Question

Somebody please explain, with examples, to me the notions of

1) Independance

2) Continuity

3) Monotonicity

In Choice Under Uncertainty concept of Microeconomic Theory.

I want to know how can one event/lottery satisfy 1 and not 2 (contin but not indep and monotonicity) or (monoton but not indep and cont), satisfy all, and neither.

Answer 1

1) independence

the definition of independence axiom states that

A preference relation ≽ on the space of lotteries P satisfies independence if for all p, p' , p" that belong to space P and α lies between [0, 1], we have

p≽p' <=> αp+(1-α)p' ≽ αp' + (1 − α)p"

where p, p' , p" are the lotteries that form the convex cobination which is also a lottery

the independence axiom states that the consumer would prefer p to p' .

in the axiom we look at the the distinction between p and p' and keep the other tems ie p" and α constant. it is also known as the subsitution axiom .

in other words if a consumer is indifferent between  two possible outcomes, then they will be indifferent between two lotteries which offer them with same probabilities, if the lotteries are same in every other way, i.e., the outcomes can be subsituted ( the reason it is called the substitution axiom )

eg: if we have p =$5 with probability 1 /4 , 0 with probability 3/4

and p' =$12 with probability 1 /10, 0 with probability 9 /10

and we have p" as the probability of getting a head. so if we get head we choose from one of the p and p' , if we get tail nothing.

so this axiom states that our choice between p and p' will remain the same whether or not p" has occured which is the independent axiom.

2) continuity

following is the definition for continuous axiom

a prefernce relation in space of P lotteries is continuous if for any p,p',p" that belong to the space P with p ≽p'≽p" and  α lies between [0, 1], we have

αp+(1-α)p"∼ p'

it means that if p is preferred to p' then any lottery close to p would also be preferred to p'

example Suppose p is a lottery where you get $10 for sure, p' is where you nothing for sure, and r is a gamble where you get killed for sure.so a rational consumer would strictly prefer p to p', which is strictly preferred to r. so there exists a probability that there is some α ∈ (0, 1) such that you would be indifferent between getting nothing and getting $10 with probability α and getting killed with probability 1 − α

3) monotonicity

it states that  a gamble or a lottery which assigns a higher probability to a outcome which is preferred to one which assigns a lower probability to a preferred outcome, as long as the other outcomes in the gambles remain unchanged( everything else is constant ). here we refer to a strict preference over outcomes and the indiffernece ones are excluded.

the following is an example of violation of independence axiom which is also called allias paradox

the independence axiom of expected utility theory may not be a valid axiom. The independence axiom states that two identical outcomes within a gamble should be treated as irrelevant to the gamble as a whole.but it ignores the idea of f complementarities, the fact your choice in one part of a gamble may depend on the possible outcome in the other part of the gamble. ie if we choose one where we are sure of winning or have a greater chance of winning , it certainly depends on all other choices of gamble which are available.

suppose we have a choice between A which wins with probability 10% and B with probability 90% . so while choosing B we would consider A as well ,. they cant be independent of each other.

one which violates both independence and continuity

. Let A = {a1, a2, a3}. Let p be the lottery [a1], p' be the lottery p'(a1) = q(a2) = 0.5, p'(a3) = 0 and p" be the lottery p"(a1) = p"(a2) = p"(a3) = 1/ 3 . Now, p" ≻ p' ≻ p. Any mixture of p" and p will either produce a lottery whose size of positive support is either 3 or 1. Hence, it cannot be indifferent to p'. So, continuity is violated.

then for independence, p" ≻ p'

But any mixture with p will produce lotteries which will have size of positive 3 and hence, will be indifferent. This violates independence.

if the prefernce over utility is an expected utility maximiser then it satisfies both continuity and indeifference.

also in terms of monotonicity , if the rpefernces are of strict order they cant be indifference.

lexicographic prefernces fail continuity

a1 ≻ a2 ≻ a3. Now, any combination of a1 and a3 will either have positive probability on a1 or probability one on a3. In the first case, such a mixture will be better than the lottery a2. In the latter case, it is the degenerate \a3, which is worse than a2