Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 205

models the various steady states of the game may have quite complicated basins
of attraction and, as a consequence, the long run outcome of the game may be
highly dependent upon initial conditions. We will also study the continuous time
version introduced by Chiarella and Szidarovszky (2001 b ) and investigate how the
asymptotic properties are altered by time delays in obtaining and implementing
information on the output of the rivals.
A weakness of the framework presented in Sect. 5.1 is that it does not take into
account the fact that players might want to change their subjective (misspecified)
demand functions. Such an approach can be justified by pointing out that the firms
do not know the cost functions of their competitors and therefore are not able to
derive the output decisions of their competitors. This implies that they are not able
to estimate the whole quantity sold in the market. So, the price they observe does not
convey sufficient information for them to realize that they are using a misspecified
demand function. An alternative to such a setup is presented in Sect. 5.2, where the
firms use a local linear approximation of the price function based on only their own
outputs. Two types of dynamics are examined. First believed best response dynamics
are examined, and then the case of adaptive adjustment processes is discussed.
Three special adaptive learning models are introduced and examined in Sect. 5.3.
Based on their beliefs of the price function each firm computes its believed equilib-
rium output and price. There the observed discrepancy between the price estimate
and the realized market price not only allows the players to conclude that they are
using an incorrect estimate of the demand function, but they are also allowed to
adaptively adjust the believed demand (or price) function. To be more precise, an
N-firm single-product Cournot oligopoly where the demand and cost functions are
linear is considered. Cost functions are completely known by all firms and, although
they know that the (inverse) demand relationship is linear, they either do not know
the slope, or the reservation price. Each firm has its own estimate of the unknown
market parameter and, by solving a static game, determines its own production quan-
tity as well as an expectation on the production quantity of the rest of the industry
(and hence, an expected industry output and an expected price). While firms will
never observe the realized industry output, they can see if the realized market price
differs from their expected price. This will make players aware that their estimate of
the market parameter is wrong, leading them to update their estimate. Our main goal
in this section is to investigate the conditions under which such a simple learning
process has a unique steady state determined by the true market parameters and if
the adjustment process converges to this steady state. In other words, we are inter-
ested in situations in which the firms learn the true demand. We will also see that
adjustment processes of these kinds are not always convergent, and we examine
their global dynamics including their basins of attraction.
Section 5.4 introduces the case of uncertain price functions, when each firm
believes in a randomized function. It is assumed that they want to maximize their
expected profits and minimize their variances. By introducing a linear utility func-
tion it is shown that the game can be reduced to deterministic oligopolies with
misspecified price functions.