Question

Briey explain the role the independence axiom plays in the expected utility theorem.

Answer 1

Expected utility implies the utility of an entity or choosing an option over a future period of time, with respect to unknowable circumstances. It is used in decision-making under uncertainty. It is the amount of utility one expects to recieve from an entity. It is calculated by multiplying the probabilities and the utility amounts.

We need to keep in mind that an expected utility representation is still an utility representation, which implies that preferences need to be 1) Complete, 2) Transitive as well as 3) Continuity.

The following standard notations are adopted in case of expected utility theory:

x y : X is preffered to y

x y : There is indifference between x and y.

x y : x is at least as preffered as y.

x # y : There is a preference gap between x and y.

{x,p;y,1-p} = A lottery with a 'p' probability of x and a '1-p' probability of y.

The Axioms of Expected Utility Theory:

Transitivity: If x y and y z, then x z , i.e. if x is at least preferred to y and y is at least preferred as z, then x is at least as preferred to z.

Completeness: x y or y z.

Continuity: If x y and y z, then there are such numbers , 0<p<1, and 0<q<1, such that {x,p;z,1-p}   y and y {x,q; z, 1-q}

However, there is another axion we further need in expected utility theory, and that is independance axion. In notations, it implies:

If x y and 0<p 1, then {x,p;z,1-p} {y,p;z,1-p}

The independence axion states that indifference is independant of context. If you are indifferent between lotteries A and B, and you put A and B inside another lottery say lottery C and D, you will still remain indifferent. So, if you are indiffferent between A and B, you should also be indifferent towards C and D.

The independance axiom is highly controversial, and assumes that two lotteries/gambles mixed with a third one that is irrelevant, will maintain the original order of preference as when the two lotteries are presented independantly of the third one. For a better idea, understand this example:

Let's say there are two different lotteries : A and B, Lets suppose A is a 25% chance of winning pineapple and a 75% chance of winning a watermelon, while B is a 25% chance of winning the watermelon and 75% of winning the pineapple. Lets say you prefer lottery A to B. Now, if you are offered the following choice between option 1 and 2 where :

Option 1: A coin is flipped. If it comes heads, then you will win lottery A , otherwise you will win the lottery which will give you banana for sure.

Option 2: A coin is flipped. If you get heads, you win lottery B, otherwise you win the lottery which will give you banana for sure.

So, what will you prefer?

Now, according to the independance axiom, as you prefer lottery A to B, you must prefer option 1 to option 2, i.e. if you prefer A to B, you must prefer a mixture of A with another lottery , say C, compared to lottery B with C.

Independance Axion: If person prefers lottery A to B, then for any other lottery C, and number 0< 1, , they must prefer:

A + (1-) C to   B + (1-) C  

So, you can see while the independance axiom seem intuitive, it actually depends on the setting. Perhaps you prefer say cake to gravy, but you dont prefer cake mixed with steak to gravy mixed with steak.

This is the role the independance axion plays in the expected utility theory.