Abstract
The previous chapters have already dealt with the behavior of boundedly rational firms in an oligopoly. Although the firms know the true demand relationship, we have assumed that they do not know their competitors’ quantity choices. Instead they form expectations about these quantities and they base their own decisions on these beliefs. In particular, we have focused on several adjustment processes that firms might use to determine their quantity selections and we have investigated the circumstances under which such adjustment processes might lead to convergence to the Nash equilibrium of the static oligopoly game. However, the information that firms have about the environment may be incomplete on several accounts. For example, players may misspecify the true demand function or just misestimate the slope of the demand relationship, the reservation price, or the market saturation point. However, if firms base their decisions on such wrong estimates, they will realize that their beliefs are incorrect, since the market data they observe (for example, market prices or quantities) will be different from their predictions. Obviously, firms will try to update their beliefs on the demand relationship and this will give rise to an adjustment process. In other words, firms will try to learn the game they are playing. Following this line of thought, in this chapter we study oligopoly models under the assumption that firms either use misspecified price functions (Sect. 5.1) or do not know certain parameters of the market demand (Sect. 5.2). The main questions we want to answer are the following. If we understand an equilibrium in a game as a steady state of some non-equilibrium process of adjustment and “learning,” what happens if the players use an incorrect model of their environment? Does a reasonable adaptive process (for example, based on the best response) converge to anything? If so, to what does it converge? Is the limit that can be observed when the players play their perceived games (close to) an equilibrium of an equilibrium of the underlying true model? Is the observed situation consistent with the (limit) beliefs of the players?
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
For c1 = 2c2 the curve F degenerates into the pair of straight lines \({a}_{1}/B = 6/(3{c}_{2} + A)\) and \({a}_{2}/B = 6/B\). For c2 = 2c1 the curve F degenerates into the pair of straight lines \({a}_{1}B = 6/A\) and \({a}_{2}/B = 6/(3{c}_{1} + A)\).
- 2.
A rigorous proof of the supercritical nature of the flip bifurcation requires a center manifold reduction and the evaluation of higher order derivatives, up to the third order (see for example Guckenheimer and Holmes (1983)). This is a rather tedious calculation for a two-dimensional map, and we prefer to rely on numerical evidence as a stable 2-cycle close to the saddle \(\overline{\varepsilon }\) is numerically detected whenever the parameters cross the bifurcation curve F.
- 3.
Also in this case, a rigorous proof of the supercritical nature of the Neimark–Hopf bifurcation requires a center manifold reduction and the evaluation of higher order derivatives, up to the third order (see for example Guckenheimer and Holmes (1983)). This is rather tedious in a two-dimensional map, and we prefer to rely on numerical evidence as a stable orbit surrounding the unstable focus \(\overline{\varepsilon }\) is numerically detected whenever the parameters cross the bifurcation curve H.
References
Baumol, W., & Quandt, R. (1964). Rules of thumb and optimally imperfect decisions. The American Economic Review, 54, 23–46.
Bischi, G. I., Carini, R., Gardini, L., & Tenti, P. (2004a). Sulle orme del caos. Comportamenti complessi in modelli matematici semplici. Torino: Bruno Mondadori Editore.
Bischi, G. I., Gardini, L., & Kopel, M. (2000a). Analysis of global bifurcations in a market share attraction model. Journal of Economic Dynamics and Control, 24, 855–879.
Bischi, G. I., Naimzada, A., & Sbragia, L. (2007). Oligopoly games with local monopolistic approximation. Journal of Economic Behavior and Organization, 62, 371–388.
Bischi, G. I., Sbragia, L., & Szidarovszky, F. (2008). Learning the demand function in a repeated Cournot oligopoly game. International Journal of Systems Science, 39(4), 403–419.
Brousseau, V., & Kirman, A. (1993). The dynamics of learning in N-person games with the wrong N. In K. Binmore, A. Kirman, & P. Tani (Eds.), Frontiers of game theory (pp. 71–93). Cambridge, Massachusetts: MIT.
Chiarella, C., & Szidarovszky, F. (2001a). The birth of limit cycles in nonlinear oligopolies with continuously distributed information lags. In M. Dror, P. L’Ecyer, & F. Szidarovszky (Eds.), Modelling Uncertainty (pp. 249–268). Dordrecht: Kluwer.
Chiarella, C., & Szidarovszky, F. (2004). Dynamic oligopolies without full information and with continuously distributed time lags. Journal of Economic Behavior and Organization, 54(4), 495–511.
Cyert, R. M., & DeGroot, M. H. (1971). Interfirm learning and the kinked demand curve. Journal of Economic Theory, 3(1), 272–287.
Devaney, R. (1989). An introduction to chaotic dynamical systems. Menlo Park, CA: The Benjamin/Cummings.
Fudenberg, D., & Levine, D. (1998). The theory of learning in games. Cambridge, MA: MIT.
Gates, D. J., Rickard, J. A., & Westcott, M. (1982). Exact cooperative solutions of a duopoly model without cooperation. Journal of Mathematical Economics, 9, 27–35.
Grebogi, C., Ott, E., & Yorke, J. (1983). Crises, sudden changes in chaotic attractors and transient chaos. Physica D, 7(1–3), 181–200.
Guckenheimer, J., & Holmes, P. (1983). Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Berlin: Springer.
Kirman, A. (1975). Learning by firms about demand conditions. In R. H. Day & T. Groves (Eds.), Adaptive economic models ( pp. 137–156) . New York: Academic.
Kirman, A. (1995). Learning in oligopoly: Theory, simulation, and experimental evidence. In A. Kirman & M. Salmon (Eds.), Learning and rationality in economics (pp. 127–178). Oxford, UK: Blackwell.
Kirman, A., & Salmon, M. (1995). Learning and rationality in economics. Oxford, UK: Blackwell.
Kopel, M. (1996). Simple and complex adjustment dynamics in Cournot duopoly models. Chaos, Solitons & Fractals7(12), 2031–2048.
Marimon, R. (1997). Learning from learning in economics. In D. Kreps & K. Wallis (Eds.), Advances in economics and econometrics (pp. 278–315). Cambridge, UK: Cambridge University Press.
Mira, C., Gardini, L., Barugola, A., & Cathala, J. (1996). Chaotic dynamics in two-dimensional noninvertible maps. Nonlinear Sciences, Series A. Singapore: World Scientific.
Negishi, T. (1961). Monopolistic competition and general equilibrium. Review of Economic Studies, 27, 136–139.
Okuguchi, K. (1976). Expectations and stability in oligopoly models. Berlin: Springer.
Puu, T. (1991). Chaos in duopoly pricing. Chaos, Solitons & Fractals, 1(6), 573–581.
Silvestre, J. (1977). A model of general equilibrium with monopolistic behavior. Journal of Economic Theory , 16, 425–442.
Szidarovszky, F. (2003). Learning in dynamic oligopolies. InInternational Conference on Complex Oligopolies, presented on May 18–21, Odense, Denmark.
Szidarovszky, F. (2004). Global stability analysis of a special learning process in dynamic oligopolies. Journal of Economic and Social Research, 9, 175–190.
Szidarovszky, F., & Krawczyk, J. (2005). On stable learning in dynamic oligopolies. Pure Mathematics and Applications, 4, 45–54.
Szidarovszky, F., & Okuguchi, K. (1990). Dynamic oligopoly: Models with incomplete information. Applied Mathematics and Computation, 38(2), 161–177.
Szidarovszky, F., Smith, V., & Rassenti, S. (2008). Cournot models: Dynamics, uncertainty and learning. CUBO, A Mathematical Journal, 11(2), 57–88.
Tuinstra, J. (2004). A price adjustment process in a model of monopolistic competition. International Game Theory Review. 6, 417–442.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bischi, GI., Chiarella, C., Kopel, M., Szidarovszky, F. (2010). Oligopolies with Misspecified and Uncertain Price Functions, and Learning. In: Nonlinear Oligopolies. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02106-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-02106-0_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02105-3
Online ISBN: 978-3-642-02106-0
eBook Packages: Business and EconomicsEconomics and Finance (R0)