WINNERS & LOSERS

Lesson 2

Some time before the earnest young man’s experiment took place, he had had a preliminary run.  During a Christmas school vaction, when he and his wife decided to take a break from teaching at the University of California at Los Angeles by spending a few days in Las Vegas, a colleague had called Thorp’s attention to an article in the Journal of the American Statistical Association, describing a strategy tips for playing blackjack.  It claimed that, following the strategy given, the hose’s advantage was limited to the tiny edge of 0.62 per cent.  This allowed the player an almost even break.  (‘Almost of course is not enough: as explained in chapter six, that percentage will wipe out a gambler’s capital in the long run, s surely as night follows day.)

  Thorp purchased ten silver dollars and tried out the strategy as recommended.  After may vicissitudes, up and down, he lost eight-and-a-a-half bucks, and quit.  No gambler he!  But the experience intrigued him, especially the evidence before his eyes that most players hadn't a clue about the fundamentals of the game or how to play their hands correctly.  (In my experience, this still holds true today.)  The visit implanted a seed in his mind.  When he got back home he studied the article (‘The Optimum Strategy in Blackjack’, baldwin, cantey, Maisel and McDermott).  The came the quantum jump, the moment – Eureka! –that Thorp changed the world.  ‘In a flash of mathematical insight I realized it must be possible for the player to beat the game.’
His idea, as a he described it, was this.  The basic strategy by Baldwin et al.  was a complete set of instructions for the player, advising him how to play his cards according to the hand he was dealt and the dealer’s upcard.  But to simplify the calculations, this analysis assumed that the deck always contained its average composition, that is that all the hands were dealt from a complete shuffled deck.  Yet that is not the case in actual play! (It would be like coming upon a large area of earth and expecting to find it completely flat.)  after the first card is dealt out, the deck is in a marginally different composition, and after the first half-dozen hands are dealt, the deck is likely to be in quite a radically different composition.  Suppose, to take an extreme case, that all four aces have been dealt out – you cannot get a blackjack.  Or again, take a very extreme case, that the deck has been dealt out all the way to the last six cards and they consist of two sevens and four eights.  How should you bet?  The answer, as Thorp explains, is the maximum – borrow from the bank if you like because you cannot lose.  You stand pat on your two cards and when the dealer looks at his hand he finds either (7.7), (7.8) or (8.8).  Being below 17 he has to draw – and whatever card he draws busts his hand.

This will never happen in practice, maybe.  But in general the proportion of cars left in the deck, after the first cards come out, will not be the same as in a complete deck, and the casino/player advantage will fluctuate.  Mathematics considerations suggested to Thorp that this fluctuation would often be larger than 0.62 per cent, and further, that the player would frequently have the advantage himself.  ‘If the player were to bet very heavily when he held the advantage and very lightly when he had the disadvantage, he would not need to have the advantage very often in order to make a handsome profit.’  This was Thorp’s great insight.  Obvious when you see how, isn’t it?
So Thorp, now an assistant professor of maths of M.I.T., set about determining when the player did have an advantage and how large it would be.  A deck of card is a complicated affair.  There are 34 million different ways of removing cards from a deck as they might be removed by a dealer.  The Baldwin calculations had taken four capable young men a total of twelve man-years of off-duty army time, working with the aid of desk calculators.  Thorp’s first step was to master every detail of these calculations; his next to analyze the effect on casino advantage when changes were made in the proportions of various cards in the deck.  The serious disadvantage to the player if all the aces are out I mentioned above.  Because greater precision was required, it was necessary to extend the Baldwin calculations many times and since ten thousand man-years at desk calculators were required, the problem could only be cracked with the aid of a computer.  Thorp had access to an IBM 704.  It was a long and tricky business writing out the program.  When he had finished it took the computer seven hours to print out the answers, enough numbers to fill an average exercise book.
The answers he got amazed Thorp.  The typical casino advantage over a player using the best complete strategy, now known everywhere as basic strategy, he found to be only 0.21 per cent.  If various groups of cards were used up in play, the advantage surged wildly back and forth between casino and online poker player (blackjack players knew this empirically but they had no idea why – winning and losing to them was just ‘luck’.)  The biggest swing was caused by the four fives.  When they were out of the deck the player had an edge of more than 3.3. per cent.  The next most important group was the four aces; with them out the casino advantage rose to about 2.7 per cent.  The effect of variations in the 16 cards of ten-value (king, queen, jack, ten) was even more dramatic.  The player or house advantage sometimes swung to plus ten per cent, occasionally much more.  (What this means, putting it in a simple way, is that when there are lots of tens in the deck, the house is likely to go bust drawing on ‘stiffs’, small hands of 12 to 16, whereas the player can stick.)
Thorp developed a detailed system of play which involved keeping count of the number of ten-value cards remaining to be played.  This technique is known- dreaded word for the casinos! – as counting.

The essence of it is that low cards are counted as + I and high cards as –I.  (If the cards are dealt out in the order A, 2,3,4,5,6,7,8,9,10 the point values would be A (-I), 2(+I) , 3(+I), 4(+I) ,5(+I), 6(+I), 7(o), 8(o), 9(o), 10(-I), giving a running total of +3.  Modern system are much more elaborate.)  The casinos hate counters.  If they detect you counting, anywhere in the world today, even playing for very small stakes like a couple of bucks a hand, the dealer and the pit boss will be on their guard.  If you look like an expert, they will usually intervene to take preventive action, quite often barring you straight away.  ‘OK fell, you ’re so smart, go count the spots on the dice!’
When Thorp emerged from his back room into the light of day, he was hailed, to his astonishment,

as a mathematical wizard, a man who had discovered the philosopher’s stone which could turn dross into gold.  What happened was that he decided to give a little talk to the annual meeting of the American Mathematical Society in Washington DC on a simplified version of his strategy based on counting fives; a few days before the meeting, the society, as is usual in academic get-togethers, published abstracts of the two hundred or so papers that were to be delivered.  Included was his short abstract titled ‘Fortune’s Formula: A Winning Strategy for Blackjack’.  After his talk he was asked to give a press conference; then he was televised by a major network and interviewed on a number of radio programmers.  Over the next few days and weeks he received literally hundreds of letters and long distance phone calls, requesting more information, and several offers to back him in a casino test of the new system.  The amounts proffered ranged from a few thousands to as much as $ 100.000. He had unwittingly touched a deep, probably universal, chord in the public’s subconscious, the desire to get rich quick at the expense of the casinos: easy money.

As I explained.  Thorp is not a gambler, he is a mathematician. He is not against making money: he later developed, as I shall relate, a ‘system ’ for beating the stock market which is probably the mot successful method ever developed, which he has employed, consistently, with spectacular success, to become a millionaire many times over.  All this was in the future.  For the moment, the question was whether to go to Nevada and put his theories to the test.  He finally decided to go: what may have clinched matters, as he confessed in the Atlantic Monthly (June 1962) were the scoffs and boasts from casinos themselves that his claims were ridiculous.  Their arrogance was succinctly summed up by a casino operator who was asked on a nationwide television programme if the customers ever walked away winners.  The man replied, in a memorable formulation: ‘When a lamb goes to the slaughter, the lamb might kill the butcher.  But we always bet on the butcher.’
Thorp took up the most attractive of the many offers pouring in, from two New York millionaires, both large-scale gamblers.  There were two approaches possible: a ‘wild’ method of playing which involved betting the casino limit whenever the advantage to the player exceeded, say, one per cent; and a conservative method of progressively increasing the stakes, as the advantage in any particular deck grew bigger.  The former method could produce the greatest gain in the shortest time, but in the run of a few days the fluctuations in the player’s capital could be violent.  His sponsors were prepared to back him beyond $ 100.000 if necessary, but Thorp did not know how he would be affected if he found himself betting more each minute than his monthly salary.  Besides, his objective wasn’t to make big money for his backers, but to conduct a serious experiment.  Accordingly, he favored the second approach, of being certain of a moderate win.  His staking method - which has become the basis of all counting techniques at blackjack in the years since then – was to double his basic stake when the player advantage was one per cent, increase it to four times when the advantage was two per cent, leveling off at ten times when the advantage increased to five per cent or more.  if his bets ranged from $ 50 to $ 500, six or seven thousand capital would probably suffice; to be on the same side they took $ 10,000.

Is it any wonder when the cutely-pie dealers at Reno saw this earnest gangling young fellow at the table, with his tedious way of peering around at all the cards, that they marked him down as another nut – and got irritated at his play?  It ’s all very different nowadays, if you ever get invited behind the scenes in a modern casino.  Up in the ‘eye in the sky’, in the tangle of cables and switches behind the one-way mirror, you will find a space-age scene: a bank of TV cameras tracking every blackjack table in the house, monitored by a card-counting expert, capable of recording on video tape every hand that is played out night and day.  What they are looking for, apart from checking on the dealer’s honesty, are Thorp’s heirs and successors.
A couple of things rapidly happened to Thorp on this field trip which are still typical of management behavior.  First, when it became clear, whether he was a nut or just lucky or whatever, that he could beat the game, the casinos took preventative action.  Their most simple resource was shuffling up the deck.  If the cards are re-shuffled early on in the deal, there is less scope for the player to pick up variations in the compositions of the deck, which naturally show up more markedly as the deck is reduced to the last few cards.  the casinos did not know what Thorp was really up to; this was simply a device to change the flow, break the pattern in some way. 
He also detected cheating.  Secondly, as soon as Thorp and his companions changed gear, to play at the high-stakes tables, the managers would arrive, wreathed in smiles.  High  rollers are the jam on the bread-and-butter of the regular small-time punters, and each casino wants as much of the pot as it can get its sticky fingers on.  Thorp and his friends were playing $ 500 maximum.  Many pleasantries and politenesses would be exchanged and hospitality proffered.  But the deck was shuffled up even earlier, 15 cards from the end, 25 cards from the end and finally 42 cards from the end!  Note that Thorp was not doing anything wrong whatever.  He was playing poker by the rules by the rules of the game.  But he was using his expertise to beat the game, and that, to put it plainly, was not acceptable.  Rien ne va plus.

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