3) A shift raising the marginal payoff curve of all players increases the
sum of all strategies (Proposition 6).
4) We provide counterexamples to all Propositions when the strong
concavity assumption is not fulfilled. Also two more examples are used to show
that in the case considered in 3) above nothing can be said about individual
strategies and utilities. Some of these examples are taken from previous work
and are included here for the sake of completeness.
Summing up, 1), 2) and 3) above show that under our assumptions, the
effects of an increase in the number of players or a shift in their payoff
function agrees with our a priori intuition. 1) above has been studied in the
Cournot case by Mc Manus (1962), (1964), Frank (1965), Ruffin (1971), Okuguchi
(1973), Seade (1980) and Szidarovsky and Yakowitz (1982). It must be noticed
that our approach not only generalizes these results but allows for simpler
proofs and does not require that the number of players can be treated as a
continuous variable. Parts 2)-3) above have been studied in the Cournot case
by Dixit (1986) and Quirmbach (1988). Besides the fact that our results apply
to a more general class of models, the motivation for our results in an
oligopolistic framework is twofold:
i) On the one hand in an imperfectly competitive market even if a firm
cares only about profits, profit maximization is not, in general, the best
policy to be pursued and moreover it does not guarantee survival (see e.g.
Vickers (1985)). Therefore the classical hypothesis of profit maximization