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:

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))

source

jl@crush.caltech.edu index