Nicolaus Bernoulli's letter
Nicolaus Bernoulli first posed the problem in a letter to the French mathematician Pierre Rémond de Montmort dated 9 September 1713. Nicolaus himself suggested that people simply ignore very unlikely events.
A fair coin is flipped until it lands tails. The pot starts at $2 and doubles with every heads. When tails comes, you take the pot. What do you win on average? In other words, what is a fair price for one game? Lock in your answer, then play, run millions of games and see how the Bernoullis dealt with it.
A fair coin is flipped until the first tails; the pot starts at $2 and doubles with every heads. The expected win is infinite: every possible outcome adds exactly $1 to the average, and there are infinitely many outcomes. Yet nobody would pay even $100 to play, and half the games pay just $2. Here you play yourself, run millions of games and see how Bernoulli, Cramer and Feller tackled the paradox.
The rules fit in one line, and you never leave empty-handed: you get at least $2. The only question is how much it is worth paying to get in.
The coin is fair: heads and tails each come up with chance 1/2, and flips are independent. It is flipped again and again until tails comes up for the first time. Then the game ends.
Before the first flip the pot holds $2. Every heads doubles it: $4, $8, $16 and so on. Tails stops the game and you take whatever is in the pot.
If the first tails comes on flip number k, you win 2k dollars:
What is the average win if you play very many times? Mathematicians call this number the expected value. It is taken as the fair price of the game: pay exactly that, and on average neither you nor the casino comes out ahead.
"A few dollars, ten at most"
In fact the average win is infinite. Every row of the table on the left adds exactly $1 to it: half the chance, but twice the pot. There are infinitely many rows, so the sum grows without bound. And at the same time half the games end on the very first flip with a win of $2.
The second assumption is unrealistic, and that is one of the keys to the paradox. Rounds 4 and 5 take it apart.
Pick an option before reading on. We mean the average win per game if you play millions of times.
Reminder: tails on flip 1 pays $2, on flip 2 $4, on flip 3 $8, and so on.
A breakdown of your answer will appear here.
You have already played 100 games at a $20 fee each. Keep playing one at a time to see the coins, or 100 or 10,000 at once to see the balance. Change the fee and compare: at what price does the game start paying off?
If the mean were finite, a player's average win would settle near it after many games. Here it never settles: it creeps upwards, and big wins kick it up now and then. The games axis is logarithmic.
The dashed log₂ n is an approximation, not an exact curve: William Feller showed that the total win over n games divided by n · log₂ n tends to 1. A typical player in the simulations runs a little above it, by about 2 to 3 dollars.
Bars: the exact chance 1/2k. Dots: the share in the 6,000,000 simulated games. The vertical scale is logarithmic, each step is 10 times smaller.
A row's contribution is its chance times its win. In exact terms it is $1 for every k. In the simulation the rare wins either did not happen at all (0) or happened once and added too much.
The calculation is simple and correct. The paradox is that it disagrees with how sensible people behave. Here are five ways to see where the gap comes from.
The chance that the first tails comes on flip k is 1/2k, and the win is 2k. The product is always $1. Row one gives $2 · 1/2, row two $4 · 1/4, row ten $1024 · 1/1024. The mean is the sum of all rows: 1 + 1 + 1 + … The more rows, the bigger the sum, with no limit.
Half the games pay $2, and the median win is $2 too. 15 games in 16 pay $16 or less. The infinity of the mean lives in rows that almost never happen: the chance of winning a million dollars or more is 1/524,288, about 2 in a million. A person who plays once is right to think in medians, not means.
| The casino has | Mean win |
|---|
If the pot exceeds what the casino owns, the player gets only what the casino has. Then the rows are cut off past a certain point and the mean becomes finite. Even a trillion-dollar bank gives only about $41: every thousandfold increase in wealth adds roughly $10. Back in 1777 Buffon calculated that after 29 heads in a row there would not be enough money in the whole Kingdom of France to pay out.
Daniel Bernoulli proposed counting not dollars but their usefulness: doubling your wealth always brings the same joy, so utility is proportional to the logarithm. Measured that way, the game's mean utility is finite and equals the utility of a sure $4. Taking the player's wealth into account, a person with $1,000 should sensibly pay up to $10.95, and a millionaire up to $20.87. Cramer reached a similar conclusion with the square root as early as 1728.
Play n games and you will typically see one win of about n dollars, two of about n/2, and so on. Each level gives roughly $1 per game, and about log₂ n levels have time to show up. So the average win per game grows like log₂ n (a typical player runs 2 to 3 dollars above it): to raise it by $1 you need twice as many games. Infinity is reached only after infinitely many games.
A similar trap with expected values sits in the two envelopes paradox: there too the "on average it pays to switch" argument breaks down on infinite means.
CE stands for "certainty equivalent": the guaranteed amount the player finds exactly as attractive as the game. For plain dollars (linear utility) it is infinite; for the logarithm and the square root it is finite, because big wins count for less and less. The bank formula explains the table above: the first L rows give $1 each, and all longer games together bring B times the chance of getting that far, 1/2L.
Three ways to make infinity finite: limit the casino, count utility instead of dollars, or ask about a series instead of one game. A simulation checks every result here.
The cut-off works like a logarithm: the mean win is log₂ B plus about one dollar. Making the bank ten times bigger adds only log₂ 10 ≈ 3.3 dollars. For the fair price to reach $100, the casino would need about 299 dollars, a number with 30 digits.
A simulation of a million games matches the exact answer well for small banks. For a billion and up it usually shows less: wins above 10 million come up less than once per 8 million games, and their dollars simply do not make it into the sample. It is the same effect as in round 3.
For Bernoulli the utility of an extra dollar is inversely proportional to how much a person already has. Hence the logarithm. Then "how much to pay" has an answer that depends on wealth: for a poor player even a $5 entry is too risky, while a millionaire can easily pay $20.
The weak spot of this answer: for any unbounded utility function you can design a game with even faster-growing wins, and the paradox comes back. Karl Menger pointed this out in 1934. For the logarithm, for example, it is enough to pay out 2 to the power 2k dollars.
An infinite mean means this: for any fixed fee, the share of players who are ahead tends to 100% as the number of games grows. But the higher the fee, the more patience you need. At $10 per game, about 4 players in 5 are ahead after 1,000 games. At $20, even after a million games about one in five is still behind.
William Feller proposed looking at exactly that, the series: a fair total fee for n games is about n · log₂ n dollars, that is log₂ n per game. Each point here comes from several hundred independent series, and for each number of games the simulation counts what share of them ended ahead.
Nicolaus Bernoulli first posed the problem in a letter to the French mathematician Pierre Rémond de Montmort dated 9 September 1713. Nicolaus himself suggested that people simply ignore very unlikely events.
Nicolaus's cousin Daniel Bernoulli published a solution based on the utility of money in the Commentarii of the St. Petersburg Academy of Sciences (Commentarii Academiae Scientiarum Imperialis Petropolitanae). Hence the paradox's name. Bernoulli wrote that any fairly reasonable man would gladly sell his chance for twenty ducats. In his version the pot started at one ducat.
Ten years before Daniel Bernoulli, the Geneva mathematician Gabriel Cramer suggested in a letter to Nicolaus Bernoulli that the value of money be measured by a square root. He remarked that mathematicians value money in proportion to its quantity, while sensible people value it by what they can do with it.
In 1777 Georges-Louis Leclerc de Buffon described 2,048 real games. The longest lasted 9 flips (there were 6 of those), and 1,061 games ended on the first flip. In our dollars that is $20,114, about $9.82 per game. Buffon's first prize was 1 rather than 2, so he got 10,057 units, about 4.9 per game. It is one of the first statistical experiments in history.
William Feller proved a law of large numbers for this game in an unusual form: the total win over n games divided by n · log₂ n tends to 1. So a fair fee for a series of n games is about n · log₂ n. Curiously, at that fee the player in simulations wins the series about 4 times in 5, so a casino would hardly accept that price.
If the utility of money is a logarithm, the game's mean utility is log₂ 4 = 2. So, ignoring other wealth, the game is worth a guaranteed $4. A person with $2 should borrow $1.35 and pay up to $3.35, and one with $1,000 up to $10.95.
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