A Heuristic Solution To The Allais Paradox

And Its Implications

Seán Muller, University of Cape Town

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

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))

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

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

Application to Allais’ Paradox: General Form

Choice #1A1: {x, 1}B1: {(0, p1); (x, p2); (y, p3)} ; y > x

Choice #2A2: {(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)

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)

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

Decision tree #1

Decision tree #2

Decision Tree Form

A1

(0,p1)

(x,p2)

(y,p3)

B1

R

(x,1)

A*1 B*1

(-x,p1)

(0,p2)

(y-x,p3)

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)

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:

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

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

 

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

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

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

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!

0 1

1

P1=prob(x1)

Increasing Preference

P3=prob(x3) P3=prob(x3)

P1=prob(x1) 10

1

Marschak-Machina Triangle with x3 > x2 >x1 ‘Fanning Out’ Indifference Curves

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”

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

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

THE END