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

Abstract

We show that in principal only a small degree of probability distortion is necessary for agents to exhibit the Allais paradox. We also show that the choices observed in the Allais experiments employing small real payoffs cannot be explained by Cumulative Prospect Theory without the assumption of low degrees of probability distortion that rule out gambling at unfair odds on all but the most extreme longshots in CPT. Given these points we show that the Markowitz model of utility supplemented by a small degree of probability distortion can explain the majority choices involved Allais experiments and other experiments as well as gambling at actuarially unfair odds.

Notes

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 ’ (pp. 154–155). 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, N  = 100. As shown by Samuelson (1963) such behaviour is inconsistent with standard expected utility theory.

³ There are 193 citations to the Markowitz model in Google scholar (Nov. 2006). This compares to 4625 to the KT 1979 paper and 1136 to their 1992 paper.

⁴ In particular, the function is postulated to fall roughly twice as fast over losses as it rises over gains, exhibiting diminishing sensitivity as the marginal impact of losses or gains diminishes with distance from the reference point (see e.g. Tversky and Kahneman, 1992).

⁵ He writes, ‘for very small symmetric bets the loss in utility from the bet is negligible and is compensated for by the fun of participation’.

⁶ Note that if all outcomes are losses, the cumulative prospect preference function in Equation 2 is identical to the rank-dependent expected utility preference function (e.g. Quiggin, 1993). Both apply the probability weighting function to the probability of the lowest outcome first. If all outcomes are gains the preference function in Equation 1 differs from the rank dependent expected utility preference function by applying the weighting function to the probability of the highest outcome first.

⁷ This conflicts with Markowitz who assumed bounded value functions. He wrote “to avoid the famous St. Petersburg Paradox, or its generalization by Cramer, I assume that the utility function is bounded from above. For analogous reasons I assume it to be bounded from below.” (p. 154).

⁸ Neilson and Stowe endeavoured to explain the Allais paradox in an experiment involving millions. It is known from the celebrated results of Rabin (2000) that a utility function calibrated for low-stakes gambles implies unreasonable behaviour for high-stakes gambles, and vice-versa. Because their experiments involved only gains the same results will hold for cumulative prospect theory preferences. However, Neilson and Stowe also show that parameter combinations that are compatible with choices expressed in gambles with small monetary outcomes are also inconsistent with parameter values that imply “reasonable” threshold probabilities between gambling and insurance.

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. Markowitz (1952), Biswanger (1980) and Hershey and Shoemaker (1980) is also inconsistent with a power value 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. Blavatsky (2005) 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. Cain et al. (2005) and Köbberling and Wakker (2004).

¹⁰ Many studies 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 is not assumed a number of studies such as Levy and Levy (2001, 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 Prelec (1998) makes no qualitative difference to the results.

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. Safra and Segal (2006) have shown that Rabin's arguments apply to many nonexpected utility theories.

¹³ We are aware that a Fechner model of random errors as in e.g. Hey and Orme (1994) is potentially relevant in any particular experiment for explaining decisions between lotteries A and B that only differ by small amounts of utility. However our purpose here is to illustrate that the Markowitz model is able, with a small amount of probability distortion, to explain majority choices in a large number of experiments without recourse to, what appear to us, extreme assumptions about probability distortion.

¹⁴ 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., δ = 0.95, n = 1.1.

¹⁵ 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 offered by bookmakers. As noted by Sauer (1998) 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), Cain et al. (2001) and Peel and Law (2007). Contrary to the assumption of Gabriel and Marsden Tote and Bookmaker returns are not the same asset. As stressed by Cain et al. (2003) bets on the Tote have uncertain payoffs whilst those with the bookmaker are essentially certain. Jensen's inequality implies that , as the agent is respectively risk loving, risk-neutral and risk-adverse over Gains, G. As a consequence as pointed out by Cain et al. (2003) 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 Cain et al. (2003), because of the assumed risk-seeking behavior over favorites and risk-aversion over long-shots, would imply tote returns could be expected to be lower for favorites and vice-versa for long shots.