Chapter 5: Utility

Question 1

Utility is represented by _____________ numbers. The utility value itself will represent something _______________ depending on which ____________ is used.

Ordinal Scales: “10 is ____________ than 5.”
Interval Scales: “The _______________ between 10 and 5 is equivalent to that of the ____________ between 5 and 0.
Ratio Scales: “10 is _____________ than that of 5.”

Answer 1

real; different; scale; better; difference; difference; twice

Question 2

To make an Ordinal Utility Scale, the decision maker must establish ________________. (e.g. if you like Death on the Nile more than Hunter of Baskervilles then you would demarcate that as Death on the Nile ___ Hunter of Baskervilles).

Answer 2

preferences; Death on the Nile ≻ Hunter of Baskervilles

Question 3

Preferences are revealed in __________ ______________. This can be formalized as: you prefer _____ over ____ if and only if you __________ ____ over ____ every time you’re given the chance.

Answer 3

choice behavior; x; y; choose; x; y

Question 4

≻ : strictly dominates (__________)
⪰ : weakly dominates (___ ________ _____ _________)
~ : equivalent (_____________)

Answer 4

prefer; at least as good; indifferent

Question 5

Completeness (preference axiom)

Answer 5

x ≻ y or x ~ y or y ≻ x

Rules out possibility that you don’t have preferences between some choices.

Question 6

For one to say they are _______________ between two objects of comparison is ______________. In fact, it’s _____________ with the _______________ axiom. HOWEVER, for one to say they ________ make comparisons between a set of _____________, is _______________ with the ______________ axiom.

Answer 6

indifferent; permissable; consistent; Completeness; can’t; objects; inconsistent; Completeness

Question 7

Asymmetry (preference axiom)

Answer 7

If x ≻ y then it’s false that y ≻ x.

Question 8

Transitivity (preference axiom)

Answer 8

If x ≻ y and y ≻ z, then x ≻ z.

Question 9

If Transitivity _______________, we won’t be able to construct an __________ ______________ Scale.

Answer 9

fails; Ordinal; Utility

Question 10

Transitivity Gone Wrong:
What’s the matter with preferring x ≻ y and y ≻ z, then z ≻ x.

Answer 10

You’ll essentially be caught in a loop (Money Pump).

Let’s say you have y, and you’re offered x for y and some amount of money (every trade). Because you prefer x to y, you take x. But you also prefer z to x, so when offered, you take z. Then of course, you also prefer y to z, so you take y, but you prefer x to y and so this process may continue without end.

Question 11

Negative Transitivity (preference axiom)

Answer 11

If it is false that x ≻ y and y ≻ z, then it is false that x ≻ z.

Question 12

for us to claim we prefer one outcome over another on an Ordinal Utility Scale, it’s important we assign ____________ values to each: ____ ≻ ____ iff. ____ > _______.

Answer 12

utility; x; y; u(x); u(y)

Question 13

Preference can be shown on an Ordinal Utility Scale iff. the x _____ y PREFERENCE relation is _____________, ________________, _____________ - _______________.

Answer 13

≻; complete; asymmetric; negatively-transitive

Question 14

While _______________, ________________,_________________, and ________________ can be used to create, and measure objects on, an _____________ utility scale, creating an ______________ utility scale will allow us to ________________ ________________ _____________.

Answer 14

Completeness; asymmetry; transitivity; negative transitivity; Ordinal; Interval; maximize expected utility

Question 15

Lotteries (Interval Utility Scale)

Answer 15

Act where outcome of the act is randomly determined, with known probabilities.

Question 16

von Nueman and Morgenstern’s Interval Utility Scale invovles the decision maker making a __________ of _________________ they would rather do than the __________ __________. (See Notes)

Answer 16

set; preferences; risky act

Question 17

von Neuman and Morgenstern’s way of constructing an Interval Utility Scale requires the decision maker’s preferences to follow ______________ of ______________ ____________. If these ________________ are followed, then preferences can be described as if applying numerical _____________ and _____________ to the lotteries (ranking them). (See Notes)

Answer 17

constraints; rational preference; axioms; utilities; probabilities

Question 18

Definition of constraints
Context: Z = set of prizes (represent risky acts)
L= Lotteries

Answer 18

1) Everything in Z is a lottery

2) If A and B are lotteries, then so is getting A from probability “p” and B from probability “1-p,” for every 0 ≤ p ≤ 1

3) Nothing else is a lottery.

Question 19

The second condition can be represented by __ ___ ___ which means…

Answer 19

ApB; “getting lottery A from p and B from 1-p.”

Question 20

Cq(ApB)

Answer 20

lottery where (0 ≤ p ≤ 1) q is a probability and C is a lottery, AND you get ApB from 1 - p.

Question 21

vNM 1 (Completeness)

Answer 21

A ≻ B, or A ~ B, or B ≻ A

Question 22

vNM 2 (Transitivity)

Answer 22

If A ≻ B, and B ≻ C, then A ≻ C

Question 23

Difference between Ordinal and Interval Utility Scales

Answer 23

In ordinal utility scales, we state preferences over OUTCOMES

In interval utility scales, we state preferences over LOTTERIES (states)

Question 24

vNM 3 (Independence) (p is strictly greater than zero)

Answer 24

A ≻ B iff. ApC ≻ BpC

Question 25

vNM 4 (Continuity) (probabilities p and q are strictly greater than zero and strictly less than 1)

Answer 25

A≻B≻C iff. there are probabilities p and q such that ApC≻B≻AqC

Question 26

The Continuity condition essentially states, that based on the ______________ stated in the antecendent (__≻__≻__), there must be some _______________ that can be assigned to _____ and _____ that make you prefer ____ to the ___/___ gamble and the ___/___ gamble to ____.

Answer 26

preferences; A≻B≻C; probabilities; A; C; B; A/C; A/C; B

Question 27

von Nueman and Morgenstern Theorem

Answer 27

The preferences relation “≻” is satisfied by vNM 1-4 only if there is a function “u” that takes us from utilities to real numbers between 0 and 1 such that…

1) A≻B iff. u(A) > u(B)
2) (p * u(A)) + (p-1 * u(B))
3) Every other function u’ is satisfied only if there are numbers c and d such that u’=(c * u) + d

Question 28

How Completeness Axiom figures into von Nueman Morgenstern Theorem

Answer 28

• Identifies best - worst lotteries in a set of lotteries.
• Sets best lottery to utility of 1.0.
• Sets worst lottery to utility of 0.

Question 29

How Transitivity Axiom figures into von Nueman Morgenstern Theorem:

Ensures the utility ________________ are mathematically appropriate, so that our utility _____________ don’t end up needing a number x that is _______________ and _______________ than y (Money Pump). Also Transitivity ensure that the _________________ lotteries are assigned utilities between ___ and ___.

Answer 29

assignments; assignment; biggger; smaller; intermediate; 0; 1

Question 30

How Independence Axiom figures into von Nueman Morgenstern Theorem:

Makes it mathematically impossible to prefer _______-__________ basic lottery more than the _________ lottery, and less than the __________ lottery.

Allows us to move from determining _____________ of intermediate lotteries given that the _____________ lottery’s utility = __, and ________ lottery’s utility = ___, to knowing that the _____________ of the intermediate lotteries must respect the _______________ already stated by the _________ and ________ lotteries.

Answer 30

non-optimal; best; worst; utility; best; 1; worst; 0; utility; preferences; best; and worst

Question 31

How Continuity Axiom figures into von Nueman and Morgenstern Theorem:

Defines the _____________ of lotteries between the ________ and ________ cases (basically ___________ lotteries). Recall the A≻B≻C situation, wherein we included lottery D (A≻D≻B≻C). It helps us to find the ____________ of the lotteries between lotteries immediately before and after it.

Answer 31

utilities; best; worst; intermediate; utility

Question 32

When making an interval utility scale, we’re oftentimes drawing _____________ between two completely _______________ objects (lotteries). So, we make a number of _____________ comparisons, again and again, until we can make an an ___________ comparison between the two lotteries.

Answer 32

comparisons; incomparable; smaller; indirect