Lecture 3: The Axioms Of Rational Choice

Question 1

Axioms

Answer 1

A property or constraint which must be satisfied.

Question 2

The preference relation

Answer 2

A ≥ B signifies “ A is preferred or indifference to B,” or “B is not preferred to A”
A = B signifies “ indifference between A and B”

Question 3

The lottery

Answer 3

(ApB) signifies a gamble in which A occurs with probability p, and B occurs with probability 1-p

Question 4

Axiom 1: Comparability

Answer 4

Given alternatives A and B, either A ≥ B, A ≤ B, or both, in which case A = B

All pairs of options can be compared; you r=either prefer one to the other, or else you are indifferent between them.

Question 5

Pathology of Axiom 1

Answer 5

Inability to make a prediction or judgment of any kind (even of indifference) for some sets

Question 6

Axiom 2: Transitivity

Answer 6

If A ≥ B and B ≥ C, then A ≥ C

Preferences can’t have “loops”; values must form a hierarchy

Question 7

Pathology of Axiom 2

Answer 7

The “money pump”; manipulation via agenda setting

Question 8

Axiom 3: Closer

Answer 8

If A and B are in S, then (ApB) is in S as well.

If I can compare A and B, then I can also compare lotteries over A and B.

Question 9

Pathology of Axiom 3

Answer 9

Inability to make a prediction or judgment for choice under risk

Question 10

Axiom 4: Distribution of Probability Across Alternatives (Frame invariance)

Answer 10

(ApB)qB = (ApqB)

Only the “final” probability of the outcomes matters, how it ex-pressing to me doesn’t matter.

My preferences should not be manipulated by the frame

Question 11

Pathology of Axiom 4

Answer 11

Manipulation by framing effects

Question 12

Axiom 5: Independence

Answer 12

For options A,B,C, A ≥ B if and only if (ApC) ≥ (BpC)
Prefer probability A otherwise C over probability B otherwise C
Lottery where I don’t get A or B, instead I get irrelevant third good C. If I like A more than B, then I should like the chance of getting A more than the same chance of getting B.
My preference of A or B shouldn’t be affect by the change of getting neither of one but C instead.

Question 13

Pathology of Axiom 5

Answer 13

Manipulation by framing effects (Stochastic money pump)

Question 14

Axiom 6: Consistency

Answer 14

A≥B if and only if A≥(ApB)≥B

ApB is mixture of A and B in the sense of sometime you get A and sometimes you get B. If you like A most or the least then some mixture of things in chance should be somewhere in between of them, not outside of two.

Question 15

Pathology of Axiom 6

Answer 15

Violation of the “sure thing” principle; “(Strong) stochastic money pump

Question 16

Axiom 7: Solvability

Answer 16

If A≥B≥C, there exists a probability p such that B=(ApC)

If you prefer A to B to C, there is some lottery over A and C which is equivalent to B

Question 17

Pathology of Axiom 7

Answer 17

Obsession with the infinitely improbable; indirect violation of completeness

Question 18

The Expected Utility Theory

Answer 18

Utility function: an abstract numerical measure of preference (outcomes)

Subjective probabilities: beliefs about states of the world, given choices (chances of

For each option, take the probabilities times outcome utilities.

Choose the option with highest expected utility.

0.6 = probability of 15 (utility)

E(u(A)) = 0.6u (15) + 0.4u (10)
= 0.6 √15 + 0.4 √10
= 0.6 (3.873) + 0.4 (3.162)
= 3.589