辛普森悖論與職業網球選手的決策─以網球四大公開賽為例

本研究在探討為何有ㄧ些職業網球選手容易發生總得分多，但卻輸了比賽。(統計學常將資料具有此類現象時稱為「辛普森悖論」，以下我們沿用此名詞。)並希望能提出一套選手的理性決策因素來解釋這些因果關係。蒐集2015-2018四大滿貫的比賽場次，將男性排名前1000名及女性排名前889名的職業網球選手作為觀察對象，主要探討的理性因素為選手平均反敗為勝次數，輸家平均反敗為勝次數越多，認為自己即使在落後的情況下也容易反敗為勝拿下此局，因此辛普森悖論下的輸家如果反敗為勝次數較多，則代表存在一套職業選手的理性決策模式來解釋這些現象。另外將可能影響辛普森悖論的因素也加進迴歸式，包括:選手的排名、年齡、身高、及場地材質等等。結論為:男性選手的每場比賽平均反敗為勝次數會正向影響比賽時的理性決策，其次數越多，會越有信心反敗為勝而不放棄每一球，因此發生辛普森悖論的機率增加，但是草地材質場地為負向影響，由於草地材質的摩擦力小、球速快，選手會選擇在接發局節省體力，保留體力至贏球率較高的發球局，使發生辛普森悖論的機率減少；關於女性選手部分，影響比賽的決策因素只有紅土及慢速硬地場地的材質，且為正向影響，因為女性普遍擊球速度不像男性的猛烈，因此在球速較慢的場地材質較容易為爭奪一分而進行多個回合的對抗，使發生辛普森悖論的機率增加，這一點也符合理性決策模式的解釋。

關鍵字

辛普森悖論；網球；理性

並列摘要

The study explored why some professional tennis players easily scored more total score, but they lost the game in the end. (Statistics usually refers to the phenomenon of “Simpson's paradox” when this phenomenon occurs. We use this term below.) I hope to propose a set of rational decision factors that explain causality to these players. We collected the four grand slam tennis from 2015 to 2018 and taking the top 1000 male and the top 889 female tennis players as observation targets. The main reason for the rationality is that the number of player's turn defeat into wins. The loser has the more player's turn defeat into wins, they think that can easily win the game even if they are behind. Therefore, if the loser has the more player's turn defeat into wins under Simpson’s paradox, it means that there is a rational decision model for professional players that explain these phenomena. Besides, we also added to the factors that may affect Simpson's paradox into the regression, including the player's ranking, age, height, and tennis court material. The conclusion is that the more players turn defeat into wins per game for male players will positively influence the rational decision-making during the game. The more times there are, the more confident they will turn defeat into wins and won't give up each ball. Therefore, the chances of Simpson's paradox increase. The grass material site has a negative impact. Due to the small friction of the grass material and the fast ball speed, the player will choose to save energy in the receiving and retain the physical strength to the service with a high winning rate. That is why the chances of Simpson's paradox are reduced. On the other hand, female players do not have the resume of rational factors in this paper. The decision-making factors affecting the game are only the materials of clay and slow hard ground, and they are positively affected because women generally do not hit the ball at a fast speed as males. Therefore, females have more likely to compete for multiple points in a field with slower ball speeds, resulting in increasing the chance of Simpson's paradox. This point is also conform to the interpretation of the rational decision-making model.