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Five Axioms 1

The document outlines five axioms of choice under uncertainty: 1) Comparability, which states that individuals can compare outcomes and express preferences. 2) Transitivity, which requires preferences to remain consistent. 3) Strong independence, which deals with indifference between gambles. 4) Measurability, which allows outcomes to be measured on a probability scale. 5) Ranking, which compares preferences between gambles with different probabilities.

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99 views2 pages

Five Axioms 1

The document outlines five axioms of choice under uncertainty: 1) Comparability, which states that individuals can compare outcomes and express preferences. 2) Transitivity, which requires preferences to remain consistent. 3) Strong independence, which deals with indifference between gambles. 4) Measurability, which allows outcomes to be measured on a probability scale. 5) Ranking, which compares preferences between gambles with different probabilities.

Uploaded by

Heart Bukkabi
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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Five Axioms of choice under uncertainty

The Axioms of cardinal utility, are the assumption that provide the minimum set of conditions for
consistent and rational behaviour.

Once they are established all remaining theory must follow.

Axiom 1 – Comparability (Completeness) For the entire set, S, of uncertain alternatives, an


individual can say either that outcome x is preferred to outcome y (X > Y) or the individual is
indifferent as to X and Y (X ~ Y)

Axiom 2 – Transitivity (consistency). If an individual prefers X to Y and Y to Z, then X is preferred to


Z. (If X > Y and Y > X then X > Z). If an individual is indifferent as to X and Y is also indifferent as to Y
and Z, then they are indifferent as to X and Z. (if X ~ Y and Y ~ Z then X ~ Z).

Axiom 3 - Strong Independence. Suppose we construct a gamble where an individual has a


probability  of receiving outcome X and a probability of (1 - Â) of receiving outcome Z. we shall
write the gamble as G (X, Z: Â). Strong independence says that if the individual is indifferent as to X
and Y, then they will also be indifferent as to a first gamble, set up between X with probability  and
a mutually exclusive outcome, Z, and a second gamble, set up between Y with probability  and the
same mutually exclusive outcome, Z.

If X ~ Y, then G (X, Z: Â) ~ G (Y, Z: Â)

Axiom 4 – Measurability. If outcome Y is preferred less than X but more than Z, then there is a
unique  (a probability) such that an individual is indifferent between Y and a gamble between X
with probability  and Z with probability (1-Â).

If X > Y ≥ Z or X ≥ Y > Z, then there exists a unique â, such that Y ~ G (X, Z: â)

Axiom 5 – Ranking. If alternatives Y and U both lie somewhere between X and Z and we can
establish gambles such that an individual is indifferent between y and a gamble between X (with
probability â1) and Z, while also indifferent between U and a second gamble, this time between X
(with probability â2) and Z, then if â1 is greater than â2, Y is preferred to U.

If X ≥ Y ≥ Z and X ≥ U ≥ Z, then if Y ~ G (X, Z: â)

And U ~ G(X, Z: â2), it follows that if â1 > â2, then Y > U

Or if â1 = â2, then Y ~ U

The axioms are a cardinal utility boil down to the following assumptions about behaviour:

First – all individuals are assumed to always make completely rational decisions

Second – people are assumed to be able to make these rational choices among thousands of
alternatives.

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