A Heuristic Solution To The Allais Paradox

1. A Heuristic Solution To The Allais Paradox And Its Implications Seán Muller, University of Cape Town

2. Introduction • Starting point: ‘Allais paradox’ (1953) • Provided great impetus to research into alternative theories of utility • Remains one of the most cited examples of violations of EUT

3. The ‘Certainty Heuristic’ (CH) Formal definition: Given a choice of the form A: (x,1) B: (y, p; 0, (1-p)) Where: for (x, p), x = outcome and p = probability an individual using the certainty heuristic will consider this as: A*: (0,1) B*: (y-x, p; -x, (1-p))

4. Why and How? • Simplifies choice problems involving certainty • Critical aspect of strategic planning: ‘if, then’ - type of reasoning • Chess • Extensive form games • Also: • Regret Theory • Hypothetical endowments

5. Assumptions on preferences • Loss aversion (Kahneman&Tversky,’79) • Utility calculations made on changes in wealth relative to reference point • S-shaped utility function: concave for gains, convex for losses • And losses weighted more heavily than gains, so: U(x)=xG, U(-x)=xL, 1 > L > G > 0

6. Application to Allais’ Paradox: General Form Choice #1 A1: {x, 1} B1: {(0, p1); (x, p2); (y, p3)} ; y > x Choice #2 A2: {(0, p2); (x, 1- p2)} B2: {(0, 1- p3); (y, p3)} Normally(EUT) choosing A1 over B1, implies: (1) U(x) > p2 U(x) + p3 U(y)

7. Application to Allais Paradox contd. • Applying the CH to choice #1, yields the following reformulation: A1*: {x-x, 1} B1*: {(0-x, p1); (x-x, p2); (y-x, p3)} • This then reduces to: A*1: {0, 1} B*1: {(-x, p1); (0, p2); (y-x, p3)} So choosing A1>B1 implies A1*>B1*, and: (2) 0 > p1U(-x) + p3U(y-x)

8. Application to Allais Paradox contd. • Can restate (2) using loss aversion as: (5) p1(x)L > p3(y - x)G • Furthermore, the majority choice of B2 over A2 provides the restriction that: (6) p3U(y) > (1-p2)U(x) • Rewriting both conditions in terms of p3 yields the overall condition that: (7) (1- p2)xG/yG < p3 < p1(xL)/(y-x)G

9. Decision Tree Form Decision tree #1 Decision tree #2

10. CH Provides Solutions to: • Allais Paradox variants • Humans • Rats (Battalio et al., ’85) • Even where independence axiom not violated (e.g. Conlisk, ’89) • ‘Common ratio’ effect • Hagen’s ‘Bergen Paradox’ (’79)

11. Framing Effects • ‘Description invariance’ and ‘procedure invariance’ • CH certainly raises questions about underlying assumptions involved in description invariance experiments • In particular: reference points & hypothetical endowments • À la ‘status quo bias’ • An example:

12. The “Asian Disease” Problem • Imagine the U.S. is preparing for the outbreak of an unusual Asian disease, expected to kill 600 people. Two alternative programs have been proposed. The exact scientific estimate of the consequences of the programs are as follows: • If Program A is adopted, 200 will be saved • If Program B is adopted, there is 1/3 probability that 600 people will be saved, and 2/3 probability that 600 people will die

13. Asian Disease Problem contd. • Second group given the same initial information, but descriptions of the programs were changed to read: • If Program C is adopted 400 people will die • If Program D is adopted there is 1/3 probability that nobody will die, and 2/3 probability that 600 people will die

14. Asian Disease Problem contd. • First group: 72% chose Program A • Second group: 78% of respondents chose D • “Although (the) statement of the problem is…identical” (Machina, ’87) • But is it? • Relies on assumption that individuals do actually internalise the hypothetical reference point

15. Theoretical Alternatives to CH • A whole field’s worth! • But notably for our purposes: • Prospect Theory • Probability weighting function • ‘Certainty effect’ • Regret Theory(’82) • Individuals anticipate ‘regret’/’rejoicing’ • Implicitly assumes CH-like process

16. Morrison(’67) • Same intuition as CH: • In the original Allais problem, “It is now as if (the subject’s) asset position were one million dollars” • However made many dubious assumptions in rationalising this • Perhaps due to being ‘pre-Prospect Theory’, and the idea of heuristics as systematic influences on choice

17. The Marschak-Machina Triangle • Key problem: relies on fixed reference points • But CH implies lotteries on triangle boundaries will cause a change in the reference point!

18. Marschak-Machina Triangle contd. • Harless (’92): “In choices over hypothetical lotteries, such as the Allais paradox, systematic violations of expected utility disappear when lotteries are nudged inside the (Marschak-Machina) triangle” • Conlisk(’89): “violations of expected utility theory are less frequent and are no longer systematic when boundary effects are removed”

19. Conclusion • Simple heuristic solution to various axiomatic violations • Based on hypothesis re: psychological selection of reference points • Feasibility of CH-like process implicitly assumed in literature

20. Implications • Probability simplex may not be appropriate for assessing theories’ consistency • Experiments, esp. framing effect experiments, may need to give greater consideration to purely psychological reference points • Allais paradox violations can be eliminated by appropriate changes in choice parameters

21. THE END