Prospect theory - Core Theory and Example

Core Concepts of Prospect Theory Introduction Prospect Theory is a framework that describes how people actually make decisions under uncertainty, often departing significantly from the predictions of traditional expected utility theory. Developed by Daniel Kahneman and Amos Tversky, it explains why people's choices sometimes seem irrational from a mathematical perspective, yet are predictable and systematic. The theory rests on several key insights into human psychology. Loss Aversion One of the most important discoveries in behavioral economics is that losses hurt more than equivalent gains delight. If you lose $1,000, the pain you experience is significantly greater than the pleasure from gaining $1,000. This asymmetry is called loss aversion. This concept is crucial because it explains many real-world behaviors that standard economic theory cannot. For example, people hold onto losing investments longer than they should, hoping to break even—they're willing to take risks to avoid realizing a loss. Similarly, people are reluctant to change status quo situations because the pain of what they might lose outweighs the pleasure of what they might gain. The strength of loss aversion varies somewhat across individuals and contexts, but empirical research suggests that losses feel roughly 1.5 to 2.5 times as intense as equivalent gains. Reference Dependence A second core principle is that outcomes are evaluated relative to a reference point, not in absolute terms. Think of your reference point as your current situation or expectations. Anything better than this reference point feels like a gain, and anything worse feels like a loss. This might sound obvious, but it's quite different from how traditional economics approaches decisions. In traditional theory, you care only about your final wealth—whether you have $50,000 or $51,000 matters in the same way regardless of how you got there. In prospect theory, what matters is whether you're ahead or behind your reference point. Here's a practical example: imagine you expect a $10,000 bonus, but receive only $8,000. In absolute terms, you're doing well—you have $8,000 more than before. But relative to your reference point ($10,000), you feel like you've lost $2,000. This "loss" relative to expectations will feel worse than the absolute gain would normally feel good. This explains why people often feel disappointed by outcomes that are objectively favorable, simply because they fell short of expectations. The reference point is typically your current wealth or status, but it can also be shaped by recent experiences, expectations, or what others are receiving. Risk Attitudes for Gains and Losses Here's where prospect theory makes a striking prediction: your attitude toward risk depends on whether you're facing potential gains or potential losses. For gains: People tend to be risk-averse. Given a choice between a guaranteed $500 gain and a 50/50 gamble to win $1,000 or nothing, most people prefer the certainty of $500. Why? Because the satisfaction from an additional $500 is smaller when you already have $500 than when you have nothing. Mathematically, this happens because the value function is concave in the gain region—imagine a curve that gets flatter as you move to the right. Each additional dollar gives you slightly less additional satisfaction. For losses: People tend to be risk-seeking. Given a choice between a guaranteed $500 loss and a 50/50 gamble to lose $1,000 or nothing, most people prefer the gamble. Why? Because the pain of losing an additional $500 is felt less intensely when you've already lost $500 than when you're losing your first $500. Mathematically, this happens because the value function is convex in the loss region—imagine a curve that gets steeper as you move further down. Each additional dollar of loss adds more additional pain than the previous dollar. This U-shaped or S-shaped pattern in the value function is fundamental to understanding why people behave differently when facing gains versus losses. Probability Distortion People are notoriously poor at understanding probabilities. Rather than treating probabilities objectively, prospect theory says that people apply a probability-weighting function that distorts how they perceive chances. The key distortions are: Small probabilities are overweighted: A 1% chance feels subjectively much more likely than 1%. This explains why people buy lottery tickets with terrible expected value—they overweight the small probability of winning. A 1-in-a-million chance gets treated like it's much more plausible than it actually is. Large probabilities are underweighted: A 99% chance feels less certain than 99%. If something has a 95% chance of occurring, people don't feel as confident as the odds suggest. This helps explain overconfidence bias and why people sometimes ignore high-probability risks. Very small and very large probabilities are less distorted than intermediate probabilities: The most extreme distortions occur for probabilities in the 5-25% range. The probability-weighting function forms an inverse-S shape: it curves upward (overweighting) for small probabilities, then curves downward (underweighting) for large probabilities, creating a shape opposite to the letter S. The Fourfold Pattern of Risk Attitudes By combining the concepts of loss aversion, the S-shaped value function, and probability weighting, prospect theory predicts a distinctive fourfold pattern in how people approach risky decisions: Risk-averse for moderate-probability gains (e.g., 50% chance of winning $1,000): People prefer a sure smaller gain over a gamble. Risk-seeking for low-probability gains (e.g., 1% chance of winning $1,000,000): People will take long-shot bets rather than accept a guaranteed smaller amount. Risk-averse for low-probability losses (e.g., 1% chance of losing $10,000): People buy insurance to avoid small-probability catastrophes, even when the expected value is against them. Risk-seeking for moderate-probability losses (e.g., 50% chance of losing $1,000): People take risks to try to avoid moderate losses rather than accepting them as certain. This fourfold pattern shows that prospect theory can explain decisions that seem contradictory from a traditional economic standpoint, but are actually quite predictable once you understand the underlying psychological mechanisms. The Formal Model of Prospect Theory The Two-Stage Decision Process Prospect theory proposes that decision-making happens in two distinct stages: Editing Stage: First, you simplify and organize the decision problem. You order the possible outcomes, establish a reference point (what counts as your baseline), and identify which outcomes are gains (better than the reference) and which are losses (worse than the reference). You might also combine probabilities or round numbers to make the problem more manageable. Evaluation Stage: Then, you compute an overall value (utility) for each possible choice and pick the option with the highest value. This evaluation uses both the value function and the probability-weighting function. This two-stage process helps explain why the same objective situation can be perceived differently depending on how it's presented—a phenomenon called framing. The editing stage is where this matters most, since how you organize and frame the decision changes your reference point and which outcomes feel like gains versus losses. The Utility Formula The overall utility of a prospect is calculated using this formula: $$U = \sum{i} \pi(p{i}) \, v(x{i})$$ Let's break down each component: $U$ is the total utility or value you perceive from the decision $x{i}$ represents each possible outcome $p{i}$ is the probability of outcome $x{i}$ occurring $v(x{i})$ is the value function applied to outcome $x{i}$—this translates the outcome into how good or bad it feels to you, taking into account loss aversion and reference dependence $\pi(p{i})$ is the probability-weighting function applied to probability $p{i}$—this is your subjectively distorted perception of the probability The $\sum$ symbol means you sum up all the weighted values across all possible outcomes The key insight is that you're not weighting outcomes by their actual probabilities; you're weighting them by your distorted perception of those probabilities. And you're not valuing outcomes at face value; you're valuing them through the lens of loss aversion and reference dependence. Value Function Characteristics The value function $v(x)$ has two critical features: It's S-shaped: This means it's concave (flattening out) in the gain region and convex (getting steeper) in the loss region. This S-shape directly encodes loss aversion—the losses side is steeper than the gains side. It's steeper for losses than for gains: This is another way of expressing loss aversion. A loss of $X$ creates a more negative change in utility than a gain of $X$ creates positive change. The slope of the curve is roughly twice as steep on the loss side as on the gain side. These characteristics mean that: Your satisfaction increases at a decreasing rate as you gain more (concave for gains) Your dissatisfaction increases at an increasing rate as you lose more (convex for losses) You feel losses roughly twice as intensely as equivalent gains Probability-Weighting Function Characteristics The probability-weighting function $\pi(p)$ has a distinctive inverse-S shape that captures how you subjectively distort probabilities: For small probabilities: $\pi(p) > p$. You overweight small probabilities. If the actual probability is 5%, you might weight it as if it's 12%. This is why people buy lottery tickets and worry about rare disasters. For large probabilities: $\pi(p) < p$. You underweight large probabilities. If the actual probability is 95%, you might weight it as if it's 87%. This is why people are sometimes overconfident or ignore high-probability risks. Boundary conditions: The function satisfies $\pi(0) = 0$ and $\pi(1) = 1$—impossible events remain impossible, and certain events remain certain. But in between, there's significant distortion. This inverse-S shape (opposite to the letter S) is crucial for explaining otherwise puzzling behaviors like why people simultaneously buy insurance (overweighting low-probability losses) and buy lottery tickets (overweighting low-probability gains). Example: The Insurance Decision Why People Buy Insurance: A Prospect Theory Explanation Consider this scenario: You have a 1% chance of losing $10,000 due to a catastrophe. An insurance company offers to eliminate this risk for a $120 premium (which exceeds the expected loss of $100). Traditional economic theory says you shouldn't buy this insurance—you're paying more than the expected loss. Yet most people do buy insurance. Why? The prospect theory explanation: First, your reference point is "no catastrophe." From this reference point: The uninsured option looks like a 1% chance of a $10,000 loss The insured option looks like a certain $120 loss Now apply the key mechanisms: Loss aversion: Losses are painful. Both options involve losses relative to your reference point, so you're comparing which loss hurts less. Probability weighting: The 1% probability of loss gets overweighted. Instead of weighting it as 1%, you might weight it as something like 3% or higher. This makes the uninsured option look much worse than its objective expected value suggests. Risk-aversion for medium-to-large losses: You prefer a certain small loss ($120) over a gamble involving a larger loss, even if the gamble has a lower expected value. When you combine these effects, the utility of buying insurance exceeds the utility of not buying it: $$U{\text{insured}} > U{\text{uninsured}}$$ This makes insurance look attractive, even though you're paying above expected value. Prospect theory thus explains a major real-world decision that traditional theory struggles with.