A Brief History of Descriptive Theories of Decision Making: Lecture 2: SWU and PT Kiel, June 10, 2005 Michael H. Birnbaum California State University, Fullerton 1
Overview • Last time, we saw that EV theory and EU theory were not descriptive of risky decision making. • This time, we review Edwards (1954, 1962) subjectively weighted utility and Kahneman and Tversky’s (1979) prospect theory. • These models account for Allais paradoxes, but they made predictions that seemed incorrect.
Models to be Reviewed • Expected Value (EV) • Expected Utility (EU) and Subjectively Expected Utility (SEU) • Subjectively Weighted Utility (SWU), including Prospect Theory (OPT) • Rank Dependent Utility (RDU), including Rank- and Sign-Dependent Utility (RSDU) and Cumulative Prospect Theory (CPT) • Configural Weighted Utility, including Rank-Affected Multiplicative Weights (RAM) and Transfer of Attention Exchange (TAX) models.
Expected Value Theory • • • Let G = (x, p; y, q; z, r) Where p + q + r = 1 EV = px + qy + rz Judged Value = f(EV) The function, f, is strictly monotonic. Hence, if EV(F) > EV(G), then F is preferred to G.
Critical Properties # 1 (Classic Paradoxes) • Risk Aversion (RA) and Risk-Seeking • St. Petersburg Paradox • Sales and Purchase of gambles and insurance • These are inconsistent with EV theory.
Expected Utility (EU) Theory Utility of money allows EU to predict risk aversion or risk seeking.
Expected Utility Theory • • Why people would buy and sell gambles Sales and purchase of insurance St. Petersburg Paradox Risk-Aversion or Risk-Seeking:
Classic Paradoxes #2: Refuted EU • Allais Common Consequence Paradox • Allais Common Ratio Paradox • Risk-Seeking and Risk-Aversion in the same person • Consequence Framing, Reflection Hypothesis • Preference Reversals: Choice versus Valuation, • Preference reversals between Buying versus Selling Prices
Edwards (1954): SWU Model • Subjective scale of probability as well as a subjective scale of money. • Utility of A = (x, p; y) is as follows: • U(A) = S(p)u(x) + S(1 – p)u(y) • Where S(p) is a weighting function on probabilities. Edwards (1954) discussed both inverse-S and S-shaped weighting functions.
Allais (1953) “Constant Consequence” Paradox Called “paradox” because preferences contradict Expected Utility. A: $1 M for sure f B: . 10 to win $2 M. 89 to win $1 M. 01 to win $0 C: . 11 to win $1 M p. 89 to win $0 D: . 10 to win $2 M. 90 to win $0
Analysis of Allais CC Paradox S(1)u(1) > S(. 1)u(2) + S(. 89)u(1) + S(. 01)u(0) [S(1) – S(. 89)]u(1) > S(. 1)u(2) [S(1) – S(. 89)]/S(. 1) > u(2)/u(1) S(. 11)< S(. 1)u(2)/u(1) > S(. 11)/S(. 1) [S(1) – S(. 89)]/S(. 1) > S(. 11)/S(. 1) 1 – S(. 89) > S(. 11) 1 > S(. 89) + S(. 11) Holds for other p + q = 1.
Analysis of Allais Common Ratio Paradox • Similarly, there is no contradiction in SWU with the common ratio problem: • S(1)u(3) > S(. 8)u(4) • S(1)/S(. 8) > u(4)/u(3) • S(. 25)u(3) < S(. 2)u(4) • S(. 25)/S(. 2) < u(4)/u(3) • S(. 25)/S(. 2) < S(1)/S(. 8)
Weighting Function
“Value” (utility) Function
Edwards (1962) Weighting Functions Perhaps 5 pages in the book of weights for these different configurations: • All consequences negative • Negative plus zero • Mixed: both positive and negative • Positive plus zero • All positive
Prospect Theory (1979) • Kahneman & Tversky (1979). Edwards theory, but just two configurations, called “regular” and “irregular” prospects (containing zero or not). One weighting function of p, “value” instead of utility. • Accounts for Allais paradoxes, 4 -fold pattern of risk-seeking and risk-aversion with positive and negative consequences.
Implications • This model violates transparent stochastic dominance. • Fishburn (1978) showed that Handa’s model (Edwards’ SEV model) violates transparent dominance. Same argument applies to SWU
Kahneman (2003) “…Our model implied that ($100, . 01; $100, . 01) — two mutually exclusive. 01 chances to gain $100 — is more valuable than the prospect ($100, . 02)… most decision makers will spontaneously transform the former prospect into the latter and treat them as equivalent in subsequent operations of evaluation and choice. To eliminate the problem, we proposed that decision makers, prior to evaluating the prospects, perform an editing operation that collects similar outcomes and adds their probabilities. ”
Editing Rules • • • Combination: collect similar outcomes Cancellation: delete common branches Simplification: round off p, x Dominance detection—spot & conform Segregation—sure thing Priority of editing—editing precedes and trumps evaluation
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