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



8 Appendix

8.1 Summary of supermodular/submodular games

We give an overview of the main definitions and results in the theory of su-
permodular games that are used in the paper, in a simplified setting that is
sufficient for our purposes. Details may be found in Topkis (1978).14

Let I1 and I2 be compact real intervals and F : I1 × I2 R. F is
(strictly) supermodular if
x1,x2 I1,x2 >x1 and y1 ,y2 I2,y2 >y1 we
have F
(x2,y2) - F(x2,y1)(>) F(x1,y2) - F (x1,y1) .Fis (strictly) submod-
ular if
-F is (strictly) supermodular.

Theorem 8.1 (Topkis’s Characterization Theorem) Let F be twice con-
tinuously differentiable. Then

(i) F12 = ∂x∂y 0 [ 0] for all х,У F is supermodular [submodular].

(ii) F12 = ∂x∂y > 0 [< 0 ] for all х,У F is strictly supermodular [sub-
modular].

The supermodularity property is not preserved by monotonic transforma-
tions of the function F. An alternative notion (ordinal) is the single crossing
property defined as follows: F has single crossing property [dual single crossing
property] in
(x, y) if x1,x2 I1,x2 >x1 and y1,y2 I2,y2 >y1 we have

F(x1,y2) - F (x1, y1) 0[0] F(x2,y2) - F(x2,y1) 0[0] .

The single crossing property does not have a correspondent differential char-
acterization and thus it is often more difficult to check. Now we present the
main monotonicity theorems.

Theorem 8.2 (Topkis’s Monotonicity Theorem) If F is continuous in y
and (strictly) supermodular [submodular] in (x, y), then arg maxyI2 F(x, y) has
(all of its) maximal and minimal selections increasing [decreasing] in
x I1 .

14 Other aspects of the theory may be found in Topkis (1979), Milgrom and Roberts (1990)
and Vives (1990).

31



More intriguing information

1. The name is absent
2. Keystone sector methodology:network analysis comparative study
3. SOME ISSUES IN LAND TENURE, OWNERSHIP AND CONTROL IN DISPERSED VS. CONCENTRATED AGRICULTURE
4. Staying on the Dole
5. Integration, Regional Specialization and Growth Differentials in EU Acceding Countries: Evidence from Hungary
6. Federal Tax-Transfer Policy and Intergovernmental Pre-Commitment
7. The name is absent
8. QUEST II. A Multi-Country Business Cycle and Growth Model
9. Implementation of Rule Based Algorithm for Sandhi-Vicheda Of Compound Hindi Words
10. Picture recognition in animals and humans