Hot Hand
Can a fair coin run cold?
For over thirty years, one famous study said the hot hand in basketball was an illusion. Then two economists flipped a coin the way the study counted shots. Do it yourself, then take it to a price chart.
Why a fair coin fails the test
Pick a row of flips and look only at the flips that come right after a heads. The flip after a heads is still a fair coin. But each row gets one vote, however many flips it holds. A row with a long run of heads packs many after-heads flips, all of them heads, into one vote. A row where a heads was followed by tails often has only one after-heads flip, and it counts as a whole row at 0%. Averaged over rows, heads after heads comes out below one half: 5 in 12 for rows of three flips, about 40.5% for rows of four. The shorter the rows and the longer the streak you ask about, the bigger the slip.
Gilovich, Vallone and Tversky asked exactly this kind of question, player by player, in their 1985 paper. They expected a player with no hot hand to shoot as well after three hits as after three misses. A player like that is a coin, and a coin taking 100 shots does about 8 points worse after three hits. The Cornell players did 4 points better. Joshua Miller and Adam Sanjurjo showed this in Econometrica in 2018. Measured against the right coin, the players' gap is about +13 points. The hot hand the study threw out was in its own data.
The same game in a trading room
David Aronson's Evidence-Based Technical Analysis (2006) repeated the 1985 verdict: "the hot hand phenomenon is an illusion. The fans have been duped by their intuition." The data were not on his side, but his rule was: "A valid trend is a streak of longer duration than those that are ordinary in random data." The 2018 correction is that rule, done properly. It asks what a coin would show under the very same count, and measures against that, not against the half everyone expects.
A chart reader who tests "after three up days" one stock at a time makes the same slip as the 1985 study. On coin-flip prices the test shows a snap-back that is not there. On real prices it can hide a trend that is there. The cure is the same in both rooms: before you trust a test, run it on prices made by flipping coins, and see what the coins score.
When someone shows you a streak, or tells you there is none, ask: what would a coin show, counted the same way?
Sources
- Thomas Gilovich, Robert Vallone and Amos Tversky, "The Hot Hand in Basketball: On the Misperception of Random Sequences", Cognitive Psychology 17 (1985), 295–314.
- Joshua B. Miller and Adam Sanjurjo, "Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers", Econometrica 86, no. 6 (2018), 2019–2047.
- David Aronson, Evidence-Based Technical Analysis (Wiley, 2006), chapter 2.
- Every coin on this page is flipped in your browser: rows of four flips, coin players of 100 shots each with a streak of three, and price charts of 100 days where each day goes up or down on a fair coin. The 40.5% is exact, from all 16 possible rows of four.