A function is a utility function if it satisfies: u:X -> R x >= y <=> u(x) => u(y)

uii=u(xii)

Existence if => a weak order:

Cantor's theorem in 1895 says:

=> a weak order on X /\ X/~ (the equivalence classes) at most numerable =>
there exists u:X->R s.t. forall x,y in X x=>y <=> u(x)=>u(y)

this is later extended to:

=> a weak order on X /\ X/~ (the equivalence classes) has a dense numerable
subset=> there exists u:X->R s.t. forall x,y in X x=>y <=> u(x)=>u(y)

Utility can be measured with a randomizing device.
let w in O = states of the world

X= a set of consequences

F= a set of acts f:O->X

Let O be finite => 2^O is an algebra.

Find a probability charge P on (O,2^O)

Find a utility function u:X->R s.t.

f=>g <=> Sum(w in O,u(f(w))p(w))=>Sum(w in O,u(g(w))p(w))

The construction:

There exists E Subset of O which is ethically neutral.

There exists x">"y,x'">"y'

(x,E;y,Ec)=>(x',E;y',Ec) <=> u(x)-u(x')=>u(y)-u(y')

Method:

find the smallest and largest X (xl,xu) , assign them u(x)=0,1 respectively.
let u(xl,E;xu,Ec)=1/2... build in from there.

source

jl@crush.caltech.edu index

state_independent_utility

Marschak_Machina_triangle

representation_theorem

rank_dependent_utility

probability_charge

seperable_utility

weighted_utility

minimax

concave

Machine

table