熱手錯覺：為什麼連諾貝爾獎得主都相信運動神話

幾十年來，熱手謬論一直被用來證明運氣和不穩定的民間故事無法與冷酷的數字相提並論。但這些數字並不能證明熱手只是憑空想像。

NBA superstar Stephen Curry is among the many athletes who insist that the "hot hand" effect is real. But a study from the 1980s claimed to show that the hot hand is a myth. Now, a new analysis of the data suggests that the original study may have been flawed.

NBA巨星史蒂芬·庫裡是眾多堅持認為「熱手」效應真實存在的運動員之一。但 20 世紀 80 年代的一項研究聲稱，「熱手」只是一個神話。現在，對數據的新分析表明，最初的研究可能存在缺陷。

The hot hand effect is the idea that players' successes come in streaks—dependent on some mysterious inner quality that ebbs and flows—and when a player’s hand is hottest and they’re “in the zone,” it can feel almost like they can’t miss.

熱手效應是指玩家的成功是連續不斷的——依賴於某種神秘的內在品質的起伏——當玩家的手最熱並且他們處於「區域」時，他們會感覺幾乎可以不要錯過。

Curry knows a thing or two about a hot hand. He holds the NBA record for making at least one three-pointer in 268 consecutive games. Following a practice one day, he sank 105 three-pointers in a row. In the Hot Ones interview, Curry said the authors of the study “don’t know what they’re talking about at all.” When Curry totaled 60 points in one game, he said, “It’s literally a tangible, physical sensation of all I need is to get this ball off my fingertips and it’s going to go in.”

柯瑞對熱手略知一二。他保持NBA連續268場比賽至少命中1個三分球的紀錄。有一天訓練結束後，他連續投進了 105 個三分球。在 Hot Ones 的訪談中，柯瑞表示這項研究的作者「根本不知道他們在說什麼」。當柯瑞在一場比賽中拿下60 分時，他說：「這實際上是一種有形的身體感覺，我所需要的就是將球從我的指尖拿開，然後它就會投進去。

The study, “The Hot Hand in Basketball: On the Misperception of Random Sequences,” was written in 1985 by eminent psychologists Thomas Gilovich, Robert Vallone, and Amos Tversky. They allegedly demonstrated through an analysis of basketball shooting data that the hot hand was a myth. Recently deceased Nobel laureate (and Tversky’s chief collaborator) Daniel Kahneman proclaimed, “The hot hand is a massive and widespread cognitive illusion.” In statistical circles, the hot hand study has taken on a metaphorical significance beyond basketball. For decades, the fallacy of the hot hand has been cited to show that folktales of luck and streakiness are no match for cold, hard numbers.

這項名為「籃球熱手：論隨機序列的誤解」的研究由著名心理學家 Thomas Gilovich、Robert Vallone 和 Amos Tversky 於 1985 年撰寫。據稱，他們透過對籃球投籃數據的分析證明，熱手是一個神話。最近去世的諾貝爾獎得主（也是特沃斯基的主要合作者）丹尼爾·卡尼曼宣稱：“熱手是一種巨大而廣泛的認知錯覺。”在統計界，熱手研究有超越籃球的隱喻意義。幾十年來，熱手謬論一直被用來證明運氣和不穩定的民間故事無法與冷酷的數字相提並論。

But the numbers do not prove the hot hand is a figment of imagination. Economists Joshua Miller of the University of Adelaide and Adam Sanjurjo at the University of Alicante in Spain, used data from multiple different sports, including the same basketball data used by Gilovich, Vallone, and Tversky, to provide robust support for a hot hand. The problem, say Miller and Sanjurjo, lies in the way the original authors analyzed their data and, in particular, a mistake they made about what random data should look like without the influence of a hot hand.

但這些數字並不能證明熱手只是憑空想像。阿德萊德大學的經濟學家Joshua Miller 和西班牙阿利坎特大學的Adam Sanjurjo 使用了多種不同運動的數據，包括吉洛維奇、瓦隆和特沃斯基使用的相同籃球數據，為熱手提供了強有力的支持。 Miller 和 Sanjurjo 表示，問題在於原始作者分析數據的方式，特別是他們在沒有熱手影響的情況下隨機數據應該是什麼樣子的問題上犯了一個錯誤。

Imagine we’re looking at a chart of hits and misses over some number of shots and trying to find where a shooter might have had a hot hand. Suppose we look for hot-handedness by considering only those attempts that came after a sequence of hits, such as three baskets in a row. These sequences are candidates for being shot with a “hot hand.”

想像一下，我們正在查看一些投籃次數的命中率和未命中率圖表，並試圖找出射手可能手感不佳的地方。假設我們只考慮一系列命中後的出手（例如連續三個籃框）來尋找手腳暴躁的情況。這些序列是用「熱手」拍攝的候選序列。

Shouldn’t years of testimony from athletes like Stephen Curry count for something?

像史蒂芬·庫裡這樣的運動員多年的證詞難道不應該有價值嗎？

However, if there’s no such thing as a hot hand, we might expect the player to have the same success rate in these attempts—the shots after three consecutive makes—as their overall average. Since our working theory is that the previous successes have no predictive power for the next shot, it would seem intuitive that our choice of shots based on what happened right before the hot streak shouldn’t matter at all.

然而，如果沒有「熱手」這樣的東西，我們可能會期望球員在這些嘗試中的成功率（連續三次投籃後的投籃）與他們的總體平均水平相同。由於我們的工作理論是，先前的成功對下一次投籃沒有預測能力，因此直觀地說，我們根據熱連勝之前發生的事情選擇的投籃根本不重要。

But that’s wrong. By selecting attempts that come after a hot streak and computing a proportion over this subset, we have unwittingly introduced a negative bias into the estimate of the rate of success that could counteract a hot-hand-induced positive effect. In other words, the observed percentage of success-following-success being equal to the rate of success-following-anything, would be evidence for the hot hand instead of against it. The way we selected the data artificially penalized the shooter; their true success rate may have been a few percentage points higher than what we tabulated. So, if our observation matched their usual average, it must be that something else was at work to offset our bias—a hot hand.

但那是錯的。透過選擇連續熱刺後的嘗試併計算該子集的比例，我們無意中在成功率的估計中引入了負偏差，這可能抵消熱手引起的正面影響。換句話說，觀察到的成功之後成功的百分比等於任何事情之後成功的比率，這將是熱手的證據，而不是反對它。我們選擇數據的方式人為地懲罰了射手；他們的真實成功率可能比我們列出的高幾個百分點。因此，如果我們的觀察結果符合他們通常的平均水平，那麼一定是有其他東西在起作用，抵消了我們的偏見——一隻熱手。

If you find yourself doubting this bias exists, you’re in good company, including the esteemed professors who first “debunked” the hot hand. Like other famously counterintuitive examples in probability, such as the Monty Hall Problem—the puzzle of whether to switch doors when searching for a prize on the game show Let’s Make a Deal—the phenomenon acts almost like an optical illusion: Our natural senses tell us something that turns out to be contradicted when we try to confirm it with hard measurement.

如果你發現自己懷疑這種偏見的存在，那麼你身邊就有很多人，包括那些首先「揭穿」熱手的受人尊敬的教授。就像其他著名的違反直覺的機率例子一樣，例如蒙蒂霍爾問題——在遊戲節目《讓我們做個交易》中尋找獎品時是否要換門的難題——這種現象幾乎就像一種視錯覺：我們的自然感官告訴我們當我們試圖透過硬測量來確認某件事時，結果發現它是矛盾的。

I’ll freely admit that I didn’t believe it either, at first. I only became convinced after I ran 100,000 simulations of 100 random coin flips and tabulated the proportions of heads following streaks of three heads. I knew the coin flips were perfectly random 50/50 processes under any conditions (coins don’t get “hot”). So, if I

我坦白承認，一開始我也不相信。我對 100 次隨機拋硬幣進行了 100,000 次模擬，並列出了連續 3 個正面出現正面的比例後，我才確信這一點。我知道拋硬幣在任何條件下都是完全隨機的 50/50 過程（硬幣不會變“熱”）。所以，如果我