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Composite Prospect Theory

Sanjit Dhami� Ali al-Nowaihiy

10 November 2010

Abstract

Evidence suggests three important stylized facts, S1, S2a, S2b. Low probabilities

are overweighted, high probabilities are underweighted (S1). Some people ignore

events of extremely low probability and treat events of extremely high probabil-

ity as certain (S2a). Others focus greatly on the size of outcomes, even for ex-

tremely low/high probabilities (S2b). We propose composite-cumulative-prospect-

theory (CCP) that accounts jointly for S1, S2a, S2b. We discuss several applications,

including insurance behavior, the class of Becker-paradoxes, the Allais-paradox and

the St. Petersburg-paradox. CCP explains everything that expected utility, rank

dependent utility, prospect theory and cumulative prospect theory do; the converse

is false.

Keywords: Decision making under risk; Composite Prelec probability weighting

functions; Composite cumulative prospect theory; Composite rank dependent utility

theory; Insurance; St. Petersburg paradox; Beckers paradox; Bimodal perception of

risks.

JEL Classication: D01 (Microeconomic Behavior: Underlying Principles); D03(Behavioral

Economics: Underlying Principles), D81(Criteria for Decision Making under Risk

and Uncertainty).

�Department of Economics, University of Leicester, University Road, Leicester. LE1 7RH, UK. Phone: +44-116-2522086. Fax: +44-116-2522908. E-mail: Sanjit.Dhami@le.ac.uk.

yDepartment of Economics, University of Leicester, University Road, Leicester. LE1 7RH, UK. Phone: +44-116-2522898. Fax: +44-116-2522908. E-mail: aa10@le.ac.uk.

... people may refuse to worry about losses whose probability is below some threshold. Probabilities below the threshold are treated as zero.Kunreuther et al. (1978, p. 182). Obviously in some sense it is right that he or she be less aware of low probability

events, ... but it does appear from the data that the sensitivity goes down too rapidly as the probability decreases.Kenneth Arrow in Kunreuther et al. (1978, p. viii). An important form of simplication involves the discarding of extremely unlikely out-

comes.Kahneman and Tversky (1979, p. 275). Individuals seem to buy insurance only when the probability of risk is above a thresh-

old... Camerer and Kunreuther et al. (1989, p. 570).

1. Introduction

In this paper we are interested in the best possible decision theory that can address the following two stylized facts on human behavior over the probability range [0; 1].

S1. For probabilities in the interval [0; 1], that are bounded away from the end-points, decision makers overweight small probabilities and underweight large probabilities.1

S2. For events close to the boundary of the probability interval [0; 1], many that are of enormous importance, evidence suggests two kinds of behaviors. S2a: A fraction � 2 [0; 1] of decision makers (i) ignore events of extremely low prob- ability and, (ii) treat extremely high probability events as certain.2

S2b: The remaining fraction 1�� places great salience on the size of the low proba- bility outcome (particularly losses). These individuals behave as ifthe magnitude of the outcomes is of paramount importance.

1.1. Stylized fact S1 in decision theory

Non-expected utility (non-EU) theories postulate a probability weighting function, w (p) : [0; 1] ! [0; 1], that captures the subjective weight placed by decision makers on the ob- jective probability, p. Such theories, e.g., rank dependent utility (RDU) and cumulative prospect theory (CP), account for S1 by incorporating a w (p) function that overweights low probabilities but underweights high probabilities. Such non-EU theories can account for S1 and S2b, but they cannot account for S2a. An example is the Prelec (1998) function: w(p) = e��(� ln p)

�

, � > 0, � > 0, which is parsimonious and has an axiomatic foundation.

1The evidence for stylized fact S1 is well documented and we do not pursue it further; see, for instance, Kahneman and Tversky (1979), Kahneman and Tversky (2000) and Starmer (2000).

2In the context of take-up of insurance for low probability natural hazards, one set of experiments by Kunreuther et al. (1978) are consistent with � = 0:8.

1

If � < 1, then this function overweights low probabilities and underweights high probabil- ities, so it conforms to S1 and S2b but not S2a. All non-EU theories, particularly RDU and CP assume a weighting function of this form. However, for � > 1 the Prelec function is in conict with S1 and S2b but respects S2a. The two cases � < 1 and � > 1 are plotted below for the case � = 1 and respectively � = 0:5 and � = 2.

0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.5

1.0

p

w

The case � < 1 (� = 0:5, � = 1)

0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.5

1.0

p

w

The case � > 1 (� = 2, � = 1)

Remark 1 : To distinguish the two cases � < 1 and � > 1, we call the former (� < 1) the standard Prelec function. This function innitely overweights innitesimal probabilities in the sense that lim

p!0 w(p)=p = 1 and innitely underweights near-one probabilities in the

sense that lim p!1

1�w(p) 1�p = 1. Most weighting functions in use in RDU and CP have this

property. We shall call these the standard probability weighting functions.

1.2. Some basics of non-linear probability weighting

Consider the lottery (x1; p1;x2; p2) where outcome xi occurs with probability pi � 0, x1 < x2, p1 + p2 = 1.3 Let u(x) be the utility of x. The expected utility of this lottery is p1u(x1)+ p2u(x2). However, under non-EU theories, decision makers use decision weights, �i, to evaluate the value of the lottery as

�1u(x1) + �2u(x2). (1.1)

Suppose that the decision weights transform point probabilities, i.e., �i = �i(pi), as in Kahneman and Tverskys prospect theory (PT). This can lead to the choice of stochas- tically dominated lotteries; see Starmer (2000). This problem was rectied by Quiggins (1982, 1993) rank dependent utility, RDU, by using cumulative transformations of proba- bility.

Remark 2 (RDU, monotonicity): Suppose that the decision maker follows RDU. Let w(p) : [0; 1] ! [0; 1] be a (strictly increasing) probability weighting function. For the

3Everything we say here can be extended to general lotteries.

2

lottery (x1; p1;x2; p2), x1 < x2, the decision weights are given by the following cumulative transformations of probabilities,

�1 = w(p1 + p2)� w(p2), �2 = w(p2). (1.2)

Notice from (1.2) that one requires that the weighting function w(p) be dened at all points in the domain [0; 1].

1.3. A discussion of the rationale for Stylized fact S2

The evidence for S2 comes from several sources, which we now briey review.

1.3.1. Bimodal perception of risk

There is strong evidence of a bimodal perception of risks, see Camerer and Kunreuther (1989) and Schade et al. (2001). Some individuals do not pay attention to losses whose probability falls below a certain threshold (stylized fact S2a), while for others, the size of the loss is relatively more salient despite the low probability (stylized fact S2b). McClelland et al. (1993) nd strong evidence of a bimodal perception of risks for insurance for low probability losses. Individuals may have a threshold below which they underweight risk and above which they overweight it. Furthermore, individuals may have di¤erent thresholds. Across any population of individuals, for any given probability, one would then observe a bimodal perception of risks; see Viscusi (1998). Kunreuther et al. (1988) argue that the bimodal response to low probability events is pervasive in eld studies.

1.3.2. Prospect theory (PT)

The only decision theory that incorporates S2 (and S1) is the Nobel prize winning work of Kahneman and Tverskys (1979) prospect theory, PT. PT makes a distinction between an editing and an evaluation/decision phase. Whilst there are other aspects of the editing phase, from our perspective the most important aspect takes place when decision makers decide which improbable events to treat as impossible and which probable events to treat as certain (stylized fact S2a). Kahneman and Tversky (1979) used the particular point transformation of probability, �i = w (pi). They drew � (p), as in Figure 1.1, which is undened at both ends to reect issues of S2a, S2b. Kahneman and Tverskys (1979, p.282-83) summarize the evidence for S2, as follows.

The sharp drops or apparent discontinuities of �(p) [w (p) in our terminology] at the end- points are consistent with the notion that there is a limit to how small a decision weight can be attached to an event, if it is given any weight at all. A similar quantum of doubt could impose an upper limit on any decision weight that is less than unity...the simplication of prospects can lead the individual to discard events of extremely low probability and to treat

3

Figure 1.1: Ignorance at the endpoints. Source Kahneman and Tversky (1979, p. 282).

events of extremely high probability as if they were certain. Because people are limited in their ability to comprehend and evaluate extreme probabilities, highly unlikely events are either ignored or overweighted, and the di¤erence between high probability and certainty is either neglected or exaggerated. Consequently