Utility and the Skewness of Return in Gambling

This paper demonstrates that the intuitively appealing argument based on the postulated trade-off between expected return, variance and skewness of return of a risk-averse gambler does not provide an explanation of observed betting behaviour. It is shown how the expected utility of a representative gambler faced with a single-prized outcome event can be expressed in terms of the mean and variance of return, the mean and skewness of return or, generally, of the mean and any other single moment of return: and the standard practice of taking a Taylor series expansion/approximation of the expected utility involving moments of return is usually incorrect. Previous analyses have suggested that a punter will accept a lower mean return for higher skewness and this work seems to have involved invalid expansions of the utility function. The upshot is that with certain utility functions which have been used in a number of studies, any analysis based on expansion and estimation of the derivatives of the utility function may be valid only for data based on odds-on favourites and not for longshots. Keywords: mean-variance frontier, Kurtosis, favourite-longshot bias, Taylor series expansion