樂透彩最適球數之探討

Abstract

本研究利用中國信託、台北富邦銀行每期「期望報酬」、「頭獎累積獎金」與「銷售量」進行迴歸分析。然後再以求出之迴歸模型推廣至不同球數下之銷售量預測，並加入成本損益之考量發現在何種球數時銷售量可以達到最大，而使發行公司能獲利最多。 在固定獎金發放額度下，不論是中國信託、台北富邦銀行、或是兩家合併後的數據，欲達到最大銷售量最適球數約為35~36顆。在不固定獎金發放額度下，以可發放獎金的改變、陸獎與普獎獎金的改變、頭獎獎金的改變分別進行觀察。一，當可發放獎金占所有獎金比例愈高時，估計銷售量與最適球數均呈現下降的趨勢。二，當陸獎與普獎獎金愈高時，估計銷售量呈現下降的趨勢，但最適球數只有當固定獎金大於現行固定獎金時呈現上升趨勢，反之則沒有任何變動。三，當頭獎獎金比例占可發放獎金愈高時，估計銷售量呈現下降的趨勢，但最適球數不管頭獎獎金比例均維持在35~36顆。 本研究最後以最新發行的威力彩為研究對象，以在樂透彩中建立的模型去估計其銷售量與球數的關係，發現現行的威力彩欲達到最高銷售量，其最適球數為20顆。

This paper extracts expected value, accumulated first prize money and sales from Chinatrust and Taipei Fubon lottery database to run a simple regression. This paper then adds in cost concerns and uses the regression results to predict expected sales under different number of balls while concluding the optimal number of balls a lottery company should play with if it aims to maximize its profits. Under current prize money distribution, the number of balls for Chinatrust, Taipei Fubon, and the mixture of two companies to maximize profits all points to 35 or 36. If the current prize money distribution is changed, this paper views the changes in three aspects, which are the change of distributable prize money, change of sixth prize and seventh prize money and the change of first prize money. For one, this paper found out when the amount of distributable money increased, expected sales and number of balls both decreased. Secondly, the higher the sixth and seventh prize money amount were the lower it was for expected sales while the number of balls rose only when the prize money were higher than before. Thirdly, when first prize money accounted for a bigger portion in distributable prize money, expected sales declined but the number of balls remained at 35 or 36. As for the new super lotto, based on the model conducted this paper concludes for super lotto to maximize its profits it needs to change its number of balls to 20.