## May 06, 2003

### Elementary Statistics

Eugene Volokh writes:

Volokh Conspiracy: People are condemning Bill Bennett, who has taken on the role of a spokesman for virtue and morality, for what seems to be a gambling habit that has lost him \$8 million over the last ten years.... Nonetheless, Bennett suggests that he's "come out pretty close to even," though others doubt this...

Suppose you play the \$500 slots 20,000 times--thus betting ten million dollars in total--on slot machines that are programmed to keep 10% of the take (which is a quite low house percentage for slot machines), and suppose that the standard deviation of payoffs is \$1350 (a large number, but then slots do give some very large payoffs). How likely is it that you have come out "pretty close to even"?

Anyone a third of the way through their first course in statistics will know that 20,000 times is enough to safely apply the central limit theorem, and thus will be able to quickly and easily figure out that:

1. Your expected loss is \$1 million even.
2. There are only five chances in a hundred that you will have lost more than \$1.3 million.
3. There are only five chances in a hundred that you will have lost less than \$686 thousand.
4. There is only one chance in a hundred that you will have lost less than \$556 thousand.
5. There is only one chance in a thousand that you will have lost less than \$410 thousand.

But the casino estimates are that Bennett has dropped eight times as much. Suppose that you had pulled the slots not 20,000 times but 160,000 times. Then the central limit theorem tells us (with a \$1350 standard deviation and a 10% house cut) that:

1. Your expected loss is \$8 million even.
2. There are only five chances in a hundred that you will have lost more than \$8.89 million.
3. There are only five chances in a hundred that you will have lost less than \$7.11 million.
4. There is only one chance in a hundred that you will have lost less than \$6.74 million.
5. There is only one chance in a thousand that you will have lost less than \$6.33 million.
6. There is only one chance in ten thousand that you will have lost less than \$5.99 million (but here the central limit theorem starts to break down: it can't be trusted when the probability of an event comes close to the inverse of the number of trials).

The thing that people who paid attention in their statistics classes know--and that people who did not, don't--is how quickly and viciously the central limit theorem bites. Either (a) William Bennett is the luckiest man who ever lived, or (b) William Bennett's claim to be "pretty close to even" is the biggest lie* Washington has seen since Richard Nixon claimed that he was not a crook.

Yet Eugene Volokh does not write "the odds are overwhelming that Bennett's claim to have gambled heavily on slots and come out pretty close to even is completely false." Instead, he writes a standard "he said, she said sentence: "Bennett suggests that he's 'come out pretty close to even', though others doubt this..." The "others" don't just "doubt" this, they know that this is false with a probability so high as to be effectively indistinguishable from certainty.

Why oh why is there such a lack of statistical literacy among such regularly intelligent people? Why oh why is Bennett's claim being taken by surprisingly many people at surprisingly close to face value? I haven't seen anything like this since... since... since... since the claims that Hillary Rodham Clinton had not received significant and valuable... "unusual"... favors from her commodities brokers.

But what really makes me want to bang my head against the wall is not my belief (say) that UCLA's senior law faculty (say) (along with many, many, many others) needs to be dragooned into an elementary statistics classroom, forced to learn the central limit theorem, and then have their tenure revoked if they cannot use it effectively. What really makes me want to bang my head against the wall is that my part of Berkeley is--right now, as we speak--failing to correct this. We make our undergraduates take statistics. Most of them hate it. Large number of them flush it from their brains even before graduation. They thus throw away one of the most powerful and effective mental tools for grappling with the world that humanity has yet invented. They don't seem to care. And this is our fault: somehow we aren't able to strike sparks, to light a mental fire, in all those students sitting there thinking, "God! This is boring! Why the f*** do they require this?"

*Virginia Postrel points out that Bennett's hypocrisy and lack of empathy--that in spite of having an enormous jones himself for the slots, he has nothing but contempt for those who are trying to deal with similar joneses for heroin, cocaine, et cetera--is not his principal vice. His principal vice is a desire to put his name on the cover of books that other people wrote. I think she is right. To be a victim of one form of bad neurochemistry and bad neural wiring and yet fail to show any empathy for others with similar forms is certainly a reason to be subjected to a two-minute public hate. To hold oneself out as a public intellectual and yet to not write one's own books is a much better and more important reason.

Posted by DeLong at May 6, 2003 12:27 PM | TrackBack

What gets me is that with the money he's funneled into his little gambling jaunts, he could have funded a number of private (and, presumably, faith-based) programs to help those at whom he preaches to kick the habits of whatever vile sins they've plumbed. And yet, Bennett's priorities are only to ejaculating page upon page, TV appearance upon TV appearance to preach -- but never, of course, to put up his money to fund any of those wonderful "thousand points of light" that conservatives profess to adore above government-funded public programs.

Posted by: Jaquandor on May 6, 2003 12:46 PM

Actually, as I remember it, the house's take on slots is closer to 1.5-2%. Why not? The point is to keep the suckers playing as long as possible - bleed them slowly rather than go for the jugular.

Otherwise, spot on. I was willing to cut him some slack on the gambling (I play craps, myself, on occaision) But his comment about "breaking even" is either a patent lie or monumental self-delusion. In either case, not a pretty sight...

Posted by: jimbo on May 6, 2003 12:50 PM

You don't need very sophisticated statistics to be pretty sure that Bennett was lying. You just have to know a few gamblers and understand how they keep track of, and report, their results. Or alternatively, you can just ask yourself, "Who is paying for all those buildings and all that free entertainment and food in Vegas, where the gambling houses have been giving away money to my gambling friends for years?"

Or you could ask someone in the biz what the house take is.

A common observation on this episode is that Bennett has chosen the second-stupidest of the stupid forms of gambling -- slot machines. (Scratch-off is the stupidest). With slots there is and extraordinarily low possibility of winning, and no skill whatsoever. Slots aren't even social and glamorous like roulette. (Hm. "Bowling alone"? Bennett didn't write that one, did he?)

Poker, blackjack, backgammon, and even betting at the track actually have a skill component. Especially in the first three, in the long run the better players win. But slots is a moron game for people who aren't happy with their lives.

Posted by: zizka on May 6, 2003 12:52 PM

Hear, hear! on the culpability of stat professors. I just finished a Stat class I was required to take for an M.S. in Comp. Sci. and it was dry as dust despite a good professor. The course amounted to memorizing which set of algebraic transformations corresponded to which word problem template.

A better class would have taught the algebra in the context of rich examples. It would have involved a lot of data gathering and reasoning about which conceptual tools to apply to real world problem. It would have been much more work for the teacher and much harder to write a generic exam for. It would have vastly increased the received value for most students at the expense of slightly decreasing the value for future PhD.s. In other words, there is no way in hell it would be taught that way...

Posted by: Sam (spenrose at well.com) on May 6, 2003 12:58 PM

Poker and backgammon, yes - but not blackjack. There are better players and worse players, but unless you're counting, the odds are against you (which is why counting gets you kicked out). Rule of thumb: anytime you play aginst the house, the house always wins...

Posted by: jimbo on May 6, 2003 12:58 PM

I is Virtue. Hear me roar about all who is less virtuous than I. Such is the virtue we would be subject to by such frauds.

Posted by: jd on May 6, 2003 01:07 PM

"A common observation on this episode is that Bennett has chosen the second-stupidest of the stupid forms of gambling -- slot machines. (Scratch-off is the stupidest). With slots there is and extraordinarily low possibility of winning, and no skill whatsoever."

Still, Billy Billy managed to break even. We just will not get to see all those marvelous winnings, wiil we?

Posted by: lise on May 6, 2003 01:17 PM

Better to be putting coins in machines than cigars in ...

Posted by: brian on May 6, 2003 01:21 PM

Let me play the spoiler here by noting the following:

1 - If the house take is indeed in the 1.5-2.0% range suggested, then Bennett's expected losses are either \$200,000 or \$1.6 million, depending on which of Brad's scenarios one buys into. Bennett may well be wealthy enough not to regard such losses as particularly serious.

2 - Given that these losses will have accrued over a time period of several years, it is easy to imagine that Bennett simply wouldn't have taken notice of them. Bleeding a little over a long timespan is very easy to do.

3 - Rather than accuse people of "lying", which Dr. DeLong seems to have an unfortunate tendency to do, might it not be the case that the man is simply in denial about the results of his actions? Who doesn't rationalize his or her own failings on occasion? As a rhetorical technique, accusing people outright of fraud or dishonesty may be emotionally satisfying, but it only serves to alienate those who might be otherwise convinced by one's arguments.

Having said this much, I'll also say that I hold no brief for Mr. Bennett, who I have considered a windbag for a very long while. I generally dislike people who appoint themselves arbiters of moral virtue, especially when they want to use the force of government to impose said "virtues" on others. It is both true that Bill Bennett's gambling shows him up as a moral hypocrite, and that most supposedly insightful intellectuals have a scandalously poor grasp of statistics.

Posted by: Abiola Lapite on May 6, 2003 01:52 PM

>>Let me play the spoiler here by noting the following: 1 - If the house take is indeed in the 1.5-2.0%...<<

Ah. But the \$8 million written-to-cover-losses number we appear to have from the casinos is not the total amount wagered. The (few) people I know who know anything about slots claim the odds are bad because casinos think slot players are stupider than roulette or blackjack players (albeit not as stupid as state lottery players). But even if the odds aren't as bad as I think (and this may well be the case), a 2% house cut implies that Bennett's total wagers are not \$80 but \$400 million... and the expected loss distribution is much the same...

Posted by: Brad DeLong on May 6, 2003 01:57 PM

>>Let me play the spoiler here by noting the following: 1 - If the house take is indeed in the 1.5-2.0%...<<

Ah. But the \$8 million written-to-cover-losses number we appear to have from the casinos is not the total amount wagered. The (few) people I know who know anything about slots claim the odds are bad because casinos think slot players are stupider than roulette or blackjack players (albeit not as stupid as state lottery players). But even if the odds aren't as bad as I think (and this may well be the case), a 2% house cut implies that Bennett's total wagers are not \$80 but \$400 million... and the expected loss distribution is much the same...

Posted by: Brad DeLong on May 6, 2003 02:00 PM

Aha! Even the host himself gets caught by his balky posting software!

Posted by: jimbo on May 6, 2003 02:02 PM

'They don't seem to care. And this is our fault: somehow we aren't able to strike sparks, to light a mental fire, in all those students sitting there thinking, "God! This is boring! Why the f*** do they require this?'

As someone who recently refreshed my knowledge on statistics after a few years, I think I know the answer: too many trees, not enough forest.

I'd estimate that, oh, maybe 10% of the stat book I have here in front of me is big-picture issues like you discuss. It's a miracle they remember anything at all, with all the matrix inversions and gruntwork equation solving they're required to do.

Posted by: Jason McCullough on May 6, 2003 02:12 PM

400 million dollars doesn't seem like a credible amount to have wagered; it's just too big. Assuming that he has spent 10 hours a month for the last 20 years gambling, that means he would have had to wager \$167,000/hour to wager 400 million. Or, if you assume that each pull on the slot machine takes 10 seconds, he would have to find a machine that lets him risk \$460 per pull. This means that either a) he didn't wager that much, and the reports of his losses are therefore wrong, or b) he didn't primarily play slots, but rather some other, higher-risk game. The news reports just don't hang together if you assume both that he was playing a game with a 2% house cut and he suffered 8 million dollars in losses.

I am willing to learn that such bizarrely expensive slot machines exist, but right now I think you need to double-check your order-of-magnitude estimates.

Posted by: Neel Krishnaswami on May 6, 2003 02:20 PM

I agree: it's genuinely *hard* to lose \$8 million in a decade on slot machines that only have a 2% house take...

He does play \$500 slots, or at least so the reports say...

Posted by: Brad DeLong on May 6, 2003 02:27 PM

This willingness of people to say that, "gee, maybe he did break even," is incredibly annoying.

There are two versions of this. First, the mindless "he said, she said" that Brad describes. Then there's the use of the break-even possibilty by Bennett's defenders, who otherwise would have to admit that whatever they think about gambling, their much-revered Bennett is either a liar or a total fool.

Posted by: Bernard Yomtov on May 6, 2003 02:32 PM

This willingness of people to say that, "gee, maybe he did break even," is incredibly annoying.

There are two versions of this. First, the mindless "he said, she said" that Brad describes. Then there's the use of the break-even possibilty by Bennett's defenders, who otherwise would have to admit that whatever they think about gambling, their much-revered Bennett is either a liar or a total fool.

Posted by: Bernard Yomtov on May 6, 2003 02:37 PM

This willingness of people to say that, "gee, maybe he did break even," is incredibly annoying.

There are two versions of this. First, the mindless "he said, she said" that Brad describes. Then there's the use of the break-even possibilty by Bennett's defenders, who otherwise would have to admit that whatever they think about gambling, their much-revered Bennett is either a liar or a total fool.

Posted by: Bernard Yomtov on May 6, 2003 02:42 PM

\$500-a-pull slots do exist; Caeser's Palce seems to offer them, so I assume other casinos do as well.

Posted by: Steve on May 6, 2003 03:00 PM

Posted by: Steve on May 6, 2003 03:05 PM

One should never argue over matters of fact.

Payout Percentages: The payback percentage on slots ranges between 80% to as high as 98%, depending on coin size and local competition. Generally, the loosest slots can be found in Las Vegas. Nationally the average house advantage is calculated to be around 10%. Be aware that payout percentages are calculated over the long term, which could mean the life of the machine. If a slot machine is programmed to pay back 95% it will pay out \$950,000 out of every one million dollars it takes in, but payouts will come in wildly fluctuating random fashion. A few lucky players will collect big jackpots, most others will lose all they put into the machine.

And from www.mastering-slot-machines.com/slot_machine_tips.html

Finding the best slot machines is often quite a task. There are many myths propagating around placement within casinos of tight and loose slots. Those are just myths. If you want to find machines that have higher payout percentages then look for banks of machines that are advertising specific percentages just for that bank. If you want to know which casinos offer the highest payout percentages then look no further. The wizard of odds has put together a list, based on actual experience, of which casinos in Vegas return the most on their slots.

5 cent machines
Rank Casino Average Return
1 Palms 93.42%
2 Gold Coast 92.84%
3 Sahara 92.81%
4 (tie) Bourbon Street 92.63%
4 (tie) Imperial Palace 92.63%
4 (tie) Slots a Fun 92.63%
7 Key Largo 92.6%
8 Western 92.57%
9 Ellis Island 92.56%
10 El Cortez 92.56%

Feel free to recalculate the distribution of expected losses.

Posted by: Jack Needleman on May 6, 2003 03:40 PM

I think that, beyond the question of the net payoff percentage (90% -- 98%), there's a skew due to the fact that much of the payoff is tied up in a few enormous jackpots. So if a million \$1 pulls pay off \$950,000 (95% payoff), but (let's say) \$200,000 is tied up in one single payoff, then the payoff for everyone but that one person is 75%.

I know scratch-off works that way, I don't know about slots.

Posted by: zizka on May 6, 2003 05:48 PM

It seems to me that the main problem with taking statistics is that the first course is always an introductory course for a statistics major, rather than a survery course for those who will not be statiticians but rather use statistics occasionally in the course of their work. A good survey course would provide the 'big laws' of statistics. It would provide ways to assess statistics - which are likely to be bogus and which are not. It would provide some of the check points - how big a sample does it take to get how much accuracy. It would provide a few statistical tools which a non-statatician can safely and accurately use. Etc.

Posted by: secular clergyman on May 6, 2003 07:16 PM

>>I think that, beyond the question of the net payoff percentage (90% -- 98%), there's a skew due to the fact that much of the payoff is tied up in a few enormous jackpots.<<

The skewness is real, but the central limit theorem does bite hard.

If you are playing a \$500 slot in which all of the prize is in a huge jackpot--an 0.05% chance of a near-million dollar prize--then by the time you get up to 20,000 pulls (so that you expect to have hit ten jackpots) almost all of the skewness is ironed out of the distribution.

Posted by: Brad DeLong on May 6, 2003 08:21 PM

...(albeit not as stupid as state lottery players).

There is a misunderstanding of gambling. The assumption expressed by the above aside is that wagers require a positive expectation of return to be sound and the greater the negative expectation the more stupid the wager.

Let's say you were allowed each week to buy one \$1 lottery ticket for a \$2 billion dollar jackpot
and in this lottery you had a one in one billion chance of winning with each ticket.

The alternative is buying one \$1 ticket each week
in a lottery paying \$10 million, each ticket having a one in 20 million chance of winning.

The reason to play the \$10 million lottery is not just the greater liklihood of winning in your lifetime, rather for most people that first \$10 million has most of the total value of that \$1 billion.

A statistician may choose to play the lottery with the overlay or not at all. But the game with
the negative expectation is the better value for some of us. Of course we would be wiser to save our wagers over the course of two weeks and invest thereafter in a sure thing like a tall Starbuck's coffee.

Posted by: CMike on May 6, 2003 08:55 PM

Bennett is up against more than the central limit theorem.

First, if BB really were breaking even over so much play, the house dicks would be swarming all over him with intensified video surveillance, electronic sniffers, read-outs and forensic tear-downs of the affected machines, and investigations of collusion with the machine makers and maintainers.

Second, BB claims to have paid taxes on his winnings, and tossed some of the rest to charity. If he's paid ~40% taxes on the winning sessions, without netting out the losing sessions, he's a bigger fool than we'e given him credit for. If he's paid taxes on the annual net, he (or his accountant) has detailed records and summaries. He KNOWS how much he lost. And he therefore knows he never broke even.

It's also unlikely he netted positive returns in any given year ... thus it's likely he's lying about his taxes (though in a truth-evasive way, not a tax-evasive way).

But Mrs. BB doesn't know. Something unvirtuous about that. \$8M went away ... \$8M that could have gone to BB's heirs, or could have gone to charity, but didn't.

In some of my past lives I've rubbed elbows with gambling addicts. Most all of them "double up to get even", and most of these end up stealing from somebody somewhere somehow (usually embezzling, and planning to pay it back out of their winnings).

And his comment that people want to talk too much in the non-solitary games. Well, no. No they don't. Not in \$500-a-throw games of skill and chance ... like the poker tables BB would frequent if he had the skills to minimize the house edge in video poker. They come to play.

BB's preference for high-stakes solitary games are telling. He knew he had a problem ... a problem with losing, and not being able to help losing, and letting others see him in this position.

An earlier comment supposed he might have spent 10 hours/month pulling slots over a period of years. Unlikely. A steaming out-of-control gambling addict is going to spend more than ten hours a month scratching his itch.

Can he break this habit cold turkey? Probably not.

For several reasons, the story is probably not over yet.

Posted by: RonK, Seattle on May 6, 2003 09:52 PM

Maybe there's even more need for "One Hundred Interesting Statistical Calculations" than "100 Interesting Mathematics Calculations." Just a suggestion...

Posted by: Andrew Boucher on May 7, 2003 12:34 AM

--------------------------------------------
If you are playing a \$500 slot in which all of the prize is in a huge jackpot--an 0.05% chance of a near-million dollar prize--then by the time you get up to 20,000 pulls (so that you expect to have hit ten jackpots) almost all of the skewness is ironed out of the distribution.
---------------------------------------------

When the expected number of successes is this small, a poisson distribution provides a much better approximation than a normal distribution. In the scenario described, it would only take 11 jackpots (one more than the expected 10) to break even. According to the Poisson approximation there is an over 40% chance of this occuring. In fact, you have an over 8% chance of hitting 15 or more jackpots and really making the big bucks. Also, I would say the distribution still has some non-negligible skewness.

I don't know what the true distribution of payoffs is for the machines Bennett played, but maybe Prof. Volokh's was right to reserve judgement on this question.

Posted by: ed johnson on May 7, 2003 02:44 AM

But \$500 slots don't put all--or even 1/10--of their payoff into a single million-dollar jackpot. And at least if the casinos have been believed Bennett has pulled the lever much more than 20,000 times...

Posted by: Brad DeLong on May 7, 2003 06:25 AM

Okay, I concede that \$500 slots exist. Color me amazed -- I guess it's clear that I just don't understand the attraction of gambling. One pull, at \$500, will buy you the best possible dinner for two at a three-star Michelin restuarant. Three will buy you the plane tickets to France. Five will let you fly first-class.

Can a slot machine really be as much fun!?

Posted by: Neel Krishnaswami on May 7, 2003 07:29 AM

The expression "\$500 slot" may be misleading in this context, if we interpret it as a machine in which the player risks \$500 on one pull of the handle.

"Nickle slots" typically give the player the option of playing one, two, or several \$.05 pieces on a pull. Payoff odds are typically maximized when the maximum number of coins are inserted.

I've never bothered to wander over and look at the \$500 machines -- slots are not interesting, just stupid -- but I would assume the same is true of these machines ... that for best return odds (and probably as a strict requirement for jackpots), the player will stake (say) \$2500 on each pull.

Can anyone confirm?

Posted by: RonK, Seattle on May 7, 2003 08:24 AM

A biology professor of mine in college was lecturing us on statistical tools one day and made a comment that I think rings with truth. He said that nobody ever really learns statistics until they have data that they care about and need to analyze.

The point being that you can require undergrads to take courses in statistics, but it all just washes away over time. If the same undergrads go on to do graduate work or hold jobs that require them to do statistical work, they manage to learn, and learn somewhat permanently, the math they need just fine. They don't get smarter after college, and the material doesn't become any more interesting, but suddenly its no longer a seemingly silly paper and pencil game about the manipulation of symbols, but a powerfool tool for making sense of the world.

So maybe the answer is to require the undergrad stats class, but then to also require the application of stats in some kind of big project (senior thesis?).

Posted by: sd on May 7, 2003 08:54 AM

"slot machines that are programmed to keep 10% of the take (which is a quite low house percentage for slot machines)"

From my experience, most slot machines in Vegas are programmed with a payout % ranging from @94%-97%. Surprisingly, a few slots actually have payout that exceeding 100% and these machines used to be placed in high traffic areas to encourage business for other slot machines. Low payout percentages of the kind described by Dr. DeLong are more likely to be found at nickle slots in Atlantic City. I would be very surprised if such low payout rates exist for the high-roller slot machines as I would expect would have higher than average payout percentages given the amounts being wagered.

Posted by: Kevin H on May 7, 2003 08:56 AM

"slot machines that are programmed to keep 10% of the take (which is a quite low house percentage for slot machines)"

From my experience, most slot machines in Vegas are programmed with a payout % ranging from @94%-97%. Surprisingly, a few slots actually have payout that exceeding 100% and these machines used to be placed in high traffic areas to encourage business for other slot machines. Low payout percentages of the kind described by Dr. DeLong are more likely to be found at nickle slots in Atlantic City. I would be very surprised if such low payout rates exist for the high-roller slot machines as I would expect would have higher than average payout percentages given the amounts being wagered.

Posted by: Kevin H on May 7, 2003 08:57 AM

I'm not eager come to Bennett's defense, but given what I've read in several places about the typical payout of high-stakes slot machines, don't you think that a 7% loss could be reasonably asserted to be "pretty close to even"? For comparison's sake, I (along with many other Americans) lost more than 40% of the money I had invested in the stock market over the last five years. In the popular mythology about gambling, I think it's expected that many gamblers lose *all* their money. In this context, losing only 7% seems to me to be "pretty close to even".

Really, if Bennett is a gambling addict, he's been pretty smart to restrict himself to a variety where his capacity to wager far exceeds the external limits of his doing so. If he were able to gamble arbitrarily large amounts of money -- like, say, speculating in the stock market -- he very easily could have lost it all.

Finally, I'm not clear on what the reported \$8M figure represents.

Posted by: Keith M Ellis on May 7, 2003 09:34 AM

keith -- It's been made clear elsewhere the \$8M represents documented NET losses at specific casinos over specific time intervals.

I suppose if he'd wagered \$1B and lost only \$8M you could call it "close to even" ... but that's not the nature of slots play. You put tokens in, you get (an average of 7% fewer) tokens out, you put these tokens back in, you get (7% fewer) tokens out, cycling and compounding until you run completely out of tokens or decide "hmmm, it's just not my day".

If you're lucky, this can take quite a while. If you're lucky and dumb, you feel entertained during this interval.

The odd thing is that the player gets every bit as much play from five-cent slots as dollar slots or \$500 slots, with much less drain on equity ... but people play the higher-denomination machines anyway.

Posted by: RonK, Seattle on May 7, 2003 09:59 AM

If they had had a stats for poets instead of just a physics for poets I probably would have taken it. Then I wouldn't need to read this website so obsessivly.

Posted by: biz on May 7, 2003 11:41 AM

Actually, I underestimated the average payout% for Vegas slots. On \$5 slot machines it was approximately 98.4% in a recent audit. And that figure is in line other recent audits. It is likely that the kind of machines Bill Bennett played had average payouts exceeding 99%. At 98-99%, he would have had to cycle through a mountain of cash to net a loss of \$8,000,000.

Posted by: Kevin H on May 7, 2003 11:41 AM

RonK,

To lose a net of \$8 mil, I think Bennett would have been cycled through anywhere from \$400,000,000 to \$800,000,000. I find the notion that high-roller slot machines payout less than 98% difficult to believe. Slot make money on volume, not the payout percentage in places like Vegas where Bennett was gambling. If he is pulling a lever in the airport or playing the nickel slots, then the payout would be lower. I don't play slots, but if I was looking for a place to play \$5 machines, I would expect a payout % of at least 98%. BTW, where did you get the 7% figure from? That sounds like the payout on a quarter slot machine in Atlantic City rather than something Bennett would be likely to use.

Posted by: Kevin Hurst on May 7, 2003 12:41 PM

An aside to Neel Krishnaswami's order of magnitude calculations: Although slot machines are still equipped with levers, all machines I've seen can be operated by buttons, and all compulsive gamblers use those, which is both faster and does not provide the moderating influence of the tennis elbow that would develop with pulling a lever thousands of times. I have not timed slot players, but 5 seconds per bet, i.e. 720 bets per hour, does not seem an unrealistic number.

BTW, I wonder whether house takes are identical across slot machine denominations. It might make some sense for them to be slightly smaller for \$500 slot machines.

Posted by: microtherion on May 7, 2003 01:41 PM

Kevin -- The 7% figure was invoked upthread. I agree returns on top-end slot are likely higher ... though "authoritative" sources are inconsistent on the exact figures.

So let's say BB sits at a 98%-return machine, loading it with five \$500 tokens -- \$2500 -- at each pull, losing an average of \$50 per pull, three times a minute, or 180 * \$50 = \$9000/hour.

111 hours -- maybe ten trips to the casino, maybe more, maybe less depending on his pattern -- equals means an expected loss of \$1M.

Posted by: RonK, Seattle on May 7, 2003 01:53 PM

Oh, btw: if you really want to beat people over the head with statistics, you're going to have to use a more sophisticated model than the Central Limit Theorem. The CLT requires that the events be IID, which slot machine pulls certainly are not. While you do not know the pre-ordained pattern that they use, slot machines use pseudo-random number generators, which essentially means that they have a long string of numbers that they cycle through.

The fact that you've moved on from one number to the next means that the events are not independent.

Of course, these things to look like they are independent; they are analyzed quite carefully to make sure that the streams of numbers have no detectable patterns in them, so you're not going to be far off claiming that the events are independent.

Also, what's your proof that Bennet is not the luckiest man in the world? Someone has to be. Frankly, while I'm more inclined to go with the idea that keeping >90% of your money on a long-term average is close to breaking even (>90% of starting money is not an unreasonable definition of close), what evidence do you have to offer that Bennet isn't the luckiest?

Statistics will not avail you here, because Bennet is only one sample point. If we had a few dozen or a few hundred moralizers with gaming losses, sure. But we have only one, and we're concerned with only one.

Assuming that he isn't because that's a safe long-term strategy reminds me of the fastest probabilistic primality test in the world:

bool isprime(BigInt n)
{
return false;
}

It's going to be right > 99% of the time for large n, and you can't get faster than that.

But cynicism is not the most useful strategy in the world...

Posted by: Chris on May 7, 2003 02:15 PM

Getting away from house odds for a moment...

"His principal vice is a desire to put his name on the cover of books that other people wrote."

Another addition to the growing list of vices. I guess Hillary "It Takes a Village" Clinton won't be getting my vote next time around.

Now ghost writers too must be added to the list of the sinful, being partners in and enablers of this vice. Though ghost writers have been around about as long as writing itself.

Hey, I've made a fair portion of my own money by ghost writing. Does this make me an intellectual prostitute or pimp or panderer or something?

Well, as long as the price was right, never mind (as I'd think a libertarian like Ms. Postrel would appreciate).

Posted by: Jim Glass on May 7, 2003 06:32 PM

Like most politicians, athletes, and CEOs, Hillary Clinton does not hold herself out to be a public intellectual--a distinction I made in my post. If, say, Daniel Patrick Moynihan had used ghosts, that would be a comparable situation. Much of Bennett's appeal to his conservative readers is not just in his message but in his status as an intellectual.

Typically, celebrity authors who use ghost writers are upfront about that use, "Joe Celebrity with Susan Ghost" being the typical credit line. (The most infamous example of being upfront about who wrote your book is probably Charles Barkley complaining about misquoted in his autobiography.) That prevents the situation from being fraudulent. As a libertarian-watcher like Jim Glass surely knows, libertarians are big on the evils of fraud.

Posted by: Virginia Postrel on May 7, 2003 07:59 PM

"... libertarians are big on the evils of fraud."

Indeed. And perhaps it's my failing here in never having thought of Bennett as being any kind of Moynihan-like public intellectual.

To me he was a political operative who smartly capitalized on political appointee jobs that gave him a marketable name by following up with a lucrative line of self-marketing cum social policy wonking. But I didn't pay much attention beyond that so maybe I don't know -- did he ever publish any serious original social research that he claimed to be his own original work that wasn't?

Refreshing my memory I look at his bio at Heritage and see those _Virtue_ books for kids and his shot at Clinton and a few lectures drawing lessons from government programs and other policy entrepreneurs, but no signs of anything much more academic than that. Using a ghost for any of that stuff would hardly be fraudulent in my mind.

But I admit I don't know his whole CV, perhaps I'm missing things? Did he really have any specific big ideas like Moynihan had that he passed off as his own that weren't?

"celebrity authors who use ghost writers are upfront about that use ... That prevents the situation from being fraudulent"

Celebrities yes, typically, because nobody expects them to be able to write, but professionals and even PhDs who use ghosts often aren't open about it at all, as contracts in my drawer would demonstrate if they didn't have confidentiality clauses.

But the fact that they use unacknowledged ghosts doesn't make them perpetrators of fraud -- usually it just means they can't write a lick either, so if they want anyone to read their ideas they need someone else to write them up for them decently.

Even when such folk have asked me to do research for them to fill gaps in their own work, I never felt like I was perpetrating a fraud on the public. Or as the actual writer was I the victim of the fraud? It's not clear to me who the victim is supposed to be if the writer gets paid fair value for the writing (and the confidentiality clause) and the public voluntarily pays fair value for the book.

Of course, I have heard horror stories about professors publishing their grad students' ideas as their own without any negotiation or compensation, but having been away from academia for many years I wouldn't know about that.

Posted by: Jim Glass on May 7, 2003 11:42 PM

Visiting Ms. Postrel's web site for further clarification ....

"Bennett is, in my opinion, lying to his readers in an important way."

That may be, but it is hardly "fraud". The essential element of fraud is that one's willful misrepresentation is relied upon by another in a way that causes harm or damages. Who suffers harm or damages as a result of Bennett using a ghostwriter for his books? (Presumably the books are better written than without the ghost, so it wouldn't seem to be the readers.)

It may be either the lawyer or would-be libertarian in me, but charges such as "fraud" are serious enough to me so that it is irksome to see them thrown around for rhetorical effect.

"Is it wrong to put your name on someone else's articles, providing you pay the someone else? That depends, I think, on whether you present yourself to the public as a writer and thinker as opposed, for example, to a politician or corporate executive. The former are supposed to do their own work;.."

This seems artificially neat. Many people in the corporate world sell themselves as thinkers and justify their high fees and salaries with their thinking power, marketing themselves through their books and articles. For them to use ghosts is less bad than for a policy wonk to do so?

OTOH, when a Jack Kennedy purportedly writes "Profiles in Courage" or Hillary writes "Village", aren't they representing that those books contain their *personal* visions? Isn't the scope for fraud here greater rather than less, when they could wind up as POTUS in part by getting credit for a book written by somebody else?

Posted by: Jim Glass on May 8, 2003 09:58 AM

"While you do not know the pre-ordained pattern that they use, slot machines use pseudo-random number generators, which essentially means that they have a long string of numbers that they cycle through."

Modern random number generators have a *really* long cycle time. For the purposes of this, they work just find as a random number source.

Posted by: Jason McCullough on May 8, 2003 12:53 PM

Amount of losses: Green said on MSNBC Tuesday night that Bennett's actual net losses were a bit more than *\$1M*, not eight.

Payout: I chose the 7% figure (as the expected loss rate) up-thread as the mid-point in the range that I had seen reported for high-stakes slots. But it looks to me like 2% is the more likely number. It should be kept in mind that this varies from place to place and machine to machine, and we don't know exactly which slots Bennett played.

Pseudo-random numbers: while it's conceivable that slot machine manufacturers could make their machines deterministic to satisfy some need of the industry, the fact is that it's trivial to make the results truly random given some real-world input. I'd be surprised if the slots weren't truly random.

Posted by: Keith M Ellis on May 8, 2003 02:58 PM

Different slot machines have quite different payout percentages, and their popularity with users only roughly correlates with payout percentages. In fact, there are other factors which correlate more.

Some friends of mine write some highly sophisticated (and highly expensive) software which helps casinos determine those factors. Their software helps casinos determine which slot machines (and other gambling games) to place where based on who is most likely to use what, how often, the different house cuts, etc. Machines that are more popular for different types of players are distributed according to some extraordinarily careful statistical analysis of usage figures, so as to try to increase the amount of play on machines with higher house takes, and thus the house's total revenue. Very impressive (and very pretty) colourful 3-D modelling is used to demonstrate to casino executives who is using what machines, when placed where, and when, and with what flow-on effects are for and from surrounding games of different types, etc. To give a trivial example: some machines do best near traffic flows of people who've just arrived at the casion, some do best with people who are just leaving. Etc, etc, etc.

To see why casinos would go to such lengths, consider this: the annual turnover of the single biggest US casino consortium is very similar to the GDP of my country. And that's just ONE of the US casino consortiums.

Sean
(New Zealand is a small country with a small country's GDP - but that's still a hell of a lot of money!)

Posted by: sean on May 8, 2003 06:19 PM

Why is anyone surprised that an inveterate gambler, as Bennett appears to be, underestimates his losses? Keeping north of 90% of what one risks, to many gamblers, is approximately breaking even in their skewed view of the world. And what if one expected to lose the average house take (a price of admission, so to speak, for the entertainment of gambling)? Then losing that amount would be breaking even, to someone newly confronted with a potentially destructive habit speaking the language of utility.

I doubt, however, that Bennett is well-versed on stats and in any event one wouldn't expect anyone in his position to trot out the CLT even if he were. But is this really the platform from which to launch a diatribe about innumeracy in America?

I'm most interested in the unseemly piling on. I can't say I enjoy listening to the guy. He's a bit dour though has a keen sense of humor. So he comes across as a pompous windage to some. And? His assessment of the public education system (circa mid-80s) stands as a shining beacon ignored by the skippers of the Valdez-like hulk that has hit the shoals with several generations of students aboard. And who can take issue with his championing of virtues in the young? That so many leap for the ad hominem jugular rather than attacking the substance of his arguments speaks volumes.

In fact, calling him a liar (though it may in fact be true) is far more reckless than a remark I heard from one of Bill Clinton's most ardent critics during the run-up to impeachment:

"The man is a serial liar, philanderer and probable felon"
-- Bill Bennett on Clinton, approx Sept. 1998

At least Bennett was prudent enough to use the qualifier "probable." Haven't seen any of that here. The first two assessments were inarguably factual, regardless of what one thought of Clinton.

No... this all reeks of the score-settling and schadenfreude of petty politics. I have no great feeling for the man either way. Which is why I'm not frothing at the mouth. I hope he gets help. It will no doubt soften his perspective on the weaknesses of others. Everyone wins.

I'm reduced here to ignoring the shrillness of tone in the rampant lip-smacking and searching (somewhat fruitlessly) for a point.

Posted by: Bill (surname not Bennett) on May 9, 2003 12:57 PM

Why is anyone surprised that an inveterate gambler, as Bennett appears to be, underestimates his losses? Keeping north of 90% of what one risks, to many gamblers, is approximately breaking even in their skewed view of the world. And what if one expected to lose the average house take (a price of admission, so to speak, for the entertainment of gambling)? Then losing that amount would be breaking even, to someone newly confronted with a potentially destructive habit speaking the language of utility.

I doubt, however, that Bennett is well-versed on stats and in any event one wouldn't expect anyone in his position to trot out the CLT even if he were. But is this really the platform from which to launch a diatribe about innumeracy in America?

I'm most interested in the unseemly piling on. I can't say I enjoy listening to the guy. He's a bit dour though has a keen sense of humor. So he comes across as a pompous windage to some. And? His assessment of the public education system (circa mid-80s) stands as a shining beacon ignored by the skippers of the Valdez-like hulk that has hit the shoals with several generations of students aboard. And who can take issue with his championing of virtues in the young? That so many leap for the ad hominem jugular rather than attacking the substance of his arguments speaks volumes.

In fact, calling him a liar (though it may in fact be true) is far more reckless than a remark I heard from one of Bill Clinton's most ardent critics during the run-up to impeachment:

"The man is a serial liar, philanderer and probable felon"
-- Bill Bennett on Clinton, approx Sept. 1998

At least Bennett was prudent enough to use the qualifier "probable." Haven't seen any of that here. The first two assessments were inarguably factual, regardless of what one thought of Clinton.

No... this all reeks of the score-settling and schadenfreude of petty politics. I have no great feeling for the man either way. Which is why I'm not frothing at the mouth. I hope he gets help. It will no doubt soften his perspective on the weaknesses of others. Everyone wins.

I'm reduced here to ignoring the shrillness of tone in the rampant lip-smacking and searching (somewhat fruitlessly) for a point.

Posted by: Bill (surname not Bennett) on May 9, 2003 01:01 PM

Why is anyone surprised that an inveterate gambler, as Bennett appears to be, underestimates his losses? Keeping north of 90% of what one risks, to many gamblers, is approximately breaking even in their skewed view of the world. And what if one expected to lose the average house take (a price of admission, so to speak, for the entertainment of gambling)? Then losing that amount would be breaking even, to someone newly confronted with a potentially destructive habit speaking the language of utility.

I doubt, however, that Bennett is well-versed on stats and in any event one wouldn't expect anyone in his position to trot out the CLT even if he were. But is this really the platform from which to launch a diatribe about innumeracy in America?

I'm most interested in the unseemly piling on. I can't say I enjoy listening to the guy. He's a bit dour though has a keen sense of humor. So he comes across as a pompous windage to some. And? His assessment of the public education system (circa mid-80s) stands as a shining beacon ignored by the skippers of the Valdez-like hulk that has hit the shoals with several generations of students aboard. And who can take issue with his championing of virtues in the young? That so many leap for the ad hominem jugular rather than attacking the substance of his arguments speaks volumes.

In fact, calling him a liar (though it may in fact be true) is far more reckless than a remark I heard from one of Bill Clinton's most ardent critics during the run-up to impeachment:

"The man is a serial liar, philanderer and probable felon"
-- Bill Bennett on Clinton, approx Sept. 1998

At least Bennett was prudent enough to use the qualifier "probable." Haven't seen any of that here. The first two assessments were inarguably factual, regardless of what one thought of Clinton.

No... this all reeks of the score-settling and schadenfreude of petty politics. I have no great feeling for the man either way. Which is why I'm not frothing at the mouth. I hope he gets help. It will no doubt soften his perspective on the weaknesses of others. Everyone wins.

I'm reduced here to ignoring the shrillness of tone in the rampant lip-smacking and searching (somewhat fruitlessly) for a point.