f:O->X = a horse race lotteries.

Let F= all horse race lotteries.

Let P= set of all simple prob. charges on X (outcomes) Let h:O->P
Let H= set of all h
F is a subset of H.

forall h,g in H, a in [0,1]: ah+(1-a)g in H

Related axioms:

• AA1
• AA2
• AA3
• AA4
• AA5

Proposition:
"=>" satisfies AA1,AA2,AA3 <=> there exists u1,u2,...,un with ui:X->R s.t. h"=>"g <=> Sum(i=1,n, X in Supp(hi), ui(x)hi(x))=>Sum(i=1,n, X in Supp(gi), ui(x)gi(x)) with ui unique up to affine transformations.

Corollary:
"=>" satsifies AA1,AA2,AA3 and {ui} represents => as in proposition above => i null <=> ui constant on X.

Theorem:
AA1-5 <=> there exists u:X->R non-constant and a prob. charge mu on O s.t.

h"=>"g <=> Sum(i=1,n,p(i)*Sum(x,u(x)h(x)))=>Sum(i=1,n,p(i)*Sum(x,u(x)g(x)))

with mu unique and u unique up to a positive affine transformation.