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)=
(13)
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.
18