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In decision theory, utility is a concept used to represent the satisfaction or value a person derives from an outcome. Instead of simply looking at monetary gains or losses, utility focuses on the subjective preferences of individuals. Here's how utility works:

1. Definition of Utility:

• Utility is a measure of the happiness, satisfaction, or benefit a person derives from a specific outcome. It is often used to compare different outcomes or choices in decision-making.
• Utility can be thought of as a numerical value that represents the desirability of an outcome from the perspective of the decision-maker.

2. Utility Functions:

• A utility function is a mathematical representation of a person's preferences. It assigns a utility value to each possible outcome based on the person's subjective evaluation.
• The utility function captures how a decision-maker values different outcomes and helps rank them in order of preference.

For example, if a person prefers $100 to $50, the utility of $100 might be 10, while the utility of $50 might be 5. The exact numbers are less important than the ranking they imply.

3. Expected Utility Theory:

• In decision theory, Expected Utility Theory is used to make decisions under uncertainty by maximizing expected utility rather than expected monetary value.
• Expected Utility is calculated by multiplying the utility of each possible outcome by its probability and summing up these products.

Formula:

This allows decision-makers to choose the option that provides the highest expected satisfaction or value, even when the outcomes are uncertain.

Example:
Suppose you have two investment options:

• Option A: 50% chance of earning $100, 50% chance of earning $0.
• Option B: Guaranteed $45.

Using expected utility, if your utility function suggests you prefer less risk, the guaranteed $45 might have higher utility, even though the expected monetary value of Option A is $50 (0.5 * $100 + 0.5 * $0). You would choose Option B because its expected utility is higher for you.

4. Risk Preferences:

• Utility functions also capture risk preferences:
  • Risk-Averse: A risk-averse person values certain outcomes more highly than risky ones with the same expected monetary value. Their utility function is concave, meaning that as wealth increases, additional wealth provides diminishing utility. They prefer security and might choose a lower but guaranteed payoff.
  • Risk-Neutral: A risk-neutral person cares only about maximizing expected monetary value, not about the risk. Their utility function is linear.
  • Risk-Seeking: A risk-seeking person derives more utility from riskier options. Their utility function is convex, meaning they value potential gains more highly than the same gains from less risky options.

Example:
If a risk-averse person is offered a 50% chance to win $100 or a guaranteed $45, they might choose the $45 because the certainty gives them higher utility, even though the expected monetary value of the gamble is higher.

5. Diminishing Marginal Utility:

• Diminishing Marginal Utility is a principle where each additional unit of something (like money) provides less additional utility than the previous unit. For example, gaining an extra $100 when you already have $1,000 feels less satisfying than gaining $100 when you only have $100.
• This concept explains why risk-averse individuals prefer less risky outcomes; the potential loss feels more impactful than the same potential gain.

Example:
Suppose you have $10,000. Winning an additional $1,000 might increase your utility slightly, but losing $1,000 might decrease your utility significantly. Therefore, the potential loss feels worse than the potential gain feels good.

6. Utility in Real-World Decision Making:

• Consumer Behavior: Utility theory explains consumer choices, such as why people might choose a more expensive product if it provides more utility (satisfaction) even though a cheaper product might fulfill the same function.
• Investment Decisions: Investors use utility to decide between different portfolios, balancing expected returns with their risk tolerance.
• Public Policy: Governments use utility-based models to evaluate the impact of policies on overall welfare, aiming to maximize the utility (well-being) of their citizens.

Conclusion:

Utility represents the value or satisfaction a person derives from an outcome. By using utility functions, decision-makers can model their preferences and make choices that maximize their overall satisfaction, taking into account their risk tolerance and the subjective value of different outcomes.