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Learning cycles in Bertrand competition with differentiated commodities and competing learning rules
Abstract
This paper stresses the importance of heterogeneity in learning. We consider a Bertrand oligopoly with firms using either least squares learning or gradient learning for determining the price. We demonstrate that convergence properties of the rules are strongly affected by heterogeneity. In particular, gradient learning may become unstable as the number of gradient learners increases. Endogenous choice between the learning rules may induce cyclical switching. Stable gradient learning gives higher average profit than least squares learning, making firms switch to gradient learning. This can destabilize gradient learning which, because of decreasing profits, makes firms switch back to least squares learning.
... In particular, the replicator dynamics-which can be derived from a process of pairwise imitation-inhibits adjustment processes to spread quickly through the population of firms. With the class of evolutionary processes studied here this is much easier, because the adoption of an adjustment process only depends upon its relative performance and not on the fraction of firms currently using 5 Corchon and Mas-Colell [13] show that any type of behavior can emerge for continuous-time gradient (or best-reply) dynamics in heterogeneous oligopoly, although Furth [23] argues that for homogeneous Cournot oligopoly there are certain restrictions as to what behavior can arise. Relatedly, Dana and Montrucchio [16] show that in a duopoly model where firms maximize their discounted stream of future profits and play Markov perfect equilibria-and therefore are rational-any behavior is possible for small discount factors. ...
... Besides these benchmark adjustment processes, many other processes obey the general form (3), such as local monopolistic approximation 13 or imitating the average (although the latter does not satisfy part (i) of Assumption A). Some other adjustment processes, such as fictitious play and least squares learning (see, e.g., Anufriev et al. [5]), cannot be represented by (3). ...
... (This case is briefly discussed in Sect. 5) leading to the emergence of two coexisting attracting period eight cycles. This sequence of period-doubling bifurcations continues, creating coexisting period 16 cycles (emerging at n ≈ 17.19) and coexisting period 32 cycles (emerging at n ≈ 17.51) and eventually leading to two coexisting four piece attractors, characterized by complicated aperiodic dynamics. ...
We consider a model of evolutionary competition between adjustment processes in the Cournot oligopoly model and investigate the effect of increasing the number of firms. Our focus is on Nash play versus a general short-memory adaptive adjustment process. We find that, although Nash play has a stabilizing influence, a sufficient increase in the number of firms in the market tends to make the Cournot-Nash equilibrium unstable. This shows that the famous result by Theocharis (Rev Econ Stud 1960), that Cournot oligopoly markets are unstable for more than three firms, is robust, although the instability threshold increases in the presence of Nash firms. We establish that both the existence and the level of this threshold depend on the information costs associated with Nash play. Moreover, the interaction between adjustment processes naturally leads to the emergence of complicated endogenous fluctuations as the number of firms increases, even when demand and costs are linear.
... With respect to Bertrand competition, [12] studies how the market with asymmetric and non-linear demand reaches a unique steady state in monopolistic price competition and finds that a simple and optimal price strategy can lead the market to a unique steady state. [4] consider a Bertrand oligopoly with firms using either least squares learning or gradient learning for determining the price. They demonstrate that convergence properties of the rules are strongly affected by heterogeneity. ...
... 3. The supplier sets the wholesale price w based on the available information. 4. The two matched retailers determine order quantities or price based on the wholesale price and the belief about market demand, respectively. ...
... It is noteworthy that both retailers are aware of this. From (4) we know that the retailers' equilibrium strategies (q 1 , q 2 ) should satisfy the conditions of ...
This paper explores the retailers’ information acquisition behavior under horizontal competition and investigates the impact of information leakage from an evolutionary game theory perspective. When information leakage does not occur, we find that firms’ acquisition behavior differ under different types of competition. Under Cournot competition, rivals prefer the competitor not to acquire information. Under Bertrand competition, however, rivals prefer the competitor to acquire information. As a result, at a moderate level of acquisition cost mixed strategies (acquire and not acquire) coexist under Cournot competition, whereas all retailers adopt the same (pure) strategy under Bertrand competition. With information leakage, two effects (collaborative forecasting and free-riding) influence firms’ decision making. When the collaborative forecasting effect dominates, all retailers would choose the acquisition strategy in the equilibrium. However, if the free-riding effect is strong, no retailer would acquire information in the equilibrium. With a moderate acquisition cost, there are still retailers willing to acquire information and mixed strategies coexist, regardless of the type of competition. Interestingly, no matter information is leaked or not, we find that Cournot competition in general better accommodates information acquisition than Bertrand competition. As a result, Cournot competition can lead to a higher total welfare than Bertrand competition when the information acquisition cost is moderate and the demand uncertainty is large, in contrast to the standard result of duopoly models.
... Gradient mechanisms are used to model several economic contexts characterized by boundedly rational agents. As an example, starting from the contribution by Bischi and Naimzada in [4], such kind of mechanism has been applied to the modelling of oligopoly competitions ( [5,6,7,8,9]). Concerning the monopoly modelling, we can mention the contribution by Puu [10], who studied the chaotic dynamics arising when a discrete gradient is considered, while Naimzada and Ricchiuti [11] investigated the effect of the continuous gradient. In both works a discrete time adjustment mechanism is considered. ...
... In Figure 6 (B) we considered a perturbation size very close to the threshold of the fold bifurcation, and we numerically find π ′′ 0 (q 1 ) = −1.56 and π ′′ 0 (q 2 ) = −4.1, for which the left hand side of (12) is equal to 0.988. We stress that looking at Figure 6 (B), we can notice that System (6) has also other steady states, which are however locally asymptotically unstable. Finally, we just want to quickly point out that the period-2 cycle arising from the equilibrium q * of the unperturbed model may not be the unique period-2 attractor of (5). ...
... For each simulation, we compute the corresponding price series and we take into account 500 prices after an initial transient of 1000 time steps. In Figure 10 we report the results of kurtosis, skewness and percent relative volatility (the percent ratio between the standard deviation of the price distribution and the mean price) obtained considering different values of the agent's reaction speed γ and of the standard deviation of the shock 6 . The values of each index are computed averaging 100 simulations, obtained with different sequences of shocks. ...
We study a family of monopoly models for markets characterized by time-varying demand functions, in which a boundedly rational agent chooses output levels on the basis of a gradient adjustment mechanism. After presenting the model for a generic framework, we analytically study the case of cyclically alternating demand functions. We show that both the perturbation size and the agent’s reactivity to profitability variation signals can have counterintuitive roles on the resulting period-2 cycles and on their stability. In particular, increasing the perturbation size can have both a destabilizing and a stabilizing effect on the resulting dynamics. Moreover, in contrast with the case of time-constant demand functions, the agent’s reactivity is not just destabilizing, but can improve stability, too. This means that a less cautious behavior can provide better performance, both with respect to stability and to achieved profits. We show that, even if the decision mechanism is very simple and is not able to always provide the optimal production decisions, achieved profits are very close to those optimal. Finally, we show that in agreement with the existing empirical literature, the price series obtained simulating the proposed model exhibit a significant deviation from normality and large volatility, in particular when underlying deterministic dynamics become unstable and complex.
... This occurrence opened the road to the modeling of oligopoly games with heterogenous …rms, which decide their outputs over time by adopting di¤erent behavioral rules. Several works deal with oligopoly models with two or more …rms following heterogeneous behavioral rules, see, just to cite a few, [51], [32], [1], [2], [5], [66], [34], [6], [16]. These papers consider duopoly and triopoly models with …rms adopting di¤erent kinds of adaptive adjustments, involving di¤erent degrees of rationality and information, whence di¤erent costs. ...
... A heuristic, or adaptive behavioral rule, can be de…ned as a rule that speci…es next period production x i (t + 1) as a function of the current quantities 6 x j (t), i; j = 1; 2, as well as of the frequency r (t), i.e. 7 ...
... However, we present the idea with only two di¤erent rules, as in the second part of this paper such an example is developed. Another example with two rules is proposed in[12].6 In general, a behavioral rule can also incorporate older information through a 'memory'term, see[12] for details. ...
... This paper belongs to the literature about heterogeneous oligopolies, in the sense that the firms adopt different decisional mechanisms. The heterogeneous framework was studied for example in the articles by Leonard and Nishimura [21], Den-Haan [22], Agiza and Elsadany [23,24], Angelini et al. [12], Tramontana [18], Dubiel-Teleszynski [25], Anufriev et al. [26]. These works concern the coupling of a best response decisional mechanism with the gradient like decisional mechanism. ...
... In a future work we intend to study the same coupling of decisional rules but for an oligopoly consisting of more than 2 firms, considering both the situation in which the fractions of the firms that choose a specific rule is fixed and the situation in which each oligopolist can adapt over time the decisional rule to follow. Concerning this last goal, we remark that Droste et al. in [34] and Anufriev et al. in [26] proposed and studied a way to allow the firms to choose between different heuristics, which we aim to investigate in our future contribution. ...
We study a heterogeneous duopolistic Cournotian game, in which the firms, producing a homogeneous good, have reduced rationality and respectively adopt a “Local Monopolistic Approximation” (LMA) and a gradient-based approach with endogenous reactivity, in an economy characterized by isoelastic demand function and linear total costs. We give conditions on reactivity and marginal costs under which the solution converges to the Cournot-Nash equilibrium. Moreover, we compare the stability regions of the proposed oligopoly to a similar one, in which the LMA firm is replaced by a best response firm, which is more rational than the LMA firm. We show that, depending on costs ratio, the equilibrium can lose its stability in two different ways, through both a flip and a Neimark-Sacker bifurcation. We show that the nonlinear, noninvertible map describing the model can give rise to several coexisting stable attractors (multistability). We analytically investigate the shape of the basins of attractions, in particular proving the existence of regions known in the literature as lobes.
... A main take-away from this strand of literature is the importance of having the 'right' amount of price experimentation. The importance of incorporating competition into these learning-and-earning models, and the potential detrimental effect of ignoring competition, has been demonstrated by Schinkel et al. (2002), Tuinstra (2004), Bischi et al. (2004Bischi et al. ( , 2007, Isler and Imhof (2008), Cooper et al. (2014), and Anufriev et al. (2013), building forth on earlier work by Kirman (Kirman, 1975, 1983, Brousseau and Kirman, 1992. ...
This paper presents the results of the Dynamic Pricing Challenge, held on the occasion of the 17th INFORMS Revenue Management and Pricing Section Conference on June 29-30, 2017 in Amsterdam, The Netherlands. For this challenge, participants submitted algorithms for pricing and demand learning of which the numerical performance was analyzed in simulated market environments. This allows consideration of market dynamics that are not analytically tractable or can not be empirically analyzed due to practical complications. Our findings implicate that the relative performance of algorithms varies substantially across different market dynamics, which confirms the intrinsic complexity of pricing and learning in the presence of competition.
... They study a route to complex dynamics that may emerge when a simple expectation rule competes with a sophisticated but costly expectation rule, using a replicator dynamics approach with mutational noise. Anufriev et al. (2013) consider a Bertrand oligopoly model in which firms switch between least squares learning and gradient leaning for determining the price. Switching between such learning rules may induce endogenous dynamics. ...
We develop a nonlinear duopoly model in which the heuristic expectation formation and learning behavior of two boundedly rational firms may engender complex dynamics. Most importantly, we assume that the firms employ different forecasting models to predict the behavior of their opponent. Moreover, the firms learn by leaning more strongly on forecasting models that yield more precise predictions. An eight-dimensional nonlinear map drives the dynamics of our approach. We analytically derive the conditions under which its unique steady state is locally stable and numerically study its out-of-equilibrium behavior. In doing so, we detect multiple scenarios with coexisting attractors at which the firms’ behavior yields distinctively different market outcomes.
... The phenomena of globalization and informatization, inherent in the post-industrial society, promote and encourage companies to constantly change in the direction of sustainable development, because non-compliance with new requirements can lead to stagnation and loss of market position. Changes require enterprises to carry out balanced investment activities, research markets, competitors, suppliers, conduct effective marketing policies, and therefore enterprises need to implement planning that will take into account the prospects of its development (Anufriev et al., 2013;Babenko et al., 2018;Bechko & Goloborodko, 2015;Bottazzi, 2014;Gonchar, 2015;Gonchar & Khachatryan 2018;Gündüz & Semercišz, 2012;Sharko, 2015;Veltyukov & Tkachuk, 2020). ...
This article examines the problems of forming a strategy for enterprise development in a post-industrial society. A characteristic feature of the contemporary post-industrial society is the constantly evolving new knowledge. Globalization processes and comprehensive digitalization affect the economic behavior of all economic entities and should be taken into account in the formation of strategies for their development. The driving force of progress in Contemporary conditions is closely related to the development of the abilities of a person, whose interests and needs are becoming one level higher. New transformation processes require enterprises to carry out balanced investment activities, research markets, competitors, suppliers, and conduct an effective marketing policy. Businesses need to implement planning taking into account the prospects for its development. The enterprise development strategy should be long-term and contain a flexible plan of action in the direction of innovation based on a detailed analysis of the internal and external environment of the enterprise, available resources and potential of the enterprise. For the successful functioning of the enterprise, the objective necessity is the assessment of its competitiveness, which is based on identifying the strengths and weaknesses of the enterprise for its maximum improvement and finding hidden opportunities for development. The implementation of the development strategy requires efficient investments and optimal use of available resources to achieve the planned results. In the process of creating a development strategy, companies need to take into account the various characteristics that affect it.
... They also investigated the dynamic characteristics of this dynamic model. Other scholars, as for example, Zhang et al. [5], Ahmed et al. [7] and Anufriev et al. [8], further studied the price competition game with differentiated products and also analyzed the dynamic characteristics of the built system. Zhang et al. [9] studied complex dynamics in a quantum Bertrand duopoly with differentiated products. ...
In this paper, taking the factor of product service provided by the manufacturers into consideration, a static Bertrand duopoly game with service factor is studied first, in which these two oligarchs produce differentiated products. A dynamic Bertrand duopoly game with bounded rationality is established by using the gradient mechanism. Keeping the adjustment speeds in a relatively small range may help the long-term stable operation of the market. It is found that there is another 2-cycle, different from the one flip bifurcated from the fixed point, which may appear through a saddle-node bifurcation. The unstable set of the saddle cycle, connecting the saddle to the node, gives a closed invariant curve. In addition, the emergence of intermittent chaos implies that the established economic system has the capability of self-regulating, where PM-I intermittency and crisis-induced intermittency have been studied. With the help of the critical curves, the qualitative changes on the basin of attraction are investigated.
... He also investigated the dynamical characteristics of this dynamic model. On the basis of Ref. [5], Ahmed et al. [7] and Anufriev et al. [8] further studied the price competition game between oligopoly that produce differentiated products. Zhang et al. [9] studied complex dynamics in a quantum Bertrand duopoly with differentiated products. ...
In this paper, taking the factor of service level provided by the manufacturers into consideration, a static duopolistic Bertrand game with service factor is studied first, in which these two oligarchs produce differentiated products. A dynamic game model of duopoly Bertrand with boundedly rational is established with using the gradient mechanism. By using numerical simulation tools, there are two paths for the system to drop into chaos, that is, flip bifurcation and Neimark-Sacker bifurcation. The symmetric structures can be found from two-parameter bifurcation diagrams. Saddle-homoclinic bifurcation also can be observed from the evolution process of phase portraits. In addition, the emergence of intermittent chaos implies that the established system has the capability of self-regulating, where PM-I intermittency, PM-III intermittency and crisis-induced intermittency have been studied. With the help of the critical curves, the qualitative changes on the basin of attraction are investigated. At last, it can be found that the values of product differentiation degree and service spillover effect are not the bigger the better. Keeping these two parameters in a relatively small range will be conducive to the long-term stable operation of the two manufacturers.
... games with heterogenous firms, which decide their outputs over time by adopting different behavioral rules. Several works deal with oligopoly models with two or more firms following heterogeneous behavioral rules; see, just to cite a few, Leonard andNishimura (1999), Den Haan (2001), Elsadany (2003, 2004), Angelini, Dieci, andNardini (2009), Tramontana (2010), Dubiel-Teleszynski (2011), Cavalli, Naimzada, andTramontana (2015), Anufriev, Kopányi, and Tuinstra (2013), and Bischi et al. (2007). These papers consider duopoly and triopoly models with firms adopting different kinds of adaptive adjustments, involving different degrees of rationality and information, whence different costs. ...
... The importance of incorporating competition into these learning-and-earning models, and the potential detrimental effect of ignoring competition, has been demonstrated by Schinkel et al. (2002), Tuinstra (2004), Bischi et al. (2004), Bischi et al. (2007), Isler and Imhof (2008), Cooper et al. (2014) and Anufriev et al. (2013), building forth on earlier work by Kirman (1975Kirman ( , 1983Kirman ( , 1995 and Brousseau and Kirman (1992). ...
This paper presents the results of the Dynamic Pricing Challenge, held on the occasion of the 17th INFORMS Revenue Management and Pricing Section Conference on June 29-30, 2017 in Amsterdam, The Netherlands. For this challenge, participants submitted algorithms for pricing and demand learning of which the numerical performance was analyzed in simulated market environments. This allows consideration of market dynamics that are not analytically tractable or can not be empirically analyzed due to practical complications. Our findings implicate that the relative performance of algorithms varies substantially across different market dynamics, which confirms the intrinsic complexity of pricing and learning in the presence of competition.
... As a final remark, it is worth underlining that in oligopoly theory there is a growing stream of literature focused on the evolutionary competition among behavioral rules built upon different assumptions of rationality and knowledge; see, e.g., Anufriev et al (2013), Bischi et al (2015), Cerboni Baiardi et al (2015) and Cavalli et al (2015). In this regard, the present model is the only one that includes a behavioral rule that leads to the Walrasian-equilibrium quantity. ...
An evolutionary oligopoly game, where firms can select between the best-reply rule and the Walrasian rule, is considered. The industry is characterized by a finite number of ex-ante homogeneous firms that, characterized by naïve expectations, decide next-period output by employing one of the two behavioral rules. The inverse demand function is linear and all firms have the same quadratic and convex cost function (decreasing return to scale). Based upon realized profits, the distribution of behavioral rules is updated according to a replicator dynamics. The model is characterized by two equilibria: the Cournot-Nash equilibrium, where all firms adopt the best-reply rule and produce the Cournot-Nash quantity, and the Walrasian equilibrium, where all firms adopt the Walrasian rule and produce the Walrasian quantity. The analysis reveals that the Walrasian equilibrium is globally stable as long as the rate of change of marginal cost exceeds the sum of residual market price sensitivities to output. If not, the Walrasian equilibrium loses stability and an attractor, representing complicated dynamics with evolutionary stable heterogeneity, arises through a bifurcation. As the propensity of firms to select the more profitable behavioral rule increases, the attractor disappears through a global bifurcation and the Cournot-Nash equilibrium can become a global Milnor attractor. To sum up, the best-reply rule can be evolutionary dominant over the Walrasian rule and this can lead an oligopoly to select the Cournot-Nash equilibrium.
... Moreover, in [22] also the possibility for the firms to switch the adjustment mechanisms was considered. We remark that, in the existing literature, the study of heterogeneous oligopolies of generic size can be found in the works by Anufriev et al. [23] (where the role of heterogeneity in learning is investigated in a Bertrand oligopoly), Banerjee and Weibull [24] (where an evolutive game with agents heterogeneous in the rationality degree is considered), Droste et al. [25] (where an infinite population of firms is studied with respect to the possibility to switch among different decisional mechanisms), Gale and Rosenthal [26] (in which experimentation and imitation behaviors are analyzed). ...
In this paper we study oligopolies of generic size consisting of heterogeneous firms, which adopt best response adjustment mechanisms with either perfect foresight (rational firms) or static expectations (naive firms). Assuming an isoelastic demand function and possibly different marginal costs for the two groups of firms, we focus on the local stability of the Nash equilibrium. We show that, with respect to the oligopoly composition, described in terms of the fraction of rational firms, different scenarios are possible. We find that a high rationality degree may not always guarantee stability, in particular when rational firms have sufficiently larger marginal costs. In fact, in this situation, increasing the fraction of rational firms can even introduce instability. Besides the usual scenarios in which replacing some naive firms with rational ones leads to a stabilization of (or at least keeps unchanged) the dynamics, we provide a family of situations, characterized by costs ratio favorable to naive firms, in which equilibrium loses its stability when naive firms are replaced by rational ones. The results we present are both analytical and simulative.
... In the existing literature, the study of heterogeneous oligopolies of generic size can be found in the works by Anufriev et al. [8], Banerjee and Weibull [9], Droste et al. [20], Gale and Rosenthal [22]. However, only the setting and the aims of the contribution by Droste et al. is similar to that investigated in the present work, as in both papers the economy is characterized by the same linear demand function and the firms may be both rational and best response. ...
We study heterogeneous Cournot oligopolies of variable sizes and compositions, in which the firms have different degrees of rationality, being either rational firms with perfect foresight or naive best response firms with static expectations. Each oligopoly can be described using its size and composition, that is, the fraction of firms that are rational. We take into account two frameworks, one in which the decisional rules are exogenously assigned and the other in which the firms may change their heuristics. We consider a switching mechanism based on a logit rule, where the switching propensity is regulated by a parameter which represents the evolutionary pressure. In the fixed fractions setting, we prove that, in general, the composition has a stabilizing effect, while increasing the oligopoly size leads to instability. However, we show that, for particular parameters settings, stability is not affected by the composition or the firms number. Similarly, in the evolutionary fractions setting, we analytically prove that when marginal costs are identical, increasing the evolutionary pressure has a destabilizing effect. Nevertheless, focusing on particular examples with different marginal costs we are able to show that evolutionary pressure may also have a stabilizing or a neutral role.
... Models with evolutionary updating of the fractions of agents, who adopt different expectations formation rules, have recently been proposed by many authors in economic and financial models, see e.g. Brock and Hommes, 1997, Hommes, 2013, Droste et al., 2002, Hommes et al., 2011and Anufriev et al., 2013 In particular, Droste et al., 2002 consider an evolutionary Cournot duopoly with homogeneous goods, linear demand and quadratic production costs. Pairs of firms, each with its own behavioral rule, are randomly matched at every time period to play the game. ...
In this paper we analyze a dynamic game of Cournot competition with heterogeneous firms choosing between two different adaptive behavioral rules in deciding output strategies. The underling oligopoly structure is standard: using a constant returns to scale technology, N firms produce homogeneous goods, which are sold in a market characterized by constant price elasticity. In this setup, we assume that a fraction of firms employs a quite rough rule of thumb, the so-called Local Monopolistic Approximation (LMA), whereas the complementary fraction plays Best Reply (BR), a more demanding strategy in terms of information and computation requirements. The model is first considered with exogenously fixed fractions of firms in the two complementary groups. Then it is generalized by considering an endogenous evolutionary switching process between the two behavioral strategies based on profit-driven replicator dynamics. The role of the number of firms, information costs and inertia (or anchoring attitude) in production decisions is analyzed, as well as the influence in the evolutionary process of random noise in the demand function and memory of past profits. Global properties of the oligopoly with evolutionary pressure between behavioral rules are discussed, with particular regard to cases in which the Nash equilibrium is unstable.
... Another form of misspecification is incorrectly assuming that there are no competitors present. Schinkel et al. (2002), Tuinstra (2004), Bischi et al. (2004Bischi et al. ( , 2007, Isler and Imhof (2008), Cooper et al. (2014), Anufriev et al. (2013) study the effect of this error on the resulting equilibria in various (linear) models, elaborating on earlier work by Kirman (Kirman, 1975, Brousseau and Kirman, 1992. In an airline revenue-management setting, Cooper et al. (2006) show that incorrectly assuming high-fare and low-fare class passengers behave independently can be detrimental for the firm's revenue. ...
The topic of dynamic pricing and learning has received a considerable amount of attention in recent years, from different scientific communities. We survey these literature streams: we provide a brief introduction to the historical origins of quantitative research on pricing and demand estimation, point to different subfields in the area of dynamic pricing, and provide an in-depth overview of the available literature on dynamic pricing and learning. Our focus is on the operations research and management science literature, but we also discuss relevant contributions from marketing, economics, econometrics, and computer science. We discuss relations with methodologically related research areas, and identify directions for future research.
... Dubiel-Teleszynski [17] studied a heterogeneous duopoly game with adjusting players and diseconomies of scale. Learning cycles in Bertrand competition with differentiated commodities and competing learning rules has been studied by Anufriev et al. [18]. Recently, behavioral rationality and heterogeneous expectations in complex economic systems has been surveyed by Hommes [19]. ...
In this foreword to the Special Issue “Nonlinear dynamics and game-theoretical modeling in economics and finance” we review the contributions in the issue highlighting the economic results and the connections with aspects of dynamic analysis. Indeed, the common theme of the contributions is the focus on system dynamics and the latest analytical techniques. This issue is devoted to celebrating the 70th birthday of Professor Laura Gardini, who inspired a generation of scholars in the study of discrete-time systems and global methods of analysis.
Past reviews of studies concerning competitive pricing strategies lack a unifying approach to interdisciplinarily structure research across economics, marketing management, and operations. This academic void is especially unfortunate for online markets as they show much higher competitive dynamics compared to their offline counterparts. We review 132 articles on competitive posted goods pricing on either e-tail markets or markets in general. Our main contributions are (1) to develop an interdisciplinary framework structuring scholarly work on competitive pricing models and (2) to analyze in how far research on offline markets applies to online retail markets.
Bao et al. (2017) show that in laboratory subjects have problems with both trading an experimental asset and forecasting its price. In this paper I explore these experimental results, by investigating a model in which financial traders individually learn how to use forecasting and/or trading anchor and adjustment heuristics, by updating them with Genetic Algorithms. The model replicates main outcomes of the Bao et al. (2017) experiment, and shows that both forecasters and traders coordinate on asset price oscillations, albeit the trading markets generate larger cycles. Both forecasters and traders learn to chase market trends, and this behavior is reinforced when agents have to perform both tasks. As a result, the famous positive feedback between financial expectations and price trends becomes enhanced and markets generate prices that oscillate between 10% and 350% of the fundamental value.
The paper studies an oligopoly game, where firms can choose between price-taking and price-making strategies. On a mixed market price takers are always better off than price makers, though the profits of both types decline in the number of price takers. We investigate and confront two possibilities of firms’ decisions about their types: forward-looking equilibrium reasoning and backward-looking individual learning. We find that the Cournot outcome is the only equilibrium prediction and it is learnable if firms are sufficiently sensitive to profit differences. However, with a larger number of firms, a unilateral deviation from Cournot behavior becomes profitable. Under learning this incentive creates a space for permanent oscillations over different markets with a positive but low number of price takers.
In this paper, a Cournot–Bertrand duopoly model with market share preference is established. Assume that there is a degree of product difference between the two firms, where one firm takes the price as a decision variable and the other takes the quantity. Both firms are bounded rational, with linear cost functions and demand functions. The stability of the equilibrium points is analyzed, and the effects of some parameters (α, β, d and v1) on the model stability are studied. Basins of attraction are investigated and the evolution process is shown with the increase in the output adjustment speed. The simulation results show that instability will lead to the increase in the average utility of the firm that determines the quantity and reduce the average utility of the firm that determines price.
In this paper we consider firms that learn about market conditions by estimating the demand function using past market data. We show that learning may lead to suboptimal outcomes even when the estimated demand function perfectly fits the observations used in the regression and firms thus perceive to have learned the demand function correctly.
We consider the Salop model with three firms and two types of consumers that differ in their sensitivity to product differences. Firms do not know the demand structure and they apply least squares learning to learn the demand function. In each period, firms estimate a linear perceived demand function and they play the perceived best response to the previous-period price of the other firms.
This learning rule can lead to three different outcomes: a self-sustaining equilibrium, the Nash equilibrium or an asymmetric learning-equilibrium. In this last equilibrium one firm underestimates the demand for low prices and it attracts consumers with high sensitivity only. This type of equilibrium has not been found in the literature on least squares learning before. Both the Nash equilibrium and the asymmetric learning-equilibrium are locally stable therefore the model has coexisting stable equilibria.
In this paper, we formulate a dynamical model of common fishery resource harvested by multiagents with heterogeneous strategy: profit maximizers and gradient learners. Special attention is paid to the problem of heterogeneity of strategic behaviors. We mainly study the existence and the local stability of non-negative equilibria for the model through mathematical analysis. We analyze local bifurcations and complex dynamics such as coexisting attractors by numerical simulations. We also study the local and global dynamics of the exclusive gradient learners as a special case of the model. We discover that when adjusting the speed to be slightly high, the increasing ratio of gradient learners may lead to instability of the fixed point and makes the system sink into complicated dynamics such as quasiperiodic or chaotic attractor. The results reveal that gradient learners with high adjusting speed may ultimately be more harmful to the sustainable use of fish stock than the profit maximizers.
In this paper, we propose an evolutionary model of oligopoly competition where agents can select between different behavioral rules to make decisions on productions. We formalize the model as a general class of evolutionary oligopoly games and then we consider an example with two specific rules, namely Local Monopolistic Approximation and Gradient dynamics. We provide several results on the global dynamic properties of the model, showing that in some cases the attractor of the system may belong to an invariant plane where only one behavioral rule is adopted (monomorphic state). The attractors on the invariant planes can be either strong attractors or weak attractors. However, we also explain why the system can be in a state of Evolutionary Stable Heterogeneity, where it is more profitable for the agents to employ both heuristics in the long term (polymorphic state).
We consider a differentiated-goods Cournot competition where each agent learns about how much to produce. There are two types of agents: ignorant and informed. Ignorants do not know about the demand function for their products and naively assume prices for their products will remain the same as the previous period in the process of learning. Informed ones, on the other hand, know about the demand function, and learn how much to produce by myopically best responding against the quantities produced by others. We show that there are situations in which ignorants are more successful than informed (in a sense that they obtain a higher payoff than the latter). This occurs because of the way two types of agents learn to behave. Of course, there are situations where the opposite, the informed being richer than the ignorants, is true. The nature of strategic interactions determines which outcome prevails.
We run a laboratory experiment that contributes to the finance literature on "return chasing behavior" studying how investors switch between mutual funds driven by past performance of the funds. The subjects in this experiment make discrete choices between several (2, 3 or 4) experimental funds in multiple periods. The time series of funds' profits are exogenously generated prior to the experiment and subjects are paid for that period according to the profit of the fund they choose. The experimental results show that the investment decision can to a large extent be explained by a discrete choice model ("switching model") with a few lags and a predisposition effect. The intensity of choice parameter/beta in the discrete choice model depends on the structure of the profit time series of the funds, and there is no evidence that it is influenced by experience.
We consider a price adjustment process in a model of monopolistic compe-tition. Firms have incomplete information about the demand structure. When they set a price they observe the amount they can sell at that price and they observe the slope of the true demand curve at that price. With this information they estimate a linear demand curve. Given this estimate of the demand curve they set a new optimal price. We investigate the dynamical properties of this learning process. We Þnd that, if the cross-price effects and the curvature of the demand curve are small, prices converge to the Bertrand-Nash equilibrium. The global dynamics of this adjustment process are analyzed by numerical sim-ulations. By means of computational techniques and by applying results from homoclinic bifurcation theory we provide evidence for the existence of strange attractors.
In recent 'learning to forecast' experiments with human subjects (Hommes, et al. 2005), three different patterns in aggregate asset price behavior have been observed: slow mono-tonic convergence, permanent oscillations and dampened fluctuations. We construct a simple model of individual learning, based on performance based evolutionary selection or reinforcement learning among heterogeneous expectations rules, explaining these dif-ferent aggregate outcomes. Out-of-sample predictive power of our switching model is higher compared to the rational or other homogeneous expectations benchmarks. Our results show that heterogeneity in expectations is crucial to describe individual forecast-ing behavior as well as aggregate price behavior.
A choice prediction competition is organized that focuses on decisions from experience in market entry games (http://sites.google.com/site/gpredcomp/ and http://www.mdpi.com/si/games/predict-behavior/). The competition is based on two experiments: An estimation experiment, and a competition experiment. The two experiments use the same methods and subject pool, and examine games randomly selected from the same distribution. The current introductory paper presents the results of the estimation experiment, and clarifies the descriptive value of several baseline models. The experimental results reveal the robustness of eight behavioral tendencies that were documented in previous studies of market entry games and individual decisions from experience. The best baseline model (I-SAW) assumes reliance on small samples of experiences, and strong inertia when the recent results are not surprising. The competition experiment will be run in May 2010 (after the completion of this introduction), but they will not be revealed until September. To participate in the competition, researchers are asked to E-mail the organizers models (implemented in computer programs) that read the incentive structure as input, and derive the predicted behavior as an output. The submitted models will be ranked based on their prediction error. The winners of the competition will be invited to publish a paper that describes their model.
Adaptively rational equilibrium is introduced, where agents adapt their beliefs by choosing from a finite set of predictor functions. Agents make a rational predictor choice, based upon a publically available performance measure such as realized past profits. This results in an adaptive belief system, where predictor choice is coupled to the market equilibrium dynamics. As a typical example, the cobweb model with rational versus naive expectations is analyzed. If the market is locally unstable and rational expectations are costly to obtain, a high intensity of choice for predictor selection leads to chaos and strange attractors.
Growing out of a conference on Expectations Formation and Economic Disequilibrium held in New York City in 1981, the papers in this volume provide a complex view of market processes in which individual rationality is no guarantee of convergence to the 'correct' model and the equilibrium coordination of agents' plans. They reject the 'optimality' argument for the rational expectations hypothesis, opening the door to other hypotheses of optimal expectations of agents in the decentralized market economy.
This paper stresses that the coexistence of different learning methods can have a substantial effect on the convergence properties of these methods. We consider a Bertrand oligopoly with differentiated goods in which firms either use least squares learning or gradient learning for determining the price for a given period. These methods are well-established in oligopoly models but, up till now, are used mainly in homogeneous setups. We illustrate that the stability of gradient learning depends on the distribution of learning methods over firms: as the number of gradient learners increases, the method may lose stability and become less profitable. We introduce competition between the learning methods and show that a cyclical switching between the methods may occur.
A crucial challenge for economists is figuring out how people interpret the world and form expectations that will likely influence their economic activity. Inflation, asset prices, exchange rates, investment, and consumption are just some of the economic variables that are largely explained by expectations. Here George Evans and Seppo Honkapohja bring new explanatory power to a variety of expectation formation models by focusing on the learning factor. Whereas the rational expectations paradigm offers the prevailing method to determining expectations, it assumes very theoretical knowledge on the part of economic actors. Evans and Honkapohja contribute to a growing body of research positing that households and firms learn by making forecasts using observed data, updating their forecast rules over time in response to errors. This book is the first systematic development of the new statistical learning approach. Depending on the particular economic structure, the economy may converge to a standard rational-expectations or a "rational bubble" solution, or exhibit persistent learning dynamics. The learning approach also provides tools to assess the importance of new models with expectational indeterminacy, in which expectations are an independent cause of macroeconomic fluctuations. Moreover, learning dynamics provide a theory for the evolution of expectations and selection between alternative equilibria, with implications for business cycles, asset price volatility, and policy. This book provides an authoritative treatment of this emerging field, developing the analytical techniques in detail and using them to synthesize and extend existing research.
Properties of Dim1 noninvertible maps, the fractal "box-within-a-box" bifurcation structure of unimodal and bimodal maps. General properties of Dim2 maps, normal forms. Embedding of a Dim1 noninvertible map into a Dim2 invertible map. Bifurcation plane of the Hénon's map, fractal foliated bifurcation plane of box-within-a- box type: cross-road area, spring area, saddle area.
An analysis of standard evolutionary dynamics adapted to extensive form games.
Evolutionary game theory attempts to predict individual behavior (whether of humans or other species) when interactions between individuals are modeled as a noncooperative game. Most dynamic analyses of evolutionary games are based on their normal forms, despite the fact that many interesting games are specified more naturally through their extensive forms. Because every extensive form game has a normal form representation, some theorists hold that the best way to analyze an extensive form game is simply to ignore the extensive form structure and study the game in its normal form representation. This book rejects that suggestion, arguing that a game's normal form representation often omits essential information from the perspective of dynamic evolutionary game theory.
The book offers a synthesis of current knowledge about extensive form games from an evolutionary perspective, emphasizing connections between the extensive form representation and dynamic models that traditionally have been applied to biological and economic phenomena. It develops a general theory to analyze dynamically arbitrary extensive form games and applies this theory to a range of examples. It lays the foundation for the analysis of specific extensive form models of behavior and for the further theoretical study of extensive form evolutionary games.
For regular oligopolies (both homogeneous and heterogeneous) the (local) Cournot-Nash equilibria are the (non-degenerate) critical points of a Morse-Smale vector field, defined on the feasible region of non-negative prices and outputs. When at the boundary of this feasible region this vector field points inwards, it follows from the Morse inequalities that there is at least one stable equilibrium. When there is a unique, non-stable, interior equilibrium, necessarily the vector field points outwards somewhere along the boundary of the feasible region. This raises to a stable boundary equilibrium.
A dynamic Cournot duopoly game, whose time evolution is modeled by the iteration of a map T:(x,y)→(r1(y),r2(x)), is considered. Results on the existence of cycles and more complex attractors are given, based on the study of the one-dimensional map F(x)=(r1∘r2)(x). The property of multistability, i.e. the existence of many coexisting attractors (that may be cycles or cyclic chaotic sets), is proved to be a characteristic property of such games. The problem of the delimitation of the attractors and of their basins is studied. These general results are applied to the study of a particular duopoly game, proposed in M. Kopel [Chaos, Solitons & Fractals, 7 (12) (1996) 2031–2048] as a model of an economic system, in which the reaction functions r1 and r2 are logistic maps.
We use simple learning models to track the behavior observed in experiments concerning three extensive form games with similar perfect equilibria. In only two of the games does observed behavior approach the perfect equilibrium as players gain experience. We examine a family of learning models which possess some of the robust properties of learning noted in the psychology literature. The intermediate term predictions of these models track well the observed behavior in all three games, even though the models considered differ in their very long term predictions. We argue that for predicting observed behavior the intermediate term predictions of dynamic learning models may be even more important than their asymptotic properties. Journal of Economic Literature Classification Numbers: C7, C92.
We present a general result on the convergence to an equilibrium of a class of dynamic adjustment procedures — which includes gradient systems and best reply dynamics as special cases — when there are two players and strategy sets are one dimensional. We also show that there are no restrictions on the form of the gradient or best reply dynamics, even under strong restrictions on the functional form of both demand and costs. This implies that we can construct examples with three players where the above dynamical procedures yield chaotic behavior.
An evolutionary game theoretic model of Cournot competition is investigated. Individuals choose from a finite set of different behavioral rules. Each rule specifies the quantity to be produced in the current period as a function of past quantities. Using more sophisticated rules may require extra information costs. Based upon realized payoffs, the fractions of the population choosing a certain behavioral rule are updated according to the replicator equation with noise. The long-run behavior of the evolutionary system consisting of the population dynamics coupled with the quantity dynamics of the Cournot game may be complicated and endogenous fluctuations may arise. We consider a typical example where firms can choose between two rules: the Nash rule and the best-reply rule. We show that a homoclinic tangency between the stable and unstable manifold of the equilibrium occurs as evolutionary pressure increases, implying bifurcation routes to complicated dynamics and strange attractors.
This paper investigates the existence of oligopoly equilibria when firms arrive, through local price experiments, at a correct estimation of their demand curves in a neighborhood of a given statu s quo. The author provides sufficient conditions for the existence of a local Nash equilibrium, defined as a point where each firm is at a local maximum of its profit function, given the prices charged by th e other firms. He also provides two examples, one of a duopoly with l ocal Nash equilibria but no Nash equilibria and the other of a duopol y with no local Nash equilibria. Copyright 1988 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.
We study a class of models in which the law of motion perceived by agents influences the law of motion that they actually face. We assume that agents update their perceived law of motion by least squares. We show how the perceived law of motion and the actual one may converge to one another, depending on the behavior of a particular ordinary differential equation. The differential equation involves the operator that maps the perceived law of motion into the actual one.
In a standard model of oligopoly with differentiated products, the existence of an equilibrium at which the first-order conditions for profit maximisation are simultaneously satisfied for all firms is proved and this is done without imposing any restrictions on the demand functions. This is an equilibrium in the following sense: although some firms may not necessarily be maximising their profits, nevertheless if each firm's knowledge of demand is limited to the linear approximation of its own demand curve, then it will believe that it is indeed maximising its profits.
We combine Nagel's “step-k” model of boundedly rational players with a “law of effect” learning model. Players begin with a disposition to use one of the step-krules of behavior, and over time the players learn how the available rules perform and switch to better performing rules. We offer an econometric specification of this dynamic process and fit it to Nagel's experimental data. We find that the rule of learning model vastly outperforms other nested and nonnested learning models. We find strong evidence for diverse dispositions and reject the Bayesian rule-learning model.Journal of Economic LiteratureClassification Numbers: C70, C52, D83.
We examine the force of three types of behavioural dynamics in quantity-setting triopoly experiments: (1) mimicking the successful
firm, (2) rules based on following the exemplary firm, and (3) rules based on belief learning. Theoretically, these three
types of rules lead to the competitive, the collusive, and the Cournot—Nash outcome, respectively. In the experiment we employ
three information treatments, each of which is hypothesized to be conducive to the force of one of the three dynamic rules.
To a large extent, the results are consistent with the hypothesized relationships between treatments, behavioural rules, and
outcomes.
Curb sets [Basu and Weibull, Econ. Letters 36 (1991), 141-146] are product sets of pure strategies containing all individual best-responses against beliefs restricted to the recommendations to the remaining players. The concept of minimal curb sets is a set-theoretic coarsening of the notion of strict Nash equilibrium. We introduce the concept of minimal strong curb sets which is a set-theoretic coarsening of the notion of strong Nash equilibrium. Strong curb sets are product sets of pure strategies such that each player's set of recommended strategies must contain all coalitional best-responses of each coalition to whatever belief each coalition member may have that is consistent with the recommendations to the other players. Minimal strong curb sets are shown to exist and are compared with other well known solution concepts. We also provide a dynamic learning process leading the players to playing strategies from a minimal strong curb set.
This paper describes a market in which firms vary their quantities of production according to a new adjustment process. Each firm bases its new production entirely upon a knowledge of its own previous productions and profits. It has no knowledge of the payoff functions of the market. Numerical analysis of the process indicates an approach to equilibrium for all initial states. The set of allowed limit points is rigorously characterized, and determined explicitly in the case of two firms. Some exact solutions are found. The process can be regarded as a way of playing a continuous game with a minimum of information.
Nonlinear Oligopolies: Stability and Bifurcations Multistability and cyclic attractors in duopoly games
- G I Bischi
- C Chiarella
- M Kopel
- F Szidarovszky
Bischi, G.I., Chiarella, C., Kopel, M., Szidarovszky, F., 2009. Nonlinear Oligopolies: Stability and Bifurcations. Springer. Bischi, G.I., Mammana, C., Gardini, L., 2000. Multistability and cyclic attractors in duopoly games. Chaos, Solitons & Fractals 11, 543–564.
Heterogeneous learning in Bertrand competition with differentiated goods Managing Market Complexity: The Approach of Artificial Economics
- D Kopányi
- A Teglio
- S Alfarano
- E Camacho-Cuena
Kopányi, D., 2013. Heterogeneous learning in Bertrand competition with differentiated goods. In: Teglio, A., Alfarano, S., Camacho-Cuena, E., Ginés-Vilar, M. (Eds.), Managing Market Complexity: The Approach of Artificial Economics, Springer-Verlag, pp. 155–166.