Utility

Question 1

What is a utility function?

Answer 1

u(x) ≥ u(y) ⇒ x ≽ y

Question 2

v is a utility represetation for ≽iff

Answer 2

There is an increasing function f: u(x) -> v(x) such that v = f ∘ u

Question 3

If ≽ admits a utility representation, then…

Answer 3

≽ is rational

Question 4

If ≽ is rational, then u is a utility representation iff

Answer 4

1) x ∼ y => u(x) = u(y)

2) x ≻ y => u(x) > u(y)

Question 5

If X is finite and ≽ is rational, then

Answer 5

≽ admits a utility representation

Question 6

If X is countable and ≽ is rational, then…

Answer 6

≽ admits a utility representation

Question 7

What is a ≽-dense set?

Answer 7

For every y, x in X there is z in S such that x ≽ z ≽ y

Question 8

What is a ≽-separable set?

Answer 8

If it contains a countable ≽-dense subset

Question 9

≽ admits a utility reresentation iff (set)

Answer 9

≽ is rational and X is ≽-dense

Question 10

What is a Ritcher-Peleg representation of a preorder ≽?

Answer 10

If for every x, y in X

1) x ∼ y => u(x) = u(y)
2) x ≻ y => u(x) > u(y)

Question 11

An advantage of having a Ritcher-Peleg representation of ≽ is that every one of its maximizes is a ≽-best element

Answer 11

False, ≽-maximal

Question 12

A disadvantage of a RitcherPeleg represetnation is that it does not allow uniquely recover the represented preorder

Answer 12

True

Question 13

What is sufficent for the existence of a Richter-Peleg representation?

Answer 13

Order Separability is sufficent but not necessary

Question 14

What is a multi-utility representation?

Answer 14

A set U contains functions X ->R is a multi-utility representation of a preorder ≽ if fore every x,y in X
x ≽ y ⇔u(x) ≥ u(y)

Question 15

A binary relation ≽ is a preorder iff…

Answer 15

It has a multi-utility representatation