The "Imperfect Casino" Analogy to Hot Hand Perceptions: A Guest Contribution by Mark Lloyd

Mark Lloyd of Seattle, Washington recently e-mailed me the following essay of his. I invited him to let me post it here on the blog as a guest contribution, and he agreed. Here it is, with light editing.

A couple of years ago I wrote a computer program simulating a basketball team with players' shooting percentages varying on different days according to distributions with a mean of 50 percent, but with different variances. I then tested the strategy of giving the ball to players who made two shots in a row compared to giving the ball to players randomly. In the long term giving the ball to the hot hand was a winning strategy.

Thinking about this recently I came up with an analogy that makes clear the difference between the casino and hot hand theory and the mistakes many statisticians make assuming that believing in the hot hand is just the same as the gambler's fallacy of believing that one should bet on a roulette wheel that has shown a streak of results.

Let’s say we have an imperfect casino with five roulette tables that, if in perfect condition, select red and black 50 percent of the time. The casino’s roulette tables however are in disrepair, and further, the casino sits above a subway line that occasionally shakes the tables, causing them to vary their percentage of hitting black or red from 40-60 to 60-40 with a mean of 50-50. There is no way to observe the condition of the roulette wheel except to observe the results of wagering on the tables.

With most visitors not having the time to do a long-term statistical analysis of the tables, is wagering on a table that has hit red or black twice in a row a better strategy than randomly betting on any color on any table? Clearly yes: A table that is hitting red 60 percent of the time will hit red twice in a row 36 percent of the time, while a table hitting 40 percent will only hit two in a row 16 percent of the time. With the tables at some range between 40-60 and 60-40, this two-in-a-row percentage will vary between 16 and 36 percent. More often imperfect tables that hit two in a row will have a higher winning percentage.

If every day you walk in to the casino, wait for a table to hit a color twice in a row, then bet on that table and color, over time you will do better than if you randomly bet on a random table and color.

A real-life non-statistician watching for a hot hand in a basketball game may well be thinking something like: “He hit two in a row, I think he is shooting better than average today.” The non-statistician has also made a correct observation that at least in the case of basketball, daily shooting percentages vary more than one would expect randomly. In the course of a game there is limited information to measure this variance, so looking for shooting streaks is an imperfect way to find players who have a higher underlying skill that day. The statistician's job is to operationalize that observation.

What statisticians will observe is that in the short term on any given day, if you measure any roulette table in the imperfect casino, the percentage of hitting a color after hitting the same color twice in a row will not be different than after any other sequence (unless a subway train passes underneath). This is the same as observing a basketball player on a given day (or a given hour depending on that shooter's pattern of consistency). This observation misleads the statistician into believing the hot hand is no different than a casino winning streak in a perfect casino, but the statistician is asking a different question than the non-statistician.

The statistician should be asking how the percentage of streaks varies from day to day; this will more closely operationalize what the non-statistician is observing and make the statistician wealthier as well.

The world of sports is a world of imperfect casinos. This confounds statisticians.

Yankees' Scoring Drought in ALCS

The Detroit Tigers have now completed a sweep of the New York Yankees in the American League Championship Series to advance to baseball's World Series. Beyond the fact of the Yankees being swept in a playoff series -- the first time this has happened since 1980 -- an additional noteworthy aspect of the ALCS is New York's extreme difficulty scoring runs.

As shown in the table below (for which I consulted the Yankees' game-by-game log), the New Yorkers put zeroes on the scoreboard for 36 of the 39 innings played. (You can click on the graphic to enlarge it.) Another way to look at the situation is that the Yankees scored in only around 8% of the innings of the ALCS.

Over the past few minutes, as I've been writing, the announcers on tonight's Giants-Cardinals National League Championship Series game have been discussing the Yankees' demise. According to these announcers, New York's team batting average against Detroit was a woeful .157.

Of course, the Tigers have excellent pitching, led by Justin Verlander. Therefore, the Yankees' poor offense in the ALCS might be understandable to some extent. An interesting comparison (to me at least) would be to look at how the Yankees did against Detroit in the regular season. Listed below are all the Yankee-Tiger regular-season games this season and the number of innings per game in which New York scored.

April 27 -- 6 of 9 innings
April 28 -- 3 of 9
April 29 -- 4 of 8

June 1 -- 4 of 9
June 2 -- 3 of 9
June 3 -- 3 of 9

August 6 -- 1 of 9
August 7 -- 3 of 9
August 8 -- 6 of 9
August 9 -- 2 of 9

So, in contrast to the ALCS, New York scored fairly readily against Detroit in the regular season. In fact, the Yankees scored in 35 of the 89 total innings (39%) in which they batted against Tiger pitching. Thus, the Yankees did not seem to be at an inherent disadvantage against Detroit pitching in the ALCS. New York just got cold.

Smith's INT-Free Streak Persists, But WVU Crushed

Here's a follow-up to my previous posting on West Virginia quarterback Geno Smith and his streak of not having his passes intercepted. He attempted 55 passes today, none of which was picked off by Texas Tech (29 of his passes were caught and 26 fell incomplete). Smith thus extends his streak to 312 interception-free throws since he was lasted intercepted, vs. South Florida on December 1, 2011. Today was hardly a good day for the Mountaineers, however, as Texas Tech dominated throughout, 49-14.

UPDATE: Smith avoided an interception for his first 13 passes against Kansas State on October 20 (9 complete, 4 incomplete), before finally being picked off by the Wildcats. His streak of interception-free throws thus ended at 325.

West Virginia QB Geno Smith Brings Long Interception-Free Streak to Game at Texas Tech

West Virginia quarterback Geno Smith, whose team plays Saturday here at Texas Tech, has an impressive interception-free streak going. Looking at Smith's player page at ESPN.com, he has played five games this season, attempting 204 passes and completing 166 of them (81.4%), without being picked off a single time.

In Smith's final game of last season, a 70-33 spanking of Clemson in the Orange Bowl, he threw the ball 43 times (completing 32 passes) without an interception. In the game before that, a 30-27 Mountaineer win at South Florida, Smith was picked off twice. Looking at the play-by-play sheet, Smith's final interception came with 9:56 remaining in the fourth quarter, a "pick-six" (interception return for touchdown). Smith rebounded, however, to drive WVU to a touchdown and game-winning field goal. During these two drives, he threw a combined 10 passes, of which 7 were completed. (One West Virginia play was listed as "Team pass incomplete," which may have involved immediately spiking the ball to stop the clock and set up the final field-goal attempt; I'm not including this in Smith's statistics.)

All told, since his last interception, he has attempted 257 passes (204 this season + 43 in the Orange Bowl + 10 at the end of the South Florida game) and completed 205 of them (166 this season + 32 in the Orange Bowl + 7 at the end of the South Florida game).

The NCAA record book for top-tier (Football Bowl Subdivision) programs features this entry:

MOST CONSECUTIVE PASSES ATTEMPTED WITHOUT AN INTERCEPTION

379—Russell Wilson, North Carolina St., 2008-09

Thus, if Smith continues to average around 40 pass attempts per game and doesn't throw any interceptions, he could surpass Wilson in three more games. For Smith's entire collegiate career, he's thrown 15 interceptions in 1,151 passing attempts (a little over 1%). If we assume, accordingly, that each future pass he throws has a .99 probability of not being intercepted, then his probability of throwing another 122 attempts without an interception (to tie Wilson) is .99 to the 122nd power. This yields roughly a .30 chance of Smith tying Wilson.

Some observers may question the value of attempts without an interception, as under this metric a quarterback receives credit for throwing a ball away when no receiver is open. I (and others) have tried to pinpoint the NCAA record for consecutive completions without an interception, without getting final resolution. According to a posting on this West Virginia fan site, Smith is not too far behind some leading figures on consecutive interception-free completions, namely Wilson at 220 and Andre Woodson (Kentucky, 2004-07) at 223. (The fan page lists Smith with 206 straight interception-free completions, a slight discrepancy with my calculation.)

When, if ever, will Smith's collegiate interception-free streak come to an end? Vote in the poll in the right-hand column.

Koji Uehara's 25 Straight Retired Batters

A few nights ago, as the Texas Rangers brought in pitcher Koji Uehara for a late-inning appearance, one of the announcers on the telecast I was watching argued that Uehara was as hot a pitcher as was currently going in Major League Baseball. Naturally, when I hear a claim like that, I have to check it out!

After a few days had gone by, I examined ESPN.com's statistical page for Uehara. I also searched for recent articles, one of which confirmed that before the streak ended (on October 1), Uehara had retired 25 consecutive batters. To put things in perspective, a perfect game involves retiring 27 straight batters (i.e., 3 batters in each of 9 innings).

In honor of Uehara's accomplishment, I have listed the 25 batters that he sat down consecutively (along with the type of out each made) to the right.

As I discuss in my book Hot Hand, then-Chicago White Sox pitcher Mark Buehrle holds the record for consecutive batters retired, at 45. The leaders in this category include starters (such as Buehrle) and relievers such as Bobby Jenks (41). In my book, I asked whether a streak of consecutive batters retired would be easier for a starter or reliever to achieve, citing the following considerations:

"A starter can... build up the streak in fewer games, but must have the stamina to keep pitching at a high level throughout the game.  A reliever, on the other hand, may know that he is only going to pitch an inning or two on a given night, which would allow him to concentrate on retiring just a few opposing batters (going “all guns a blazing” with each one). A reliever, though, would have to keep this up for perhaps 20, 30, or more appearances to contend for this record" (p. 93).

I invite readers to share their views on this matter, in the Comments.