Michael Cain

David Law
email: d.law@bangor.ac.uk

David A Peel

Bounded Cumulative Prospect Theory: Some Implications for Gambling Outcomes

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

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 favored outcomes, where by construction the greatest volume of money is bet. This paper 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 small-stake 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.

Introduction

Original Prospect theory (OPT), proposed by [START_REF] Kahneman | Prospect Theory: An Analysis of Decision under Risk[END_REF], and Cumulative Prospect theory (CPT) as proposed by Starmer and Sugden (1989) and Tversky and Kahneman (1992) is able to resolve the Allais paradox [see e.g. [START_REF] Allais | Expected Utility Hypotheses and the Allais Paradox[END_REF]] and also explains a variety of experimental evidence which is inconsistent with standard expected-utility theory [see e.g. Starmer (2000), [START_REF] Rabin | Risk Aversion and Expected-Utility Theory: A Calibration Theorem[END_REF], [START_REF] Rabin | Risk Aversion and Expected-Utility Theory: A Calibration Theorem[END_REF][START_REF] Rabin | Risk Aversion[END_REF]Thaler (1985)]. ii It is also claimed that CPT can explain outcomes in gambling markets For instance. [START_REF] Kahneman | Prospect Theory: An Analysis of Decision under Risk[END_REF], 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. 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 [START_REF] Prelec | The Probability Weighting Function[END_REF]Vaughan Williams (1999) for comprehensive surveys]. 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 a model specification that can explain the outcomes in actual gambling markets seems perhaps more important than one that can explain the risk attitudes of small samples of students v who, as we show below. Given this background, the purpose in this paper 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 [START_REF] Köbberling | An Index of Loss Aversion[END_REF], Schmidt and Zank (2005) and Law and Peel (2005)). Consequently any optimal model of gambling based on power value functions (e.g. [START_REF] Bradley | The representative bettor, bet size and prospect theory[END_REF]) 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 also not consistent with other empirical evidence. As a consequence we specify a bounded value function. (Also suggested by [START_REF] Köbberling | An Index of Loss Aversion[END_REF] and Giorgi and Thorsten Hens (2005) 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 bounded ness of the value function will imply a minimum in the indifference curve in expected return-win probability space, for a given sise 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 0 -1 (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.

(f) Our parameterizations can simultaneously account for gambling on unlikely gains and the Allais paradox behavior. This is not the case with the standard parametric specification.(see Neison 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. (see e.g. see Sauer (1998) andVaughan Williams (1999), [START_REF] Cain | Bias and Insider Trading in a Variety of Betting Markets[END_REF])

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.

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: vi

- ( ) ( ) w (1-) ( ) 0 r l EU w p U so p U s + = (1)
where the win-probability is given by p, and the functions w p + ( )and w p ( ) 1 are non-linear s-shaped probability weighting functions. U so r ( ) is the value derived from a winning gamble, where o are the odds and s the stake. U s l ( ) is the disutility derived from a losing gamble.

From (1) the optimal stake is such that

= EU s 0 so that ' ' ( ) ( ) ( ) ( ) (1 ) ( ) r l w p s p p U w p U s p p µ µ + = (2)
where the expected return from a unit gamble, µ , is defined as

µ = + p o ( ) 1 (3)
A bet is defined to be actuarially fair when µ = 1 .

From (2) we have that s s p = ( , )

µ if 0 EU . Substituting s s p = ( , ) µ into (1)
gives expected value, EU , as a function of µ and p , and hence an indifference map in ( , )

µ p space may be obtained by differentiating (1) with respect to p and equating to zero, in order to find the combinations of expected return, µ , and probability, p , between which the bettor is indifferent.

= + + = + + + + ( ) ( ) ( ) ( ) ( ) ( ) { ( ) ( )} [ ( ) ( ) ( ( ) ) ( ) ( )] 1 1 0 2 µ µ µ µ (4)
and hence, in view of ( 2), (4) reduces to

(1 ) ( ) {1 } ( ) ( )
= = = + + ' ( ) ( ) , ( ) ( ) , ( ) ( ) 1 1
where u is the elasticity ofU (.) , 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 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 µ is observed to be positive over much of its range in real markets.

We recall that in the early literature the explanation of the favourite-longshot bias assumed that agent's utility functions were convex. and so exhibited risk-loving behaviour. (see e.g. Ali (1987) ). In the CPT model agents will act as if riskloving, so that d dp µ >0, when the probability distortion is strong enough to dominate factors which militate against gambling, namely the concavity of the value function, loss aversion, and the fact that gambles are typically actuarially unfair at low probabilities.

From inspection of (5) (and ( 7)) we observe the precise form that this has to take.

For the agent to appear as if risk-loving it is necessary that the sum of the elasticity's of the probability weighting functions are less than the elasticity of the value function over gains. In many of the experimental studies, based on student responses, the estimates of parameters of the elasticity's of power utility and weighting function violate this condition for gambling. (see e.g. [START_REF] Camerer | Violations of the betweeness axiom and nonlineraity in probability[END_REF] u =0.225, gp =0.56 (for small probabilities), Wu and Gonzalez (1996) u =0.5, gp =0.71, (for small probabilities).

The size of stake will also influence the slope of the indifference curve by, in general, changing the elasticity of the value function over gains, so that a lower (higher) stake size increases (decreases) the elasticity. It will also change the propensity to gamble, (from 5) as changing stake size changes the ratio of the utility loss to the utility gain from the gamble at given odds.

One interesting feature of the CPT model, as shown below in a parametric example, is that the slope of the indifference curve, d dp µ , can be negative at actuarially unfair odds, µ < 1. This is not possible in the standard expected-utility model, where d dp µ <1 and µ > 1 everywhere.

A Parametric Example of the Kahneman -Tversky Model

K and T assumed a power value function in their empirical work. However this functional form has a variety of limitations that make it unsuitable for analysis of gambling in general. The limitations of the power function are both empirical and theoretical and are as follows.

1. K and T (1992) assume that 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

In the case of a power value function, the assumption of loss aversion is violated for small-stake gambling, and the agent becomes infinitely gain loving as the stake approaches zero. Let the value function be

F o r P e e r R e v i e w 9 0 (0< 1 ( ) (7) ( ) 0 ( 0, 0< <1) x x v x x x < = < >
Now > is a necessary assumption in order to ensure that stake size is determinate Loss aversion, defined for a symmetrical gamble as the ratio of the gain to the loss is given for the power utility as

We observe from ( 8) that as stake size approaches zero the ratio becomes infinitely large so that the agent violates the assumption of loss aversion ( See also e.g. Köbberling and Wakker (2004) and Law and Peel (2007))).

2. Some important experimental evidence is inconsistent with a power value function. Markowitz (1952), Biswanger (1980), [START_REF] Hershey | Prospect theory's reflection hypothesis: a critical examination[END_REF] and [START_REF] Holt | Risk Aversion and Incentive Effects[END_REF] report evidence that increases in payoff levels increase risk aversion. In these experiments the probabilities, in a sequence of gambles, are kept fixed as agents choose between a gamble and its certainty equivalent (kept fixed) or between a "safer" and "more risky" gamble. Choices change significantly with size of payoff. With a power value function this would not occur. This result is particularly important in the case of Holt and Laury who use real payoffs as well as hypothetical payoffs. They find that risk aversion increases sharply as payoffs are scaled up.

3. [START_REF] Blavatsky | Back to the St. Petersburg Paradox?[END_REF] shows that the Kahneman-Tversky parameterization cannot resolve the St. Petersburg paradox unless the power coefficient of the utility function is less than that of the probability weighting function. However, this implies that the agent cannot exhibit risk-loving behaviour, as shown above.

x x ( ) (8)
3. [START_REF] Neilson | A Further Examination of Cumulative Prospect Theory Parameterizations[END_REF] show that parameters required of the power specification and weighting function cannot simultaneously resolve the Allais paradox and account for gambling at actuarially unfair odds.

. Given these objections In order to generate further predictions from the analytical framework set out above, we specify a parametric form for the Kahneman -Tversky model. Rather than the power form used by Kahneman and Tversky, this paper employs the expo-power function ( see [START_REF] Saha | Expo-Power Utility: A Flexible Form for Absolute and Relative Risk Aversion[END_REF]) with the advantage that it nests both the exponential and the power function..

where r, and are positive constants.

For n 1 the agent is everywhere risk-averse over gains and risk-seeking over losses as postulated by KT. Also as 0 we obtain the power specification of K and T.

The value function in (9) has upper and lower bounds as is commonly assumed, e.g. Markowitz (1952) and [START_REF] Machina | Expected Utility" Analysis without the Independence Axiom[END_REF]. This is a sufficient condition for the resolution of the St. Petersburg Paradox [see, e.g. [START_REF] Menger | The Role of Uncertainty in Economics[END_REF] and [START_REF] Bassett | Myopic Loss Aversion and the Equity Premium Puzzle[END_REF]].

EU w p e w p e r s o s (  )

n n n = + ( )( ) ( ) ( ) 1 1 1 (9)
1 1 (10)
Unlike the power value function the degree of loss aversion varies as stake size changes.

As stake size approaches zero, the assumption of loss aversion requires that r < 1, and as it becomes large that 1 1 < .

In order to ensure that LA s 0 , so that the degree of loss aversion does not decrease with an increase in stake size, which would be counter intuitive, we require in general

that d U s U s ds r l ( ) ( ) < 0 , which for LA<1, implies that < U s s U s s r l
( ) ( ) for all s, which is consistent with the definition of loss aversion of Benartzi and Thaler (1995) and [START_REF] Köbberling | An Index of Loss Aversion[END_REF]). For the exponential function this implies the additional requirement that r 1.

Differentiation of (9) with respect to stake size gives us the optimal stake size, s, as

s w p ro w p ro n n n = L N M M M M O Q P P P P + ln ( ) ( ) ( ) 1 1 1 (11)
with the second order condition From the numerator of (11) a necessary condition for optimal stake size to be positive is that

ro n > 1 0 (12)
w p ro w p n + > ( ) ( ) 1 1 ( 13
)
This condition is precisely that required to undertake gambles with the power value formulation of KT, but in that model optimal stake size is indeterminate. Peel and Law (2007) pointed out that ( 13) can hold for odds on favourites when the gamble is actuarially unfair if n is small enough. As n becomes smaller the agent becomes more risk-averse over gains and more risk-loving over losses.

We also note that there are important implications of varying stake size. As stake size approaches zero, employing L'Hopitals rule, the agent will gamble if

o w p w p r n + ( ) ( ) 1 (14)
This condition will be met for standard specifications of the weighting function as odds become large. Consequently the model implies agents will gamble at small enough stakes without violating the assumption of loss aversion.

In Figures 1(a) and 1(b) we plot the probability weighting function employed and its elasticity, ( gp ) over gains for the parameter values of Tversky and Kahneman (1992)], to illustrate their numeric values over the probability range, (employing alternative functional forms such as that of [START_REF] Prelec | The Probability Weighting Function[END_REF]or Wu and Gonzalez (1996) made no qualitative difference to our analysis). The ratio of the weighting functions, w p w p

+ ( ) ( ) 1
, is a key determinant of whether to gamble from (1) and their impact, relative to the zero probability distortion-free case, is given by ( ) . ( 1 We observe that the elasticity of the probability weighting function becomes infinitely large as p approaches 1, and is less than one when p=0.

We observe from Figure 2(a) that the probability weighting function enhances the attraction of longshot gambles per se, but diminishes the attraction of more favoured outcomes, with the cross-over occurring at probabilities of around 0.45.

When probability distortion over gains exceeds that over losses, the cross-over can occur at much higher probabilities, illustrated in Figure 2

µ = + + + { ( ) ( ) ( ) ( ) } ( ) 1 1 1 1 1 1 (15)
Note from (15) that as o , then d dp µ will ultimately become negative (due to the third and fourth terms increasing at a greater rate than the second-recall e lp is negative). In this case the bounded ness of the value function ultimately dominates the probability weighting function, so that there is a range of behaviour not obtainable in previous analysis (see e.g. Prelec's (2000, p.90)).

Also, note that increasing stake size will ultimately lead to d dp µ < 0 , (due to the third and fourth term) and this may occur over the whole of the probability range so that the agent will again appear to behave as a expected utility maximizer, so long as µ >1 which will be the case for large enough stakes.

As stake size becomes small the elasticity of the value function approaches n so that d dp µ > 0 from (15) assuming any feasible elasticities of the weighting function.

There are two further implications of the property of the expo-power function that it is bounded from above and below.

Let the upper bound limit be unity (so U so r n ( ) 1 as so n ). The agent would then turn down an infinite gain gamble if

w p w p U s l n + < ( ) ( ) ( ) 1 (16)
Consequently, there is a win-probability threshold beyond which infinite gain bets will be turned down. The precise threshold will depend on particular parameter values, as illustrated below, but can occur even with small stakes. This is one implication of boundedness. A second one is that if bet size becomes very large, so that U so r n ( ) ) 1 and U(s n so that the agent will gamble if

w p w p + > + ( ) ( ) 1 1 (17)
As a consequence, the agent could gamble at actuarially unfair odds at some large enough stake. Clearly such gambles are less likely to be observed if the degree of loss aversion for large stake gambles is itself large. µ can be negative, when expected returns are less than unity, a feature that cannot occur in an expected-utility model, and that the indifference curve has a maximum in the favourite end of the spectrum at better than actuarially fair odds.

From Figures 3(c) and 3(d) we observe that the indifference curve exhibits both a minimum, and an asymptote, so that the agent turns down a gamble with infinite expected return, at an extremely small probability. In this case, the boundedness of the value function ultimately "overpowers" the probability weighting function, contrary to previous models in the literature.

In Figure 4 we plot the indifference curve between expected return and winprobability for a large s . The key features are that the asymptote now occurs at a higher win-probability, and the indifference curve is negatively sloped , and the probability weighting functions of Kahneman and Tversky (1992), with parameters of 0.61 for gains and 0.69 for losses, we calculate that in order to bet $10, with win-probability of 0.5, the agent would need to win at least $18.13 (plausibly less than that of the students in the Kahneman-Tversky experiments). In addition, this agent would accept a bet to win infinity or lose $100 at win probability of 0.5 ( EU = 0 093 .

), unlike the expected-utility maximizer, who would (absurdly) reject this gamble, as demonstrated by [START_REF] Rabin | Risk Aversion and Expected-Utility Theory: A Calibration Theorem[END_REF]. Indeed, our agent would accept this $100 gamble if the potential gain were more than $336.1. However, this Kahneman-Tversky agent would reject a bet to win infinity or lose $10 at a win-probability of 0.02 or less. Consequently seemingly reasonably well calibrated CPT models can also exhibit seemingly absurd behaviour similar to that of the expected-utility maximiser.

Kahneman-Tversky agents with less loss aversion

The degree of loss aversion assumed by Kahneman He questions who would actually turn down a bet to win $11 or lose $10 at a winprobability of 0.5 (as the Kahneman-Tversky students do), noting that such gambles have riskQreturn characteristics superior to those of the daily returns on common stocks, which individuals generally find acceptable.

With this point in mind, we relax the degree of loss aversion over small stakes.

x

In addition we allow the probability distortion over losses to be slightly greater than over gains, as suggested by the empirical work of [START_REF] Jullien | Estimating Preferences under Risk: The Case of Racetrack Bettors[END_REF].

In Figures 5(a) we plot over the win-probability range typically observed in horseracing (0.01 -0.7), the indifference curve between expected return and win probability for the agent making a 10 unit stake. It has the shape of the typical favourite-longshot bias reported for the US and UK horse -racing markets. In addition, the probability weighting function induces a maximum in the indifference curve for extreme favourites, requiring rates of return greater than unity.

The parameter values chosen for this agent are such that she will exhibit the typical pattern of preferences outlined by [START_REF] Allais | Le Comportement de l'Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l'Ecole Américaine[END_REF] for gambles involving millions as well as those found by Kahneman and Tversky who employed moderate (thousands) rather than large scale gains xi . Whilst exhibiting the "Allais

Paradox" the agent will nevertheless will gamble at even odds at roulette and horse racing or bet on the NFL. In Figure 5(b) we illustrate the expected return-win probability indifference curve that might depict a typical student. Whilst she too will exhibit the preferences found by Allais and Kahneman and Tversky in experiments she will not gamble on favourites. She exhibits the degree of loss aversion suggested by K and T over small stakes and requires exactly to win 30 to gamble 10 on a 50-50 chance. Nevertheless she will buy lottery tickets.

Finally in Figure 6 we illustrate the agent betting on an odds on favourite when the agent becomes very risk-averse over gains and very risk-seeking over losses although everywhere loss averse. We illustrate this point assuming no probability distortion. The parameters are r n = = = 25 50 0 25 , , . . . We suppose for illustrative purposes that µ =0.94737, the expected return to a one-unit gamble at roulette.

In figure 6 we plot the relationship between expected utility and the objective win probability when stake size is optimal and given by (11).

.

Conclusion

Whilst the standard parametric specification of CPT by KT can provide an explanation of gambling on longshots (low probability bets) at actuarially unfair odds, gambling on more favoured outcomes is inexplicable. Given that in actual economies a sizeable proportion of agents gamble and most of the money bet is on favourites this is a major limitation of the model. Whilst K and T assumed power value functions in their experimental work this is unsuitable for the simultaneous analysis of gambling outcomes and the Allais paradox as noted by [START_REF] Neilson | A Further Examination of Cumulative Prospect Theory Parameterizations[END_REF], who stated that the most obvious conclusion of our work is that alternative functional forms are needed. In this paper we assume an exp-power value function and show that gambling over all outcomes at actuarially unfair odds can be explained. In addition we can simultaneously explain the Allais paradox with the chosen parameter values, as shown in our calibrated examples in the last section.

We show that boundedness of the value function in CPT theory implies that the indifference curve between expected-return and win-probability will exhibit a minimum at low win-probabilities, because the shape of the value function dominates the probability weighting function. Also, a maximum will occur at high win-probabilities.

The analysis in the paper shows how KT agents with different degrees of loss aversion, probability distortion or risk-aversion or risk-loving over gains and losses can optimally bet at actuarially unfair odds on probabilities spanning the whole probability range. Naturally an aggregate model of gambling outcomes based on KT agents would have to embody heterogeneous agents rather than a representative frame work as in [START_REF] Ali | Probability and Utility Estimates for Racetrack Bettors[END_REF]. However such a model is consistent with a major feature of experimental work. For example [START_REF] Harrison | Naturally Occurring Preferences and Exogenous Laboratory Experiments: A Case Study of Risk Aversion[END_REF] who suggest that one should not readily assume homogenous risk preferences for the population. In fact aggregate models based on heterogeneous agents would seem necessary to simultaneously explain the favourite long shot bias or reverse favourite long shot bias observed in different counties or between different gambling activities. Peel and Law (2007a) is one such attempt. ii 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). [START_REF] Rabin | Risk Aversion and Expected-Utility Theory: A Calibration Theorem[END_REF] 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=100, as shown by [START_REF] Samuelson | Risk and Uncertainty: A Fallacy of Large Numbers[END_REF]. Again, this appears absurd a priori. As a consequence of these implications, Rabin suggests that economists should reject standard expected-utility theory in favor of some version of the non-expected utility model, such as that proposed by Kahneman and Tversky. iv Strumpf (2003), in his study of six illegal bookmakers in New York City over the period 1995-2000 (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.

v Of course, it is still the case that some economists explain gambling by invoking nonpecuniary returns such as excitement, buying a dream or entertainment [see e.g. [START_REF] Clotfelter | Selling Hope: State Lotteries in America[END_REF]]. However, there are convincing a priori and empirical reasons for giving little weight to this rationalization in general. [START_REF] Friedman | The Utility Analysis of Choices Involving Risk[END_REF] 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. 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. xi This possibility receives some support from the empirical analysis of race-track betting by Jullien and Sallanie (2000).

xii There is an observational equivalence between increasing (decreasing) s and decreasing (increasing) in this model. is a scale parameter.

ix The postulated degree of loss aversion is based partly on experimental evidence in which students required a "substantial" win of approximately $30 in order to induce them to bet $10 on a 50/50 chance.

x Note that in the exponential value function we can redefine s as the percentage of wealth (by deflating by total wealth and redefining the parameters).

xi In particular the agent prefers gamble A to B, C to D, E to F, G to H, I to J, K to L.

where A is a certain 1 million, B is 5 million with probability 0.1, 1 million with probability 0.89 and zero with probability 0.01. C is 5 million with probability 0.1, D is

  typical parameter estimates reported not gamble at actuarially unfair odds.

  d), in the absence of probability distortion agents will, paradoxically, ultimately accept very large bets on odds-on chances at actuarially unfair odds ;

+

  This is plotted in Figures2(a) and 2(b).

  (b). vii For the exponential value function the slope of the indifference curve is given by

  above possibilities are illustrated in Figures 3(a) -4 with n=1 for convenience . In Figure 3(a) expected utility is plotted against win-probability when the stake is set at its optimal value so = EU s 0 , the degree of loss aversion is as postulated by Kahneman and Tversky, and the probability weighting function has the parameter estimates suggested in some of the experimental literature. The agent is observed gambling on a longshot where the expected loss per unit staked is 0.45, so µ = 0 55 . . The distortion to probabilities caused by the probability weighting function overcomes the disinclination to gamble caused by the degree of loss aversion, so that the agent bets on longshots. In Figures 3(b), 3(c) and 3(d) we plot the indifference curves between expected return and win-probability for a small constant s . viii Figure 3(b) illustrates that d dp

  . The interaction of high stakes and the curvature of the value function dominate the influence of the probability weighting function so that the agent has an indifference curve which is observationally equivalent to an expected utility maximiser. By choice of s and other parameter values we can position the asymptote at any win-probability. For example, using the exponential value function,

  Figures 1 (a) -1(b)

  Figure 4Asymptote in Expected Return and Probability

Figure 6

 6 Figures 5 (a) and (b)Indifference curves for a Less Loss Averse Gambler and a "student"

  g.Cornis (1978), and The Wager (2000 b)].vi 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 increases in wealth above W , where W x + is wealth measured from W to . We require that the marginal utility and the second derivative for an increase in wealth,

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  For the expo-power value function in (9), the degree of loss aversion, (LA), is defined by the ratio of the utility gain to the utility loss from a symmetric gamble,

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  Leroy (2003) makes the point in the context of more traditional asset markets.

	and Tversky, based on student responses, arguably seems to large to be widely applicable to other F o r P e e r agents. Submitted Manuscript R e v i e w
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  Endnotesi We are grateful to Robert Sugden for comments on an earlier version.

												Submitted Manuscript
			Eu		0.2
				0.15
					0.1
				0.05
						0	0.76	0.78	0.8	0.82	p 0.84	0.86	0.88	0.9	0.92	0.94
			F o r µ = 0 94737 . F o , r = 25, = 50,n = 0.25 r
												P P
	EU w p		e	r so	w		p	e e s e r e e r
	s w p w note	p	p	r p	p		p	r p	p			R e v i e w R e v i e w
												31 32

Page 32 of 35 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

	Submitted Manuscript
	iii 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)].
	F o
	r
	P
	e e r
	R e v i e w
	33

Page 34 of 35 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  probability 0.11, E is 2,400 with certainty, F is 2500 with probability 0.33, 2400 with probability 0.66 and zero with probability 0.01, G is 2500 with probability 0.33, H is 2400 with probability 0.34., I is 3000 with certainty, J is 3000 with probability 0.8, K is 4000, with probability 0.2, L is 3000 with probability 0.25.

	1 million with Page	Submitted Manuscript
	F o
	r
	P
	e e r
		R e v i e w
		35

35 of 35 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

, .

.

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:

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and Quantitative Methods

Keywords: C72 -Noncooperative Games &lt, C7 -Game Theory and Bargaining Theory &lt, C -Mathematical

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Economists have long been interested in the economics of betting markets as exemplified by the excellent survey articles by [START_REF] Sauer | The Economics of Wagering Markets[END_REF][START_REF] Vaughan Williams | Information Efficiency in Betting Markets: A Survey[END_REF]. 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 the 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 the 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 11 papers 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 non-pecuniary 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 apriori reasoning 2 , with the fact that a majority say they gamble to make money 3 and with other experimental evidence 4 .

1 For instance, [START_REF] Bruce | Toward an Explanation of Betting as a Leisure Pursuit[END_REF] report average stakes of £22.63 on race horse favorites. [START_REF] Strumpf | Illegal Sports Bookmakers[END_REF] reports that average bet size averaged in excess of $1000 in a study of six illegal bookmakers. 2 Friedman and Savage (1948) and [START_REF] Markowitz | The Utility of Wealth[END_REF] in their seminal papers provide a critique of the entertainment rationale. For instance, Markowitz notes 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. Cumulative Prospect theory (CPT) developed by [START_REF] Kahneman | Prospect Theory: An Analysis of Decision under Risk[END_REF] and [START_REF] Tversky | Advances in Prospect Theory: Cumulative Representation of Uncertainty[END_REF] can explain a variety of experimental results inconsistent with expected utility theory and also optimal gambling on long shots.

In the first paper 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 favorites.

One possibly 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 [START_REF] Allais | Le Comportement de l'Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l'Ecole Américaine[END_REF] paradox it is necessary to assume probability distortion. In paper two Peel, Zhang and Law introduce a small degree of probability distortion into the [START_REF] Markowitz | The Utility of Wealth[END_REF] 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 paper three Bhattacharyya and Garrett develop a model based on an extension of that of [START_REF] Friedman | The Utility Analysis of Choices involving Risk[END_REF] model to explain state lottery games. The utility function proposed by [START_REF] Bhattacharyya | From Mean Variance Space to Mean Skewness Space -Implications for Simultaneous Risk Seeking and Risk Averting Behavior[END_REF] 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. 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 non-expected utility approach.

In paper five Farrell and Forrest consider the interesting issue of the extent of displacement effects across gaming products. Employing Australian data they estimates 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 neighborhood hotels (pubs) and clubs. Their evidence suggests a potential for local casinos to divert significant sums from lotto and the causes it 5 See [START_REF] Brockett | A Re-examination of the Relationship between Preference and Moment Orderings by Rational Risk Averse Investors[END_REF] and [START_REF] Cain | Utility and the Skewness of Return in Gambling[END_REF]. For instance for a 1 unit bet with odds o win probability p the relationship between the moments is given by In paper six 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 paper seven Winter and Kukuk develop an innovative model that can explain the favorite longshot bias. The show that the favorite-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 to the family of neural networks) in forecasting using IX2 and Asian Handicap odds data from the English Premier league. The modeling 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 paper eleven Forrest and Simmons examine whether sentiment plays in a part in the fixed odds setting process. They report new results for both Spanish and Scottish football that bookmaker prices appear to be 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

3

  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 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.

  = p(1+ o), variance = skewness , s, is negatively related to expected return but there is no behavior implied.

  results are of relevance to the current policy debate in other countries such as the UK.

  the horse directly. It would be interesting to incorporate such elements into the Winter and Kukuk model. Numerous papers have addressed the issue of whether gambling markets are efficient. The final four papers provide further evidence on various aspects of this hypothesis. In paper eight 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 U.S. racetracks. They find a favorite-longshot bias in exacta wagers but results are unclear for trifecta wagers. In paper nine Graham and Stott develop a results-based 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 favorites but this favorite-longshot bias cannot be exploited by their statistical model. In paper ten Vlastakis, Dotsis, and Markellos evaluate the performance of a Poisson count regression and that of a Support Vector Machine (SVM) (belonging

  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 type of matches in the same manner.

When surveys ask gamblers why they gamble, a majority (approximately

42%-70%) cite financial reasons: 'to make money' (see, for example, The[START_REF] Wager | Stress, Anxiety, and Why Gamblers Gamble[END_REF].)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).


D A Peel

Jie Zhang

D Law

Design of Experiments < C -Mathematical and Quantitative Methods, D84 -Expectations|Speculations < D8 -Information and Uncertainty < D -Microeconomics

Keywords: Allais Paradox, Markowitz model of Utility Allais Paradox, Markowitz model of Utility, Cumulative Prospect Theory, Gambling. JEL classification: C72, C92, D80, D84

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.

Introduction

In a seminal paper [START_REF] Markowitz | The Utility of Wealth[END_REF] proposed a new model of utility. Unlike in standard expected utility theory he assumed that from agents customary or normal level of wealth the agent was initially risk-loving then risk-averse over gains whilst initially risk-adverse then risk-seeking over losses. Though perhaps not widely appreciated the agent was also assumed to be loss averse. 1 Apart from explaining simultaneous gambling and insurance the Markowitz model can also explain a variety of experimental evidence not consistent with standard expected utility theory. 2 1 Markowitz writes, "Generally people avoid symmetric bets. This suggests that the curve falls faster to the left of the origin than it rises to the right. We may assume that U X U X X ( ) ( ), > >0 where X = 0 is customary wealth. " (pp154-155)". His diagram did not appear to exhibit this feature but the text p(155) and discussion makes it very clear that he assumes loss aversion.

2 This is because often the experimental evidence that conflicts with standard expected utility is explicable in terms of reference point effects or loss aversion and not probability distortion per-se. The Markowitz model through changes in the reference point, due to an increase in wealth, can explain the apparent preference of some agents for segregated gains reported by Thaler (1985, p. 203). The reference point effect can also explain why agents might turn down one gamble such as lose $100 or win $150 with win-probability 0.5, but accept a sequence of N such bets, say, is the Allais paradox, (see e.g. [START_REF] Allais | Le Comportement de l'Homme Rationnel devant le Risque: Critique des Postulats ET Axiomes de l'Ecole Américaine[END_REF]). As is well known resolving this paradox requires that agents' subjective probabilities differ from objective probabilities. In the Markowitz model they are assumed equal.

Cumulative Prospect Theory of [START_REF] Kahneman | Prospect Theory: An Analysis of Decision under Risk[END_REF] and Tversky and Kahnerman (1992) (KT) is now perhaps become the major alternative to expected utility theory having superseded Markowitz's model. 3 Cumulative Prospect Theory (CPT) embodies the reference point hypothesis of Markowitz but restricts preferences to be solely risk-averse over gains and solely risk-loving over loses. KT also assume loss aversion but suggest it is of higher order of magnitude4 over small stakes than that that Markowitz would accept. 5 To explain both the Allais paradox over large hypothetical returns and gambling on actuarially unfair outcomes KT assume that the probabilities of events are subjectively distorted by agents, via an inverted s-shaped probability weighting function so that small probabilities are over-estimated and large probabilities are understated. The probabilities are assigned to outcomes employing a N=100. As shown by [START_REF] Samuelson | Risk and Uncertainty: A Fallacy of Large Numbers[END_REF] unfair odds. However it is now known from analysis by [START_REF] Neilson | A Further Examination of Cumulative Prospect Theory Parameterizations[END_REF] that power value functions 7 , assumed by KT in their parametric specification, cannot explain with the same parameter values both the Allais paradox over large returns and gambling over unlikely outcomes. [START_REF] Neilson | A Further Examination of Cumulative Prospect Theory Parameterizations[END_REF], state that the most obvious conclusion of their work is that alternative functional forms are needed. 8

6 Note that if all outcomes are losses, the cumulative prospect preference function in

(2) is identical to the rank-dependent expected utility preference function (e.g. [START_REF] Quiggin | Generalized Expected Utility Theory: The Rank-Dependent Model[END_REF] [START_REF] List | A Simple Test of Expected Utility Theory Using Professional Traders[END_REF], using small real stakes, reports statistical significance of the Allais paradox in responses from both professional traders and students. However the majority of professional traders or students choose the risky option in both Allais questions so that the majority response does not reveal the Allais paradox. The choice patterns observed in the List and Haigh experiments is a common, though not universal finding, in Allais experiments involving small real stakes where often the choices do not reveal the Allais paradox. E.g. Conlisk (1989), [START_REF] Harrison | Expected Utility Theory and the Experimentalists[END_REF] and [START_REF] Burke | An Experimental Note on the Allais Paradox and Monetary Incentives[END_REF] and also [START_REF] Ballinger | Decisions, error and heterogeneity[END_REF], Battalio etal. (1990) Bernasconoi (1994), [START_REF] Camerer | An experimental test of several generalized utility theories[END_REF], [START_REF] Kagel | Tests of "Fanning Out" of Indifference Curves: Results from Animal and Human Experiments[END_REF], [START_REF] Starmer | Probability and juxtaposition effects: An experimental investigation of the common ratio effect[END_REF] and [START_REF] Wu | Curvature of the Probability Weighting Function[END_REF].

Choice of the risky options in the Allais experiments employing small real stakes contrasts with choices in the Allais experiments involving large, hypothetical, stakes.

In these experiments the great majority of agents choose the certain outcome in one of the two questions (see e.g. [START_REF] Tversky | Advances in Prospect Theory: Cumulative Representation of Uncertainty[END_REF]). Accepting the validity of these hypothetical experiments we can make two points. First if the differing choices in Allais experiments employing small and larger stakes are to have a consistent choice theoretic explanation then, ceteris paribus, the power specification of the value function is invalid since the choice pattern is dependent on the size of the payoffs. 9

9 The recent empirical results of Holt and Laury are also particularly important in this context. They use real payoffs as well as hypothetical payoffs and find that risk aversion increases sharply as payoffs are scaled up as agents choose between a "safer" and "more risky" gamble. Both sets of experiments suggest that that increases in payoff levels increase risk aversion. With a power value function this would not occur. Since these experiments use small stakes the issue is not about the validity of the power function as an approximation over large or small stakes. We also note that some early experimental evidence, e.g. [START_REF] Markowitz | The Utility of Wealth[END_REF]), Biswanger (1980) and [START_REF] Hershey | Prospect theory's reflection hypothesis: a critical examination[END_REF] is also inconsistent with a power value Second agents may make choices consistent with the expected utility model even though they may suffer from probability distortion. Probability distortion is a necessary condition to exhibit the Allais paradox but not sufficient. This point has not apparently been fully appreciated in the literature. We show why in Allais experiments employing small the relative returns employed in components of the gamble will be expected to elicit preferences that display very different evidence for the paradox even when agents are assumed to exhibit the same degree of probability distortion.

The question arises as to the degree of probability distortion that is required to explain the preferences exhibited by agents in the Allais paradox over small and large stakes.

In fact, the degree of probability distortion required to explain the Allais paradox can, as we show, be epsilon small. However we also show that the degree of probability distortion required in the KT model to induce agents to act as if risk-seeking, and thus engage in actuarially unfair gambles on long shots, translates into subjective expected function. In these experiments the probabilities, in a sequence of gambles, are kept fixed as agents choose between a gamble and its certainty equivalent. Choices change significantly with size of payoff. Finally we note some theoretical objections to the power assumption. [START_REF] Blavatsky | Back to the St. Petersburg Paradox?[END_REF] shows that the Kahneman-Tversky parameterization cannot resolve the St. Petersburg paradox unless the power coefficient of the utility function is less than that of the probability weighting function. However, this implies that the agent cannot exhibit risk-loving behavior and so gamble. Also the agent becomes infinitely gain loving over small enough stakes violating the loss aversion assumption, see e.g. [START_REF] Cain | Cumulative Prospect Theory and Gambling[END_REF] and Köbberling and Wakker (2004). 7 rates of return that can run into hundreds of percent. Regardless of ones view of learning these sorts of subjective return appear too large to be systematic. Even assuming a small degree of probability distortion-in the third decimal point-we show that the subjective expected returns differ from the objective by a few percent.

Given these points we illustrate with a parametric specification how the Markowitz model, supplemented with a small degree of probability distortion, is able to parsimoniously explain, unlike CPT, majority choices in both experimental evidence such as the Allais paradox, other important experiments such as [START_REF] Holt | Risk Aversion and Incentive Effects[END_REF] also involving real stakes, as well as gambling at actuarially unfair odds.

The rest of the paper is structured as follows. Our analysis is set out in section 1 and the final section offers our conclusions.

Section1

We assume that the value function over gains, UG , is given by the expo-power function of [START_REF] Saha | Expo-Power Utility: A Flexible Form for Absolute and Relative Risk Aversion[END_REF].

UG e rG

n = 1 (1)
where G is gains and r, n are positive constants.

For n 1 the agent is everywhere risk-averse over gains as postulated by KT. For We assume in our examples that the probability weighting function, w p ( ) , has the form employed by KT and is given by

w p p p p ( ) [ ( ) ] = + 1 1 (2)
where p is the objective probability and is a positive constant less than unity 11 .

We assume that probability weighting over gains and losses is that of CPT.

Consider the two standard forms of the Allais experiments. In experiment one agents are offered the choice between the alternatives A and A* and the alternatives B and B* where A: a certain $X. or A*: $X with probability p 1 , $Y with probability p 2 and $0 with probability1 1 2 p p .

B: $X with probability 1 1 p or B*: $Y with probability p 2 .

For instance [START_REF] Tversky | Advances in Prospect Theory: Cumulative Representation of Uncertainty[END_REF] is not assumed a number of studies such as [START_REF] Levy | Testing for Risk Aversion: a Stochastic Dominance Approach[END_REF],2002) ,Pennings and Smidts (2000,2003) report empirical evidence that many agents do not have the shape of value function conjectured by K and T. We discuss in the paper some experiments which can, in principle, provide tests between the competing value functions. Much of the empirical evidence is "observationally equivalent".

11 Employing the weighting function of [START_REF] Prelec | The Probability Weighting Function[END_REF] makes no qualitative difference to the results. 3 were h is typically set equal to 4 as in for example [START_REF] Tversky | Advances in Prospect Theory: Cumulative Representation of Uncertainty[END_REF] with G=3000 , H=4000 and p 3 0 8 = . .

Under Cumulative Prospect Theory the choice between A and A* is given by

U (X) or w p U Y w p p w p X ( ) ( ) [ ( ) ( )]U( ) 2 2 1 2 + + (3)
Recalling that Y>X and between B and B* by

w p U X p U Y ( ) ( ) ( ) ( ) 1 1 2 or w (4)
where Y X as we let payoff size X 0 (or r 0) by L'Hopitals rule we obtain from ( 8) that the choice is given by the same relationship as with a power specification of the value function, namely a choice between .

Consequently for large enough payoffs we would choose A and B and act as an expected utility maximiser regardless of the degree of probability distortion. We note from ( 9) the important point that, ceteris paribus, the larger the more likely the agent to make the choices A* and B*. The Conlisk experiment had a value of of 5 whilst that of List and Haigh had a value of of 10 7 .

In Table 1 we tabulate the values of the distortion parameter required to make the choice A>A* and B>B* for various values of n assuming a power value function.

We can deduce from Table1 that in List and Haigh's experiments, given by 1a and 1b, if all the traders had a power utility parameter of n=0.88 and a distortion parameter of =0.835 then they would choose A* and B* and behave as expected utility maximisers.

Suppose students had the same probability distortion parameter of =0.835 but were more risk-averse with n=0.5. They would choose A>A* and B*>B and exhibit the Allais paradox. Consequently List and Haigh's experimental outcomes could reflect the fact that some students are more risk-averse over gains than traders not that they suffer greater probability distortion.

One other interesting feature of their experiment is that the choice of A* by many respondents implies a higher value of than postulated by KT and one which will mitigate against gambling at actuarially unfair odd as we show below.

If we consider the Conlisk experiment, 2a and 2b we observe that in order not to make the choices A* and B* respondents would have to exhibit extreme probability distortion with <0.44.

In Table 2 where we report some values of r n and , for which an agent with the expo- which is $2x with probability p and $1.6x with probability 1-p and the "more risky gamble N: which is $ 3.85x with probability p and $0.1x with probability 1-p. p takes values between 0.1 and 1. x is set at 1 or 20 in the experiments we consider. Their experiments suggest that the increases in payoff levels increase risk aversion with the switch between M and N occurring at different probabilities. With a power value function this switch could not occur. Since these experiments use relatively small stakes the issue is not simply about the validity of the power function as an approximation to utility over large or small stakes. Whilst Holt and Laury did not investigate their outcomes from the perspective on non-expected utility we can employ their carefully crafted experiments in order to calibrate our non-expected model.

The important point that emerges from Table2 is that an agent with an expo-power value function with r>0.00007 from experiment 4a and r< l 0.00016 from experiment 3b, and =0.95 and n=1.1, so that it takes the Markowitz form, will make the majority responses observed in the six experiments. There may, of course, be other parameter combinations with even less probability distortion and the Markowitz form of value function, which can also explain the experimental evidence. However our example illustrates that the Markowitz model, with a much smaller degree of probability distortion than assumed by KT, can be consistent with important experimental evidence. From an apriori perspective we think this important. It is apparently not widely appreciated how small differences between the objective and subjective probabilities imply large differences between the objective and subjective expected rate of returns of a gamble.

F

= ( )( ) ( ) ( ) ( ) 1 1 1 ( 14
)
where we set r above in (1) as r = a convenient parameterization for the analysis below. , and k are constants and other terms are as defined earlier.

For the utility function in ( 14), the degree of loss aversion, (LA), is defined by the As stake size approaches zero, the assumption of loss aversion requires that k < 1 , (by L'Hopital's Rule), and as it becomes large that ( ) ( ) for all s, which is consistent with the definition of loss aversion of [START_REF] Benartzi | Myopic Loss Aversion and the Equity Premium Puzzle[END_REF] and [START_REF] Köbberling | An Index of Loss Aversion[END_REF]). For the expo-power function this implies the additional constraint that 1.

The parameter values we employ are = = = 6 3 10 0 000071 . , , . k and n = 1.1

With these parameter values the Markowitz agent makes the majority choices in the experiments involving gains reported above. How does the agent behave over gambles involving potential losses? Calculation informs us that the agent is happy to gamble at some actuarially unfair odds. For instance the agent will (a)

Bet $1 on a number at roulette wheel, objective p = 1 38 , objective rate of return equal -0.05263, bet $1 on a horse with objective p =0.01 and objective rate of return >-0.18, bet $1 on a lottery ticket with objective p=0.001 and objective rate of return>-0.38.

In addition the agent will (b) Bet $10 at win probability, p= 0.5 if she wins in excess of $15.23 12 12 We recognise that our parametric Markowitz specification is still subject to the Rabin criticism, (as originally applied to expected utility preferences). The level of risk aversion observed with respect to the small gamble ( turn down a win $11 lose $10 at p=0.5) still leads to a high, though perhaps less absurd, level of risk aversion in that our agent would reject a win infinity lose in excess of $1513 gamble at p=0.5 as opposed to infinity and $100 if an expected utility maximiser. However, e.g. [START_REF] Safra | calibration Results for Non-Expected Utility Theories[END_REF] have shown that Rabin's arguments apply to many non-expected utility theories. 16 (c) Will not accept a bet of lose $100 or win $150 at win probability 0.5, but will accept a sequence of two or more such bets. An outcome inconsistent with expected utility theory as shown by [START_REF] Samuelson | Risk and Uncertainty: A Fallacy of Large Numbers[END_REF].

Cumulative prospect theory can, of course, also produce outcomes qualitatively similar to (a) to (c). However betting at actuarially unfair odds is rapidly curtailed as the degree of probability distortion is reduced. With the parameter values = = 0 0 6 9 and n = 0.88, k = 2.25, = 0.61 and .

we have the model set out in Tversky and Khanerman (1992). If we decrease the degree of probability distortion over gains and losses, so that = 0.84 and = 0 85 . , the agent will make the majority responses found in the List and Haigh experiment. However we now find that the agent will not accept any of the actuarially unfair gambles in (a) above and needs as a matter of fact to receive an objective expected rate of return of in excess of 103% to gamble at objective odds of p=0.01 and in excess of 184% at objective p=0.001.

Conclusions

We show that in principal only a small degree of probability distortion is necessary to exhibit the Allais paradox. Also that in Allais experiments involving small real payoffs, even when agents exhibit probability distortion, majority preferences will often not reveal the Allais paradox if relative payoff ratios in the different components of the small payoff experiment are large. We also show that the choices observed in the Allais experiments employing small real payoffs, such as List and Haigh, cannot be explained by CPT without the assumption of much lower degrees of probability distortion than usually assumed in that model. However lower degrees of probability Given the above points we conduct a similar exercise as that Nielson and Stowe (2002) and ask if a calibrated version of the Markowitz function can be consistent with the majority outcomes observed in number of experiments. We show that this is possible.

The Markowitz model of utility, supplemented by a small degree of probability distortion, can explain the majority choices involved in some Allais experiments over large and small payoffs, the experiments of Holt and Laury as well as gambling at actuarially unfair odds 13 . Given that the subjective expected returns required to induce gambling on long shots in the KT model are, at least from our perspective, apriori, too high to be credible, we consider this a major advantage of the modified Markowitz model. 14 13 We are aware that a Fechner model of random errors as in e.g. [START_REF] Hey | Investigating Generalisations of Expected Utility Theory Using Experimental Data[END_REF] A second set of experiments which deserve further study are those of [START_REF] Battalio | Testing Between Alternative Models of Choice Under Uncertainty: Some Initial Results[END_REF]. They reported that the majority of their respondents displayed risk seeking behaviour in choices over gambles involving small real gains at probabilities of 0.5

This finding is not explicable in the KT framework since it contradicts the subcertainty principle used to explain the Allais paradox (see KT, 1979,pp281-282).

Of course such an outcome can be consistent with a Markowitz specification 15 .

Finally, as first pointed out by [START_REF] Cain | The Favourite-Longshot bias and the Gabriel and Marsden Anomaly: An Explanation Based on Utility Theory[END_REF], the Markowitz model is consistent with the stylised properties of winning bets in the pari-mutuel pools to those obtained in the parallel bookmaking market whilst, as pointed out first by [START_REF] Paya | Testing for Market Efficiency in Gambling Markets: Some observations and new Statistical Tests based on a Bootstrap Method chapter 17[END_REF], the KT model is apparently not. 16 . I particular agents betting on odds on favourites on the Tote 15 For instance the majority preferred $25 with p=0.5 or $5 with p=0.5 to a certain $15. With our parameters the risky gamble is preferred if r<0.0048., = = 0 95 11 . , . . n 16 Gabriel and Marsden (1990) in an innovative analysis compared the returns to winning bets in the British Tote with those offered by bookmakers at starting prices.

They reported the striking finding that tote returns to winning bets during the 1978 British horseracing season were higher at all bookmaker odds , on average, than those [START_REF] Sauer | The Economics of Wagering Markets[END_REF] this result calls for explanation. In fact it appears from subsequent analysis that the relationship between tote returns and bookmaker returns for winning gambles is more complicated than reported by Gabriel and Marsden. Tote pay outs for a unit stake appear higher than bookmakers for long shot winners with the reverse apparently the case for favored horses: see Blackburn and Pierson (1995), [START_REF] Cain | The Incidence of Insider Trading in Betting Markets and the Gabriel and Marsden Anomaly[END_REF] and [START_REF] Peel | Non-Expected Utility Models and Heterogeneity in Risk Attitudes: Towards an Explanation of Gambling Outcomes for Individual and Markets[END_REF].

Contrary to the assumption of Gabriel and Marsden Tote and Bookmaker returns are not the same asset. As stressed by [START_REF] Cain | The Favourite-Longshot bias and the Gabriel and Marsden Anomaly: An Explanation Based on Utility Theory[END_REF] bets on the Tote have uncertain payoffs whilst those with the bookmaker are essentially certain.

Jensen's inequality implies that

EU G U EG ( ) ( ) >
, EU(G) = U(EG) and EU(G) < U(EG) , as the agent is respectively risk loving, risk-neutral and risk-adverse over Gains, G.

As a consequence as pointed out by [START_REF] Cain | The Favourite-Longshot bias and the Gabriel and Marsden Anomaly: An Explanation Based on Utility Theory[END_REF] because the Tote pay-out is uncertain, ex-ante, whilst bookmaker returns are essentially known, ex-ante, expected returns would be expected to be equal on average only if the representative punter is risk-neutral, an assumption implicit in Gabriel and Marsden's analysis. If the agent is assumed everywhere risk-averse over gains, as in the Cumulative prospect theory of Kahneman and Tversky, it would appear that ex-ante they must expect a higher return from betting on the Tote on either favorites or longshots. Since the empirical evidence shows that Bookmaker payouts are higher on average for favourites than those on Tote the model of KT cannot, apparently, explain this key empirical finding.

On the other hand the Markowitz model, as pointed out by [START_REF] Cain | The Favourite-Longshot bias and the Gabriel and Marsden Anomaly: An Explanation Based on Utility Theory[END_REF]

  evidence that the Markowitz model cannot explain

n

  > 1 the agent is risk-loving (risk-verse, <, risk-neutral =assume that the value function takes the form assumed by KT and given this solve for the value of the weighting function parameters and value of the probability distortion parameters consistent with their choices in experiments. (see e.g.Stott (2005) for a comprehensive survey). Where the KT form of value function power function has the convenient property that as r 0 the function approximates the power value function, G n .

  agents are offered the choice between the alternatives C and C* and D and D* where C: a certain $G or C* $H with probability p 3 D: $G with probability p h 4 1 = or D* $H with probability, p h

  power value function would make the majority response in six experiments. These are the three experiments above plus the Allais experiments involving the choices between C and C* and D and D*, also set out above, and in addition two of the important experiments of Holt and Laury (2002) who used real stakes.

  and Laury experiments agents choose between the "safer" gamble, M:

  ratio of the utility gain to the utility loss from a symmetric gamble,

  gambling at unfair odds on all but the most extreme longshots in CPT.

  number of other experiments that would offer potential for differentiating between CPT and the Markowiz model. The first are Allais experiments employing real stakes in which the relative payoffs are closer together. Consistent choice of the risk-seeking alternatives in such experiments would imply, from the perspective of the KT model, that any probability distortion had to be minimal and consequently render the KT model unable to explain gambling at actuarially unfair odds.

  function to that developed byQuiggin. 6 CPT is in principle able to explain the Allais paradox, a variety of other experimental evidence inconsistent with standard expected utility theory as well as gambling at actuarially

	similar cumulative weighting
	F o r
	P
	e
	e r
	R e v
	i e
	w
	4

such behaviour is inconsistent with standard expected utility theory 3 There are 193 citations to the Markowitz model in Google scholar (nov2006). This compares to 4625 to the KT 1979 paper and 1136 to their 1992 paper.

  Substitution also reveals that this is the case for the violation C>C* and D<D*.

	1	1 w p p + (	2	)	>	w	1 (	p	1	)	2 ( ) w p	(6)
	which, for example, for the Conlisk (1989) experiment is
	1-w (0.99)>w (0.11)-w (0.1)	(7)
	Substitution of the weighting function (2) into (7) reveals that any value of the
	distortion parameter less than unity is consistent with (7). Consequently epsilon
	probability distortion is consistent with the Allais paradox preferences A>A* and B<
	B*. However we stress this is only a necessary condition. F Consider the comparison between A and A* using the expo-power function (1). We o r choose between
	1	e	rX	n	1 or w(p 2 )(		e	P + w p p 2 + r X n n ) [ (	1	)	1 ( )]( 2 w p	e	rX	n	)
														e
	That between C and C* by	e r
	U G ( ) and that between D and D* by p U H ( ) ( ) or w 3 w h U G p h U H ( ) ( ) ( ) ( ) 1 3 or w The typical majority choice patterns observed in experiment one when payoffs are R e v (5) i e large is A>A* and B<B* and for small payoffs A<A* and B<B*. The typical choices w
	observed for large payoffs in experiment two are C>C* and D<D*.
	Consistent choice under expected utility requires the choice patterns A>A* and B>B*
	or A<A* and B<B* in experiment one and C>C* and D>D* or C<C* and D<D* in
	experiment two.

Manipulation of the choice pattern A>A* and B<B* informs us that a necessary condition for the Allais paradox choice pattern A>A* and B<B* is that

Table1: Values consistent with choices in Allais Paradox of A>A*, B>B*

  to be explicable in terms of the KT model so that they would appear to be prime agents for inclusion in experiments on risk-attitudes.Notes; power value G n , where G is gain and n is the constant exponent.

	F o r No.. Experiment 1a A: 7 or A*: 0.2 of 10, 0.75 of 7 and 0.05 of zero 1b B:0.25 of 7 or B*: 0.2 of 10 2a A: 5 or A*: 0.1 of 25, 0.89 of 5.and 0.01 of zero do not appear F o r F o r F o r F o r F o r F o r F o r 2b B : 0.11 of 5 or B*: 0.10 of 25 3a A:2400 or A*: 0.33 of 2500, 0.66 of 2400 and 0.01 of zero F o r	,n n=0.5 any n=0.5 >0.841 n=0.5 <0.44 n=0.5 none n=0.5 any	,n n=0.88 <0.832 n=0.88 none n=0.88 <0.31 n=0.88 none n=0.88 <0.96	,n n=1.5 <0.64 n=1.5 none n=1.5 <0.22 n=1.5 none n=1.5 <0.85
	3b	P P P P P P P P B: 0.34 of 2400 or B*: 0.33 of 2500 P	n=0.5 >0.74	n=0.88 >1.2		n=1.5 >1.87
		e e e e e e e e e
	e r R e v e r R e v e r R e v e r R e v e r R e v e r R e v e r R e v e r R e v p p ( ) ] + 1 e r 1 , where p=probability and is solution of 1 n -w(0.1) 5 n -[ w(0.99)-w(0.1)] 1 n >0 Weighting function w p p ( ) [ = constant. e.g in 2a R e v	is a
		i e i e i e i e i e i e i e i e i e
			w w w w w w w w w
					,
	because of the assumed risk-seeking behavior over favorites and risk-aversion
					20

Table 2 : Values of r Consistent with choices in Allais and Holt and Laury Experiments for A>A* and B>B*

 2

	No. Experiment

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

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.[START_REF] Tversky | Advances in Prospect Theory: Cumulative Representation of Uncertainty[END_REF]].

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

µ =0, p=0.5 µ s = -0.03348 w(0.5) = 0.42065 w(1-0.5) = 0.454