Heterogeneous Duopoly Research Articles

Analysis of heterogeneous duopoly game with information asymmetry based on extrapolative mechanism

Abstract Information plays an important role in decision-making process in oligopoly market. This paper establishes two Cournot duopoly games with information asymmetry based on extrapolative mechanism, and focus on the impacts of information asymmetry from the perspective of stability, complexity and profit. The results show that the extrapolative mechanism plays a different role for heterogeneous expectation firms. In general, proper prediction of rival’s output information is conducive to the improvement of firms’ profit and system stability. However, for firms adopting the adaptive adjustment mechanism, the optimal profit does not occur for the most perfect prediction. Firms who adopt gradient adjustment mechanism are worth to improve prediction accuracy to promote the system stability and the profit. In addition, it is more important for all firms to control their own output adjustment mechanism to gain more profits.

A Heterogeneous Duopoly Game under an Isoelastic Demand and Diseconomies of Scale

In this paper, we investigate a duopolistic market with heterogeneous firms under the assumptions of an isoelastic demand and quadratic costs. We obtain the sufficient and necessary condition of the local stability of the Cournot–Nash equilibrium and analytically compare it with that of the analogue model under linear rather than quadratic costs. By approaches of symbolic computation, we prove that diseconomies of scale have an effect of stabilizing the game provided that the cost parameters are large enough. Moreover, by means of numerical simulations, we find that our model loses its stability only through a period-doubling bifurcation, which is different from its analogue having two possible routes to chaotic dynamics.

Patent Licensing and Technological Catch-Up in a Heterogeneous Duopoly

A cost-reducing innovation is available to Cournot duopolists through licensing. The firms are ex ante heterogeneous and adoption of the innovation impacts them differently. Is it possible for the inefficient duopolist to catch up with its efficient rival? Under certain conditions, yes. The conditions, however, are stringent: It is not sufficient, for instance, that the innovation promotes a change in the efficiency rank of the firms.

A continuous-time heterogeneous duopoly model with delays

In this paper, the continuous-time version with delays of the model of Tramontana (Econ Model 27(1):350–357, 2010) is presented. The stability of the equilibrium point is discussed by analyzing the distribution of roots of associated characteristic equation. It is found that combining marginal costs with time delays gives rise to different economic scenarios, where stability switches may appear and Hopf bifurcations occur within certain range of parameters.

Positioning–pricing problem of heterogeneous duopoly with uncertain consumer preferences

Small- and medium-sized firms in China have competed passively with advantaged enterprises since 2000, and the prosperity indexes have always been in the recession. This paper aims to find the optimal positioning–pricing strategies for disadvantaged small- and medium-sized firms and explore the complex influence of consumer preference uncertainty in the optimal strategies in duopoly market. In order to do so, the classic Hotelling model is modified to be a heterogeneous duopoly game model under consumer preference uncertainty. By using the backward recurrence algorithm, we first get the optimal equilibrium and then figure out the expected equilibrium solution through mathematical expectation method. The computational results show that disadvantaged enterprises could survive themselves in fierce competition as long as they choose appropriate strategies. In addition, consumer preference uncertainty plays a significant role in affecting the optimal strategy decisions of positioning and pricing in duopoly since producers can hardly obtain complete and perfect information about the market. Specifically, the increased uncertainty of consumer preference generally raises the expected equilibrium price and profit of duopoly enterprises. However, the expected equilibrium profit of the advantaged enterprise in a duopoly decreases with consumer preference uncertainty increasing under certain conditions.

Competition and The Operational Performance of Hospitals: The Role of Hospital Objectives

We examine the effect of a hospital's objective (i.e., non‐profit vs. for‐profit) in hospital markets for elective care. Using game‐theoretic analysis and queueing models to capture the operational performance of hospitals, we compare the equilibrium behavior of three market settings in terms of such criteria as waiting times and patient costs from waiting and hospital payments. In the first setting, a monopoly, patients are served exclusively by a single non‐profit hospital; in the second, a homogeneous duopoly, patients are served by two competing non‐profit hospitals. In our third setting, a heterogeneous duopoly, the market is served by one non‐profit hospital and one for‐profit hospital. A non‐profit hospital provides free care to patients, although they may have to wait; for‐profit hospitals charge a fee to provide care with minimal waiting. A comparison between the monopolistic and each of the duopolistic settings reveals that the introduction of competition can hamper a hospital's ability to attain economies of scale and can also increase waiting times. Moreover, the presence of a for‐profit sector may be desirable only when the hospital market is sufficiently competitive. A comparison across the duopolistic settings indicates that the choice between homogeneous and heterogeneous competition depends on the patients' willingness to wait before receiving care and the reimbursement level of the non‐profit sector. When the public funder is not financially constrained, the presence of a for‐profit sector may allow the funder to lower both the financial costs of providing coverage and the total costs to patients. Finally, our analysis suggests that the public funder should exercise caution when using policy tools that support the for‐profit sector—for example, patient subsidies—because such tools may increase patient costs in the long run; it might be preferable to raise the non‐profit sector's level of reimbursement.

Equilibrium stability of a nonlinear heterogeneous duopoly game with extrapolative foresight

We make a further attempt to investigate equilibrium stability of a nonlinear Cournot duopoly game with adaptive adjustment toward best reply by assuming heterogeneous firms where one firm only uses naive expectations whereas the other employs a simple forecast technology to form sophisticated expectations. More precisely, based on the knowledge of actual production of the competitor and its actual rate of change, the clever firm is able to evaluate its opponent's output in the near future by virtue of straightforward extrapolative foresight. We finally arrive at a conclusion that this seemingly rational mechanism takes a positive effect on convergence to equilibrium behavior. Inconsistent with common intuition, we demonstrate that stronger foresight ability is not always better to stabilize the equilibrium. Particularly, perfect foresight dose not give rise to the best stabilizing factor.

Analysis of Triopoly Game with Isoelastic Demand Function and Heterogeneous Players

We analyze a triopoly game model with fully heterogeneous players when the demand function is isoelastic. The three players were considered to be bounded rational, adaptive, and naïve. Existing equilibrium points and their locally asymptotic stability conditions are studied. Complexity of the dynamical system is examined by means of numerical simulations, such as period cycles, bifurcation diagrams, strange attractors and sensitive, dependence on initial conditions. This paper extends the result of Tramontana (2010) who considered a heterogeneous duopoly with isoelastic demand function. Comparisons with respect to the heterogeneous triopoly model of Elabbasy et al. (2009) assuming linear demand function are performed.

The dynamics of a differentiated duopoly with quantity competition

We analyse the dynamics of a Cournot duopoly with heterogeneous players to investigate the effects of micro-founded differentiated products demand on stability. The present study, which indeed modifies and extends Zhang et al. (2007) (Zhang, J., Da, Q., Wang, Y., 2007. Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling 24, 138–148) and Tramontana, F., (2010) (Tramontana, F., 2010. Heterogeneous duopoly with isoelastic demand function. Economic Modelling 27, 350–357), reveals that a higher degree of product differentiation may destabilise the market equilibrium. Moreover, we show that a cascade of flip bifurcations may lead to periodic cycles and ultimately chaotic behaviours. Since a higher degree of product differentiation implies weaker competition, then a theoretical implication of our findings, that also constitute a policy warning, is that a fiercer (weaker) competition tends to stabilise (destabilise) the unique positive Cournot–Nash equilibrium.

Heterogeneous duopoly, spatially discriminatory pricing, and optimal location

It is found in the literature that if duopolists produce differentiated products and engage in price competition in a linear city model with elastic demand, the two firms necessarily agglomerate at the market center and this is the unique locational equilibrium, irrespective of the pricing policies charged by the firms. Utilizing a more reasonable market-serving assumption, this paper finds that the firms can be either centrally agglomerated or dispersed depending on the magnitude of the transport rate and the degree of product differentiation. Moreover, if the two firms choose quantity instead of price as their decision variables, the two firms become less likely to stay apart. But if they do, the distance of their locations necessarily shrinks. This paper also examines the locational configuration in the absence of the market-serving assumption and finds that spatial dispersion could be the only location pattern.

Price-taking behavior versus continuous dynamic optimizing

Abstract In a quantity-competition oligopoly, previous studies have shown that a price-taking firm can outperform any rival with identical technology that produces at a different output level at any intertemporal equilibrium. This research seeks to examine this seemingly counter-intuitive fact in a heterogeneous duopoly consisting of an adaptive price-taker and a dynamic optimizer who dynamically optimizes its either absolute or relative profit stream over an infinite planning horizon. With conventional economic assumptions, the Hamiltonian system always converges to a saddle point. If goal of the dynamic optimizer is absolute profit, the price-taker produces more than the dynamic optimizer at the equilibrium, around which there exists a long-run relative profitability portion for the price-taker. Consequently, the price-taker unconsciously makes higher relative profit than the dynamic optimizer in the long-run. In contrast, if the dynamic optimizer pursues the relative profit, both firms produce at the competitive level at the equilibrium while the whole optimal path lies in the relative profitability regime of the dynamic optimizer so that it will enjoy a higher profit than the price-taker before reaching equilibrium. These results provide economical justification for ever-growing evolutionary game theoretical literatures in appreciating price-taking behavior and additionally clarify possible misunderstandings in interpreting their conclusions.

Nonlinear dynamics in a heterogeneous duopoly game with adjusting players and diseconomies of scale

A repeated, discrete time, heterogeneous Cournot duopoly game with bounded rational and adaptive players adjusting the quantities of production is subject of investigation. Linear inverse demand function and quadratic cost functions reflecting decreasing returns to scale are assumed. The game is modeled with a system of two difference equations. Evolution of outputs over time is obtained by iteration of a two dimensional nonlinear map. Existing equilibria and their stability are analyzed. In face of diseconomies of scale, bounded rational and adaptive duopolists are shown to experience a decrease in the latitude of their output adjustment decisions with respect to the market stability compared to constant returns to scale and ceteris paribus. Chaotic dynamics is confirmed to depend mainly on the adjustment behavior of the bounded rational player, who if overshoots leaves the adaptive player with limited opportunities to stabilize the market again, hence industries facing diseconomies of scale are found to be less stable than those with constant marginal costs. Complexity of the dynamical system is examined by means of numerical simulations, where the paper extends the results of other authors who considered analogous games assuming linear cost functions. Intermittent transition to chaos and attractor merging crisis are shown among others.

Cournot–Nash and Bertrand–Nash equilibria for a heterogeneous duopoly of differentiated products

Under Cournot---Nash and Bertrand---Nash equilibria, analytical expressions are found for outputs, prices, and profits in the case when all duopoly parameters of differentiated products are different. It is shown that the signs of differences between such outputs, prices, and profits depend on the complementarity or substitutability of products. The concepts of exogenous, endogenous, and asymmetric differentiated products are introduced.

Heterogeneous duopoly with isoelastic demand function

In this paper we analyze a duopolistic market with heterogeneous firms when the demand function is isoelastic (Puu, T., 1991. Chaos in duopoly pricing. Chaos, Solitons and Fractals 1, 573–581.). We consider the same heterogeneous firms of Zhang et al. (Zhang, J., Da, Q., Wang, Y., 2007. Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling 24, 138–148.) introducing a nonlinearity in the demand function instead of the cost function. Stability conditions of the Nash equilibrium and complex dynamics are studied. In particular we show two different routes to complicated dynamics: a cascade of flip bifurcations leading to periodic cycles (and chaos) and the Neimark-Sacker bifurcation which originates an attractive invariant closed curve. Comparisons with respect to the Puu model and the model of Zhang et al. are performed.

Bifurcation analysis of a dynamic duopoly model with heterogeneous costs and behavioural rules

We develop a nonlinear dynamic Cournot duopoly model in discrete time, where two bounded rational quantity-setting firms, both endowed with naïve expectations, are heterogeneous as to their cost functions and output strategies. One of the two competing firms adopts best-reply behaviour, whereas the other adjusts its production according to the gradient of its profit. Although excessive reaction of the latter firm generally results in destabilization of the ‘Cournot–Nash’ equilibrium and brings about complex dynamic scenarios, this occurs via qualitatively different bifurcation routes, depending on the ratio between firms’ marginal costs. The paper also provides a preliminary exploration of the long-run sustainability of the heterogeneous duopoly competition, by investigating numerically the impact of fixed costs on firms’ survival, along different output trajectories.

Simultaneous and sequential price competition in heterogeneous duopoly markets: experimental evidence

We investigate simultaneous and sequential price competition in duoply markets with differentiated products and random matching of symmetric firms. We find that second movers gain from the sequential structure in comparison to simultaneous-move markets whereas first movers do not. As predicted by the theory, there is a significant first-mover disadvantage in the sequential game. Finally, we report the results of control treatments varying the matching scheme and the mode of eliciting choices (strategy method vs. standard sequential play).

Stackelberg price leadership in the linear heterogeneous duopoly

We consider for the linear heterogeneous duopoly with substitutive goods and with price strategies the situations in which one firm, firm 1 say, prefers to be price leader, whereas firm 2 prefers to be price follower. Conditions that necessarily hold in that case are derived and interpreted in economic terms. These conditions characterize the powerful position of firm 1 relative to firm 2 on the market.