Economics of Betting Markets

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2 Economics of Betting Markets During the last few decades, commercial gambling has increased substantially throughout the Western world. More people than ever before have access to sources of legalised gambling, leading to significantly increased revenues over the last decade or so for the institutions involved. Naturally enough, this has led to an increased interest in the area of the economics of betting. This book addresses the issues raised by the continued growth of the gambling sector. How can we model the behaviour of people who seemingly act irrationally? What are the implications of different tax policies with regard to gambling? Are casinos capable of taking money away from state-run lotteries and the causes they fund? Can bookmakers odds be influenced in such a way as to make the gambling market inefficient? The authors in this volume provide insights based on data from many different countries, including England, the USA, Australia, Spain and Cyprus. This volume brings together work which addresses the economic impact of the huge growth of commercial gambling in the Western world, as well as trying to model the cognitive processes which can explain why individuals are prepared to behave in such apparently irrational ways. This book was published as a special issue of Applied Economics. The academic editor of this journal is Mark P. Taylor. David Peel is a Professor in Economics at Lancaster University Management School. He has held previous posts at the University of Liverpool, Aberystwyth and University of Cardiff. Professor Peel has published widely on empirical and theoretical issues in macroeconomics and the economics of gambling markets.

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4 Economics of Betting Markets Edited by David Peel

5 First published 2010 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN Simultaneously published in the USA and Canada by Routledge 270 Madison Avenue, New York, NY Routledge is an imprint of the Taylor & Francis Group, an informa business 2010 Taylor & Francis Typeset in Times by Value Chain, India Printed and bound in Great Britain by TJI Digital, Padstow, Cornwall All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN10: ISBN13:

6 CONTENTS Introduction: economics of betting markets 1 D. A. Peel 1. Bounded cumulative prospect theory: some implications for gambling outcomes 4 Michael Cain, David Law and David A. Peel 2. The Markowitz model of utility supplemented with a small degree of probability distortion as an explanation of outcomes of Allais experiments over large and small payoffs and gambling on unlikely outcomes 15 D. A. Peel, Jie Zhang and D. Law 3. Why people choose negative expected return assets - an empirical examination of a utility theoretic explanation 25 N. Bhattacharya and T. A. Garrett 4. The relative regressivity of seven lottery games 33 Kathryn L. Combs, Jaebeom Kim and John A. Spry 5. Risk attitudes in large stake gambles: evidence from a game show 38 Cary Deck, Jungmin Lee and Javier Reyes 6. Measuring displacement effects across gaming products: a study of Australian gambling markets 50 Lisa Farrell and David Forrest 7. The economics of casino taxation 60 Hasret Benar and Glenn P. Jenkins 8. Do horses like vodka and sponging? - On market manipulation and the favourite-longshot bias 71 Stefan Winter and Martin Kukuk 9. The efficiency of exotic wagers in racetrack betting 84 Marshall Gramm, C. Nicholas McKinney and Douglas H. Owens 10. Predicting bookmaker odds and efficiency for UK football 93 I. Graham and H. Stott 11. Nonlinear modelling of European football scores using support vector machines 104 Nikolaos Vlastakis, George Dotsis and Raphael N. Markellos 12. Sentiment in the betting market on Spanish football 112 David Forrest and Robert Simmons Index 121

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8 Introduction: economics of betting markets D. A. Peel Department of Economics, Management School, Lancaster University, Lancaster, LA1 4YX, UK Economists have long been interested in the economics of betting markets as exemplified by the excellent survey articles by Sauer (1998) and Vaughan Williams (1999). This is perhaps not surprising given that in many countries the majority of persons gamble often with large stakes. 1 For instance 68% of the population of the UK had participated in some form of gambling activity within the past year in UK (48% excluding people who had only gambled on the National Lottery Draw) (see British Gambling Prevalence Survey, 2007). Also, the amount of money spent on gambling activities is increasing in many countries. For example, the amount retained by gambling operators in UK after the payment of winnings, but before the deduction of the costs of the operation was 9.8 billion, a 36% increase in nominal terms since 1999 (see British Gambling Prevalence Survey, 2007). The last few years has seen an increase in academic literature on the economics of gambling with new specialist journals formed. This probably reflects the availability of data sets and the increasing importance of the gambling sector. This issue of Applied Economics is given up to 12 articles that reflect current work in this area. The standard expected utility model is of course inconsistent with gambling at actuarially unfair odds. Some still add a nonpecuniary motive (such as excitement or entertainment), that compensates for the expected negative pecuniary returns to that model to explain gambling. However, whilst it is undoubtedly the case that nonpecuniary returns are relevant for some, who engage in gambling; the hypothesis per se is unattractive, because it is inconsistent with a priori reasoning, 2 with the fact that a majority say they gamble to make money 3 and with other experimental evidence. 4 Cumulative Prospect theory (CPT) developed by Kahneman and Tversky (1979) and Tversky and Kahneman (1992) can explain a variety of experimental results inconsistent with expected utility theory and also optimal gambling on long shots. In the first article Cain, Law and Peel show that an alternative parametric specification of CPT enables the CPT model to explain gambling on all outcomes including odds on favourites. One possible problematic feature of CPT is that betting on long shots at unfair odds is induced by extreme probability distortion. This probability distortion implies that the subjective expected returns sometimes run into hundreds of percent. In order to explain the famous Allais (1953) paradox it is necessary to assume probability distortion. In article two, Peel, Zhang and Law introduce a small degree of probability distortion into the Markowitz (1952) 1 For instance, Bruce and Johnson (1992) report average stakes of on race horse favourites. Strumpf (2003) reports that average bet size averaged in excess of $1000 in a study of six illegal bookmakers. 2 Friedman and Savage (1948) and Markowitz (1952) in their seminal articles provide a critique of the entertainment rationale. For instance, Markowitz noted that if the utility of a gamble is the expected utility of the outcomes plus the utility of playing the game then, for given fair odds, the smaller the amount bet, the higher the expected utility. This implies millionaires should play poker for pennies and no one should purchase more than one lottery ticket. 3 When surveys ask gamblers why they gamble, a majority (approximately 42 70%) cite financial reasons: to make money (see, for example, The Wager, 2000). 4 Many experimental results conflict with the standard expected utility model. An excellent discussion of some of this experimental evidence can be found in Starmer (2000). 1

9 2 Introduction model of expected utility, of which the value function in CPT is a special case. They show that this model can explain the Allais experiments as well as gambling outcomes without assuming subjective rates of return run into hundreds of percent. In article three, Bhattacharyya and Garrett develop a model based on an extension of that of Friedman and Savage (1948) model to explain state lottery games. The utility function proposed by Bhattacharyya (2003) is concave for wealth below the current wealth of the agent and it is convex above the current wealth of the agent. Their model implies that lottery players trade-off expected return for skewness of return. Carefully carried out empirical analysis of two interesting data sets appears to give support to their model. However, the precise structure of a lottery ticket implies a relationship between expected return and the higher moments. 5 Such a relationship can also be derived from the utility function. Both should perhaps be explicitly included in future analysis. Also, their model is similar to a restricted version of the Markowitz model and appears to imply unbounded stakes and preference for one-prize lottery tickets. It would surely be interesting to build on their analysis employing alternative models of utility, which do not have these implications. In paper four, Combs, Kim and Spry investigate the relative regressivity of one lottery product versus another. They test hypotheses of regressivity for individual lottery products and of differences in regressivity between products offered by a single lottery agency. Their results based on a careful statistical analysis demonstrate significant differences in the implicit tax incidence between same games. In article five, Deck, Lee and Reyes estimate the degree of risk aversion of contestants appearing on Vas o No Vas, the Mexican version of Deal or No Deal. They find substantial evidence of risk aversion. However, their estimates are lower than previous estimates based on game shows. They also find a considerable variation in risk attitudes, with a few people being extremely risk averse while others are risk loving. It would perhaps be interesting to rework the analysis employing a nonexpected utility approach. In article six, Farrell and Forrest consider the interesting issue of the extent of displacement effects across gaming products. Employing Australian data they estimated a state level (fixed effects) panel data model, exploiting the intra-state differences in the portfolio of gaming products available, to estimate the extent of displacement effects across the gaming sector. In particular they examine whether sales of lotto and lotto-style products in Australia were displaced by the introduction and growth of large casinos and by the spread of casino-style machine gaming within neighbourhood hotels (pubs) and clubs. Their evidence suggests a potential for local casinos to divert significant sums from lotto and the causes it funds. Their results are of relevance to the current policy debate in other countries such as UK. In article seven, Benar and Jenkins develop a model that focuses on the implications for economic welfare of different taxation schemes for casinos, though it is assumed that casinos cater exclusively to foreign tourists. The model is applied to the situation in North Cyprus. They find that a tax on the turnover of funds gambled is an equally efficient to one that taxes the annual fixed costs of the casinos. However, because of the relative collection costs it might be welfare improving to maintain a low turnover tax; and use a tax on the annual fixed costs of the casinos to tax away the rest of the economic rents. Extension of their model relaxing the foreigner assumption would be of interest. In article eight, Winter and Kukuk develop an innovative model that can explain the favourite longshot bias. They show that the favourite-longshot bias may be the rational answer of an honest audience to a simple, but highly lucrative cheating opportunity of insiders. The cheating takes the form of knowing that a horse will not win. Employing a large scale German data set they demonstrate that the pattern of the favourite-longshot bias changes as the opportunity of cheating vanishes. In countries such as the UK insiders with such information have the option of laying the horse 5 See Brockett and Garven (1998) and Cain and Peel (2004). For instance, for a 1 unit bet with odds o win probability p the relationship between the moments is given by 2 2 þ s ð 2 Þ 2 ¼ 0 where expected return ¼ ¼ pð1 þ oþ, variance ¼ 2 ¼ 2 ð1 pþ, p skewness ¼ s ¼ 3 ð1 pþð1 2pÞ p 2 skewness, s, is negatively related to expected return but there is no behaviour implied.

10 Introduction directly. It would be interesting to incorporate such elements into the Winter and Kukuk model. Numerous articles have addressed the issue of whether gambling markets are efficient. The final four articles provide further evidence on various aspects of this hypothesis. In article nine, Gramm, Mckinney and Owens employing a much larger data set than has been employed in many previous studies examine efficiency in multihorse exotic wagers using data from US racetracks. They find a favourite-longshot bias in exacta wagers but results are unclear for trifecta wagers. In article ten, Graham and Stott develop a resultsbased probit model and an odds forecasting model to compare with the odds of UK bookmaker. They explicitly allowed for the impact of home advantage. They found that the bookmakers offer better odds on favourites but this favourite-longshot bias cannot be exploited by their statistical model. In article eleven, Vlastakis, Dotsis and Markellos evaluate the performance of a Poisson count regression and that of a Support Vector Machine (SVM) (belonging to the family of neural networks) in forecasting using IX2 and Asian Handicap odds data from the English Premier league. The modelling results show that while the SVM is only marginally superior on the basis of statistical criteria, it manages to produce out-of-sample forecasts with positive out-of-sample profits and thus, suggests inefficiency. However, the sample size employed by the authors is relatively small and there is a need for replication of this methodology on larger samples and other data sets. In article twelve, Forrest and Simmons examine whether sentiments play a part in the fixed odds setting process. They report new results for both Spanish and Scottish football that bookmaker prices appear to be influenced by the relative number of fans of each club in a match and that returns can be increased by employing this information. They point out that this mispricing could be a commercial decision. Again, there is a need for replication on other data sets. Also, the measure of sentiment might be sharpened. At the moment it is proxied by the difference in average home attendance in the previous season. In principle, two well supported or two badly supported teams could have the same difference. It is not clear why the bookmaker should treat these types of matches in the same manner. References Allais, M. (1953) Le comportement de l homme rationnel devant le risque: critique des postulats et axiomes de l ecole Ame ricaine, Econometrica, 21, Bhattacharyya, N. (2003) From mean variance space to mean skewness space implications for simultaneous risk seeking and risk averting behavior, ssrn.com/author¼ Brockett, P. L. and Garven, J. R. (1998) A re-examination of the relationship between preference and moment orderings by rational risk averse investors, Geneva Papers on Risk and Insurance Theory, 23, British Gambling Prevalence Survey (2007) National Centre for Social Research, Northampton Square: London. Bruce, A. C. and Johnson, J. E. V. (1992) Toward an explanation of betting as a leisure pursuit, Leisure Studies, 11, Cain, M. and Peel, D. (2004) Utility and the skewness of return in gambling, The Geneva Papers on Risk and Insurance Theory, 29, Friedman, M. and Savage, L. (1948) The utility analysis of choices involving risk, Journal of Political Economy, 56, Kahneman, D. and Tversky, A. (1979) Prospect theory: an analysis of decision under risk, Econometrica, 2, Markowitz, H. M. (1952) The utility of wealth, Journal of Political Economy, 56, Strumpf, K. S. (2003) Illegal sports bookmakers, mimeo, University of North Carolina at Chapel Hill. Sauer, R. D. (1998) The economics of wagering markets, Journal of Economic Literature, 36, Starmer, C. (2000) Developments in non-expected utility theory: the hunt for a descriptive theory of choice under risk, Journal of Economic Literature, 38, Tversky, A. and Kahneman, D. (1992) Advances in prospect theory: cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5, The Wager (2000) Stress, anxiety and why gamblers gamble, Vol. 5, p. 27, Massachusetts Council on Compulsive Gambling. Vaughan Williams, L. (1999) Information efficiency in betting markets: a survey, Bulletin of Economic Research, 5l,

11 Bounded cumulative prospect theory: some implications for gambling outcomes Michael Cain a, David Law a and David A. Peel b a School for Business and Regional Development, University of Wales, Bangor, Gwynedd LL57 2DJ, UK b Department of Economics, Management School, University of Lancaster, Lancaster LA1 4YX, UK Standard parametric specifications of Cumulative Prospect theory (CPT) can explain why agents bet on longshots at actuarially unfair odds. However, the standard specification of CPT cannot explain why people might bet on more favoured outcomes, where by construction the greatest volume of money is bet. This article outlines a parametric specification than can consistently explain gambling over all outcomes. In particular we assume that the value function is bounded from above and below and that the degree of loss aversion experienced by the agent is smaller for smallstake gambles (as a proportion of wealth) than usually assumed in CPT. There are a number of new implications of this specification. Boundedness of the value function in CPT implies that the indifference curve between expected-return and win-probability for a given stake will typically exhibit both an asymptote (implying rejection of an infinite gain bet) and a minimum, 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. I. Introduction Original Prospect theory (OPT), proposed by Kahneman and Tversky (1979), and Cumulative Prospect theory (CPT) as proposed by Starmer and Sugden (1989) and Tversky and Kahneman (1992) is able to resolve the Allais paradox (Allais and Hagen, 1979) and also explains a variety of experimental evidence which is inconsistent with standard expected-utility theory (Rabin, 2000; Starmer, 2000; Rabin and Thaler, 2001 and Thaler, 1985). 1 1 For instance the apparent preference of some agents for segregated gains reported by Thaler (1985, p. 203), whose survey evidence indicated that most people believe that a person would be happier to win $50 plus $25 in separate lotteries rather than $75 in a single lottery. An excellent discussion of this experimental evidence can be found in Starmer (2000). Rabin (2000) provides further indirect support for CPT, in demonstrating that the assumption of global risk-aversion has implications for agents preferences with respect to small and large gambles that appear untenable a priori. In particular, he shows that if an agent turns down a gamble to win $11 or lose $10, each with probability 0.5, at all prevailing wealth levels, then she will also turn down a bet to win infinity or lose $100, each with probability 0.5. In addition, Rabin notes that the assumption of global risk-aversion implies that agents who turn down a gamble to lose $100 or win $200 with win-probability 0.5, would turn down a sequence of N such bets, say, N ¼ 100, as shown by Samuelson (1963). Again, this appears absurd a priori. As a consequence of these implications, Rabin suggests that economists should reject standard expected-utility theory in favour of some version of the nonexpected utility model, such as that proposed by Kahneman and Tversky. 4

12 M. Cain et al. It is also claimed that CPT can explain outcomes in gambling markets for instance. Kahneman and Tversky (1979), 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. This appears to be the case, with the standard parametric specifications, gambling on longshots at actuarially unfair odds can optimally occur, but betting on 50/50 and odds-on chances cannot occur. Given that most of the money bet is on favourites many would argue CPT as currently constructed has not provided an explanation of outcomes observed in gambling markets (see Sauer (1998) and Vaughan Williams (1999) for comprehensive surveys). Given that the great majority of people in developed countries participate in gambling, at least occasionally; 2 and that gambles often involve large stakes, 3 a model specification that can explain the outcomes in actual gambling markets seems perhaps more important than the one that can explain the risk attitudes of small samples of students 4 who, as we have shown below, would not gamble at actuarially unfair odds according to some typical parameter estimates report. Given this background, the purpose of this article is to consider the implications of a different parametric specification of CPT for gambling over mixed prospects. This is because, the standard parametric specification of CPT based on power value functions is not appropriate for explaining gambling outcomes. In particular power utility violates the assumption of loss aversion for low enough stakes, (also pointed out by Ko bberling and Wakker, 2005; Schmidt and Zank, 2005 and Law and Peel, 2005). Consequently, any optimal model of gambling based on power value functions (Bradley, 2003) will imply violation of the assumption of loss aversion. There are also a number of other reasons outlined below to suggest that the power function is not consistent with other empirical evidence. As a consequence we specify a bounded value function. (Also suggested by Ko bberling and Wakker (2005) and Giorgi and Thorsten Hens (2006) in different contexts.) This specification has a number of interesting and new implications in the gambling context. 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, for a given size of stake; (b) the indifference curve will exhibit an asymptote, typically at very small probabilities, indicating that the agent would turn down a bet involving the possibility of an infinite gain; (c) though depending on the degree of risk aversion assumed over gains, the asymptote can occur at any probability in the range of 0 1; (d) in the absence of probability distortion, agents will paradoxically and ultimately accept very large bets on odds-on chances at actuarially unfair odds; (e) if the agent is sufficiently risk-averse over gains and risk-seeking over losses they can bet on odds-on favourites although everywhere loss averse and (f) our parameterizations can simultaneously account for gambling on unlikely gains and the Allais paradox behaviour. This is not the case with the standard parametric specification (Neilson and Stowe, 2002). Our parametric specification of the assumption of boundedness of the utility function allows us to choose the degree of loss aversion exhibited by agents over small and large stake symmetric gambles. Varying the different parameters, namely the degree of risk-aversion over gains and risk-seeking over losses, the degree of loss aversion or the probability weighting function over gains and losses, allows us to explain both gambling on favoured outcomes and longshots and hence contribute towards an explanation of the stylized empirical outcome of the favourite-longshot bias observed in most gambling markets (Sauer, 1998; Vaughan Williams,1999; Cain et al., 2003). 5 2 The proportion of people reported, as gambling is high and varies little between countries. For instance, in 1998, 68% of respondents in the United States reported gambling at least once in the previous year. Legal gambling losses in America totalled over $50 billion, and illegal gambling has been estimated at over $100 billion greater than the estimated expenditure on illegal drugs (see e.g. Strumpf, 2003; Pathological Gambling, 1999 and The Wager, 2000a). 3 Strumpf (2003), in his study of six illegal bookmakers in New York City over the period (two of which had turnover in excess $100 million per annum), reports that average bet size was relatively large for these firms, averaging in excess of $1000. We also note that observation of high rollers on odd/even bets at roulette is folklore. 4 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 (Clotfelter and Cook, 1989). However, there are convincing a priori and empirical reasons for giving little weight to this rationalization in general. Friedman and Savage (1948) provide one convincing a priori critique of the entertainment rationale. 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, 2000b).

13 6 Bounded cumulative prospect theory The rest of the article is structured as follows. In Section II 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 III develops further implications by assuming a particular parametric form of the Kahneman Tversky function, and Section IV contains a brief conclusion. II. 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 nonnegative 5 : EU ¼ w þ ðpþu r ðsoþ w ð1 pþu l ðsþ 0 where the win-probability is given by p, and the functions w þ (p) and w (1 p) are nonlinear s-shaped probability weighting functions. U r (so) is the value derived from a winning gamble, where o are the odds and s the stake. U l (s) is the disutility derived from a losing gamble. From Equation 1 the optimal stake is such ¼ 0 so that w þ ðpþ p sð pþ ð pþu r0 ¼ w ð1 pþu l0 ðsþ p ð1þ ð2þ where the expected return from a unit gamble,, is defined as ¼ pð1 þ oþ ð3þ A bet is defined to be actuarially fair when ¼ 1. From Equation 2 we have that s ¼ s(, p) if EU 0. Substituting s ¼ s(, p) into Equation 1 gives expected value, EU as a function of and p, and hence an indifference map in (, p) space may be obtained by differentiating Equation 1 with respect to p and equating to zero, in order to find the combinations of expected return,, and probability, p, between which the bettor is indifferent. This produces deu dp ðpþ U r ð1 pþ U wþ ðpþs p 2 U r0 ðsoþþ s wþ ðpþ d U r0 ðsoþ p dp þ wþ ðpþ sð pþ ð pþu r0 p p ds w ð1 pþu l0 ðsþ dp ¼ 0 and hence, in view of Equation 2, the Equation 4 reduces to d dp ¼ 1 þ o "gp o " u þ "lp w ð1 pþu l ðsþo " u w þ ðpþu r ð5þ ðsoþ where " u ðsoþ ¼ sour0 U r, " gp ðpþ p w þ ðpþ, " lp ð1 pþ w ð1 pþ where " u is the elasticity of U(), " gp is the elasticity of the probability weighting function over gains (strictly positive), and " lp is its elasticity over losses (strictly negative). Equation (5) also holds for any arbitrary fixed level of stake. We can simplify (5), for the purposes of exposition by noting from (1) that in order to gamble we require 1 w 1 (1 p)u 1 (s)/w þ (p)u r (so). So that for the expected-probability frontier, (, p), where the agent is just indifferent between gambling and not gambling, Equation 5 simplifies to d dp ¼ 1 þ o o "gp " lp " u ð4þ ð6þ The expected return-probability frontier has to exhibit a positive slope in order to be able to explain gambling at unfair odds or to explain the favourite-longshot bias, since d/dp is observed to be positive over much of its range in real markets. We recall that in the early literature the explanation 5 If we define the current level of wealth as W, and the level of utility associated with W as U then the exponential utility function U ¼ U þ UðW þ xþ defines utility for increases in wealth above W, where w þ x is wealth measured from W!1. We require that the marginal utility and the second derivative for an increase in 4 2 U=@x For a decrease in wealth below W, we define the utility function as U ¼ U UðW xþ where W x is wealth measured from 0 to W. We require that the marginal utility and the second derivative for a decrease in wealth are both positive, as postulated by Kahneman and Tversky. ðaþ ðbþ