On the Optimality of Belic’s Lottery Designs

Abstract

Consider a lottery scheme consisting of randomly selecting a winning $t$-set from a universal $m$-set, while a player participates in the scheme by purchasing a playing set of any number of $n$-sets from the universal set prior to the draw. The player is awarded a prize if $k$ or more elements in the winning $t$-set match those of at least one of the player’s $n$-sets in his playing set ($1\le k\le min\{n,t\}\le m$). This is called a $k$-prize. The player may wish to design a \emph{smallest} playing set which guarantees the player a $k$-prize, no matter which winning $t$-set is chosen from the universal set.

In this paper, we consider the optimality of the 302 cardinality 7 (or less) lottery design listings in \verb”BELIC” R: \emph{Lotto Systems and Toto Systems to win Wheel Game}, [online], [cited 2003, October 31], available from: {http://www.xs4all.nl/$\sim$ rbelic/}, for which $m>20$. It is shown, by means of a computerized search technique, that 192 of these designs are optimal, whilst 78 are not, in which case we provide optimal designs.

Then, an additional 429 upper bounds in the tables of Belic (not necessarily of cardinality 7 or less) are improved; 126 of which are optimal. Thus, apart from the 192 designs that we show to be optimal, 204 new lottery numbers are established in this paper, and a further 304 upper bounds are improved. Finally, the optimality of 54 designs of cardinality 7 or less could not be established; however, in each of these cases, a hitherto best known lower bound is provided.