The Percentage Game

The action takes place in London, UK, approached via Runyon's Broadway, NY.

Note for pedants: The author is well aware that the St Petersburg paradox was first noted by Nicolaus Bernoulli in the 18th century, not by Blaise Pascal in the 17th.

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I am walking down the street wondering whether to spend a bob or two in Sally Lunn's Bijou Caff and Grill on the offchance that congenial company may be found therein, or whether to invest it in a nag called Wingless Wonder that my milkman informs me that very morning is a sure thing for the 2.30, on the solemn oath of his brother-in-law who is acquainted with a girl whose sister's boyfriend cleans stables in Epsom, when whom do I see approaching but System Sid, wearing a big cigar between his teeth, a broad smile on his mug, and a buxom blonde on his arm. I am very surprised indeed to see Sidney in this condition, because when we last meet he is stony broke and most downhearted, so I greet him cordially but not so loudly as to be audible to any passing fuzz, as I figure that sudden affluence is often an attribute of characters whose assistance is required by the Old Bill in connection with their enquiries.

"Things all right then, are they, System?" I ask, sotto voce, to which he replies in a hearty manner, "Never better, squire, never better. Perhaps you are wondering about my change of circumstances since our last encounter? You recall that on that occasion I am still licking my wounds after my grand challenge match with Educated Ernie?" I nod my head, for I do indeed recollect just that, the match in question being regarded in these parts as most noteworthy, the circumstances being as follows.

System Sid is well known as a regular punter who bets on any game of chance, and indeed upon some in which chance is not allowed to interfere. He is always following some system or other, which is how he acquires his monicker, but with infrequent success, perhaps because his idea of a system is apt to verge upon the eccentric. Last year, for example, he loses an unwarrantable proportion of his dole money backing second favourites who are wearing blinkers for the first time.

Sidney has dreams of a big win someday, but his chief ambition is to put one over on Educated Ernie, a character who is so called because when he is last a guest of Her Majesty's Government he enrols in the Open University, thereby acquiring both a higher education and maximum remission. He puts his name down for Sociology like all the other cons, but a clerical error in the Governor's office results in his taking a degree in Statistics, which is close enough, alphabetically speaking. To his surprise, he finds that a large part of Statistics concerns probability theory, and that a large part of probability theory concerns the outcome of games of chance. Upon his release, Ernie puts his new-won knowledge to good use, and soon has a reputation for knowing the odds against drawing to a bobtail straight, and suchlike useful intelligence. His studies obviously include Applied Statistics, because in next to no time Ernie is winning more bets than Sidney loses, which is saying something.

The rivalry between them reaches a head when Sidney challenges Ernie to a series of personal wagers, which Ernie gladly accepts. This is the match to which Sidney now refers, and he reminds me of its details.

"We agree," he says, "to play three different games, each game to be played continuously for a whole day. The first day, Ernie suggests a simple dice game. I stake £6 and throw three dice. If any of them shows a five, Ernie pays me £13. I figure that I have a fifty-fifty chance of getting a five, so the even money payout is £12. With the extra pound as my edge, how can I lose? To my surprise, at the end of the first day, I am out of pocket.

"The second day, Ernie bets me even money that in any group of thirty people, two or more of them are having the same birthday. I figure that as there are 365 days in a year, the odds are more than ten to one against, so I happily accept this bet. To play this game, we go to the Library, where there is a book with details of famous people. I choose which page to start on, and we look at the next thirty birthdays. To my sorrow, I find I am paying out to Ernie far more often than I am winning.

"The third day, Ernie suggests that we sit in the window of Sally Lunn's and watch the traffic go by. He bets me even money that whenever a blue van goes past, the next bus is a number 57. Naturally, I check with the bus company before I take this bet. They tell me that there are three bus routes passing Sally Lunn's: the 57, the 23, and the 79. The frequency of all three is exactly the same: one every five minutes throughout the day. I also observe that blue vans are by no means uncommon in that street; one passes every few minutes. Naturally I figure that the odds are two to one against the next bus being a 57, and I take the bet. Would you believe that in five hours, Ernie wins 35 times and I win only 25 times and am out of cash? Ernie is by no means magnanimous in victory. He not only refuses to lend me the bus fare home, but he calls me a mug for accepting such bets. He claims that he figures the odds for each game and that they favour him. Furthermore, he says, he looks forward to meeting more mugs like me, for he is always willing to wager on bets where the expected value of a win exceeds his stake.

"Naturally, I am very disheartened by this debacle," says Sidney, "but a few days later my luck changes." He goes on to tell me that he pops into the betting shop to pass the time of day with his coterie, and there meets a peculiar little man who projects such a receptive aura that soon Sidney is confiding his unhappy experience to him, whereupon the stranger clucks sympathetically.

"I fear that this Ernest out-calculates you," he says. "In the first game, the fair payout is nearer £14 than £13. The odds are not fifty-fifty, as you suppose. You are forgetting that sometimes you are getting more than one five at a time, and for these you get only one payout. The probability of not getting a 5 with three dice is five-sixths times five-sixths times five-sixths, or approximately four out of seven. That is Ernie's edge."

"The birthday bet is somewhat similar. The probability of thirty people having birthdays all on different days is 364 divided by 365, times 363 divided by365, times 362 divided by 365, and so on up to 336 divided by 365, which works out to a little less than thirty percent, so you see Ernie is on to a good thing with an even money bet."

"As to the buses," he says, "you are overlooking that what counts is not how often they run, but when, in relation to each other. As a matter of fact," he continues, "I happen to know that the 57 runs on the hour (and every five minutes thereafter), while the 23 runs one minute later, and the 79 one minute after that. So you see," he says, "in any five minute period, there are three minutes when the next bus is a 57, and only two minutes when it is a 23 or a 79."

Sidney is very sad to receive this information, as he begins to appreciate that he is indeed a mug, and he says despondently to the stranger, "So there is no way of betting successfully against a person who figures the percentages as accurately as Educated Ernie?"

"I am not saying that," says the stranger pensively. "You tell me, I recall, that Ernest promises faithfully to bet on any wager where the odds are in his favour? Then listen carefully," and he gives Sidney some very detailed instructions.

The next day, Sidney tells me, he calls in some favours to raise a little betting money, and seeks out Educated Ernie. He reminds Ernie of his promise to take any bet with favourable odds, tells him that he now has such a proposition for him, and asks him to make good. "Certainly," says Ernie, "on condition that you prove that the odds are favourable, and that I am allowed to play the game as often as I wish." Sidney agrees, and then speaks as follows.

"First," says Sidney, "let us be sure we agree on how to calculate your expectation of a game. For example, if you are to throw a single dice, and I am to pay you one pound per spot showing, you have a one in six chance to win £1, plus a one in six chance to win £2, plus a one in six chance to win £3, and so on. So your expectation is one-sixth of £1, plus one-sixth of £2, plus one-sixth of £3, plus one-sixth of £4, plus one-sixth of £5, plus one-sixth of £6, which adds up to £3.50. So the odds are in your favour if I offer to play such a game with you for a stake of less than £3.50." Ernie assents to this method of calculation.

"Very well," says Sidney, "here is the game I propose we play: you pay a stake and then toss a coin until it comes down heads, which is the end of the game. If you get a head on the first throw, I pay you £2; if you do not get a head until the second throw, I pay you £4; if you do not get a head until the third throw, I pay you £8; and so on, doubling the payout for every successive tail that you throw. Now what is your expectation for this game? Obviously you have a chance of one in two of winning £2, plus a chance of one in four of winning £4, plus a chance of one in eight of winning £8, and so on, so your expectation is a half of £2, plus a quarter of £4, plus an eighth of £8, plus one-sixteenth of £16, and so on ad infinitum, which equals £1 + £1 + £1 + £1 and so on ad infinitum. Since your expectation is an infinite sum of money, it follows that no matter how large your stake, the odds are in your favour. I am willing to let you play this game as often as you like, for a stake of £10,000 per game, which is infinitely less than your expectation. When are you wanting to begin?"

"So you win a large sum of money from Ernie playing this game?" I ask, hoping to expedite Sidney's narrative.

"Not exactly," says Sidney. "At first Ernie goes a strange green colour as he checks the calculations and can find nothing wrong with them. Then he offers me £5,000 to release him from his promise to wager with me, which I accept, as I am by no means as small minded in victory as he is. Then he asks me where I am learning of this game, and I tell him about the stranger. He asks his name, and I remember that the stranger tells me that his name is Blaze Paskle. Educated Ernie goes from green to white when he hears that name, for it appears that not only is this Paskle gent a Frenchman very well known for being adept at calculating probabilities in wagers, he is also dead for the past three hundred years.

"To tell you the truth," adds Sidney, "I am wishing that I know earlier that Blaze is not of this world, for then I might have more confidence in him and save myself the fifty quid that I bung Fingers McShane for the loan of his Irish penny, which he positively guarantees to turn up heads three times out of four."

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