Chapter 5: Utility Flashcards
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.”
real; different; scale; better; difference; difference; twice
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).
preferences; Death on the Nile ≻ Hunter of Baskervilles
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.
choice behavior; x; y; choose; x; y
≻ : strictly dominates (__________)
⪰ : weakly dominates (___ ________ _____ _________)
~ : equivalent (_____________)
prefer; at least as good; indifferent
Completeness (preference axiom)
x ≻ y or x ~ y or y ≻ x
Rules out possibility that you don’t have preferences between some choices.
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.
indifferent; permissable; consistent; Completeness; can’t; objects; inconsistent; Completeness
Asymmetry (preference axiom)
If x ≻ y then it’s false that y ≻ x.
Transitivity (preference axiom)
If x ≻ y and y ≻ z, then x ≻ z.
If Transitivity _______________, we won’t be able to construct an __________ ______________ Scale.
fails; Ordinal; Utility
Transitivity Gone Wrong:
What’s the matter with preferring x ≻ y and y ≻ z, then z ≻ x.
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.
Negative Transitivity (preference axiom)
If it is false that x ≻ y and y ≻ z, then it is false that x ≻ z.
for us to claim we prefer one outcome over another on an Ordinal Utility Scale, it’s important we assign ____________ values to each: ____ ≻ ____ iff. ____ > _______.
utility; x; y; u(x); u(y)
Preference can be shown on an Ordinal Utility Scale iff. the x _____ y PREFERENCE relation is _____________, ________________, _____________ - _______________.
≻; complete; asymmetric; negatively-transitive
While _______________, ________________,_________________, and ________________ can be used to create, and measure objects on, an _____________ utility scale, creating an ______________ utility scale will allow us to ________________ ________________ _____________.
Completeness; asymmetry; transitivity; negative transitivity; Ordinal; Interval; maximize expected utility
Lotteries (Interval Utility Scale)
Act where outcome of the act is randomly determined, with known probabilities.
von Nueman and Morgenstern’s Interval Utility Scale invovles the decision maker making a __________ of _________________ they would rather do than the __________ __________. (See Notes)
set; preferences; risky act
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)
constraints; rational preference; axioms; utilities; probabilities
Definition of constraints
Context: Z = set of prizes (represent risky acts)
L= Lotteries
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.
The second condition can be represented by __ ___ ___ which means…
ApB; “getting lottery A from p and B from 1-p.”
Cq(ApB)
lottery where (0 ≤ p ≤ 1) q is a probability and C is a lottery, AND you get ApB from 1 - p.
vNM 1 (Completeness)
A ≻ B, or A ~ B, or B ≻ A
vNM 2 (Transitivity)
If A ≻ B, and B ≻ C, then A ≻ C
Difference between Ordinal and Interval Utility Scales
In ordinal utility scales, we state preferences over OUTCOMES
In interval utility scales, we state preferences over LOTTERIES (states)
vNM 3 (Independence) (p is strictly greater than zero)
A ≻ B iff. ApC ≻ BpC
vNM 4 (Continuity) (probabilities p and q are strictly greater than zero and strictly less than 1)
A≻B≻C iff. there are probabilities p and q such that ApC≻B≻AqC
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 ____.
preferences; A≻B≻C; probabilities; A; C; B; A/C; A/C; B
von Nueman and Morgenstern Theorem
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
How Completeness Axiom figures into von Nueman Morgenstern Theorem
- Identifies best - worst lotteries in a set of lotteries.
- Sets best lottery to utility of 1.0.
- Sets worst lottery to utility of 0.
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 ___.
assignments; assignment; biggger; smaller; intermediate; 0; 1
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.
non-optimal; best; worst; utility; best; 1; worst; 0; utility; preferences; best; and worst
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.
utilities; best; worst; intermediate; utility
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.
comparisons; incomparable; smaller; indirect
