## Differentiated oligopolistic markets with concave
cost functions via Ky Fan inequalities

Giancarlo Bigi and Mauro Passacantando
### Summary

An oligopoly is a market structure with a small number of competing firms that produce the same kind of commodity.
Since Cournot introduced and analysed the celebrated duopoly, models for oligopolies have been widely studied in economics.
The analysis of the competition typically exploits the tools of game theory to predict the behaviour of the firms.
In turn, if no cooperation between rms is allowed, game theoretic models can be formulated as variational problems
and optimization techniques can be exploited to solve them.

Generally, these game theoretic models shares two features that seem a
bit restrictive. The costs increase at least linearly with the produced quantity,
while usually the cost per unit does decrease as the production increases. As a
consequence, costs should generally increase less than linearly. The commodity
is supposed to be homogeneous, i.e., all the firms produce the identical product
with even no slight difference, so that a unique unitary price is set for all the
firms (by the market) through an inverse demand function. Commodities of
different firms, though similar, have their own characteristics and are rarely
identical. Moreover, firms generally try to differentiate their products from
those of their competitors to improve their market share. As a consequence,
the dynamics of the price should depend also on the considered firm. To the
best of our knowledge, a first attempt to analyse product differentiation with
concave costs can be found in Muu et al (2008) where piecewise linear cost
functions have been considered modeling the market through mixed variational
inequalities.

This paper provides another step in this direction. In Section 2 a Nash-Cournot model for oligopolistic markets with concave quadratic cost functions
and a differentiated commodity is introduced and formulated as Ky Fan inequalities. Section 3 describes a general algorithmic scheme for Ky Fan
inequalities based on the minimization of a suitable merit function. Moreover,
two concrete descent algorithms based on gap and D-gap functions are provided and their global convergence is deduced from the general scheme under
suitable convexity and monotonicity assumptions. The uniqueness of the solution is also investigated. Finally, Section 4 provides numerical tests of the two
algorithms for randomly generated markets. The sensitivity of the algorithms with respect to their parameters are reported and some market scenarios are
analysed.

The original paper is available at http://dx.doi.org/10.1007/s10203-017-0187-7.

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