Utility & Preference Functions in Decision-Making
What Are Utility Functions?
Utility measures satisfaction or value derived from an outcome. A utility function mathematically represents how individuals rank different outcomes based on their preferences.
Core Idea: People choose options that maximize their expected utility.
Key Concepts
Cardinal vs Ordinal Utility:
| Type | Meaning | Example |
|---|---|---|
| Ordinal | Ranking only (A > B > C) | Ice cream better than salad |
| Cardinal | Magnitude matters (A = 100 utils, B = 50 utils) | Ice cream gives twice the satisfaction |
Traditional finance assumes cardinal utility to quantify preferences.
Properties of Utility Functions
Standard Assumptions
Completeness: Can compare any two options
- Given A and B, prefer A, B, or indifferent
Transitivity: Logical consistency
- If A > B and B > C, then A > C
Continuity: No abrupt jumps in preferences
Non-Satiation: More is better (diminishing marginal utility)
Risk Attitudes Reflected
Risk-Averse: Utility function is concave
- U(certain ₹100) > Expected U(gamble averaging ₹100)
- Most investors are risk-averse
Risk-Neutral: Utility function is linear
- Only expected value matters
Risk-Seeking: Utility function is convex
- Prefer risky gambles over certainty
Expected Utility Theory (EUT)
Formula: EU = Σ [p_i × U(x_i)]
Where:
- p_i = probability of outcome i
- U(x_i) = utility of outcome i
Example:
- Gamble: 50% chance ₹1,000, 50% chance ₹0
- Risk-averse utility: U(₹1,000) = 10, U(₹0) = 0
- EU = 0.5×10 + 0.5×0 = 5
- Certain ₹400: U(₹400) = 6
- Choose certain ₹400 (higher utility despite lower expected value of ₹500)
Behavioral Violations
Traditional utility theory assumes rationality, but people systematically violate its axioms:
Allais Paradox: Preference reversals violate independence axiom
Ellsberg Paradox: Ambiguity aversion violates expected utility
Framing Effects: Same choice, different frames, different decisions
Behavioral finance response: Prospect Theory replaces EUT to better describe actual behavior.
Preference Functions
Preferences aren't always stable:
Context-Dependent:
- Choosing between A and B depends on whether C is available (decoy effect)
Reference-Dependent:
- Value judged relative to reference point, not absolute levels (Prospect Theory)
Time-Inconsistent:
- Present bias makes preferences reverse over time (hyperbolic discounting)
Key Takeaways
- Utility functions quantify satisfaction from outcomes
- EUT maximizes expected utility given probabilities
- Assumptions: Completeness, transitivity, continuity
- Risk attitudes: Reflected in function shape (concave = risk-averse)
- Reality: People violate EUT axioms systematically (Prospect Theory explains better)
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