Prospect Theory: The Mathematical Model of How Humans Make Decisions Under Risk

Prospect Theory: The Mathematical Model of How Humans Make Decisions Under Risk

In 1979, Kahneman and Tversky published a paper titled “Prospect Theory: An Analysis of Decision under Risk.” This paper later helped Kahneman win the Nobel Prize in Economics—it used a mathematical model to prove that human decision-making under risk differs completely from what economics textbooks assume.

For traders, prospect theory isn’t academic decoration—it’s the psychological engine manual behind every position you open, every stop loss, every stubborn hold.

Core Principles

Point 1: The Value Function Is Reference-Point Dependent

Traditional economics assumes people care about absolute wealth—whether you have $100,000 or $100,010, the utility difference is minimal. Prospect theory says: people don’t care about absolute values, they care about changes relative to a reference point.

The reference point is usually your current status. If you just deposited $10,000, the reference point is $10,000. Making $1,000 is a “win”; losing $1,000 is a “loss.” But if you’ve already made $5,000, your reference point might have moved to $15,000—then a $1,000 drawdown isn’t a “small loss,” it’s “winning less”—the pain is lighter than losing $1,000 from the starting point.

In crypto: a trader with 50% unrealized profit tolerates a 10% pullback much better than one with no unrealized profit who gets anxious at a 2% pullback. Different reference points make the same波动 feel completely different.

Point 2: Loss Aversion—Loss Pain Is ~2.5x Gain Pleasure

The value function is much steeper in the loss domain than the gain domain. Experimental data shows the loss aversion coefficient λ is approximately 2.0–2.5—meaning the psychological pain of losing $100 roughly equals the pleasure of gaining $250.

This directly explains why traders prefer holding losing positions rather than stopping out: the pain of stopping out (承认 loss) far exceeds the pleasure of an equivalent-sized gain. Holding is the psychological state of “haven’t lost yet,” while stopping out is the现实 confirmation of “have lost.” The brain avoids that 2.5x pain.

Point 3: Probability Weighting Function’s Nonlinear Distortion

Prospect theory doesn’t directly multiply objective probability by value—it uses a probability weighting function w(p) to扭曲 probability. Core features:

• Extremely low probabilities are overestimated: A 1% event is perceived as having 5%–10% importance. This explains why crypto traders chase small-probability narratives—“what if this coin goes 10x?”
• Medium-low probabilities are roughly正常: 20%–50% probability perception is basically accurate.
• High probabilities are underestimated: A 90% event is perceived as 70%–80% certainty. This explains why traders don’t set stop losses—“I’m 90% sure it won’t drop there,” but 90% isn’t 100%, and after underestimation it becomes “almost impossible.”

Point 4: Certainty Effect

People异常放大 weight for the last few percentage points approaching 100% certainty. The psychological jump from 95% to 100% is far greater than from 50% to 55%. This causes two behaviors:

• Risk aversion: Locking in certainty when大概率 profitable (taking profit too early)
• Risk seeking: Chasing certainty when小概率 avoiding loss (holding instead of stopping out)

A trader who rushes to take profit when in profit (locking in certainty) and refuses to stop out when losing (chasing the小概率 “might rebound”)—this is prospect theory’s精确 prediction.

Point 5: The Fourfold Pattern

Prospect theory summarizes four typical decision patterns:

	Low Probability	High Probability
Gain Domain	Risk seeking (buying lottery tickets)	Risk aversion (early profit-taking)
Loss Domain	Risk aversion (fearing even small losses)	Risk seeking (gambling on rebound after big loss)

This four-cell grid precisely maps crypto traders’ typical behaviors: chasing小概率 moonshots (risk seeking), taking profit at 5% unrealized gain (risk aversion), getting anxious at 2% pullback (risk aversion), adding positions after a 30% drop (risk seeking).

Crypto Applications

Crypto is prospect theory’s amplified laboratory.

Leverage放大了 reference-point dependence. A 10x leveraged trader experiences 10% capital波动 from a 1% price move. The reference point shifts from “principal” to “leveraged预期收益,” and gain-loss perception gets极度放大. The value function becomes steeper under leverage—10x leverage doesn’t just放大 risk, it放大 loss aversion.

Crypto’s high volatility creates more “low-probability events.” BTC rising 20%+ in a single day is nearly impossible in traditional markets but happens periodically in crypto. Prospect theory predicts these小概率 surges will be overestimated—traders will过度 chase moonshot opportunities while underestimating the probability of normal pullbacks after surges.

Social narratives扭曲 probability perception. “Someone turned 100U into 100,000U” stories make a 1% probability feel like 10%. Meanwhile, “someone lost 100U to zero” stories, equally frequent, are selectively ignored—because loss aversion makes people回避 losing-money narratives.

Practical Scenarios

Scenario 1: Diagnosing Your Profit-Taking and Stop Loss Habits Using Prospect Theory

Check your last 30 trades:

• What’s your average profit-taking幅度? If most are at 3%–8%, you may be driven by the certainty effect—rushing to lock in gains.
• What’s your average stop loss幅度? If most stop losses are above 15% or nonexistent, you may be driven by loss aversion—逃避 the pain of stopping out.
• The ratio between these two numbers is an approximate indicator of how much prospect theory is操控 your decisions.

Scenario 2: Setting a Risk Control Framework Against Prospect Theory

Prospect theory predicts you’ll make certain系统性 errors—you can use rules to对抗 them:

• Profit-taking rule: Set target profit幅度 (like 15%–25%), not “feeling enough then running.” Rules对抗 the certainty effect.
• Stop loss rule: Set hard stops (like 5%–8%), don’t allow manual cancellation. Rules对抗 loss aversion.
• Position rule: Fix per-trade risk at no more than 1%–2% of total capital. Rules对抗 probability weighting扭曲 (prevent过度 committing to小概率 events).

Scenario 3: Watch for Reference Point Migration When Adding to Winning Positions

Adding to profitable positions is common, but注意 your reference point has migrated. After adding, the total position’s risk ratio needs recalculation—you can’t use “I already have unrealized profit cushioning” as an excuse to loosen risk control. Unrealized profit isn’t free—it’s an illusion created by reference point migration.

Common Misapplications

Misapplication 1: Using prospect theory to解释 all wrong decisions. Prospect theory describes statistical倾向, not every trade’s唯一 cause. Sometimes not stopping out is because the strategy本身 allows larger drawdowns; sometimes early profit-taking is because market conditions确实 require fast exits. Distinguishing “bias-driven” from “strategy-driven” requires回顾 original decision records.

Misapplication 2: Believing “understanding prospect theory means I won’t make mistakes.” Knowing about loss aversion ≠ no longer experiencing loss aversion. Prospect theory describes人类大脑’s hardware bias—not software you can uninstall by reading a paper. You need rules and systems, not just knowledge.

Misapplication 3: Equating the certainty effect with “shouldn’t take profit.” The certainty effect makes you take profit too early, but this doesn’t mean all profit-taking is biased. There’s a difference between reasonable profit-taking rules and bias-driven early exits—the distinction lies in whether you had预设 targets.

Summary

Prospect theory uses mathematical models to揭示 three layers of human decision-making distortion under risk: reference-point dependence makes gain-loss perception vary with context, loss aversion makes loss pain far exceed gain pleasure, and the probability weighting function makes小概率 events overestimated and大概率 events underestimated.

These three distortion layers精确 predict traders’ typical错误 behaviors: early profit-taking, late stop losses, chasing小概率 moonshots, holding through大概率 losses. Understanding prospect theory can’t消除 these biases, but can help you识别 them, then use rules and systems to替代 bias-driven decisions.

Trading isn’t about fighting the market—it’s about fighting your own prospect theory hardware. You need not just analytical ability, but institutional design to对抗 your own brain.

For more practical methods, see Dimen Trading.