Dynamic Duopoly Game Research Articles

R&D investment and network effects: A dynamic model of the search engine market

ABSTRACT This paper considers R&D investment for quality improvement in a dynamic duopoly game in the search engine market. The aim is to determine the circumstances under which asymmetry between search engines (in terms of quality) tends to increase or decrease over time. I show that when the future sufficiently matters, the low-quality search engine can catch up with the high-quality engine by investing in quality. In contrast, when only the present matters, the high-quality search engine wins and the structure of the market transitions to a monopoly. I also show that a shift in asymmetry between firms can result in either an increase or decrease in consumer welfare, depending on the degree of substitutability between the two search engines.

Strategic investment under uncertainty: why multi-option firms lose the preemption run

We consider a dynamic duopoly game where firms choose both the timing and size of their investments. The existing real options literature predominantly consists of contributions where firms have a single option to invest. This paper relaxes this assumption by giving Firm A multiple options to undertake further investments with the purpose to expand whereas Firm B only holds the option to enter the market. In this asymmetric setting we get the surprising result that, in equilibrium, Firm B invests first. If Firm A invests first, Firm A and Firm B keep on being involved in preemption games for subsequent investments until Firm B enters the market, which leads to inefficiently early investments of Firm A. When Firm B invests first, then only one preemption game is played, which leads to Firm A being free to choose its unrestricted optimal investment moments.

When should grocery stores adopt time‐based pricing? Impact of competition and negative congestion externality

Consumers dread shopping during peak hours, and the Covid‐19 pandemic has created additional safety concerns about overcrowding in addition to long waiting times. In view of consumer's congestion aversion, should competitive brick‐and‐mortar grocery stores charge higher prices during congested peak hours to smooth demand? To examine “whether and when” stores should adopt intraday time‐based pricing under competition, we examine a 2‐stage dynamic duopoly game. At the beginning of each stage, each store can make an irreversible decision to adopt time‐based pricing by setting the peak‐hour and normal‐hour prices. We also endogenize consumer's shopping decisions (i.e., when and which store to shop) by incorporating the issue of negative congestion externality. Our equilibrium analysis reveals that time‐based pricing is always beneficial for the stores, and both stores would adopt it eventually in equilibrium. As such, only two equilibria can sustain: either both firms adopt time‐based pricing immediately in stage 1, or only one firm adopts in stage 1 while the other postpones its adoption until stage 2. Interestingly, due to the competitive dynamics, it is less likely for both firms to adopt immediately when consumers are more averse to congestion. Moreover, although the adoption of time‐based pricing leads to differentiated price competition, it can “soften” price competition, causing both peak‐hour and normal‐hour prices to rise above the status quo equilibrium uniform prices. We find that time‐based pricing can always induce demand smoothing and reduce congestion. Although time‐based pricing creates value for the stores (through higher prices), it offers no benefit to consumers.

A Dynamic Multi-Objective Duopoly Game with Capital Accumulation and Pollution

This paper studies a discrete-time dynamic duopoly game with homogenous goods. Both firms have to decide on investment where investment increases production capacity so that they are able to put a larger quantity on the market. The downside, however, is that a larger quantity raises pollution. The firms have multiple objectives in the sense that each one maximizes the discounted profit stream and appreciates a clean environment as well. We obtain some surprising results. First, where it is known from the continuous-time differential game literature that firms invest more under a feedback information structure compared to an open-loop one, we detect scenarios where the opposite holds. Second, in a feedback Nash equilibrium, capital stock is more sensitive to environmental appreciation than in the open-loop case.

Stability, Multistability, and Complexity Analysis in a Dynamic Duopoly Game with Exponential Demand Function

In this paper, a discrete-time dynamic duopoly model, with nonlinear demand and cost functions, is established. The properties of existence and local stability of equilibrium points have been verified and analyzed. The stability conditions are also given with the help of the Jury criterion. With changing of the values of parameters, the system shows some new and interesting phenomena in terms to stability and multistability, such as V-shaped stable structures (also called Isoperiodic Stable Structures) and different shape basins of attraction of coexisting attractors. The eye-shaped structures appear where the period-doubling and period-halving bifurcations occur. Finally, by utilizing critical curves, the changes in the topological structure of basin of attraction and the reason of “holes” formation are analyzed. As a result, the generation of global bifurcation, such as contact bifurcation or final bifurcation, is usually accompanied by the contact of critical curves and boundary.

A Dynamic Duopoly Game with Content Providers’ Bounded Rationality

This paper investigates the dynamical behaviors of a duopoly model with two content providers (CPs). Competition between two CPs is assumed to take place in terms of their pricing decisions and the credibility of content they offer. According to the CPs’ rationality level, we consider a scenario where both CPs are bounded rational. Each CP in any period uses the marginal profit observed from the previous period to choose its strategies. We compute explicitly the steady states of the dynamical system induced by bounded rationality, and establish a necessary and sufficient condition for stability of its Nash equilibrium (NE). Numerical simulations show that if some parameters of the model are varied, the stability of the NE point is lost and the complex (periodic or chaotic) behavior occurs. The chaotic behavior of the system is stabilized on the NE point by applying control.

Dynamic Effects Arise Due to Consumers' Preferences Depending on Past Choices.

We analyzed a dynamic duopoly game where players adopt specific preferences. These preferences are derived from Cobb–Douglas utility function with the assumption that they depend on past choices. For this paper, we investigated two possible cases for the suggested game. The first case considers only focusing on the action done by one player. This action reduces the game’s map to a one-dimensional map, which is the logistic map. Using analytical and numerical simulation, the stability of fixed points of this map is studied. In the second case, we focus on the actions applied by both players. The fixed points, in this case, are calculated, and their stability is discussed. The conditions of stability are provided in terms of the game’s parameters. Numerical simulation is carried out to give local and global investigations of the chaotic behavior of the game’s map. In addition, we use a statistical measure, such as entropy, to get more evidences on the regularity and predictability of time series associated with this case.

Measurement on the Complexity Entropy of Dynamic Game Models for Innovative Enterprises under Two Kinds of Government Subsidies

In this paper, setting the high-tech industry as the background, we build a dynamic duopoly game model in two cases with different government subsidies based on the innovation inputs and outputs, respectively. We analyze the equilibrium solution and stability conditions of the system, and study the dynamic evolution of the system under the conditions of different system parameters by the numerical simulation method. The simulation results show that both innovation subsidy policies have positive effects on firms’ innovation activities. Besides, improving the level of innovation can encourage firms to innovate. It also shows that an exaggerated adjusting speed of innovation outputs may cause complicated dynamic phenomena such as bifurcation and chaos, which means that the system has relatively higher entropy than that in a stable state. The degree of the government innovation subsidies is also shown to impact the stability and entropy of the system.

The complexion of dynamic duopoly game with horizontal differentiated products

In this paper, firms are considered on the hypothesis of having incomplete rationality expectation and incomplete information of the market to get the dynamic development of price competition behavior in the Hotelling model (Hotelling, 1929). Under the assumption of the heterogeneous expectations of two firms, we have observed that the Nash equilibrium price can be a dynamic equilibrium to realize when the speed of price adjustment is lower. However, the numerical simulation shows that the system may present a periodic and chaotic status when the speed of price adjustment is higher. The effect of the degree of horizontal differentiation on the stability of Nash equilibrium of the system is also discussed. We have a different conclusion from Lucino Fanti and Luca Gori's (2012); that is the greater the degree of product horizontal differentiation is, the more stable the Nash equilibrium of the system is.

Chaotic Characteristics and Application of Cooperative Game and Evolutionary Game

According to a dynamical multiteam Cournot game in exploitation of a renewable resource, a new dynamic Cournot duopoly game model with team players in exploitation of a renewable resource is built up in this paper. Based on the theory of bifurcations of dynamical systems, the stability of the system is studied and the local stable region of Nash equilibrium point is obtained. The effect of the output adjustment speed parameters and the weight parameter of the system on the dynamic characteristics of the system are researched. The complexity of the system is described via the bifurcation diagrams, the Lyapunov exponents, the phase portrait, the time history diagram, and the fractal dimension. Furthermore, the chaos control of the system is realized by the parameter adjustment method. At last, an evolutionary game as a special dynamic system is constructed and analyzed which is more useful and helpful in application. The derived results have very important theoretical and practical values for the renewable resource market and companies.

Analysis of Price Stackelberg Duopoly Game with Bounded Rationality

The classical Stackelberg game is extended to boundedly rational price Stackelberg game, and the dynamic duopoly game model is described in detail. By using the theory of bifurcation of dynamical systems, the existence and stability of the equilibrium points of this model are studied. And some comparisons with Bertrand game with bounded rationality are also performed. Stable region, bifurcation diagram, The Largest Lyapunov exponent, strange attractor, and sensitive dependence on initial conditions are used to show complex dynamic behavior. The results of theoretical and numerical analysis show that the stability of the price Stackelberg duopoly game with boundedly rational players is only relevant to the speed of price adjustment of the leader and not relevant to the follower’s. This is different from the classical Cournot and Bertrand duopoly game with bounded rationality. And the speed of price adjustment of the boundedly rational leader has a destabilizing effect on this model.

Dynamics analysis and chaos control of a duopoly game with heterogeneous players and output limiter

In actual economies, players sometimes would offer an upper limiter to their output due to capacity constraints, financial constraints and cautious response to uncertainty in the world, or offer a lower limiter to their output due to economies of scale or break-even consideration. In this paper, we discuss a dynamic duopoly game with heterogeneous players by assuming that one of them imposes an upper limiter on output, and the other one imposes a lower limiter. We analyze how the limiter affects the dynamics of output and the performance of players, and explore the number of the equilibrium points and the distribution of conditioned equilibrium points of the model. We then discuss the stable region of conditioned equilibrium. The theoretical results and numerical experiments show that adding appropriate limiter can make the system more robust, and even get rid of its chaos. The numerical results show that chaotic output dynamics can be beneficial to one firm but harmful to the other, and can also be harmful to both of them, and also show that adding appropriate limiter can improve the performance of the player.

Numerical solution of dynamic oligopoly games with capital investment

This paper discusses how numerical techniques may be used to solve the simultaneous functional equations that arise in general dynamic stochastic games. Unlike the conventional linear-quadratic approach, our methods may be used to address general model specifications that may include non-quadratic objective functions, non-linear equations of motion, and constraints on decision variables. As an illustration, we apply our methods to a dynamic duopoly game in which competing firms play short-run quantity game subject to production cost that can be lowered through investment in capital stock in the long run.

Synchronization, intermittency and critical curves in a duopoly game

The phenomenon of synchronization of a two-dimensional discrete dynamical system is studied for the model of an economic duopoly game, whose time evolution is obtained by the iteration of a noninvertible map of the plane. In the case of identical players the map has a symmetry property that implies the invariance of the diagonal x 1= x 2, so that synchronized dynamics is possible. The basic question is whether an attractor of the one-dimensional restriction of the map to the diagonal is also an attractor for the two-dimensional map, and in which sense. In this paper, a particular dynamic duopoly game is considered for which the local study of the transverse stability, in a neighborhood of the invariant submanifold in which synchronized dynamics takes place, is combined with a study of the global behavior of the map. When measure theoretic, but not topological, attractors are present on the invariant diagonal, intermittency phenomena are observed. The global behavior of the noninvertible map is investigated by studying of the critical manifolds of the map, by which a two-dimensional region is defined that gives an upper bound to the amplitude of intermittent trajectories. Global bifurcations of the basins of attraction are evidenced through contacts between critical curves and basin boundaries.

Empirical Analysis of a Dynamic Duopoly Model of Competition

Empirically validating and testing the specification of game theoretic models has received limited attention in the marketing literature. The authors provide an econometric framework for estimating the parameters of response functions when the observed data in the market place are the Nash equilibrium outcomes of an underlying dynamic duopoly game specification. Specifically, the estimation procedure accounts for the joint endogeneity of market shares and marketing efforts of market rivals using a system of simultaneous equations that included the market response function and the Nash equilibrium conditions. A formal statistical test is used to detect model misspecification. The empirical analysis is carried out using data from four product markets: pharmaceutical, soft drink, beer, and detergent. Comparisons are provided with conventional estimation of the response function parameters in which the equilibrium conditions are ignored in the estimation. Managerial implications of the empirical results are discussed.

Dynamic Duopoly with Best-Price Clauses

This article investigates best-price clauses as a strategic devise to facilitate collusion in a dynamic duopoly game. Best-price clauses guarantee rebates on the purchase price if a finds a better price after his purchase. Two different price clauses are distinguished: most favored customer and meet or release. I examine the collusive potential of both clauses in a finite-horizon duopoly model with homogeneous durable goods. In each period, new consumers enter the market. I show that in this context, meet-or-release clauses have a greater anticompetitive potential than most-favored-customer clauses.

Optimal pricing in a dynamic duopoly game model

This paper deals with optimal pricing of new products over a finite planning period in a duopolistic market. Modelling saturation effects and no cost-side learning effects optimal pricing strategies for different kinds of demand functions are determined. In this direction the paper extends some results known from the monopolistic case. It turns out, that the optimal prices are decreasing functions of time, where the prices at each moment of time are higher than the marginal costs. Thus the optimal pricing strategies can be characterized as skimming policies.

Dynamic Duopoly Game: Differential Game Theoretic Approach

Output variation type Cournot duopoly game is studied within the framework of differential game theory. Necessary conditions for a general case are developed in their typical coupled form and the difference between an open-loop control (supply adjustment) and a closed-loop control is pointed out. Linear dynamic market demand function and quadratic cost functions case is then studied in detail and analytic expressions for the optimal Cournot controls are obtained in terms of the solution of a set of differential equations for a finite horizon case (t ε [0, T)) and, a set of algebraic equations for an infinite horizon case (t ε [0, ∞]).