When You're Hot You're Hot... Maybe

The Times has an intriguing piece on streaks, hot hands, and the human inclination to spot patterns where (perhaps!) none exist:

Gamblers, Scientists and the Mysterious Hot Hand

By GEORGE JOHNSON   OCT. 17, 2015

...

Psychologists who study how the human mind responds to randomness call this the gambler’s fallacy — the belief that on some cosmic plane a run of bad luck creates an imbalance that must ultimately be corrected, a pressure that must be relieved. After several bad rolls, surely the dice are primed to land in a more advantageous way.

The opposite of that is the hot-hand fallacy — the belief that winning streaks, whether in basketball or coin tossing, have a tendency to continue, as if propelled by their own momentum. Both misconceptions are reflections of the brain’s wired-in rejection of the power that randomness holds over our lives. Look deep enough, we instinctively believe, and we may uncover a hidden order.

To which I feel obliged to add - the ability to spot real patterns, such as moon phases, tides, weather cycles, planting cycles, animal migration, feeding and hunting habits was surely vital to the survival of early humans (not to mention other species.)

But, although out of season, we are talking basketball:

Recent studies show how anyone, including scientists, can be fooled by these cognitive biases. A working paper published this summer has caused a stir by proposing that a classic body of research disproving the existence of the hot hand in basketball is flawed by a subtle misperception about randomness. If the analysis is correct, the possibility remains that the hot hand is real.

...

In a study that appeared this summer, Joshua B. Miller and Adam Sanjurjo suggest why the gambler’s fallacy remains so deeply ingrained. Take a fair coin — one as likely to land on heads as tails — and flip it four times. How often was heads followed by another head? In the sequence HHHT, for example, that happened two out of three times — a score of about 67 percent. For HHTH or HHTT, the score is 50 percent.

Altogether there are 16 different ways the coins can fall. I know it sounds crazy but when you average the scores together the answer is not 50-50, as most people would expect, but about 40-60 in favor of tails.

There is an attached table laying out the sixteen alternatives and explaining that calculation. The gist - although there are sixteen possible results for a test sample of four consecutive coin flips, two of them - TTTT and TTTH - are dropped from the analysis due to an insufficiency of heads.

What that implies for an analysis based on longer runs is not explained here. My guess is that the 40-60 ratio he mentions converges towards 50-50, but showing that may complement my afternoon football.

We do hear from one expert:

In an interesting twist, Dr. Miller and Dr. Sanjurjo propose that research claiming to debunk the hot hand in basketball is flawed by the same kind of misperception. Studies by the psychologist Thomas Gilovich and others conclude [LINK] that basketball is no streakier than a coin toss. For a 50 percent shooter, for example, the odds of making a basket are supposed to be no better after a hit — still 50-50. But in a purely random situation, according to the new analysis, a hit would be expected to be followed by another hit less than half the time. Finding 50 percent would actually be evidence in favor of the hot hand. If so, the next step would be to establish the physiological or psychological reasons that make players different from tossed coins.

Dr. Gilovich is withholding judgment. “The larger the sample of data for a given player, the less of an issue this is,” he wrote in an email. “Because our samples were fairly large, I don’t believe this changes the original conclusions about the hot hand. ”

Let's close with the words of wisdom from noted Latin American socialist Lefty Gomez - "I'd rather be lucky than good".

NO MORE FUZZY MATH: There is a mini Reader Revolt in the comments section at the Times. One example:

There is an obvious mistake in the coin toss example. The reason that the authors came up with a 40.5% probability that a heads follows a head is because they took a simple average rather than a weighted average. Each sequence has a different number of opportunities and this is information that needs to be included in the calculation. For example, HHHH has 3 opportunities for heads to follow heads (HH=3 + HT=0) while HHTH has only 2 (HH=1 + HT=1). If you weight each sequence by the number of opportunities, you will get the 50% probability that you expect by intuition.

Or put it this way: The table shows 16 possibilities for 4 coin flips, totaling 64 flips, half heads and half tails. The final flip in each group has no successor, so set that aside. Of the remaining 48 flips, 24 are heads and 24 tails. And, as noted in the table, 12 times a head was followed by a head, and 12 times it was followed by a tail. 50/50, as expected.

The paper is here. Eventually I hope to see whether it was the professors or the reporters who got this wrong, but I know how I'm betting.

THAT WAS EASY. The opening paragraph of the paper, my emphasis:

Jack takes a coin from his pocket and decides that he will flip it 4 times in a row, writing down the outcome of each flip on a scrap of paper. After he is done flipping, he will look at the flips that immediately followed an outcome of heads, and compute the relative frequency of heads on those flips. Because the coin is fair, Jack of course expects this empirical probability of heads to be equal to the true probability of flipping a heads: 0.5. Shockingly, Jack is wrong. If he were to sample one million fair coins and flip each coin 4 times, observing the conditional relative frequency for each  coin, on average the relative frequency would be approximately 0.4

Huh. That was pretty much the Times take-away too. And as noted above, for the 48 tosses (in 16 trials of four flips) that had a successor, there were 24 heads, half of which were followed by heads. However, by adopting an equal-weighting scheme for each of the sixteen possible 4-flip outcomes the results become tilted. Why? Because HHHH scores as 100% confirmation of the notion that H follows H, just as TTHH does. However, HHHH walked a tightrope, since any H among the first three could have been followed by a T. TTHH only had one opportunity to do or die. So Jack could very plausibly have decided to use a weighted average, which would have brought him back to concluding that HH and HT are 50/50.

However, this discussion by Steve Landsburg (which preceded the Times article) leaves me believing that the new paper is refuting the actual weighting scheme used by the basketball people [Gilovich et al]. But leave Jack out of it! And the WSJ has lots more on the availability of new data in all sports and its use in apparently confirming a 'hot hand' effect.

This footnote leaves me wondering whether we are all on the same page:

Similarly, it is easy to construct betting games that act as money pumps. For example, we can oer the following lottery at a $5 ticket price: a fair coin will be flipped 4 times. If the relative frequency of heads on those flips that immediately follow a heads is greater than 0.5 then the ticket pays $10; if the relative frequency is less than 0.5 then the ticket pays $0; if the relative frequency is exactly equal to 0.5, or if no flip is immediately preceded by a heads, then a new sequence of 4 flips is generated. While, intuitively, it seems like the expected payout of this ticket is $0, it is actually $-0.71 (see Table 1).

Table 1 is similar to the table in the Times, but let me note that I, at least, have no obvious "intuition" about whether this bet is fair. Since the payout is either $0 or $10 regardless of the probability of a specific outcome, it means the game is not fair if the underlying distribution is skewed. "Intuitively" I would worry that if one side of the expected value represents longshots, i.e., low probability of a seemingly valuable result, then the capped payouts might be an issue. E.g., a string of four straight heads has more effect on the overall expected value calculation than it does on the payout for a hypothetical ticket. Put another way, HHHH and HHHT pay off the same even though my intuition is that HHHH "ought" to be worth more, since a 100% success rate "ought" to be worth more than a 67% success rate. Looking at the table, as they describe this game, there are only 4 ways out of 16 to win but 6 ways to lose (with 6 tickets leading to a do-over).

JUST WHAT WAS BEING REFUTED?

This is from the 1985 Gilovich et al paper (p. 7/20 .pdf), and is as close to grouping the data in blocks of four as I can find:

To obtain a more sensitive test of stationarity, or a constant hit rate, we partitioned the entire record of each player into nonoverlapping sets of four consecutive shots. We then counted the number of sets in which the player’s performance was high (three or four hits), moderate (two hits), or low (zero or one hit). If a player is occasionally hot, then his record must include more high-performance sets than expected by chance.

The number of high, moderate, and low sets for each of the nine players were compared to the values expected by chance, assuming independent shots with a constant hit rate (derived from column 5 of Table 1). For example, the expected proportions of high-, moderate-, and low-performance sets for a player with a hit rate of 0.5 are 5/6, 6/16, and 5/16, respectively. The results provided no evidence for nonstationarity, or streak shooting, as none of the nine x2 values approached statistical significance.

This analysis was repeated four times, starting the partition into consecutive quadruples at the first, second, third, and fourth shot of each player’s shooting record. All of these analyses failed to support the nonstationarity hypothesis.

As I understand this (and using the Heads/Tails notation), HHHT, HHTH, HTHH and THHH would all score equally as a high success rate. Yet in the refuting paper, those four results produce quite different scores, and are the basis for the refutation. Que pasa?

Comments

[And use a two-headed coin.]

Always call the turn.

[I dunno what tail to tell.]

Who's on first?

[Jeff Dobbs]

Well, I was involved in six coin flips last night at the pool tournament. Four heads and two tails.

[Jeff Dobbs]

When You're Hot You're Hot.

Jerry Reed, 1971

https://www.youtube.com/watch?v=0rdF7o08KXw

[Thank you, Jerry Jeff Walker back.]

Who's gonna collect my welfare? Pay for my Cadillac.

[jimmyk]

Landsburg has a good discussion of this at his site:

http://www.thebigquestions.com/2015/10/07/boys-girls-and-hot-hands/

It's not an issue of perception, it's an objective result having to do with the fact that the mathematics of probability the expected value of a ratio is not the same as the ratio of expected values. I know, you were promised no math.

[Beasts of England]

Heads!

There's no gambling at Bushwood, sir!

[maryd]

Thanks, jimmyk! This English major will treasure your explanation forever!

[maryd]

Thanks, jimmyk! This English major will treasure your explanation forever!

[maryd]

ok, jimmyk. pls explain the probability of the above.

[MarkO]

Ok. Two out of three, then.

[GMax]

I wouldn't bet on that...

[squaredance]

I do not know if someonrposted this before, as it is a couple of days old, but Kissinger has a great op-ed piece--almost an essay, really--on the ME over at the WSJ. See here.

Well worth reading.

[hrtshpdbox]

I don't know of any gambler suffering the fallacy that a bad streak must, somehow, be followed by a good one. Nor the other stated "fallacy", that winning streaks possess momentum that keeps them going. Both ideas seem like nonsense that no self-respecting gambler would have any part of. But that's not saying there's no such thing as luck, which is a completely different and undeniable animal - ask any pit boss nervously lighting matches in hopes of stopping a long run at the craps table where crowds of cheering gamblers are riding the coattails of a grinning shooter bathed in luck's light. Heck, public gaming companies that report bad earnings will sometimes cite customer's good luck as the culprit.

[Rocco]

Will the Pats score 60? Brady's Dad hopes so

http://profootballtalk.nbcsports.com/2015/10/17/tom-brady-sr-wants-the-pats-to-score-60-against-the-colts/

[jimmyk]

I don't know of any gambler suffering the fallacy

hrtshpdbox, I don't think the debate is over pure luck 'streaks' like coin tosses, roulette, etc. But there's no law of probability that rules out hot streaks or slumps in sports. That's why one has to look at the data, and use correct statistical analysis. In the analysis, the 'null hypothesis' is that free throw shooting, say, is random like a coin toss, versus the alternative that players have streaks. To do the comparison you have to understand what randomness implies for the statistics you examine. That's where some studies have gone wrong.

The interesting thing about the Landsburg link above is that apparently the jeenyuses at Google got this wrong in its test question for applicants.

[boatbuilder]

Why do they throw out two possibilities for "an insufficiency of heads?" I'm pretty much a math illiterate but if anyone told me that was an element of the analysis I'd say no thanks I'm not buying whatever it is you're selling.

[Roy Lofquist]

I get a real kick out of these guys. I'll bet their favorite sport is tiddlywinks. These are human beings we're talking about, not robots. Anybody who's ever played the game knows that some days are diamonds, some days are stone. The ball can look like a pea or a dinner plate. The basket can look like a teacup or a washtub. But I guess if it keeps them off the streets there not much harm done.

[Beasts of England]

Are we gonna have a Patriots v. Colts thread?

[boatbuilder]

Why did the line open at only Pats minus 7.5? That seems crazy to me. The bookies always seem to know something, though. A buddy says that the NFL is going to screw Brady for saying bad stuff about Coke. At this point anything seems plausible.

[Roy Lofquist]

The bookies don't know nuthin'. Eleven gets you ten. They just balance the books. If they start with Oshkosh+6 and get 10 bets on one side they change the line to get 10 bets on the other side. They don't care who wins. They collect their 10% either way.

[Miss Marple 2]

The only local news I can report is that Jim Irsay requested fans not fly a blimp referring to the Brady incident over the stadium today.

[Miss Marple 2]

Second bit of local news is that they are letting it be known that Luck will probably play today.

If he doesn't, that big investment in him will be shown to have been very foolish. You will remember that Peyton Manning was odds on not to get through another season or two without serious injury.

So much for odds.

[boatbuilder]

Was Irsay concerned that the cold temperatures might deflate the blimp?

[Miss Marple 2]

boatbuilder,

HA! I think he was concerned that Indianapolis would be poking the Patriots in the eye with a stick.

What fans would spend money for a blimp?

NUTS, I say!

[Jane]

Rocco,

What time and channel are the Pats on? I can't for the life of me figure out how to find the programming on Comcast.

[Tom Maguire]

NO MORE FUZZY MATH: OK, there is a Reader revolt in the comments section at the Times. One example:

There is an obvious mistake in the coin toss example. The reason that the authors came up with a 40.5% probability that a heads follows a head is because they took a simple average rather than a weighted average. Each sequence has a different number of opportunities and this is information that needs to be included in the calculation. For example, HHHH has 3 opportunities for heads to follow heads (HH=3 + HT=0) while HHTH has only 2 (HH=1 + HT=1). If you weight each sequence by the number of opportunities, you will get the 50% probability that you expect by intuition.

That was my halftime insight, too late to make the Times comments.

[jimmyk]

I'm not sure that comment is a refutation. The exercise, as I understand it, is to look at a sequence of four coin tosses, and calculate the expectation of the percentage of times that an H is followed by an H (meaning that you throw out the observation if the first three tosses are T). Each of the 16 outcomes is equally likely, as is each of the 14 with an H in the first three tosses. So weighting isn't the right thing to do for that exercise.

[Dave (in MA)]

Jane, Sunday Night Football, 8:30 NBC
Next week, your Jaguars are on at 9:30 AM, live from London.

[Strawman Cometh]

https://www.google.com/webhp?source=search_app&gws_rd=ssl#safe=off&q=A+Path+Out+of+the+Middle+East+Collapse
Kissinger

[sbw]

The Kissinger article was a clear and useful read.

[Man Tran]

My favorite 'luck' story was from a Montecarlo Night at the Cowlumbus Porsche Club that was hosted by a buddy in his photo studio. We were sitting at the blackjack table and having a 'random' luck experience all around. The dealer was eventually replaced by Martha, a local business owner of some success and a decent event driver. Not a wallflower by any means. Well, she cleared the table in about 20 min. and killed the game for the evening. I've never seen anything like it, except for in the movies. (No money involved.)

[clarice]

jimmyk and the other math heads here???????????????

[Jane]

I have Jaquars? Who knew?

[Dave (in MA)]

You get to pick from among 3 teams while we're stuck sharing ours with 5 other states.

[Sweet little puddytats.]

It's OK, Jane, these ones don't eat dogs.

[Jane]

I pick the Pats. Amy just sent me a pix of it snowing in Sturbridge. It's only 79 here.

[Extraneus]

You get to pick from among 3 teams while we're stuck sharing ours with 5 other states.

Too bad one of those states isn't New Jersey, because the fans of both NY teams who play in NJ get to call those teams as their own.

[glasater]

But.." Two out of three"..

♪♫•*¨*•.¸¸ain't bad.. ;-)

[mockmook]

jimmyk,

I would submit that Landsburg has a terrible discussion of the Girl/Boy problem.

He gives the Google problem and then "refutes" it with a totally different problem.

Can you "simply" explain why the ratio for Boys and Girls in the Google problem is not 50%?

(If a small population makes the difference stark, how about an explanation using just 4 couples.)

[Red Zone.]

Time out the hot hand. Can't have it overinflating the ball.

[jimmyk on iPhone]

Mockmook, will try later. Busy watching Mets-Cubs and Homeland. The boy-girl thing is the same idea as the coin toss--the average of a ratio is not the same as the ratio of the average.

[Thomas Collins]

See Miss Maple, 2? I told you I was being generous. In any event, it's a great game, and I have enjoyed the treats you sent me, so no hard feelings no matter what happens!

[mockmook]

jimmyk,

Appreciate the consideration of my dilemma -- I'm sure I'm missing something "basic" (dare I say, some nuance).

[mockmook]

jimmyk,

BTW, I'm sure part of my problem is I don't understand what you are getting at in this sentence:

"the average of a ratio is not the same as the ratio of the average"

[Mahon]

The existence of a hot hand implies the existence of a cold hand, so the test should be how many times is an H or T followed by another instance of the same result. I'm pretty sure this would even things out. As widely noted, the reason they are getting skewed results is that they are working with short strings.

[mockmook]

Mahon,

I believe that is the point -- hot hands are followed by cold hands -- they must even out.

But a coin is always 50/50, no hot nor cold hand (but, of course, they still even out).

[stan]

The notion that shooting a basketball is a random event is ridiculously stupid.

There are days an athlete is on and days when he is off. When Tiger Woods was the best player in the world, he hit shots closer and made putts more often than he does today. It ain't random.

There are days when a pitcher has better command and days when he doesn't. Try to tell him that his pitches are all random on both kinds of days and he will laugh his ass off at you. Because mastering the fundamentals and techniques of a skill is not something that occurs equally at all times.

[mockmook]

stan,

Yes, a more interesting question is the time period for these "hot" periods (and perhaps that is what the studies were touching on).

Does your typical basketball player have hot streaks within a game? Or, are his hits and misses mostly random within a game (or better yet, within a half)?