On skewness of return and buying more than one ticket in a lottery

Abstract

The purpose in this article is to demonstrate that buying more than one ticket in a lottery is readily explicable in models of utility that permit gambling at actuarially unfair odds. However, contrary to popular view, we show this choice cannot be explained in terms of a variance–skew trade-off.

Notes

¹ This is because the third derivative of the utility function of a globally risk-averse agent is positive and therefore appears to imply a valid trade-off between mean, variance and skewness of return, that is, preferring a higher third moment for two random variables having equal means and variances.

² They prove and demonstrate with examples that expected utility preferences never universally translate into moment preferences. Cain and Peel (2004) also illustrate the potential fallacy in the context of simple gambles.

³ We can obtain the same outcomes in Cumulative Prospect theory proposed by Kahneman and Tversky (1979) and Tversky and Kahneman (1992) or Rank-Dependent expected utility of Quiggin (1993). However probability distortion, a key element of their models, demand more space than is available here.

⁴ σ² (A) = 4.049 × 10⁷ σ₃(A) = 3.644 × 10¹³. σ²(B) = 4.049 × 10⁷ σ₃(B) = 1.82232 × 10¹⁵. EU(A) = 1.465 × 10⁻⁵, EU(B) = −7.99 × 10⁻⁸.