Figure 4: Quasi-convexity of the payoffs implies that players prefer corner solu-
tions.
a linear demand function and for some parameterization of the cost functions,
profits are convex in own quantity. In both papers it is then concluded that if
some conditions on the parameters hold, asymmetric equilibria might arise. It
is easily shown that the conditions presented in the papers can be deduced from
the assumptions of Theorem 5.1.
6 Extensions
Most of the results of the previous sections depend on some form of monotonic-
ity of the reaction curves. We used the cardinal notions of complementarity
and substitutability to obtain this property mainly due to its convenient char-
acterization through the cross partial derivatives of the payoff functions. Super-
modularity and submodularity are cardinal notions that are not preserved by
monotone transformations of the objective function. The usefulness of ordinal
properties under which comparative statics results are invariant is clear. Mil-
grom and Shannon (1994) proved that the result of Topkis holds when single
crossing property substitutes supermodularity of the objective function. In this
section we show that our results can be generalized to the ordinal definitions of
complementarity.
27
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