Behavioral Economics & Decision Analysis
Prospect Theory Value & Prelec Weighting Formula
Evaluate reference-dependent gains and losses via Kahneman and Tversky’s (1992) asymmetric value function v(x) and Drazen Prelec’s (1998) inverse-S probability weight w(p) = EXP(-((-LN(p))^gamma)).
Explain simultaneously why consumers buy BOTH lottery tickets AND overpriced extended warranties!
Expected Utility Theory evaluates final wealth linearly in probabilities p, failing to explain why the same household buys low-probability lottery tickets AND pays steep markups to insure small deductibles! Daniel Kahneman and Amos Tversky’s (1992) Cumulative Prospect Theory pairs Reference-Dependent Value v(x) = IF(x >= 0, x^alpha, -lambda * ((-x)^beta)) (diminishing sensitivity 0 < alpha, beta <= 1 around reference point 0, plus Loss Aversion lambda > 1 where losses hurt ~2.25x harder than equal gains!) with Inverse-S Probability Weighting! Drazen Prelec (1998) axiomatized the cleanest 1-parameter probability weight w(p) = EXP(-((-LN(p))^gamma)) (0 < gamma < 1), which overweights tail events p < 1/e = 0.3679, crosses the diagonal at exact fixed point p* = 1/e = EXP(-1) regardless of gamma, and underweights moderate/high probabilities p > 1/e!
Verify on Kahneman-Tversky’s canonical parameters (alpha = 0.88 in B3, beta = 0.88 in B4, lambda = 2.25 in B5) and Prelec curvature gamma = 0.65 (B7): a -$100 Loss (x = -100 in B2) inflicts Subjective Value -2.25 * (100^0.88) = -129.47, whereas an equal +$100 Gain yields +57.54 (-2.25x asymmetry!); meanwhile, a p = 0.10 (10.0%, B6) tail risk receives Prelec Decision Weight EXP(-((-LN(0.10))^0.65)) = 0.1791 (79.1% overweighting above objective 10%!), and at p = EXP(-1) = 0.367879, Prelec weight equals exact 0.3679!
| Behavioral Decision Metric (alpha=0.88, lambda=2.25, gamma=0.65) | Formula | Verified Output |
|---|---|---|
| Subjective Value of -$100 Loss | -lambda * ((-x)^beta) | -129.47 utils |
| Subjective Value of +$100 Gain | x^alpha = 100^0.88 | +57.54 utils (Loss/Gain = 2.25x) |
| Prelec Decision Weight at p = 0.10 | EXP(-((-LN(0.10))^0.65)) | 0.1791 (vs 0.1000 Objective!) |
| Prelec Invariant Fixed Point (p = 1/e) | EXP(-((-LN(1/e))^gamma)) | 0.3679 (Exact 1/e for All gamma) |
Kahneman-Tversky (1992) Reference-Dependent Value v(x) Formula
Validate diminishing-sensitivity exponents 0 < alpha <= 1, 0 < beta <= 1, and loss-aversion coefficient lambda >= 1.
=LET(x_ch, B2, al_g, B3, bt_l, B4, lm_la, B5, IF(OR(al_g <= 0, al_g > 1, bt_l <= 0, bt_l > 1, lm_la < 1), "Invalid Prospect Input", ROUND(IF(x_ch >= 0, IF(x_ch = 0, 0, x_ch ^ al_g), -lm_la * ((-x_ch) ^ bt_l)), 2)))Prelec (1998) Compound-Invariance Probability Weight w(p) Formula
Handle boundary probabilities p = 0 (-> 0) and p = 1 (-> 1) explicitly before calling LN(p) on interior 0 < p < 1!
=LET(p_ob, B6, gm_cv, B7, IF(OR(p_ob < 0, p_ob > 1, gm_cv <= 0, gm_cv > 1), "Invalid Probability Input", IF(p_ob = 0, 0, IF(p_ob = 1, 1, ROUND(EXP(-((-LN(p_ob)) ^ gm_cv)), 4)))))Try the example
Use the original values or change them to test your understanding.