Consider the decision problem illustrated below. 1 / 2 1 / 4 1 / 4 a1 49

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Consider the decision problem illustrated below. 1 / 2 ...

Consider the decision problem illustrated below. $$ \begin{array}{llll} \hline & 1 / 2 & 1 / 4 & 1 / 4 \\ \hline a_1 & \$ 49 & \$ 25 & \$ 25 \\ a_2 & \$ 36 & \$ 100 & \$ 0 \\ a_3 & \$ 81 & \$ 0 & \$ 0 \\ \hline \end{array} $$ (a) The decision maker's utility $u$ of money is linear. Which act should be chosen according to the principle of maximising expected monetary value? (b) The decision maker's utility $u$ of money $x$ is given by the formula $u(x)=\sqrt{x}$. Which act should be chosen according to the principle of maximising expected utility?

Consider the decision problem illustrated below.
$$
\begin{array}{llll}
\hline & 1 / 2 & 1 / 4 & 1 / 4 \\
\hline a_1 & \$ 49 & \$ 25 & \$ 25 \\
a_2 & \$ 36 & \$ 100 & \$ 0 \\
a_3 & \$ 81 & \$ 0 & \$ 0 \\
\hline
\end{array}
$$
(a) The decision maker's utility $u$ of money is linear. Which act should be chosen according to the principle of maximising expected monetary value?
(b) The decision maker's utility $u$ of money $x$ is given by the formula $u(x)=\sqrt{x}$. Which act should be chosen according to the principle of maximising expected utility?

Key Concepts

Expected Utility

Expected utility is the sum of the utilities of all possible outcomes, each weighted by its probability. This concept extends the idea of expected monetary value by incorporating the decision maker's risk preferences, allowing for more realistic decision-making under uncertainty, especially when the utility function is non-linear.

Expected Monetary Value

This concept involves calculating the weighted average of monetary outcomes across different states, where each outcome is multiplied by its probability. It is used as a decision rule in scenarios where the decision maker is assumed to be risk-neutral, meaning they care only about the average payoff and not about the variability in outcomes.

Utility Function

A utility function serves as a tool to represent a decision maker’s preferences over outcomes, mapping raw monetary values to a level of satisfaction or utility. It reflects how the decision maker values different amounts of money, with the shape of the function (linear, concave, etc.) indicating their attitude towards risk.

Risk Attitude

Risk attitude describes an individual's predisposition towards risk, which can be risk-neutral, risk-averse, or risk-seeking. It is crucial in decision-making since it influences how a decision maker evaluates uncertain outcomes. For instance, a concave utility function, such as the square root function, indicates risk aversion by valuing incremental gains less as monetary amounts increase.

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