May 07, 2003

The Law of Large Numbers

Eugene Volokh attempts to explain why he is "agnostic" on the question of whether William Bennett is, over a long period of intensive high-stakes gambling at the slots, (a) "nearly even" (as Bennett claims), or (b) has lost a fortune. The fact that anybody who gambles a lot of money over a long time at the slots loses a fortune with extremely high probability, is, to Volokh, not relevant: "Some casinos are estimating the total losses at over $8 million, but Bennett explicitly says otherwise.... This has little to do with statistics -- it's a question of fact."

This strongly reaffirms my belief that Eugene Volokh--along with many, many, many others--simply does not understand statistics, does not understand the force of statistical arguments, does not understand how overwhelmingly unlikely it is that any long-time heavy gambler at the slots could be "about even."

From my perspective (and from the perspective of everybody here on the third and sixth floors of Evans Hall), it is simply impossible to write a paragraph like Volokh's unless you are--in some powerful sense--ignorant of statistics. When the background probability of some state of affairs is overwhelmingly improbable, claims that the state of affairs actually took place need to be regarded with great skepticism, not taken at face value. To dismiss the background probabilities and turn it into a simple "he said, he said" problem about which one should be "agnostic" is a major analytical mistake. It's simply wrong: the quantitative reasoning equivalent of the philosophical-logic mistake of arguing that "all men are mortal," and "Socrates is a man" implies that "all men are Socrates."

So: everyone reading this: I command you to get a sense of how quickly the law of large numbers and the central limit theorem set in: go to a computer located 30 feet below this one, http://stat-www.berkeley.edu/~stark/Java/lln.htm, and play with Berkeley Statistics Professor Phillip Stark's Java Applet. Set the number of trials--the number of times the slot lever is pulled--to 10,000 or more. Set the probability of success--50% minus half the house percentage--to 45% (for a 10% house edge) or 46% (for an 8% house edge) or 47% (for a 6% house edge)--and see how much money you lose, and how extraordinarily low are the chances of your winding up "about even." (To wind up even, you need to be +500 above the expected line if the house edge is 10%, +400 above if the house edge is 8%, +300 if the house edge is 6%.) (Slot machine payoffs are skew, which Phillip's Applet does not take into account: but skewness of payoffs is irrelevant at the scales at which Bennett was playing.)

Go do it now. This is something you really need to understand. Phillip Stark is doing great work with the teaching materials he is developing. I think he's the best hope we have for actually doing--rather than failing to do--our professorial job of enhancing statistical literacy over the next generation.


The Volokh Conspiracy: Sometimes, statistics isn't the answer... here's the full quote, without the ellipses, which may explain why I'm agnostic on the matter: "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. (UPDATE: Several people suggested the $8 million might be gross losses, without any offset for winnings; I checked several press accounts, and they've generally said that he's lost over $8 million over the last decade, which in normal parlance means net losses. The Newsweek article, for instance, says 'Some casino estimates put his total losses over the past decade at more than $8 million.' Nonetheless, Bennett suggests that he's 'come out pretty close to even,' though others doubt this, so it's possible that the reported losses don't include winnings. See here for the New York Times account, which reflects this uncertainty.) Suddenly, the real issue becomes a bit clearer than it was when the intermediate sentences were removed. Some casinos are estimating the total losses at over $8 million, but Bennett explicitly says otherwise; instead, he's saying that he's come out pretty close to even (whatever exactly that means), and thus (returning to the previous paragraph) that the supposed casino estimates are mistaken or highly incomplete. This has little to do with statistics -- it's a question of fact...

Posted by DeLong at May 7, 2003 10:56 AM | TrackBack

Comments

This is doubly curious because Eugene is very bright, and in fact quite mathematically inclined (He has an undergrad deg. in math/cs). I remember from my undergrad probability course that we set up the problem thusly;
suppose you have $100, and you sit down at the blackjack table, with a ~1.5% advantage to the house. You play perfectly, making no errors in basic strategy, and you decide beforehand to get up when you have doubled your $$, or lost it all. As I recall (dimly), the probability of getting up with $200 is something like 0.05. It was a very good lesson in the power of percentages. It did not, however, stop me from going to Vegas and gambling!

Posted by: Paul Orwin on May 7, 2003 11:17 AM

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This is doubly curious because Eugene is very bright, and in fact quite mathematically inclined (He has an undergrad deg. in math/cs). I remember from my undergrad probability course that we set up the problem thusly;
suppose you have $100, and you sit down at the blackjack table, with a ~1.5% advantage to the house. You play perfectly, making no errors in basic strategy, and you decide beforehand to get up when you have doubled your $$, or lost it all. As I recall (dimly), the probability of getting up with $200 is something like 0.05. It was a very good lesson in the power of percentages. It did not, however, stop me from going to Vegas and gambling!

Posted by: Paul Orwin on May 7, 2003 11:18 AM

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William Bennett has caused enormous damage to this country. His anti-drug war nonsense is responsible for much suffering. Virginia Postrel also perceives Bennett as something of a plagiarizer who fails to credit those who assist him with the writing of his books.

I could care less if William Bennett lost far less than $8,000,000 in the Las Vegas casinos. Even 10% of that figure is a hell of a lot of money. He is obviously an out of control addict. Itís time to pull the rug from underneath this clown.

Posted by: David Thomson on May 7, 2003 11:28 AM

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We live in a time of too little systematic thought. Of course, either William Bennett was the luckiest person alive or the losses were horrid. Of course, if you are making ten of millions of dollars telling others how to be moral what difference does losing $8 million make. Of course, if you have made of career of intimidating others because they are not as moral as moral as you then gamboling and lying about gamboling should be meaningful. We are dealing with another supreme fraud. Of course, William Bennett is a right wing fraud and so all will be forgiven. Remember, Rickky Santorum?

Posted by: lise on May 7, 2003 11:29 AM

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It's been noted for a while that human intuition does not have a base rate correction; Tversky and Kanhman were among those pointing it out. Taking the prior probability into account is a learned response. It seems that even some people one would think have learned this have not.

Posted by: Jonathan Goldberg on May 7, 2003 11:48 AM

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It's been noted for a while that human intuition does not have a base rate correction; Tversky and Kanhman were among those pointing it out. Taking the prior probability into account is a learned response. It seems that even some people one would think have learned this have not.

Posted by: Jonathan Goldberg on May 7, 2003 11:53 AM

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I've always though Bennett very tiresome, but I just can't muster any concern about a rich guy blowing $8 million in casinos,just like I can't muster any concern about rich people blowing millions on yachts, airplanes, or anything else. Slot machine manufacturer employees gotta eat, just like yacht manufacturer employees. Of course, I also maintain that Peruvian coca farmers gotta eat too, so Bennett and I didn't see much the same way to begin with.

In any case, I do think that Volokh does not give enough weight to the statistical unlikelyhood of someone who has pushed gambling machine buttons for a long time being anywhere close to even. That's what strikes me as odd about this affair; the insipid way in which Bennett chose to gamble. There are games available in Las Vegas in which your primary competior is not the house, which is sure to make it exceedingly difficult to win anything substantial (blackjack) if not nearly certain to crush you outright (most other games). The sports books in Las Vegas, and the horse betting parlors around the country, allow the gambler to have the betting public as the competitor, which is far more favorable, in that, like some schools of contrarian investing, it allows the well studied individual to spot situations where crowd sentiments have become wildly at odds with reality, and thus profit handsomely. It really can be quite fun, and profitable, often offers more transparency than finincial investing, while being far more aesthetically pleasing; a thorougbred thundering down the stretch, or a wide receiver leaping to make a catch, is a helluva lot more visually interesting than a stock ticker.

For the most part, however, first time visitors to Las Vegas would be better served if they took a cab ride down the strip after leaving the airport, and had the cabbie turn to face them, about 80% down the road, and say "They're a few hundred thousand hotel rooms in this town, and that's because everybody loses."

Posted by: Will Allen on May 7, 2003 12:09 PM

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Actually what most interests me in this brouhaha is that it seems quite lilkely that the executives of publicly traded casino interests have harmed their shareholders by participating in the outing of a whale like Bennett, and thereby discouraging him from losing further millions, while also striking fear into other whales that they could be subjected to similar treatment, and thus discouraging them as well.

Big losers like Bennett are very important to the profit margin of expensive to run, 5 star, casinos, so assisting in his public embarassment really is against the interest of the casino shareholders, which the executives of the casino have a duty to protect. Maybe one of virtue posses that Bennett associates with was making too much noise about anti-gambling legislation, and they decided it was a net positive to cut this whale loose, while discrediting the posse by association. Otherwise, well, shareholder lawsuits have been filed on flimisier pretexts.

Posted by: Will Allen on May 7, 2003 12:22 PM

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Will,
Slot machine makers aren't raking in that $8 mil, it's casinos that do that. The fraction of money actually going to support real production of real goods (like slot machines) is low. Whatever the utility of Maybachs, Rolls Royces, Rolexes, or yachts, the fact is that most of the money spent on those luxuries does support a lot of highly skilled and well paid workers.

Posted by: fastback on May 7, 2003 12:26 PM

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The casino owners aren't stupid, and they have a pretty clear bottom line. I'm sure they made some precise calculations on how much they stood to lose from outing BB, and how much from not doing so. His Empower America group had decided to attack the casino industry. Not a wise decision, perhaps, in retrospect.

Posted by: John Isbell on May 7, 2003 12:41 PM

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fastback, entertainment and diversion are real goods. Casinos provide that. Blackjack dealers, cocktail waitresses, bartenders, cooks, etc. etc. gotta eat to, and before you deride those occuaptions, you should check out the mortgage- free homes of some of the booze-slingers I know. The fact that some people spend a ton of money in casinos, and some people ruin their lives spending money in casinos, is no more important than the fact some people spend a ton of money collecting wine (yes, sometimes millions) while others ruin their lives guzzling Thunderbird.

Posted by: Will Allen on May 7, 2003 12:41 PM

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This has got to be good for the Internet betting exchanges like Flutter and Tradesports, all those right-wing scolds out there will not make sure their dalliances are totally anonymous.

Posted by: Chad on May 7, 2003 12:41 PM

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Hey, I didn't say they were stupid, at least not when to comes to making financial calculations (although we may be in the midst of an overbuilding bubble in Vegas which once again proves that nobody figures the odds correctly all the time). What they might be, however, like a lot of people occasionally, is spiteful, even to the detriment of the shareholders they have a duty to serve. Sending the message to all the whales they harpoon that there is no expectation of confidentiality cannot be good for business, and one would expect that the other high-roller casinos that compete with the Bellagio would use this breach of confidentiality to steal some whales from the Bellagio; it's a very, very competitive market.

Posted by: Will Allen on May 7, 2003 12:56 PM

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Will,
Of course you're right services are goods too, but those services are in general provided by low wage workers rather than the skilled workers at DaimlerChrysler, BMW/Rolls Royce, or Rolex. My point was that 8 million spent in a casino doesn't enrich slot machine manufacturers as much as it does those casino owners.

Posted by: fastback on May 7, 2003 12:57 PM

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If we are now going to pass moral judgement on people's spending preferences based upon which parties profit most from the spending, well, we have reached a level of busybodyness that, ironically, I found so tiresome in Bennett.

Posted by: Will Allen on May 7, 2003 01:07 PM

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In 00.00001% (or whatever) of cases, it's a question of fact.

In the other 99.99999% (whatever) of cases, it's a question of character.

Posted by: RonK, Seattle on May 7, 2003 01:35 PM

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Just a couple of observations.

First, it is possible to have a very, very tiny house edge on some video poker machines (as opposed to non-poker slot machines) - low enough to be measured in basis points. Proper machine selection (based on payouts for poker hands like flushes and full houses) and perfect basic strategy are needed. Adding in comps from the casino also impacts the calculation of winning or losing.

Second, as a general rule house edges on slot machines decrease as the denomination of play increases. The house edge on a $500 per pull slot machine shouldn't be anywhere near 10%, or even 6%. A couple of percentage points would be more reasonable.

I have no idea if Bill Bennett lost $8mm or not and I don't really care. But I always take information from annonymous sources with a grain of salt, especially when I don't know if they have an agenda. An $8mm loss, assuming even an overall 2% house edge, would mean $400mm of total action. That's a lot of slot play, even at $500 a throw - 800,000 pulls.

Posted by: Ron Wilson on May 7, 2003 01:45 PM

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Thanks.The 8 million dollar figure seemed a little fishy to me.

Posted by: Will Allen on May 7, 2003 01:50 PM

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The average payout % on the slots that Bennett was playing were almost certainly at least 98% and probably higher than that. Not to mention that he was playing video poker apparently where a skilled player who pays close attention to odds on each machine has a better chance of winning over a longer-term. Bottom line is that at the roughly 98-99.5% payouts that I assume for the machines he played, Bennett's claim to have come close to breaking even is not outlandish. Volokh is correct in to adopt an agnostic stance regarding the $8 million net-loss assertion. What is clear is that the 10%, 8%, or 6% house takes asserted by Dr. DeLong bear no relationship to the payout percentages programmed into slot machine software of the $5 (or even $1, for that matter) and above variety

Posted by: Kevin H on May 7, 2003 03:22 PM

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I think this shows just how much money is available if you're willing to shill for the far right.

Posted by: IssuesGuy on May 7, 2003 03:51 PM

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I've been pushing the idea that the reason Bennett got outed is that he's in arrears in his payments, and that the next step will be cement overshoes at the bottom of the Atlantic.

Let me dream, OK?

But we should watch out for a flurry of moralizing lectures, two a day, seven days a week.

As with other failing businesses, of course, his profitability may have taken a hit, so he may end up having to lecture at old folks' homes, store openings at malls, and county fairs (last time I went through Montana I had a chance to see Blue Oyster Cult, BTW).

But in that case he'll never be able to pay up. He'll be like Argentina. Poor guy.

Posted by: zizka on May 7, 2003 04:18 PM

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Brad, why do you keep attacking Volokh? Yes, it's true that if you know (1) the number of times played (2) the amount wagered per play, and (3) the distributions of payouts, you can calculate the distribution of net winnings. So can I, and I expect so can Prof. Volokh.

But you clearly don't know any of these things. I'm not sure why you're so confident in your guesses about (1) and (2) from the sketchy news reports we've seen. And your examples show that you are _WAY_ off on your assumptions about (3) as well.

Plus, it's not clear what is meant by "about even." If the machines pay 98%, and Bennett in fact lost exactly 2% of the total amount wagered, a reasonable person might call this "coming out about even." If there is a chance for large jackpots (and I don't know if there is), coming close to 100% is not out of the realm of possibility.

(BTW, I'm not defending Bennett, just Volokh).

Posted by: ed johnson on May 7, 2003 04:47 PM

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Brad, why do you keep picking on Volokh? We all know that if you have (1) the number of times played, (2) the amount wagered per play, and (3) the distribution of payouts, you can calculate the distribution of net losses. So can I, and I expect so can Prof. Volokh.

But you clearly don't have any of these things. I don't know why you would be confident in your guesses about (1) and (2), from the sketchy news reports we've seen. And your examples show that you are way off on (3) as well.

Plus, it's not clear what is meant by "coming out about even." If the machines paid 98%, and if Bennett in fact lost exactly 2% of the amount wagered over the years, a reasonable person might call that "coming out about even." Throw in the possibility of a lucky big jackpot or two (and I don't know how big they are, do you?), then coming out close to 100% is not out of the realm of possibility.

(BTW, I'm not defending Bennett, just Volokh.)

Posted by: ed johnson on May 7, 2003 05:00 PM

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Isn't it clear from Volokh's post that he's agnostic about how much Bennett has wagered, not the likihood of his having won? Green et al claim that he wagered enough to lose $8,000,000, and Bennett is insunuating, anyway, otherwise. The Central Limit Theorem has nothing to do with it.

Besides, people have already dug up the fact that slot machines have as high as a 98% payout, and some video poker a potential positive payout (a fact that blows my mind...I still don't quite believe it). If I made a $100 worth of bets and returned with $98, I don't think I'd be lying to anyone be saying I came out "about even."

Posted by: Scott Wood on May 7, 2003 05:02 PM

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I think you all should leave Mr. Bennett alone.

I don't think it is immoral to gamble away all that money. It is the right thing to do if you are rich, bad if you are poor and hurt someone by losing money. In anthropology I read about a tribe in NW Canada that every once in a while if someone got too rich, they would have a "potlatch" party where they would get together and destroy all their stuff. The one who destroyed the most had the most status. So, it must be that having wealth distributed unequally is inherently an unstable thing that has to be corrected.

My brother in law works for the Indian casinos doing service work on the machines. He depends a lot on casinos. My niece was born with a stroke and only has use of one side of her brain, so she needs a lot of health care and therapy. Since the company he works for is small they don't offer any health insurance, so they are on MN care the Minnesota health plan. It costs them about $4000 a year for insurance and expenses to drive her down to the Cities to get treatment. Minnesota is having severe budget problems so the health care plan is going to be cut. He only makes about $30,000 yr so with a family of 4 that is quite an expense.

Also, I know the casinos do a lot of work with the difficult to employ, such as ex prisoners, who have a really hard time getting a job anywhere else.

So, I conclude that yes, you should gamble if you are rich, preferably at an Indian casino, and as much as your budget will allow. It's the right thing to do.

Posted by: nkirsch on May 7, 2003 06:17 PM

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The question isn't whether Bennett lost $8 million, it's whether he came out "about even" after years of gambling millions of dollars. I've got a lot of gamblers in my family, and every one of the comes out "about even" in the long run. In fact, out of the hundreds of people I know who gamble, all of them either win or are "about even." It's amazing those suckers are building those huge casinos in the desert. Don't they know, as a matter of "fact", that nobody loses money gambling.

Posted by: pj on May 7, 2003 09:51 PM

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A little correction to your Socrates sillogism. It should read: All men are mortal, Socrates is mortal, All men are Socrates.

As you wrote it does not make sense even as a false sillogism.

Posted by: Mooraq on May 8, 2003 03:49 AM

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PJ nails it. This whole discussion is pretty academic for anyone who's known any gamblers. Gamblers never tell you they lose most of the time. But -- they lose most of the time.


There are exceptions for skilled players in poker, backgammon, and a few other games, but they aren't really gamblers. But the people who play AGAINST them, who think they're gambling, almost always lose.

Posted by: zizka on May 8, 2003 07:34 AM

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I think that understanding statistics does tend to reduce gambling.

Around 1986, the American Physical Society meeting had to be moved from San Diego because construction was still being done on the hotel. The only place that could accomodate the APS on such short notice was Las Vegas. Las Vegas did not like the APS; the attendees did far less gambling than most conventioneers. The APS didn't like Las Vegas much, either.

I bet law conventions are no problem, though.

Posted by: Bob Hawkins on May 8, 2003 09:13 AM

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Las Vegas now has enough excellent dining opportunities and other diversions that it is not nearly as gambling-centric for visitors as it once was. On my last visit, I was alone on business, and had great time without gambling. After finishing work one day, I wandered over to the Commander's Palace, the Las Vegas outpost of the New Orleans institution. The maiter d' went out of his way to make a single diner feel welcome, gladly granted my request to dine at the bar, which I prefer when dining out alone, and made excellect recommendations regarding the menu. The bartender was knowledgeable with those wine labels I did not recognize, suggested an excellent pairing, and was a terrfic conversationalist, as was evidenced by the obvious regulars who stopped by in the ensuing two and a half hours, and shared her gift for gab. Walking back to my car, I noticed an outpost of the Blue Note jazz club, and stopped in for a set. Altogether, it was a great night, and I never actually set foot in the gambling areas. In fact, the only wager I made was on Sunday, waiting for my evening flight, when I went down to the sports book at the Las Vegas Hilton and and placed $50 on a ball game I wanted to watch. The dog failed to cover, but I didn't mind too much.

Posted by: Will Allen on May 8, 2003 11:11 AM

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Your numbers are plain wrong!

I actually tried playing that java script with a 48.5% probability. (A 97% odds, "loose" machine) It will take more than 10,000 plays to guarantee 100% that you are losing. since

(sqrt(10000) = 300) ~= (.03*10000 = 100)

a significant fraction of players will be winning after 10,000 plays. (~10%)

Of course, once you get past 10,000 the odds start getting worse very fast indeed!

By the time you hit 30,000 plays your chances of winning really are down at the 9 sigma level, the proverbial snowball in hell.

Posted by: pmac on May 8, 2003 08:47 PM

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Your numbers are wrong!

I actually tried playing that java script with a 48.5% probability. (A 97% odds, "loose" machine) It will take more than 10,000 plays to guarantee 100% that you are losing. since

(sqrt(10000) = 300) ~= (.03*10000 = 100)

a significant fraction of players will be winning after 10,000 plays. (~10%)

Of course, once you get past 10,000 the odds start getting worse very fast indeed!

By the time you hit 30,000 plays your chances of winning really are down at the 9 sigma level, the proverbial snowball in hell.

Now, at 1 play every 20 seconds, that's a total playing time of 166 hours, or 7 days nonstop playing until you really are sure to be losing.

So in a sense, Bennett is right, he really did win nearly as much as he lost.

Say he played his stated 4 times per year, for 6 hours a day, 4 day weekends. That's a total of 4 days straight per year, or 17000 pulls per year, for an expected loss of

17,000*.03 * 500 = $260,000 /year.

Over 8 years, that's a total loss of about $2 million, +/- about $60,000.

Blow a little of that my way!

Posted by: pmac on May 8, 2003 08:58 PM

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I'll put my money on the trifecta.

I'll bet Bennett is a degenerate gambler, a giant loser, and a liar.

Any takers?

Posted by: arslan on May 9, 2003 05:49 PM

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FOAF Warning. But still interesting. An aquaintance of mine reported that several years ago he had dinner with a man who claims to be a very close friend of Bennett's (and this claim is believable since it is generally known that he is at least a friend of Bennett's -- how close is another matter) and this man said that Bennett told him that he chose not to run for President because he'd previously had an affair and he knew the press would dig it up were he to run. True? I dunno. It's fun to believe it, though.

Posted by: Keith M Ellis on May 10, 2003 06:34 AM

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Can someone point me to a reference to the math behind this sentence "skewness of payoffs is irrelevant at the scales..."? Couldnt understand why is it.

Posted by: S. Rocha on November 18, 2003 04:34 PM

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