L1 - Expected Utility

Question 1

What is the representation theorem?

Answer 1

The preference relation $\succsim$ admits an expected utility representation if and only if $\succsim$ satisfies axioms 1-3.

1. Rationality
2. Continuity
3. Indipendence

Basically, If there conditions are satisfied a preference relation can be represented with a utility function

Question 2

Which are the three axioms?

Answer 2

1. Rationality
2. Continuity
3. Independence

Question 3

When is a preference relation Rationality (Axiom 1)?

Answer 3

When the preference relation \succsim is complete and transitive.

Question 4

What do we mean by a preference relationship being complete?

Answer 4

When it is total and reflexive:

• Total: We can compare each lottery to any other lottery
  
  • if for all$L_1, L_2 \in \Delta (C)$ such that $L_1 \neq L_2$ or/and $L_2 \succsim L_1$
• Reflexive: Each lottery can be compared to it self
  
  • $L\succsim L$ for each $L \in \Delta(C)$

Question 5

What do we mean with “transitive”?

Answer 5

The preferences are not circular.

Question 6

What does Continuity (Axiom 2) mean?

Answer 6

Axiom 2, or Continuity, means that small changes in probabilities should result in small changes in preferences. For example, if you slightly increase the probability of winning a lottery, your preference for that lottery should also slightly increase. Continuity ensures that preferences are not too sensitive to small changes in probabilities. To check if the preference relation satisfies Continuity, we can check if two sets of probabilities are closed.

Remember the example with poisoned skittles or safety first preferences. E.g., if $\alpha$ is the probability of surviving a trip. With safety first preferences we would never go on a trip as long as $\alpha< 1$, but if $\alpha = 1$ we would completely flip and go on the trip. This safety first behaviour is ruled out with continuity.

Question 7

What do we mean with Independence (Axiom 3)

Answer 7

See definition.

If I prefer lottery $L_1$ over $L_2$, then adding the lottery $L_3$ equally to both $L_1$ and $L_2$, should not change that I prefer $L_1$ over $L_2$. My preference regarding $L_1$ and $L_2$ is independent of other lotteries.

Question 8

A utility function U has an expected utility form if and only if (IFF) ……..

Answer 8

It is linear

Question 9

What is FOSD? Formally and intuitively…

Answer 9

See Definition.

Intuitively, the proposition states that $F$ is better than $G$ if $F$ puts less probability mass on low monetary outcomes and more mass on high monetary outcomes than $G$.

FOSD defines a relationship between lotteries, or equivalently their CDFs, such that regardless of their risk attitude, rational agents should prefer one lottery to another for all increasing utility functions

FOSD basically means that the dominated CDF always is above the dominant CDF.

With FOSD the mean of f(x) is higher than for g(x). Where f(x) is the pdf of F(x) etc.