You are indifferent between receiving A for sure and a...

You are indifferent between receiving $A$ for sure and a lottery that gives you $B$ with a probability of 0.9 and $C$ with a probability of 0.1 . You are also indifferent between receiving $A$ for sure and a lottery that gives you $B$ with a probability of 0.6 and $D$ with a probability of 0.4 . All of your preferences satisfy the von Neumann-Morgenstern axioms. (a) What do you prefer most, $C$ or $D$ ? (b) Calculate the (relative) difference in utility between $B$ and $C$, and between $B$ and $D$. (c) If we stipulate that your utility of $B$ is 1 and your utility of $C$ is 0 , what are then your utilities of $A$ and $D$ ?

You are indifferent between receiving $A$ for sure and a lottery that gives you $B$ with a probability of 0.9 and $C$ with a probability of 0.1 . You are also indifferent between receiving $A$ for sure and a lottery that gives you $B$ with a probability of 0.6 and $D$ with a probability of 0.4 . All of your preferences satisfy the von Neumann-Morgenstern axioms.
(a) What do you prefer most, $C$ or $D$ ?
(b) Calculate the (relative) difference in utility between $B$ and $C$, and between $B$ and $D$.
(c) If we stipulate that your utility of $B$ is 1 and your utility of $C$ is 0 , what are then your utilities of $A$ and $D$ ?

Key Concepts

Expected Utility Theory

This theory posits that when faced with risky outcomes, a decision maker evaluates lotteries by calculating the weighted average of the utilities of the outcomes, with the weights being the probabilities of the corresponding outcomes. It provides a framework to compare uncertain prospects by reducing them to a single, comparable value, thus allowing decisions to be made based on maximizing expected utility.

von Neumann-Morgenstern Axioms

These axioms are the foundational principles that justify the use of expected utility in decision-making under risk. They include properties like completeness, transitivity, independence, and continuity, ensuring that an individual's preferences over lotteries can be represented by a utility function that is linear in probabilities. These axioms are essential for establishing consistency and rationality in choices under uncertainty.

Utility Normalization

Utility normalization involves setting specific outcomes to fixed utility values (commonly 0 and 1). This approach does not affect preference rankings because the expected utility representation is unique up to positive affine transformations. Normalizing utilities simplifies the analysis and interpretation of relative differences in utilities across different outcomes.

Lottery Indifference Analysis

This concept involves equating the expected utilities of different lotteries when a decision maker is indifferent between them. By doing so, one can derive relationships between the utilities of individual outcomes. This method is used to solve for unknown utility values and to understand the relative attractiveness of various outcomes, facilitating a deeper analysis of risk preferences.

Transcript

Please give Ace some feedback