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Lancaster University Management School Working Paper

2005/034

Cumulative prospect theory and gambling

David Peel, Michael Cain and D Law

The Department of Economics

Lancaster University Management School Lancaster LA1 4YX

UK

© David Peel, Michael Cain and D Law All rights reserved. Short sections of text, not to exceed

two paragraphs, may be quoted without explicit permission, provided that full acknowledgement is given.

The LUMS Working Papers series can be accessed at http://www.lums.lancs.ac.uk/publications/

LUMS home page: http://www.lums.lancs.ac.uk/

http://www.lums.lancs.ac.uk/publications/ http://www.lums.lancs.ac.uk/

Cumulative Prospect Theory and Gambling M. Cain, D. Law (University of Wales Bangor) and D. A. Peel (University of Lancaster) Corresponding author. E-mail address: d.peel@lancaster.ac.uk Telephone: 0152465201, Lancaster University Management School, Lancaster, LA1 4YX, United Kingdom.

Abstract Whilst Cumulative Prospect theory (CPT) provides an explanation of gambling on longshots at actuarially unfair odds, it cannot explain why people might bet on more favoured outcomes. This paper shows that this is explicable if the degree of loss aversion experienced by the agent is reduced for small-stake gambles (as a proportion of wealth), and probability distortions are greater over losses than gains. If the utility or value function is assumed to be bounded, the degree of loss aversion assumed by Kahneman and Tversky leads to absurd predictions, reminiscent of those pointed out by Rabin (2000), of refusal to accept infinite gain bets at low probabilities. Boundedness of the value function in CPT implies that the indifference curve between expected-return and win-probability will typically exhibit both an asymptote (implying rejection of an infinite gain bet) and a minimum at low probabilities, as the shape of the value function dominates the probability weighting function. Also the high probability section of the indifference curve will exhibit a maximum. These implications are consistent with outcomes observed in gambling markets.

Keywords: Cumulative Prospect Theory; Exponential Value Function; Gambling JEL classification: C72; C92; D80; D84

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mailto:d.peel@lancaster.ac.uk

Cumulative Prospect Theory and Gambling There is, however, one common observation which tells against the prevalence of risk aversion, namely, that people gamble ...I will not dwell on this point extensively, emulating rather the preacher, who, expounding a subtle theological point to his congregation, frankly stated: "Brethren, here there is a great difficulty; let us face it firmly and pass on': Kenneth Arrow (1965) 1. Introduction

The non-expected utility model proposed by Kahneman and Tversky (1979) and

Tversky and Kahneman (1992), which they called Cumulative Prospect theory

(CPT), has three key features. The first is that from a given reference point

agents are risk-averse over potential gains but risk-loving over potential losses.

Second, the utility or value function exhibits loss aversion so that the slope

changes abruptly at the reference point. In particular, the function is postulated

to fall roughly twice as fast over losses as it rises over gains, exhibiting

diminishing sensitivity as the marginal impact of losses or gains diminishes with

distance from the reference point [see e.g. Tversky and Kahneman (1992)].

Third, the probabilities of events are subjectively distorted by agents, via an

inverted s-shaped probability weighting function so that small probabilities are

exaggerated, and large probabilities are understated. The CPT model is able to

resolve the Allais paradox [see e.g. Allais and Hagen (1979)] and also explains a

variety of experimental evidence which is inconsistent with standard expected-

utility theory [see e.g. Starmer (2000), Rabin (2000), Rabin and Thaler (2001),

and Thaler (1985)]. i

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Of particular importance is the probability weighting function, which can generate

what Tversky and Kahneman (1992) call the most distinctive implication of CPT,

namely the four-fold pattern of risk attitudes. This may arise because the normal

risk-averse and risk-seeking preferences for gains and losses respectively may

be reversed by the overweighting of small probabilities.

Prelec (2000) notes that for the four-fold pattern to emerge in general,

probability weighting must over-ride the curvature of the value function;

sometimes it works in favour and sometimes against the patternii.

He suggests that the purchase of lottery tickets, for instance, indicates that

probability over-weighting is strong enough to compensate for three

factors which militate against such purchases, namely the concavity of the

value function (which diminishes the value of the prize relative to the

ticket price), loss aversion, and the fact that lottery tickets sell at an

actuarially unfair price.

It is interesting that Prelec refers to outcomes in gambling markets as supporting

CPT. This is also true of Kahneman and Tversky (1979), who note that CPT

predicts insurance and gambling for small probabilities but state that “the present

analysis falls far short of a fully adequate account of these complex phenomena”.

In fact there has been little discussion of whether CPT can provide a coherent

explanation of gambling at actuarially unfair odds. Given that the great majority

of people in developed countries participate in gambling, at least occasionally, iii

and that gambles often involve large stakes,iv many would argue that an ability to

explain outcomes observed in gambling markets [see Sauer (1998) and Vaughan

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Williams (1999) for comprehensive surveys] is at least as important a test of a

theoretical model as consistency with experimental evidence on the risk attitudes

of small samples of students.

Of course, it is still the case that some economists explain gambling by invoking

non-pecuniary returns such as excitement, buying a dream or entertainment [see

e.g. Clotfelter and Cook (1989)].v However, there are convincing a priori and

empirical reasons for giving little weight to this rationalisation in general.

Friedman and Savage (1948) provide one convincing a priori critique of the

entertainment rationalevi. Subsequently a number of surveys of gamblers have

been conducted in which respondents are asked to cite the main reasons why

they gamble. The predominant response, usually by 42%-70%, is for financial

reasons - “to make money” [see e.g. Cornis (1978), and The Wager (2000 b)].vii

Given this background, the purpose in this paper is to consider the implications of

CPT for gambling over mixed prospects. With the standard assumptions,

gambling on longshots at actuarially unfair odds can optimally occur, but betting

on 50/50 and odds-on chances cannot. We show the conditions in which the

curvature of the value function can modify these results. In particular, (a) if

stakes are not too large the assumption of ultimate boundedness of the value

function will imply a minimum in the indifference curve in expected return-win

probability space, (b) the indifference curve will typically exhibit an asymptote at

very small probabilities, indicating that the agent would turn down a bet involving

the possibility of an infinite gain; (c) depending on the degree of risk aversion

assumed over gains, the asymptote can occur at any probability in the 0 -1

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range; (d), in the absence of probability distortion agents will, paradoxically,

ultimately accept very large bets on odds-on chances at actuarially unfair odds .

Finally, we illustrate how modification of the CPT model, such that agents are

less loss averse over small- stake gambles than over large ones, and that

probability distortions over gains are less than over losses, can explain both

gambling on favoured outcomes, and also the favourite-longshot bias observed in

most gambling markets.

The rest of the paper is structured as follows. In Section two we consider the

implications of the CPT model for the shape of the indifference curve between

expected-return and win-probability for mixed prospects. Section three develops

further implications by assuming a particular parametric form of the Kahneman-

Tversky function, and Section four contains a brief conclusion.

2. The Indifference Curve between Expected-return and Win-probability

Defining reference point utility as zero, for a gamble to occur in CPT we require

expected utility or value to be non-negative:viii

-( ) ( ) w (1- ) ( ) 0 r lEU w p U so p U s+= − ≥

p)

(1)

where the win-probability is given by p, and the functions and are

non-linear s-shaped probability weighting functions. U s is the value derived

from a winning gamble, where are the odds and the stake. U is the