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?
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Notes
- 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.
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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
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