# Utility Functions and Risk Attitudes in Decision Analysis

Candice H. Huynh (University of California, Irvine, USA) and Wenting Pan (Saint Mary's College of California, USA)
DOI: 10.4018/978-1-4666-5202-6.ch233

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## Background

When a person considers whether to take a gamble, a typical first step is to calculate the expected value – which is the average amount one would win (or lose) if a gamble is played an infinite number of times. For example, let us consider an example in a fair coin heads or tails gamble where a player receives \$1000 dollars if the coin lands heads up and \$0 otherwise. Then the expected value of this gamble is

.

Let represent the monetary gain (or loss) incurred with probability of occurring.

The formula for computing the expected value for discrete gambles is:

.

Referring back to the fair coin flip gamble with the possibility of gaining \$1000 on heads, and \$0 on tails, a risk-neutral person would be indifferent between receiving \$500 and taking the gamble. A risk-averse person would prefer receiving the \$500 with certainty to taking the gamble, while a risk-seeking person would prefer taking the gamble to receiving the \$500 with certainty. The three risk behaviors (risk-neutral, risk-averse, and risk-seeking) can be modeled with utility functions over all gamble amounts.

Consider the following gamble:

• Which do you prefer?

• A: \$500 for sure;

• B: 50% chance to win \$0;

• 50% chance to win \$1000.

## Key Terms in this Chapter

Expected Value: The average amount one would win (or lose) if a gamble is played an infinite number of times.

Risk Premium: Refers to the minimum amount of money the player must receive in addition to the gamble in order to be indifferent between the gamble option and the sure-thing option.

Decision Tree: A graphical representation for a decision problem that specifies the decisions that must be made, the chance events the decision maker faces, and the possible outcomes that result from the decisions and chance events that occurs.

Risk-Seeking: A term used interchangeably with risk-loving, describes the risk attitude of a person who prefers to take a gamble of the same expected dollar amount over the amount itself without a gamble, or equivalently, the utility function of the individual is convex.

Certainty Equivalent: The amount of money the player assigns as the value of the gamble.

Risk-Averse: Describes the risk attitude of a person who prefers to take the amount of money over a gamble with the same expected dollar amount, or equivalently, the utility function of the individual is concave.

Exponential Utility Function: A concave utility function commonly used to depict the utility function of a risk-averse individual, , where is the dollar amount and R is the risk tolerance parameter.

Risk-Neutral: Describes the risk attitude of a person who is indifferent between any gamble with the same expected dollar amount and the amount itself without a gamble, or equivalently, the utility function of the individual is linear.

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