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

3.1 The results

Different subsets of the following assumptions will be needed for our conclusions
below. The notation is as laid out in Section 2.⁵ A full discussion of the
assumptions and results is presented at the end of the section. Most of the
proofs can be found in the appendix.

A1 U, L are submodular

A2 U1(x,x) >L1 (x, x), ∀x ∈ (0, c)

A3 U1(0,0) > 0,L1(c,c) < 0

A1 says that on either side of the diagonal, but not necessarily globally,
each player’s marginal returns to increasing his action decrease with the rival’s
action. A2 holds that each player’s payoff, though globally continuous in the
two actions, has a kink along the diagonal in the shape of a ”valley”. The role
of A3 is simply to rule out PSNEs at (0, 0) or (c, c).

These assumptions form a sufficiently general framework to encompass many
of the studies mentioned in Section 1 as illustrated below. Furthermore, all three
assumptions are easy to check in a particular model.

Theorem 3.1 Assume that A1 - A3 hold. Then the game Γ is of strategic sub-
stitutes, has at least one pair of asymmetric PSNEs and no symmetric PSNEs.

The idea of the proof is that overall submodularity of the payoff function is
inherited from the submodularity of its components U and L in the presence
of assumption A2. We know from Topkis’s Monotonicity Theorem that global
submodularity of the payoff function implies globally decreasing best replies.
Assumptions A2 - A3 imply that the best replies have a downward jump that
crosses over the diagonal. This situation is depicted in figure 1.⁶ (We caution the
⁵ In addition, throughout the paper, partial derivatives are denoted by a subindex corre-
sponding to the relevant variable, i.e. Uι(x,y) = ^dU∂(X^,y) and U2(x, y) = ^dU∂y,,y).

⁶ Notice that unusually x is the variable in the vertical axis. This corresp onds to analyzing
the game from the point of view of player 1 that chooses x as a resp onse to y. We m aintain
this convention throughout.

11

<<   8   9   10   11   12   13   14   >>

More intriguing information