Kahneman & Tversky's prospect theory replaces expected-utility with two distortions of objective inputs: outcomes are evaluated by a reference-dependent, concave-for-gains / convex-for-losses value function, and probabilities are transformed by a weighting function that overweights small probabilities and underweights large ones.
v(x) = x^α if x ≥ 0 (gain)
v(x) = -λ·(-x)^β if x < 0 (loss)
w(p) = p^γ / [p^γ + (1-p)^γ]^(1/γ)
The green tube (z=0) plots v(x) over outcomes from -100 to +100; the orange tube (back row) plots w(p) against the dashed identity line w=p. Both spheres mark your current slider values.
- Outcome x — moves the green marker along the value curve and sets v(x).
- Loss aversion λ — steepens the loss side; λ>1 means losses hurt more than equal gains feel good.
- Curvature α=β — diminishing sensitivity; lower values flatten the curve further from the reference point.
- Probability p — moves the orange marker along the weighting curve, showing w(p) vs the true probability.
- Weighting curvature γ — controls how strongly small probabilities are overweighted (γ<1 gives the classic inverse-S shape).
This is the mechanism behind nudges, insurance purchases, lottery ticket demand, and loss-framed marketing copy: identical expected values produce different choices because of how x and p are subjectively transformed before any decision is made.