Endogenous Heterogeneity in Strategic Models: Symmetry-breaking via Strategic Substitutes and Nonconcavities

The main difference is that agents’ strategies form (partially) strategic comple-
ments. This occurs when a more aggressive strategy from one agent rises the
other player’s marginal returns to increasing his own strategy. Consequently, an
increase of the opponent’s choice is responded to by an increase of own choice
variable. This property implies that best-replies are increasing in own action.
As stated above, strategic complementarities are partial in the sense that they
are not observed overall. Reaction curves are piecewise increasing, however due
to a symmetry breaking nonconcavity in the payoff function, they possess a
jump down across the diagonal. The common aspect to the whole analysis is
the canyon shape of the agents’ payoff functions along the 45^o line. Once again,
a player would never optimally respond to an action of the rival by playing that
same action. Whereas in the previous section the submodularity of the payoff
function (or alternatively the strategic substitutability) was a global feature, in
this section we cannot state global supermodularity (or strategic complemen-
tarity). The main consequence is that we cannot guarantee without further
assumptions the existence of a PSNE. Nevertheless, when it exists, it will never
be symmetric.

When strategies are strategic complements, there is no need to assume the
quasiconcavity of the payoff function in order to guarantee the existence of a
PSNE. The reason is that existence might be based on the fact that reaction
curves are increasing and continuity plays no role. Likewise, when payoff func-
tions are quasiconcave supermodularity is not crucial for arguing the existence of
a PSNE. When player’s payoff function is partially quasiconcave and possesses a
nonconcavity along the diagonal, we obtain the same type of symmetry-breaking
already discussed. In this case, reaction curves are continuous (not necessarily
increasing), except for the jump across the diagonal. Existence of PSNE can be
assured (as before) via added conditions, however it is clear that no PSNE can
be symmetric.

We provide now the main results of this section, first for the case in which
strategic complementarities are present and then for the case of partially qua-
siconcave payoff functions.

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