Study Notes on Behavioral and Experimental Economics: Prospect Theory

Concept originated to resolve paradoxes in decision-making under uncertainty.
Notably addressed the St. Petersburg Paradox presented by Daniel Bernoulli in 1738.

Option A: You receive a certain amount of money, denoted as $X$.
Option B: A coin is flipped until tails first appears; the starting pot is $1, and it doubles every time heads appears. The pot's total is kept once tails appears.
Question: What is the minimum value of $X$ for which one would prefer Option A?

Key terms: Expected value, expected utility.
Expected Monetary Value ($E$): The average of all possible outcomes weighted by their probabilities.
Expected Utility ($EU$): A measure of satisfaction or benefit derived from uncertain outcomes.
EU = ext{Expected utility value}

Economists conceptualize decision-making under uncertainty through lotteries defined as probability distributions over a finite set of outcomes.
Notation:
- Let $g_i$ represent a lottery.
- Let $A = ig a1, a2, ext{…,} a_N ig$ denote the set of outcomes from which probabilities are drawn.
- Let $p_{i,n}$ denote the probability of outcome $n$ in lottery $i$.

Monotonicity: More is better (i.e. if $g ext{ is preferred to } g'$, then $U(g) > U(g')$).
Completeness: Individuals can rank all lotteries.
Transitivity: If $g ext{ is preferred to } g'$ and $g' ext{ is preferred to } g''$, then $g ext{ is preferred to } g''$.
Continuity: Upper and lower contour sets of a preference relation over lotteries are closed, indicating smooth transitions in preferences.
Substitution or Independence Axiom: If indifferent between alternatives in a lottery, this indifference remains under equal substitutions (i.e. if indifferent between $aj$ and $aj'$, then indifferent between the lotteries containing them with equal probabilities).

Substitution (Independence Axiom): Identify common consequences.
- If $p(a; (1-p)X) ext{ is equivalent to } p(b; (1-p)X)$, then $p(a; (1-p)Y) ext{ is equivalent to } p(b; (1-p)Y)$.
Mathematical Representation:
U(gi) = ext{E}[U(a1, a2, ext{…,} aN)]
where $U(gi)$ denotes the utility of lottery $gi$.

EUT does not convincingly account for certain real-life decisions, leading to documented paradoxes such as the Allais Paradox.

Choice 1:
- Option A: $500 with 10% probability; $100 with 89% probability; $0 with 1% probability.
- Option B: $100 with 100% probability.
Choice 2:
- Option C: $500 with 10% probability; $0 with 90% probability.
- Option D: $100 with 11% probability; $0 with 89% probability.
Observations: People tend to prefer options that contradict the expected utility calculations.

Choice 3: Compare probabilities where certainty is altered, revealing behavioral inconsistencies.
- $A: 25 @ 33%; 24 @ 66%; 0 @ 1%
- $B: 24 @ 100%$
Common Ratio Effect: A prevalent discrepancy in decision-making behavior, marked by inconsistency when varying probability ratios.
Isolation Effect: Details where initial outcomes significantly impact choices, as seen in loss framing versus gain framing.

Certainty Effect: People overweight certain outcomes compared to uncertain ones.
Reflection Effect: Greater weight on losses than on gains, indicating a psychological bias towards risk.
Framing Effects: How information is presented influences choices (isolation effect).

Addressed shortcomings of EUT by incorporating a more psychologically accurate reflection of decision-making under risk.

Probability Weighting Function: Transformation of objective probabilities into subjective weights, altering perceptions of risk.
Value Function: Reference-dependent utility where satisfaction is derived from changes relative to a baseline (initial endowment).

Fourfold pattern of risk attitudes includes:
1. Risk seeking over low-probability gains: Preference for buying lottery tickets.
2. Risk aversion over low-probability losses: Individuals ensure to protect against unlikely but damaging events.
3. Risk aversion over high-probability gains: Standard behavior observed; under-weighting prevailing events.
4. Risk seeking over high-probability losses: When facing significant loss, taking extreme risks for potential gain becomes prevalent.

Final consideration on whether the explanations and behavior outlined by prospect theory can be deemed rational, particularly related to regret avoidance in decision-making contexts.