Towards a Characterisation of Lottery Set Overlapping Structures

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, and is awarded a prize if $k$ or more elements of the winning $t$-set occur in at least one of the player’s $n$-sets ($1\le k\le \{n,t\}\le m$). This is called a $k$-prize. The player may wish to construct a playing set, called a lottery set, which guarantees the player a $k$-prize, no matter which winning $t$-set is chosen from the universal set. The cardinality of a smallest lottery set is called the lottery number, denoted by $L(m,n,t;k)$, and the number of such non-isomorphic sets is called the lottery characterisation number, denoted by $\eta (m,n,t;k)$. In this paper, an exhaustive search technique is employed to characterise minimal lottery sets of cardinality not exceeding six, within the ranges $2\le k\le 4$, $k\le t\le 11$, $k\le n\le 12$, and $max\{n,t\}\le m\le 20$. In the process, $32$ new lottery numbers are found, and bounds on a further $31$ lottery numbers are improved. We also provide a theorem that characterises when a minimal lottery set has cardinality two or three. Values for the lottery characterisation number are also derived theoretically for minimal lottery sets of cardinality not exceeding three, as well as a number of growth and decomposition properties for larger lotteries.