3 Endogenous heterogeneity with strategic sub-
stitutes
In this section, we consider a two-player symmetric normal-form game charac-
terized by two key properties. The first is that actions form strategic substitutes.
This means that an increase in one player’s strategy lowers the other player’s
marginal returns to increasing his own strategy. As a result, players respond
optimally to an increase of the opponent’s choice with a decrease of their own
variable. In other words, the best reply correspondences are downward-sloping
and a pure-strategy Nash equilibrium exists (see, Vives, 1990 and Milgrom and
Roberts, 1990).
The second key property is that each player’s payoff, though jointly contin-
uous in the two actions, admits a fundamental nonconcavity along the 45o line,
giving rise to a canyon shape along the diagonal. A key consequence of this
feature is that a player would never optimally respond to an action of the rival
by playing that same action.
Taken together, these two properties imply that each best reply is a de-
creasing correspondence with a (downward) jump over the 45o line.4 Hence,
no PSNE could ever be symmetric. At any of the possibly multiple equilib-
ria, which obviously occur in pairs due to the symmetry of the game, ex ante
identical agents will necessarily take different equilibrium actions.
While all three results presented in this paper share this same flavor, the
main result in terms of the generality of the assumptions and thus of the scope
of applicability is this section’s.
4 In this paper, we will say that a function f : R → R is increasing (strictly increasing) if
x0 > x implies f(x0) ≥ (>)f (x). A correspondence is increasing if its maximal and minimal
selections are increasing functions (as in the conclusion of Topkis’s Monotonicity Theorem).
10
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