

Testing the algebraic modeling of competitive impacts from wage premium shocks: a case study from a booming airline market

Labor relations issues such as collective bargaining and strike threat power have recently gained increased attention in the Brazilian air transportation market due to a short-term shortage of qualified workforce, leading stronger pilot unions to demand higher wages and better working conditions. Exacerbated by the notable expansion of the Brazilian economy between 2020 and 2023, where air travel demand surged by 60%, this pressure has driven labor costs to grow more rapidly than other cost categories. This paper examines exogenous flight crew cost shocks, treating these as increases in wage premiums resulting from stronger unionization and higher labor market rents. Using a differentiated duopoly model, we analyze the impact of these wage increases on competition between a major network carrier and a small low-fare carrier, introducing the hypothesis of non-exhausted economies of density. Our findings indicate that smaller airlines are more adversely affected by labor cost shocks, though differentiated wage increase schemes can mitigate these impacts, reducing market concentration and preserving consumer welfare. This study contributes to the literature by incorporating wage premium shocks into existing models and suggesting that labor regulations in Brazil should be liberalized to address the rapid growth in air transportation demand and the resultant workforce shortage.

Comparison of Cournot Duopoly Model and Stackelberg Model

Comparison of Cournot Duopoly Model and Stackelberg Model

Bifurcation and Chaos in a Fractional-Order Cournot Duopoly Game Model on Scale-Free Networks

In this study, a Cournot duopoly model describing Caputo fractional-order differential equations with piecewise constant arguments is discussed. We have obtained two-dimensional discrete dynamical system as a result of applying the discretization process to the model. By using the center manifold theory and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip bifurcation about the Nash equilibrium point. Phase portraits, bifurcation diagrams, and Lyapunov exponents show the existence of many complex dynamical behaviors in the model such as the stable equilibrium point, period-2 orbit, period-4 orbit, period-8 orbit, period-16 orbit, and chaos according to changing the speed of the adjustment parameter [Formula: see text]. The discrete Cournot duopoly game model is also considered on two scale-free networks with different numbers of nodes. It is observed that the complex dynamical networks exhibit similar dynamical behaviors such as the stable equilibrium point, flip bifurcation, and chaos depending on changing the coupling strength parameter [Formula: see text]. Moreover, flip bifurcation and transition chaos take place earlier in more heterogeneous networks. Calculating the largest Lyapunov exponents guarantees the transition from nonchaotic to chaotic states in complex dynamical networks.

Cournot Policy Model: Rethinking centralized training in multi-agent reinforcement learning

This work studies Centralized Training and Decentralized Execution (CTDE), which is a powerful mechanism to ease multi-agent reinforcement learning. Although the centralized evaluation ensures unbiased estimates of Q-value, peers with unknown policies make the decentralized policy far from the expectation. To make progress in more stabilized and effective joint policy, we develop a novel game framework, termed Cournot Policy Model, to enhance the CTDE-based multi-agent learning. Combining the game theory and reinforcement learning, we regard the joint decision-making in a single time step as a Cournot duopoly model, and then design a Hetero Variational Auto-Encoder to model the policies of peers in the decentralized execution. With a conditional policy, each agent is guided to a stable mixed-strategy equilibrium even though the joint policy evolves over time. We further demonstrate that such an equilibrium must exist in the case of centralized evaluation. We investigate the improvement of our method on existing centralized learning methods. The experimental results on a comprehensive collection of benchmarks indicate our approach consistently outperforms baseline methods.

Stability and Chaos of the Duopoly Model of Kopel: A Study Based on Symbolic Computations

Stability and Chaos of the Duopoly Model of Kopel: A Study Based on Symbolic Computations

Dynamic Price Dispersion of Seasonal Goods in Bertrand–Edgeworth Competition

The primary distinguishing characteristics of various revenue management systems include fixed capacities, uniform products, and customers' sensitivity to pricing when making purchase decisions. Even with only two or three competitors, the competition remains intense. This study examines the sub-game perfect Nash equilibrium of price competition within an oligopoly market where goods are perishable assets. Each seller possesses a single unit of a non-replenishable good and competes by setting prices to sell their inventory over a finite sales period. Each period, buyers seek to purchase one unit of the good, and the number of buyers entering is random. All sellers' prices are visible to buyers, and there is no cost associated with searching for goods. Through stochastic dynamic programming methods, sellers' optimal responses can be determined by treating the competition as a one-time price-setting game, considering the remaining sales periods and the current demand pattern. Assuming a binary demand model, it is demonstrated that the duopoly model exhibits a unique Nash equilibrium. In addition, it is shown that the oligopoly model does not result in price dispersion based on a discussed metric. Moreover, when a generalized demand model is considered, the duopoly model features a unique Nash equilibrium with mixed strategies, whereas the oligopoly model exhibits a unique symmetric Nash equilibrium with mixed strategies.

The role of asymmetric innovation’s sizes in technology licensing under partial vertical integration

In this paper, we compare the scenarios of exclusive licenses and cross-licenses under the existence of partial vertical integration. To do this, a successive duopoly model is proposed, with two owners and two firms competing in a differentiated product market. Each technology owner has a share in one of the competing firms, so that competition is also extended to the upstream R&D sector. We propose a novel analysis where differences in the size of their innovation process are allowed, extending the results in Sánchez et al. (2021). We find that the cross-licensing scenario is preferred when the size of the innovation is small; this occurs regardless of the participation in the competing companies and how many innovate. If the innovation is very large, the owners may be better off with exclusive licenses.

Bifurcations and complex dynamics in a banking duopoly model with macroprudential policy

We consider a banking duopoly model with a macroprudential policy in Indonesia called loan-to-deposit ratio-based reserve requirement (LDR-RR). The objective of the policy is to control the banking loans growth using a LDR-based penalty scheme that requires banks to save more money in the central bank to maintain their liquidity ratio. Following recent studies on banking models, we analyze a piecewise discrete-time model with two banks. We assume that the dynamics of the deposits follows the discrete logistic growth. Moreover, our model has two borders, hence we examine the resulting border-collision bifurcations. From the local stability analysis we find that, according to the parameter values, only the border-collision bifurcation or the flip bifurcation occurs. Finally, we perform several numerical simulations to confirm the stability analysis’ results. Our discrete dynamical system offers various possibilities of development for future research perspectives.

Dynamic Analysis of a Simple Cournot Duopoly Model Based on a Computed Cost

The paper is organized to study some mathematical properties and dynamics of a simple Cournot duopoly game based on a computed quadratic cost. The time evolution of this game is described by a two-dimensional noninvertible discrete time map using the bounded rationality mechanism. For this map, some dynamic characteristics such as multistability and synchronization are investigated. Its equilibrium points are obtained for the asymmetric case, and their conditions of stability are obtained. Our results investigate that the Nash equilibrium point may be unstable due to flip bifurcation and under certain parameter values, and Neimark–Sacker bifurcation is born after the period-4 cycle. Through some restrictions, the coordinate axes of the map construct an invariant manifold, and therefore, their dynamics can be analyzed by using a one-dimensional map. In the symmetric case, both firms behave identically, and this implies that the diagonal set forms an invariant manifold, and hence the synchronization phenomena take place. Furthermore, the global bifurcation of the map is confirmed through contact between critical curves and the boundaries of infeasible domains.

Fast or stable: Research on dynamic behaviors of duopoly automakers under the background of energy transformation in the automobile industry

New energy vehicles (NEVs) are crucial way to solving environmental and energy dilemmas. However, the transition from traditional energy vehicles (TEVs) to NEVs is extremely intricate. If the process for TEVs switching to NEVs falls into out order, the automobile industry will be shaken. This paper takes electric vehicles (EVs), representing NEVs, and internal combustion engine vehicles (ICEVs), representing TEVs, as research objects, and constructs a dynamic Gounod duopoly model for the production decisions of two oligopolistic automakers under the dual credit policy. Combining the nonlinear dynamic theory and Cournot game, the influence of policy, market, technology, and cost on the stability of duopoly automakers’ dynamic production game equilibrium and the nonlinear evolution characteristics of Cournot-Nash equilibrium are analyzed systematically. Results show that: (1) In the transition from ICEVs to EVs, if the speed of production adjustment is too fast, the automobile system would trap into chaos and shakeouts, and the social welfare would show volatility. (2) The revised dual credit policy increases the credit ratio and decreases the credit coefficient which enhances the stability of automobile system. This revision may accelerate the transformation from ICEVs to EVs in the automobile industry. Especially, the credit coefficient plays a more significant role. (3) The consumer acceptance for vehicle prices and the range of EV are positively correlated with the production of EVs and the social welfare, and negatively correlated with the production of ICEVs. Nevertheless, the cost of the two types of vehicle had the opposite effect compared price acceptance and range of EVs.

Impact of the New International Land–Sea Transport Corridor on Port Competition between Neighboring Countries Based on a Spatial Duopoly Model

The development of international land–sea transport corridors has provided more convenient access to the sea for inland areas and promoted the improvement of transportation efficiency, environmental improvement, and the strengthening of international cooperation. However, the construction of international land–sea transport corridors has also intensified competition among the ports, which has extended from the local and regional to the national and even international levels. This paper explores the impact of international land–sea transport corridors on oligopolistic port competition between neighboring countries using the Hotelling model. By setting up the utility of the shipper’s port selection, the equilibrium price, market share, and profit of duopoly ports in neighboring countries are analyzed under different conditions of cross-border land transportation and maritime transportation. It is found that the high cross-border transportation cost of the international land–sea transport corridor is not conducive to increasing the market share of the overseas oligopolistic ports in the region. If the maritime transportation cost of overseas oligopoly ports is too high compared with domestic oligopoly ports, it will offset the land transport advantages brought by international land–sea transport corridors. The findings in this paper could provide support for strategic decision making in port markets and cross-border transport corridor development.

Cournot–Bertrand Duopoly Model: Dynamic Analysis Based on a Computed Cost

In this paper, some mathematical properties and dynamic investigations of a Cournot–Bertrand duopoly game using a computed nonlinear cost are studied. The game is repeated and its evolution is presented by noninvertible map. The fixed points for this map are calculated and their stability conditions are discussed. One of those fixed points is Nash equilibrium, and the discussion shows that it can be unstable through flip and Neimark–Sacker bifurcation. The invariant manifold for the game’s map is analyzed. Furthermore, the case when both competing firms are independent is investigated. Due to unsymmetrical structure of the game’s map, global analysis gives rise to complicated basin of attraction for some attracting sets. The topological structure for these basins of attraction shows that escaping (infeasible) domain for some attracting sets becomes unconnected and the rise of holes is obtained. This confirms the existence of contact bifurcation.

Comparative dynamics analysis of the environmental policy efficacy in a nonlinear Cournot duopoly with differentiated goods and emission charges

The price and the exchanged quantity volatility observed in real-world markets may be explained, according to the existing empirical literature, in terms of the endogenous fluctuations generated by the presence of nonlinearities. We then replace with a sigmoid adaptive best response mechanism the linear partial adjustment best response rule considered in Mamada and Perrings (2020), where the effect produced by quadratic emission charges on the dynamics of a Cournot duopoly model with homogeneous goods was investigated. Moreover, the sigmoid nonlinearity, in addition to being well suited to describe the bounded output variations caused by physical, historical and institutional constraints, makes the model able to generate interesting, non-divergent dynamic outcomes, despite the linearity of the demand function and of marginal costs. Additionally, following the suggestion in Mamada and Perrings (2020), we deal with the more general case of differentiated products. Beyond analytically studying the stability of the unique steady state, coinciding with the Nash equilibrium, and the effect produced by the main parameters on the stability region, we perform two comparative dynamics investigations which allow to evaluate the environmental policy efficacy when the Nash equilibrium is not stable and thus the standard comparative statics technique does not fit for the purpose. In particular, the former analysis, which is based on a comparison of emissions for different levels of charges, shows that, also in case the Nash equilibrium is not stable, the considered environmental policy may be effective both with complements and with substitutes. The latter investigation, consisting in a comparison of emissions along non-stationary trajectories and along the equilibrium path, in the proposed experiments highlights that emissions are larger along non-stationary trajectories. This gives us the opportunity to show how to act on the level of the asymptotes of the sigmoid adjustment mechanism to reduce output variations, reaching at one time a complete stabilization of the system and limiting pollution.

Sequential licensing with several competing technologies

We assume a multistage oligopoly wherein a given number of innovators compete by selling their substitutive technologies. Each innovator sequentially and independently chooses how many licenses to sell, and subsequently, all licensees compete à la Cournot in the product market. We show that, in equilibrium, the total number of licensees grows exponentially with the number of innovators. In addition, this sequential outcome is also obtained as a subgame perfect Nash equilibrium in pure strategies of a game with endogenous timing. Interestingly, by extending the duopoly model of Badia et al. (Math Soc Sci 108:8–13, 2020) to the case of more than two innovators and exploring pure strategy equilibria instead of mixed strategy equilibria, we derive drastically different policy implications, in terms of patent regulations. Our results suggest that more competition in the upstream market (e.g., by relaxing patent protection against the appearance of similar technologies) tends to increase downstream competition and welfare instead of discouraging or delaying technology adoption. In addition, our analysis is extended to explore the strategic role of public investment in basic R &D.

Collusive Outcomes Without Collusion: Algorithmic Pricing in a Duopoly Model

Collusive Outcomes Without Collusion: Algorithmic Pricing in a Duopoly Model

Can an E-commerce Platform and its Third-Party Sellers Benefit From Each Other’s Market Entry?

E-commerce platforms have an informational advantage over their third-party sellers, leading to the common belief that a platform's market entry would harm sellers with similar products. However, unlike traditional retail competition, the platform and its sellers have aligned incentives: the platform's commission depends on the seller's revenue, and sellers rely on the platform to strengthen their online presence. Hence, the platform has no incentive to enter the market to harm the seller's revenue severely. This paper introduces a duopoly model where the seller is the "incumbent" and the platform is the "potential entrant". Our model captures two salient features: (a) the "reputation effect" that enables the platform to obtain a higher consumer valuation than the seller, and (b) the "spillover effect" that expands the market size when an additional entity (e.g., the platform) enters the market. Our equilibrium analysis debunks the prevailing belief about platform's entry, showing that platform's entry can enable both the platform and the seller to obtain a higher profit when the unit cost is sufficiently low and the spillover effect is sufficiently high. For robustness checks, we consider three different extensions: an alternative duopoly model with reversed roles where the platform is the incumbent and the seller is the potential entrant, a scenario with an endogenously determined spillover effect, and a simultaneous market entry/exit decision-making process. We find a consistent result across all three extensions that both seller and platform entries can mutually benefit under similar market conditions, fostering a symbiotic relationship.

Knowledge Management: High-Quality Production Decisions in Partial Volume Flexibility to Create Value for Customers

Uncertainty in the demand of individual customers increases demand volatility in the entire market. In this context, partial volume flexibility can help hedge against demand uncertainty and competition pressure, and subsequently help create value by means of high-quality production decisions and market responsiveness, within the assumption of transformation and applied knowledge incorporated into the production stages. The article aims to investigate this phenomenon. To achieve this, as part of the methodology, a unified measure, the flexibility degree, is taken to represent the level of partial volume flexibility. We find that under demand uncertainty, a firm goes through a process with four stages—capacity decision, stable production, flexible production, and pricing. In a monopoly model, we find that a slight change in the flexibility degree could significantly increase a firm's adjustable production range to respond to demand changes. Furthermore, we develop a general asymmetric duopoly model. We characterize the equilibrium and find that a firm with a higher flexibility degree is more sensitive to demand changes. Accordingly, in response to specific demand, a higher flexibility firm obtains more profit only when the demand is sufficiently high or low, otherwise, the lower flexibility firm wins. Finally, we observe that a firm's capacity increases in its flexibility degree and decreases in its rival's flexibility degree; this is because the firm balances the market share and product profitability.

Comparison between Simultaneous and Sequential Movements in a Duopoly Game

In general, a Stackelberg competition benefits the leader and the consumer due to making the first move in a sequential game. However, in this paper, sequential movement does not offer an advantage over simultaneous movement in a static duopoly under quantity competition with an isoelastic demand function. By introducing dynamics into the duopoly model through numerical simulations of simultaneous and sequential movements towards the Cournot equilibrium, there is evidence of first-mover advantage. However, simultaneous movement is advantageous over sequential movement in terms of better profits for both firms and lower prices for the consumer.

Insights on the Theory of Robust Games

AbstractA robust game is a distribution-free model to handle ambiguity generated by a bounded set of possible realizations of the values of players’ payoff functions. The players are worst-case optimizers and a solution, called robust-optimization equilibrium, is guaranteed by standard regularity conditions. The paper investigates the sensitivity to the level of uncertainty of this equilibrium focusing on robust games with no private information. Specifically, we prove that a robust-optimization equilibrium is an $$\epsilon $$ ϵ -Nash equilibrium of the nominal counterpart game, where $$\epsilon $$ ϵ measures the extra profit that a player would obtain by reducing his level of uncertainty. Moreover, given an $$\epsilon $$ ϵ -Nash equilibrium of a nominal game, we prove that it is always possible to introduce uncertainty such that the $$\epsilon $$ ϵ -Nash equilibrium is a robust-optimization equilibrium. These theoretical insights increase our understanding on how uncertainty impacts on the solutions of a robust game. Solutions that can be extremely sensitive to the level of uncertainty as the worst-case approach introduces non-linearity in the payoff functions. An example shows that a robust Cournot duopoly model can admit multiple and asymmetric robust-optimization equilibria despite only a symmetric Nash equilibrium exists for the nominal counterpart game.