How the S-Shaped Value Function Makes People Risk-Averse for Gains and Risk-Seeking for Losses

The human brain evaluates financial risk not by calculating absolute wealth, but by measuring the immediate distance from a baseline. Every time an investor, a consumer, or a policymaker faces a choice involving uncertainty, their cognitive architecture applies an asymmetrical filter to the potential outcomes. They are able to lock in guaranteed gains or gamble to avoid losses, and they make this calculation the moment a new reference point is established.[6]

Before 1979, the dominant framework for understanding these choices was expected utility theory. Developed by John von Neumann and Oskar Morgenstern in 1944, this model assumed that individuals act as rational agents who calculate the mathematical expected value of any gamble and apply a consistent level of risk aversion based on their total absolute wealth. Under this classical model, a person with $100,000 in the bank should feel the exact same utility whether they arrived at that number by gaining $10,000 or losing $10,000.[1][2]

That assumption failed to match actual human behavior. Psychologists Daniel Kahneman and Amos Tversky dismantled the classical model with the publication of their 1979 paper, introducing what Britannica describes as "prospect theory, a descriptive model of decision making under risk." Their work proved that people do not evaluate outcomes based on final asset positions, but rather as gains and losses relative to a neutral reference point.[1]

The core engine of this behavioral model is the value function, which maps objective financial outcomes to subjective psychological value. Unlike the smooth, continuous utility curve of classical economics, the prospect theory value function is S-shaped. It passes through the reference point, dividing the psychological landscape into a domain of gains and a domain of losses, with distinct behavioral rules governing each side.[4]

In the domain of gains, the curve is concave. As objective gains increase—from $10 to $100 to $1,000—the marginal subjective value diminishes. The psychological difference between gaining $0 and $100 feels massive, while the difference between gaining $1,000 and $1,100 feels negligible. This concavity mathematically guarantees risk aversion: individuals will consistently choose a guaranteed $500 over a 50 percent chance to win $1,000, because the second $500 provides less psychological utility than the first.[2][4]

The architecture flips entirely in the domain of losses. Here, the S-curve becomes convex. The psychological pain of losing the first $100 is acute, but the marginal pain decreases as losses mount. Because the curve flattens out, individuals become reliably risk-seeking when facing negative outcomes. They will reject a guaranteed loss of $500 and instead accept a 50 percent chance of losing $1,000, gambling to get back to their reference point.[2][6]

The two halves of the S-curve are not symmetrical. The function is significantly steeper for losses than for gains, a phenomenon known as loss aversion. Empirical studies referenced by Cambridge University Press in their analysis of "psychological aspects beyond expected value" consistently show that the loss aversion coefficient hovers around 2.25. This means a loss hurts 2.25 times more than an equivalent gain provides pleasure.[2]

This mathematical asymmetry drives specific, measurable market behaviors. Financial Modeling Prep notes in their 2024 analysis that "investors are much more distressed by prospective losses than they are happy about equivalent gains." This manifests directly as the disposition effect, where retail traders sell winning stocks quickly to lock in the concave utility of a gain, while holding onto losing stocks for months, hoping a risky rebound will rescue them from the convex pain of a realized loss.[5]

The value function does not operate in isolation; it is modified by a probability weighting function. Kahneman and Tversky found that human beings systematically misunderstand probability, overweighting highly unlikely events and underweighting highly likely events. This weighting function bends the S-curve at its extremes, explaining why the same person might buy a lottery ticket with a 0.0001 percent chance of winning while simultaneously purchasing insurance against a 0.0001 percent chance of a house fire.[1][2]

Combined, the S-shaped value function and the probability weighting function create the fourfold pattern of risk attitudes. People are risk-averse for high-probability gains and low-probability losses, but risk-seeking for low-probability gains and high-probability losses. This matrix accurately predicts choices that expected utility theory classifies as irrational.[4][6]

The mathematical definitions of these curves continue to evolve. A 2022 paper published on arXiv proposed a "new concept for the value function," attempting to refine the exact inflection points where risk aversion transitions into risk-seeking behavior. These mathematical adjustments aim to capture edge cases where the standard 2.25 loss aversion ratio fluctuates based on the size of the initial stake and the speed at which the loss occurs.[3]

Corporations actively weaponize the S-curve in their pricing and marketing strategies. AdviceOnly's 2026 glossary entry on the topic highlights how marketers frame choices to manipulate the reference point. By presenting a standard price as a "discount" rather than a base rate with a surcharge, companies force consumers into the domain of gains, triggering risk-averse purchasing behavior that locks in the sale.[4]

Despite its predictive power, the S-shaped curve has structural limitations. It struggles to model decisions where the reference point is ambiguous or constantly shifting, such as in hyper-volatile cryptocurrency markets. Furthermore, the exact steepness of the loss aversion slope varies across different cultures and income brackets, meaning the curve must be calibrated to specific populations rather than applied as a universal biological constant.[2][6]

The next frontier for behavioral economists involves mapping the S-curve dynamically in real-time trading environments. As algorithmic trading platforms integrate biometric data and behavioral profiling, the exact moment an investor's reference point shifts from a gain to a loss can be quantified. The models will only change when researchers can predict not just that a trader will seek risk to avoid a loss, but the exact dollar amount that triggers the gamble.[6]