🎱Game Theory Unit 2 – Utility Theory and Preferences

What's Utility Theory All About?

• Utility theory provides a framework for analyzing and understanding decision-making under uncertainty
• Focuses on how individuals make choices based on their preferences and the expected outcomes of those choices
• Assumes that individuals aim to maximize their utility, which is a measure of satisfaction or well-being derived from a particular outcome
• Utility is often represented by a utility function, which assigns a numerical value to each possible outcome
• Utility theory helps explain how people make trade-offs between different attributes of a decision (price vs. quality)
• Incorporates the concept of risk aversion, where individuals tend to prefer certain outcomes over uncertain ones with the same expected value
• Provides a foundation for various economic and game-theoretic models, such as expected utility theory and prospect theory

Key Concepts in Preferences

• Preferences refer to an individual's relative valuation of different outcomes or choices
• Completeness assumes that individuals can compare and rank all possible outcomes or choices
• Transitivity states that if an individual prefers A to B and B to C, then they must also prefer A to C
  • Helps ensure consistency in decision-making
• Monotonicity implies that more of a desirable attribute is always preferred to less (higher income is preferred to lower income)
• Convexity suggests that individuals prefer a mix of goods rather than extreme quantities of a single good
• Continuity assumes that small changes in the attributes of a choice do not lead to drastic changes in preferences
• Independence of irrelevant alternatives means that the presence of an additional choice should not affect the relative preferences between the original choices

Types of Utility Functions

• Cardinal utility functions assign a specific numerical value to each outcome, allowing for the comparison of utility differences between outcomes
• Ordinal utility functions only provide a ranking of outcomes without specifying the magnitude of the differences in utility
• Constant absolute risk aversion (CARA) utility functions imply that an individual's risk aversion remains constant regardless of their wealth level
  • Exponential utility function is an example of a CARA utility function: $U(x) = -e^{-ax}$, where $a$ is the risk aversion coefficient
• Constant relative risk aversion (CRRA) utility functions suggest that an individual's risk aversion depends on the proportion of their wealth at stake
  • Power utility function is an example of a CRRA utility function: $U(x) = \frac{x^{1-r}}{1-r}$, where $r$ is the relative risk aversion coefficient
• Quasi-linear utility functions are linear in one attribute (usually money) and non-linear in another attribute
  • Useful for analyzing decisions where one attribute is considered a "numeraire" or benchmark for value

Rational Decision Making

• Rational decision-making assumes that individuals make choices that maximize their expected utility
• Expected utility is calculated by multiplying the utility of each possible outcome by its probability and summing these products
• Individuals are assumed to have consistent preferences and to use all available information when making decisions
• Rational decision-makers consider opportunity costs, which are the potential benefits foregone by choosing one option over another
• Sunk costs, which are costs that have already been incurred and cannot be recovered, should not influence rational decision-making
• Bounded rationality acknowledges that individuals may not always make perfectly rational decisions due to cognitive limitations, time constraints, and incomplete information
• Satisficing is a decision-making strategy where individuals choose the first satisfactory option rather than seeking the optimal choice

Indifference Curves Explained

• Indifference curves represent all combinations of two goods that provide an individual with the same level of utility
• Points along an indifference curve are considered equally desirable to the individual
• Indifference curves are typically downward-sloping, as individuals are willing to trade off one good for another while maintaining the same utility level
• The slope of an indifference curve at any point is called the marginal rate of substitution (MRS), which measures the rate at which an individual is willing to trade one good for another
• Indifference curves cannot intersect, as this would violate the transitivity assumption of preferences
• Higher indifference curves represent higher levels of utility, as they contain combinations of goods that are preferred to those on lower indifference curves
• The shape of indifference curves reflects an individual's preferences and risk attitudes (convex curves indicate risk aversion)

Risk and Uncertainty in Utility

• Risk refers to situations where the probabilities of different outcomes are known, while uncertainty refers to situations where the probabilities are unknown
• Expected utility theory assumes that individuals make decisions based on the expected utility of each choice, considering both the utility and probability of each outcome
• Risk aversion implies that individuals prefer a certain outcome to a risky one with the same expected value
  • Risk-averse individuals have concave utility functions
• Risk neutrality suggests that individuals are indifferent between a certain outcome and a risky one with the same expected value
  • Risk-neutral individuals have linear utility functions
• Risk-seeking behavior occurs when individuals prefer a risky outcome to a certain one with the same expected value
  • Risk-seeking individuals have convex utility functions
• The certainty equivalent is the guaranteed amount an individual would accept instead of a risky prospect
• The risk premium is the difference between the expected value of a risky prospect and its certainty equivalent

Real-World Applications

• Consumer choice theory uses utility functions to analyze how individuals make purchasing decisions based on their preferences and budget constraints
• Portfolio theory applies utility theory to understand how investors make decisions about allocating their wealth among different assets based on their risk preferences
• Insurance markets rely on utility theory to explain why risk-averse individuals are willing to pay premiums to protect against potential losses
• Game theory uses utility functions to model strategic interactions between players and to predict the outcomes of these interactions
• Behavioral economics incorporates insights from psychology to understand how individuals make decisions that may deviate from the predictions of standard utility theory (loss aversion, framing effects)
• Public policy decisions often involve trade-offs between different objectives, and utility theory can help policymakers evaluate the relative merits of different options
• Utility theory is used in the design of incentive schemes, such as employee compensation plans, to align the preferences of agents with those of the principal

Common Pitfalls and Misconceptions

• The independence axiom of expected utility theory, which states that the preference between two options should not be affected by the presence of a third option, is often violated in practice (Allais paradox)
• The endowment effect suggests that individuals tend to value items they own more than identical items they do not own, contradicting the assumptions of standard utility theory
• Framing effects occur when the way a decision is presented influences the choice made, even if the underlying options remain the same
• The sunk cost fallacy is the tendency to continue investing in a project or decision because of past investments, even if it is no longer rational to do so
• The availability heuristic is the tendency to overestimate the likelihood of events that are easily remembered or imagined, leading to biased decision-making
• The representativeness heuristic is the tendency to make judgments based on how similar an event or object is to a typical example, rather than considering the underlying probabilities
• The gambler's fallacy is the belief that past events influence the probability of future independent events (believing a coin is more likely to land on heads after a series of tails)