Representation theorems typically say something about how a set of axioms can
be turned into a utility function.
Ex: thm: "=>" on a finite X with completeness /\ transitivity <=> there exists
u: X->R which represents "=>" i.e. forall x,y in X: x=>y <=>
u(x)=>u(y). u is unique up to monotonic increasing transformations.
variants include the <=> only being a =>. Also the utility function may be
linear or additive.
Here are some representation theorems:
Theorem: X countable => "=>" a weak order <=> there exists u:X->R which
represents "=>"
Theorem:
"=>" a waek order on X and u:X->R represents "=>" => v:X->R represents "=>" <=> there exists phi:R->R s.t
* phi always nondecreasing /\ strictly increasing on {r in R: r=u(x) forall x in X}
* v=phi(u(x))
Proof:
Note: I use ^ to denote inverse.
u /\ v represent "=>"
=> u(x)>u(y) <=> v(x)=v(y) /\ u(x)=u(y) <=> v(x)=v(y) for u and v on X/~.
Suppose u(a)=c /\ u(b)=d => a=u^(b), b=u^(d).
c>d <=> v(u^(c)) > v(u^(d)) => v(u^()) is strictly increasing with phi=v(u^()).
Theorem:
X subset of R^n => "=>" continuous weak order <=> there exists u:X->R a continuous representation of "=>". This theorem is generalizable to an arbitrary topology.
If a topology has a countable base => the above theorem with topology t instead of R^n
Examples:
X=(0,1)
t=topology={empty set,(0,1)}=indiscreet topology
The only continuous function in t are f(x)=constant forall x in (0,1).
=> there exists no representation of "=>" on topology t.
X=I1 x I2 x ... x In = cartesian product of intervals
x => y <=> x !=y /\ xi=>yi forall i.
Theorem:
* "=>" a weak order
* x=>y => x>y (strong monotonicity)
* (x>y) /\ (y>z) => there exists a,b in [0,1] s.t. (ax+(1-a)z > y)/\(y>bx+(1-b)z)
=> there exists u:X->R which represents "=>" with u increasing x"=>"y => u(x)>u(y)
Theorem:
The extension "=>'" is representable by u:X->R =>
* x~y => u(x)=u(y)
* (x"=>"y => u(x)=>u(y))
This can be changed to (x"=>"y <=> u(x)=>u(y)) forall possible u.
Theorem:
> a strict partial order on X /\ X/~~ countable => there exists u:X->R s.t. x>y => u(x)>u(y) /\ x~~y => u(x)=u(y).
Theorem:
If there exists u:P->R satisfying S1,S2,S3 /\ J4 => U on X defined by u(x)=U(delta(x)) is bounded.