Random Environment	Upper boundary function
CG CGn	fG(x) = x − x ln(x) gn(x) = nx − (n − 1) xn/(n−1)
Ciid Ciidn	x φn(x) strictly increasing, strictly concave, differentiable
CI, CIn	fI (x) = 2x − x2
Cαn	2x − x2 if x < α; and x + (1 − x)α if x ≥ α
Cβ, Cβn	fβ (x) = 2x − x2/β if x < 1 − , and fβ (x) = x + (1 − x)(2(1 − ) − β) if x ≥ 1 − √1 −