Von Neumann Morgenstern Expected Utility Under Risk Quiz

The Von Neumann-Morgenstern utility theorem provides a framework for understanding how individuals can assign numerical values to their preferences, allowing for a systematic approach to decision-making under uncertainty. It identifies specific conditions that ensure preferences can be represented consistently, facilitating the analysis of choices involving risk and utility.

Transitivity is a fundamental axiom in preference theory, stating that if an individual prefers option A over B and B over C, then it logically follows that A should also be preferred over C. This consistency in preferences helps to establish a coherent ranking of choices.

The expected value is calculated by multiplying each outcome by its probability and summing the results. In this case, $100 with a probability of 0.5 contributes $50, while $0 with a probability of 0.5 contributes $0. Therefore, the total expected value is $50.

A person who consistently prefers a certain outcome over a lottery with the same expected value demonstrates risk aversion. This behavior indicates a preference for certainty and a reluctance to engage in risky situations, even when the potential rewards are equal to the expected value of the gamble.

The independence axiom asserts that preferences between options should remain consistent when mixed with a third option, as long as the probability of that third option remains unchanged. This means that if one option is preferred over another, introducing a common element should not affect that preference, thus validating the statement as true.

A risk-neutral individual evaluates choices based solely on expected value, showing indifference between a guaranteed outcome and a lottery with the same expected value. They do not prefer certainty over risk when the expected payoff is equal, indicating a balanced approach to risk and reward without favoring one over the other.

In the Von Neumann-Morgenstern framework, an ordinal utility function merely ranks preferences without conveying the strength of those preferences. In contrast, a cardinal utility function not only ranks preferences but also reflects the intensity of those preferences, enabling meaningful comparisons and calculations of expected utility. Thus, the statement is true.

The expected utility of a lottery represents the average outcome of various possible results, taking into account their likelihood. By calculating the weighted sum, each utility value is multiplied by its corresponding probability, allowing for a comprehensive assessment of the overall expected benefit or satisfaction derived from the lottery.

Completeness ensures that for any two outcomes, a decision-maker can express a preference for one over the other or indicate indifference. This axiom guarantees that preferences are well-defined across all possible outcomes, allowing for consistent decision-making. Without completeness, choices may remain ambiguous or undefined, complicating the evaluation of alternatives.

A concave utility function reflects diminishing marginal utility, meaning that as a person's wealth increases, each additional unit of wealth provides less additional satisfaction. Consequently, the incremental benefit derived from increases in wealth diminishes, leading to a decrease in marginal utility as wealth rises.

The continuity axiom in Von Neumann-Morgenstern theory asserts that if one option is preferred over a second, and the second over a third, then there must be a combination of the first and third options (a lottery) that is equally preferred to the second. This reflects a smooth transition in preferences among choices.

A risk-seeking individual exhibits a convex utility function, indicating that their marginal utility of wealth increases with more wealth. This preference leads them to favor risky options, such as gambling, over guaranteed outcomes, as they derive greater satisfaction from the potential for higher returns despite the associated risks.