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

Figure 2: Reaction curves are constant except for a jump down, which precludes

symmetric equilibrium.

The number of equilibria may be obtained by finding the explicit form of
the reaction curves.

1 ify ≤ 1/2                     1 ifx ≤ 1/2

r1 (y)=                     r2 (x)=

I  1/2 if y> 1/2              I  1/2 if x> 1/2

Figure 2 illustrates that there exists exactly one pair of pure strategy Nash
equilibria, namely (1, 2 ) and ( 2, 1) .

We can compare equilibria from the point of view of player 1, using the dual
to the Theorem 3.3. The payoff function of player 1 when x ≤ y, is constant
along his best reply, i.e. L2(r1 (y) ,y)=0. Therefore we conclude that player 1
prefers the equilibrium where he is more active.

4 Endogenous heterogeneity without monotonic
best replies

In the previous section we discussed symmetry breaking via strategic substitutes
and nonconcavity of the payoff function. In this section we extend the analy-
sis from the previous section to encompass other forms of strategic interaction.

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