The dummies' guide to lottery design

This paper outlines the issues relevant to the operation of lottery games. We consider how such games should be designed, what a portfolio of games might look like, how the operator should be regulated, how spending on lottery games should be taxed, and what considerations are relevant to the use of the revenue from such games. Our research suggests that the lottery tickets sales depend positively on the proportion of revenue returned as prizes (i.e the mean of the prize distribution), the skewness in the prize distribution (e.g how much of the prize money goes to the jackpot), and negatively on the variance in the prize distribution. Thus good causes revenue might be higher if the game were meaner (less of the stakes used as prize money), or if more of the prize money was used for the jackpot, or if the variance in the expected prizes were reduced. In practice, it is difficult to change one aspect of the design of the prize distribution without having a counterveiling effect on another aspect. Thus, it is difficult to make judgements about the merits of alternative game designs without looking at all of the parameters being proposed. We find no empirical evidence to suggest that there is any merit in having much of the take-out (the revenue that is not returned as prizes) dedicated to good-causes, and no evidence that the nature of the operator might make any difference. The current “beauty contest” process of choosing an operator is fraught with risk and we suggest that, subject to a probity check, the license should be auctioned

NLTE 1.5D Modeling of Red Giant Stars

Spectra for 2D stars in the 1.5D approximation are created from synthetic spectra of 1D non-local thermodynamic equilibrium (NLTE) spherical model atmospheres produced by the PHOENIX code. The 1.5D stars have the spatially averaged Rayleigh-Jeans flux of a K3-4 III star, while varying the temperature difference between the two 1D component models ($\Delta T_{\mathrm{1.5D}}$), and the relative surface area covered. Synthetic observable quantities from the 1.5D stars are fitted with quantities from NLTE and local thermodynamic equilibrium (LTE) 1D models to assess the errors in inferred $T_{\mathrm{eff}}$ values from assuming horizontal homogeneity and LTE. Five different quantities are fit to determine the $T_{\mathrm{eff}}$ of the 1.5D stars: UBVRI photometric colors, absolute surface flux SEDs, relative SEDs, continuum normalized spectra, and TiO band profiles. In all cases except the TiO band profiles, the inferred $T_{\mathrm{eff}}$ value increases with increasing $\Delta T_{\mathrm{1.5D}}$. In all cases, the inferred $T_{\mathrm{eff}}$ value from fitting 1D LTE quantities is higher than from fitting 1D NLTE quantities and is approximately constant as a function of $\Delta T_{\mathrm{1.5D}}$ within each case. The difference between LTE and NLTE for the TiO bands is caused indirectly by the NLTE temperature structure of the upper atmosphere, as the bands are computed in LTE. We conclude that the difference between $T_{\mathrm{eff}}$ values derived from NLTE and LTE modelling is relatively insensitive to the degree of the horizontal inhomogeneity of the star being modeled, and largely depends on the observable quantity being fit.Comment: 46 pages, 14 figures, 7 tables, accepted for publication in ApJ on April 5, 201