Megan McArdle uncovers one of the strategy secrets of ENRON:

Asymmetrical Information: Question of the day: I've heard of a supposedly foolproof system of winning at roulette, as follows:

1) Put down your stake on either even, odd, black, or red

2) If you lose, double what you had on the table and bet again on the same thing.

3) If you lose again, double again.

4) Repeat until you win

5) Then stop.You can only win the amount of your original stake, of course. Can any budding mathematicians find the hole in this theory?

Seriously, this is what ENRON did. Have a quarter in which prices moved against you and you would have to report a loss? Stuff the loss into a special-purpose entity (so you don't have to report it for a while) and place a bigger bet. Keep doing this. Sooner or later one of the bigger bets will come through, right?

The example of ENRON tells you what's wrong with this strategy. If you look ahead five rounds, you see that you have a 31/32 chance of coming up ahead $1, and a 1/32 chance of behind behind by $31. If that 1/32 chance doesn't cause you more discomfort than the 31/32 chance causes pleasure, take your brain in for a rebuild before you wind up like William Bennett.

But suppose you're willing to last ten rounds? Then you have a 1023/1024 chance of being ahead $1, and a 1/1024 chance of being behind $1023. Twenty rounds? A 1048575/1048576 chance of being up $1, and a 1/1048576 chance of being down $1048576. The chance that you lose falls astronomically fast, but the amount that you lose if you do falls just as fast.

Ultimately, what is wrong with this game is that you can't be certain that you will keep playing it until you win. Perhaps (as in the case of ENRON) somebody will finally figure out what you are doing. Perhaps someone at the casino will politely remind you that you cannot place that large a bet. Perhaps people in dark suits will ask to see your money when you're down--just to be certain you have it, you understand.

Posted by DeLong at October 16, 2003 05:02 PM | TrackBack

Comments

Isn't this more like Long Term Capital Management than Enron?

LTCM had made a lot of different bets that it thought were uncorrelated... and even then it would have survived had it been adequately capitalized...

ENRON, by contrast, did keep stuffing bigger and bigger losses into special-purpose entities...

Posted by: Brad DeLong on October 16, 2003 07:00 PMAnd, of course, the comparison between the roulette game and Enron also depends on the idea that Enron's chances of winning each "bet" were actually 50-50 (or almost 50-50, since of course they aren't actually exactly 50-50 in Roulette either what with the 00). No way of calculating that in any exact way, and the chances of losing may still have been pretty large even with multiple bets ...

Posted by: sdf on October 16, 2003 07:01 PMAnd, of course, the comparison between the roulette game and Enron also depends on the idea that Enron's chances of winning each "bet" were actually 50-50 (or almost 50-50, since of course they aren't actually exactly 50-50 in Roulette either what with the 00). No way of calculating that in any exact way, and the chances of losing may still have been pretty large even with multiple bets ...

Posted by: sdf on October 16, 2003 07:03 PMI once tried this with a computer casino game. The idea was to double down any loss, then revert to the original small bet after a win. Start with a $1 bet, then move forward. Keep doing this, and you can inexorably ratchet forward, $1 at a time.

At first it worked, if being a trifle tedious. However, I found that even starting with a fairly large stake, at some point the losses always became too great to cover. Doubling down puts you in the grip of a geometrically increasing debt, and geometric increases become very large suprisingly fast.

Through the years I'd imagine customers coming in using this strategy have made casinos a lot of money.

Posted by: jimBOB on October 16, 2003 07:05 PMapologies for the double post

Posted by: sdf on October 16, 2003 07:05 PMAll this assumes that the game is purely one of chance. However, Enron tried not to leave matters to chance when politicians could be bought. There has been some speculation about Arnold's meeting with Ken Lay in 2001 and that Arnold might derail the CA Enron settlement. My thought is that CA will not be able to pass up the cash.

http://www.consumerwatchdog.org/utilities/nw/nw003559.php3

http://www.gregpalast.com/printerfriendly.cfm?artid=284

Cheney's secret energy plan meetings. Arnold's clandestine meeting with Enron. The secrecy is just begging for nutcase conspiracy theories to emerge that explain their actions.

Posted by: bakho on October 16, 2003 07:55 PMITS TIME WE STARTED TO LISTEN TO ARAFAT

Arafat Invokes 1974 Phased Plan Calling for Israel's Destruction

Compares Oslo to Temporary Truce

PA Chairman: "I Envy the Martyrs and Hope to Be One"

Arafat's Secret Agenda Is to Wear Israelis Out

Arafat estimates that the final-stage agreements between the Palestinians and Israel will ultimately bring about Israel's collapse. He reportedly told the diplomats that a migration of Arabs to "the West Bank and Jerusalem" and the psychological warfare the Palestinians would wage against the Israelis would cause a massive emigration of Jews to the United States.

Arafat: Jihad to

Liberate Jerusalem

The Jihad [Islamic holy war] will continue, and Jerusalem is not [only] for the Palestinian people, it is for all the Muslim nation.

You are responsible for Palestine and for Jerusalem before me [applause], the land which had been blessed for the whole world.

Now after this agreement you have to understand our main battle.

Our main battle is Jerusalem. Jerusalem. The first shrine of the Moslems.

This has to be understood for everybody and for this I was insisting before signing to have a letter from them, the Israelis, that Jerusalem is one of the items which has to be under discussion and not the state, the permanent State of Israel! No! It is the permanent State of Palestine [applause]. Yes, it is the permanent State of Palestine.

"When we stopped the Intifada we did not stop the Jihad [Islamic holy war] to establish Palestine with Jerusalem as our capital.... We know only one word: Jihad, Jihad, Jihad.... We are in a conflict with the Zionist movement, the Balfour Declaration, and all imperialist activity...."

IF WE LISTEN, WE WILL UNDERSTAND

ITS TIME WE STARTED TO LISTEN TO ARAFAT

Arafat Invokes 1974 Phased Plan Calling for Israel's Destruction

Compares Oslo to Temporary Truce

PA Chairman: "I Envy the Martyrs and Hope to Be One"

Arafat's Secret Agenda Is to Wear Israelis Out

Arafat estimates that the final-stage agreements between the Palestinians and Israel will ultimately bring about Israel's collapse. He reportedly told the diplomats that a migration of Arabs to "the West Bank and Jerusalem" and the psychological warfare the Palestinians would wage against the Israelis would cause a massive emigration of Jews to the United States.

Arafat: Jihad to

Liberate Jerusalem

The Jihad [Islamic holy war] will continue, and Jerusalem is not [only] for the Palestinian people, it is for all the Muslim nation.

You are responsible for Palestine and for Jerusalem before me [applause], the land which had been blessed for the whole world.

Now after this agreement you have to understand our main battle.

Our main battle is Jerusalem. Jerusalem. The first shrine of the Moslems.

This has to be understood for everybody and for this I was insisting before signing to have a letter from them, the Israelis, that Jerusalem is one of the items which has to be under discussion and not the state, the permanent State of Israel! No! It is the permanent State of Palestine [applause]. Yes, it is the permanent State of Palestine.

"When we stopped the Intifada we did not stop the Jihad [Islamic holy war] to establish Palestine with Jerusalem as our capital.... We know only one word: Jihad, Jihad, Jihad.... We are in a conflict with the Zionist movement, the Balfour Declaration, and all imperialist activity...."

IF WE LISTEN, WE WILL UNDERSTAND

The problem with doubling down is that zero exists. This is also the problem with Arafat.

Posted by: Joshua Halpern on October 16, 2003 09:20 PMAnd if you've only got a finite amount of money to put down, you'd lose in the long run at this kind of thing even if the house odds were slightly in your favor, which of course they aren't. Eventually you get busted and you just can't double any more.

so what does it say about the chicago mba program that mba and economist student mcardle had to ask her readers to explain this to her?

jebus, mba programs must really be suckin

Posted by: jebus on October 16, 2003 11:19 PMThe strategy is called a martingale and it *does* work (assuming that you follow it to the letter, which people with finite resources might find irksome). The concept of a martingale is fundamental to a huge amount of probability work and finance theory.

Posted by: dsquared on October 16, 2003 11:33 PMThis strategy does not "work" if "work" means "produces a positive expected value." This strategy does "work" if "work" means "you almost surely end up ahead."

It seems that a more precise definition of "work" is called for...

Posted by: Brad DeLong on October 16, 2003 11:47 PMAppreciate you're sense of humour, but dont't underestimate the value of the serious lesson here: Before even beginning to count the numbers, you have to separate fininte amounts from unbounded amounts!

Casiono strategy "works" for infinitely wealthy player. Then again *everything* works for her!

Grass is greener paradox (remember?) only works for infinetely wealthy game organiser. You can always strip down these things into problems that is only about keeping track of what is finite and what is not: pick a random positve integer number, tell me which, then pick anotherone. This second number almost certainly has to be larger than the first one: how can the second of two *random* numbers be always larger than the first?

This is the gambling bondholder's game: expect that most payoffs will be small wins, and hope not to hit the astronomical loss. I find it interesting that we often see it played in reverse, in the stockholder's variant, where most of the players wind up with small losses, but one or two hit astronomical gains. There was an HBR article recently which pointed out that (as the opposite of cats which once sat on a hot stove and now won't sit on another stove, even a cold one) the danger with promoting people who have made those gains is they may believe it was because of something they did, and prove not only insufficiently risk-averse in the future, but insufficiently risk-averse in a future in which they have more power and responsibility.

Posted by: Dave Long on October 17, 2003 02:34 AMThere is a zero and double zero on the roulette wheels, these ensure that the house will win.

Posted by: Big Al on October 17, 2003 03:58 AMThis is a homework problem in Grimmett & Stirzaker's Probability and Random Processes, who provide illustrative quotations from Cassanova's Memoirs.

Posted by: Cosma on October 17, 2003 05:03 AMLoved the very funny way of so succinctly capturing the math of the basic issue. Just wanted to add one bit of Enron realism. These SPEs were joint ventures when Andy Fastow had the outside partners share in the upside potential but not in the downside risk. So it's like placing a $100 bet on the first coin toss with $50 of your money and $50 of my money - which you lent to me. Heads, we both win $50 but tails, but tails, I leave the casino before repaying my loan to you. Now Fastow was supposed to be serving your interests (aka the Enron shareholders) but he forgot to tell you he was getting a cut from me.

Posted by: Harold McClure on October 17, 2003 06:15 AMI agree with jebus - it's pathetic that McArdle doesn't know this stuff. Even if you don't know Lebesgue integration and measure theory and all that jazz, this is pretty basic probability...

Also, the upshot here is that you are risking a huge amount of money - in fact, your entire amount of wealth on every bet - for measly payoffs. You can get a much better return by just putting that into some low-risk financial instrument.

Posted by: engineer on October 17, 2003 06:27 AMSuppose you're a resident of a non-Powerball state and you're planning a visit to a participating Powerball state at some time when the lottery's payout is exceptionally large -- so large that the ratio of potential maximum payout to ticket price is one half. (This approximates the roulette problem.)

SO, all your neighbors and coworkers ask you to buy a ticket for them while you're visiting. Twenty five "players" each give you a dollar to make the buy, and you make one buy for yourself.

So the game (for your pool) has players A thru Z and after the buy you're holding 26 chances to win.

One ticket hits for some modest sub-jackpot amount, say fifty bucks.

1) Is the money yours, for being the one who played the game?

2) Is the money split among all the friends, so "everybody's a winner", (leaving yourself out)?

3) Is the money split 26 ways -- each friend getting $1.92 and you keeping $2.08.

4) Do you "assign" one particular ticket of the 26 to one favorite or selected friend (perhaps a boss or the Person of Opposite Sex you most wish to win later favors from) -- "Here Hillary, you won $50?"

Does it make any difference if the lottery is replaced by futures trading in some commodity, such as pork bellies?

Prof DeLong,

I don't understand your: 'This strategy does not "work" if "work" means "produces a positive expected value.'?

This is really a version of the Petersburg game/paradox, whose expected value is infinite. What am I missing?

Posted by: maciej on October 17, 2003 07:19 AMNo. The way this is set up, the expected value is zero: it's a series of fair bets.

It is related to the St. Petersburg paradox, however...

Posted by: Brad DeLong on October 17, 2003 07:33 AMYou're right. I should have thought 60 seconds longer. Thanks.

Posted by: maciej on October 17, 2003 07:40 AMThe moral of the Saint Petersburg game is that averages are bad measures of asymmetric distributions. This applies to Republican tax cuts.

Posted by: Joshua Halpern on October 17, 2003 07:55 AMWhy would you care about the expectation of the game when you know that you'll win a positive amount in finite time with probability 1? :)

Posted by: dsquared on October 17, 2003 08:41 AMThis is really a version of the Petersburg game/paradox, whose expected value is infinite. What am I missing?

~~~

"It is misleading to consider the payoff without taking into account the amount lost on previous bets, as can be shown as follows...."

http://mathworld.wolfram.com/SaintPetersburgParadox.html

This is really a version of the Petersburg game/paradox, whose expected value is infinite. What am I missing?

~~~

"It is misleading to consider the payoff without taking into account the amount lost on previous bets, as can be shown as follows...."

http://mathworld.wolfram.com/SaintPetersburgParadox.html

Maybe the strategy wasn't so stupid after all if, through their insider knowledge/market rigging, they could ever so slightly tip the balance of probabilities away from 50/50. If they thought their chances of winning was 0.51, their expected gain after 5 rounds would be positive 6.16 cents. Can you be a major energy trading concern and not think you have any information advantage? :)

Posted by: maciej on October 17, 2003 08:54 AMThere's another issue here, independent of the need for an infinite house limit. I don't have my old probability and statistics books handy-- but IIRC, the distribution of runs in a 'conditionally terminated' random process is quite different from the distribution in a regular old random process that terminates at a predetermined time. This is a source of subtle biasses in 'ESP' experiments & I recall that Feller had some choice comments about it in his books.

In any case, it means that the gambler who uses the doubling strategy will have to sit through very long runs of losing money before squeaking through with a small profit at the end.

Posted by: Matt on October 17, 2003 09:46 AM>>Why would you care about the expectation of the game when you know that you'll win a positive amount in finite time with probability 1? :)

Why would you care about the game? Either you can't play it because your wealth is finite, or you don't care to play it because your wealth is infinite.

Posted by: Mats on October 17, 2003 10:02 AMI can't believe that McArdle was that dumb. Wasn't it a rhetorical question? (I say this as someone with little respect for her).

Almost all gamblers are idiots. (The few exceptions make their livings off the idiots). I used to suggest the Martengale strategy to gambler acquaintances as a joke, but I quit because I saw them taking me seriously. (Isn't it obvious that you can only win a little bit, and that if the bet gets too high for you you lose everything?)

On the other hand, some gambler superstitions have a degree of truth. I knew a guy who dreamed of the number seven, so the next day he put $777 on the 7th horse in the 7th race (a 7-1 shot, of course). Sure enough, the horse came in seventh.

Posted by: Zizka on October 17, 2003 10:40 AMMatt,

This game is a martingale, and optional stopping doesn't affect the martingale property (Feller, vol.2, VI, 12)

Posted by: maciej on October 17, 2003 01:58 PMJust to increase the freakiness quotient, to point out that the 0 and 00 don't make a difference. You could have a roulette wheel on which you doubled your money but only with a 1 in 100 chance, and by changing your bet size, there would still be a martingale which provided you with a $1 profit in finite time with probability 1.

Posted by: dsquared on October 18, 2003 03:38 PMWhat Daniel Davies means is...

Suppose we have a six sided die. If it comes up one, you win the amount of your bet. If it comes up anything else, you lose the amount of your bet.

Then follow the same strategy--if you win, quit. If you lose, double and place another bet. Then after...

1 throw you have 1/6 chance of having won $1, and of having stopped playing the game.

After 10 throws you have an 83.8% chance of having won $1, and of having stopped playing the game.

After 30 throws you have a 99.6% chance of having won $1, and of having stopped the game.

After 60 throws you have a 99.998% chance of having won $1, and of having stopped the game.

And after 100 throws you have a 99.999999% chance of having won $1, and of having stopped the game.

Of course, if you haven't rolled a "1" yet, after one throw you are down $1, after 10 throws you are down $512, after 39 throws you are down $536,870,912, after 60 throws Excel chokes on the exact number, but it is close to $576000000000000000, and after 100 throws--if you haven't thrown a "1" yet--you are down approximately $633800000000000000000000000000.

Roll the die three times a minute, and you can get to this point in half an hour.

Posted by: Brad DeLong on October 18, 2003 09:33 PMPost a comment