Savage's model is built from the following tools.
* S= states of the world
* X= set of consequences
* A=2^S = algebra
* F=set of all acts f:S->X
* "=>" a binary relation on f
There are a set of 7 axioms:
* P1
* P2
* P3
* P4
* P5
* P6
* P7
Step 1:
Proposition: P1-5 => "=>*" is a qualitative probability
Proposition: P1-6 => "=>*" is a qualitative probability and satisfies partition
axiom 2.
corollary: P1-6 => there exists p, a probability charge on (S,A) s.t.
* forall B,C in A B"=>*"C <=> p(B)=>p(C)
* forall B in A forall rin [0,1] there exists C subset of B s.t. p(B)=rP(A)
with P unique.
Step 2:
get the result for finite acts.
Let Pf = distribution on a finite X induced by f.
Proposition: P1-6 /\ f,g in Fs s.t. Pf=Pg => f~g
Step 3: representation.
Proposition: given P1-6, "=>" on Ps satisfies J1,J2,J3 => there exists U:X->R s.t. p"=>"q <=> Sum(P(x)u(x))=>Sum(q(x)u(x))
step 4: Extend result to t~ by defining integral.
Theorem: "=>" satisfies P1-P7 <=> there exists P a probability charge on (S,A)
which is convex randed /\ there exists u:X->R bounded and nonconstant s.t.
f"=>"g <=> Integral(u(f(s))dP(s))=>Integral(u(g(s))dP(s))