13 / 2024-04-04 09:58:58

Picking Lotto Winning Numbers Using Mixed Integer Programs: Beyond Finite Geometry & Random Pick

U.K. National Lottery; Pair-Wise Similarity; Symmetry; Moment-Based

Picking lottery numbers has long captivated the public imagination, evolving from a mere game of chance into a significant research challenge that intertwines mathematics, statistics, and even psychology. This study reevaluates betting strategies for the UK 6/59 Lotto game. Our objective is to develop a strategy for creating m tickets to maximize the potential of the highest-performing ticket, increasing the odds of winning.



We begin by contextualizing the problem within a moment-based distributionally robust optimization model and exploit the symmetry in the model formulation to extract the importance of the notion of ``pairwise similarities" to the performance of the strategy. This leads to a mixed-integer programming model to minimize overall similarity while ensuring balanced pairwise similarities among tickets. This new method surpasses both random pick and finite-geometry strategies, boosting the odds of matching three winning numbers by 2-3%, using 27 lotto tickets.



Using this new strategy, we demonstrate that it is optimal to buy 15 lotto tickets, to maximize the break-even probability. This challenges the conventional belief that more bets increase losses - while buying one ticket has around 10.83% chance of breaking even, buying 15 tickets in the optimal manner has around 15.88% chance of breaking even. Increasing the size of bets beyond 15 tickets, however, leads to a diminishing return in terms of the break-even probability. This highlights a tipping point where the cost of additional tickets starts to outweigh the proportional increase in winning odds.