1
Each term is the same
A prize that doubles is exactly cancelled by a probability that halves, so every toss contributes 1 to the expected value. The partial sums are 1, 2, 3 and so on.
Lesson · Decision and Rationality · Puzzles of expected value
The St Petersburg Game
A coin is tossed until it lands heads, and the prize doubles with each toss. Every term of the expected value is 1, so the sum grows without bound, and yet nobody would pay much to play. Capping the bankroll makes the series finite, and the lab computes the price for a bank of any size.
prize 2^k if the first head is on toss k
chance of that: 1 / 2^k
every term: 2^k · 1/2^k = 1
n terms sum to n, with no upper bound
bankroll cap: the series has a finite sum
Infinite on paper, small in practice
Core ideas
What this lesson establishes
The puzzle sets expected value against common sense, and the sum that creates it can be written down in one line.
2
The sum has no ceiling
However large a number you name, enough terms pass it. The expected value is not a large number but no number, and the rule says to pay anything for the right to play.
3
A cap turns the series finite
A real bank can pay only what it has. Once the prize is capped at the bankroll, late terms shrink instead of staying at 1, and the series adds up to a figure you can compute.
The material
Terms that never shrink, and a cap that makes them
The sum first, then what stops it, then what the stopping does and does not show.
A casino offers a game. A fair coin is tossed until it first lands heads. If that happens on toss k, the player is paid 2^k dollars. So heads on the first toss pays 2, heads on the second pays 4, heads on the third pays 8, and a run of twenty tails before the first head pays over a million. The question, posed in Nicolaus Bernoulli's letter of 1713, is what the player should pay to enter.
The expected value is the sum, over k, of the prize times its probability. The first head falls on toss k with probability 1 / 2^k, so each term is the prize times that chance.
k prize chance term
1 2 1/2 1
2 4 1/4 1
3 8 1/8 1
4 16 1/16 1
Theorem · Every term is 1
The term for toss k is 2^k · 1/2^k = 1. After n tosses the partial sum is exactly n, and a larger n passes any number you name. The expected value is not finite.
The lab, with no cap and twenty terms, prints a sum of 20 and no limit. Add terms and the sum rises by 1 each. If expected value is the price to pay, the player should pay any amount to enter, and no ticket is too dear. Nobody believes that. Ask people what they would pay and the answer is usually a few dollars.
The first reply is that no bank has an infinite purse. Suppose the bank has a bankroll of 1024 dollars, which is 2^10. If the first head arrives on toss 10 or earlier the bank pays in full. If it arrives later the bank pays only 1024. The term for toss k then becomes 1024 divided by 2^k whenever k is above 10.
k 1 to 10 11 12 13
term 1 1/2 1/4 1/8
The first ten terms add to 10. The terms after them are a half, a quarter, an eighth, and so on, which add to exactly 1. The whole series therefore sums to 11, and 11 dollars is the expected value of the game against a bank with 1024. The lab prints the limit 11, and for twelve terms the partial sum 43/4, a quarter short of it.
The cap grows slowly. A bank with 1,048,576 dollars, which is 2^20, gives twenty terms of 1 and then a tail of 1, so the game is worth 21. A bank with a trillion dollars would still be worth only a little over forty. The price rises with the logarithm of the bank, which is why even a very rich casino cannot make the game look expensive.
So the capped game is no puzzle, but it is not a full answer either. The puzzle was meant to show that expected value and sensible price come apart, and the cap shows one reason: the infinity was in the bank. Another reason is in the utility, and it is Daniel Bernoulli's reply of 1738. If the player values each extra dollar less, as in “Risk Aversion and the Value of Information”, the late prizes are worth little to her, and the sum of the utilities can be finite without any cap. The lab computes the first reply only.
Both replies leave something unsaid. A cap is a fact about the bank. A utility curve is a fact about the player, and a concave one closes this game without closing the puzzle: a curve that keeps rising, however slowly, can be outrun by a game whose prizes grow faster, and the infinite sum returns. Only a utility with a ceiling closes every such game, and a ceiling on how good things can get is a strong thing to assert. A reader who finds either reply too convenient has the harder position: that the uncapped game is a coherent thing to evaluate, and that the rule gives the wrong answer on it.
Interactive
Exact partial sums
n terms added as fractions, against the limit when there is one
Method
Summing the game, with and without a cap
Four steps, and the cap decides which sum you are asked for.
1
Write the term for each toss
Prize times probability. Without a cap this is 2^k · 1/2^k, which is 1 for every k.
2
Add the terms
The partial sum after n tosses is n, which has no upper bound. Any price is below the expected value.
3
Apply the bankroll
Replace the prize by the smaller of 2^k and the bankroll. Terms up to the cap stay 1; later ones are the bankroll over 2^k.
4
Add the tail
Beyond the cap the terms halve each time, so they sum to a number you can write down. Add it to the count of terms of 1 to get the value of the game.
Terms, sums and caps
Score 0 / 0Choose the answer you can justify.
The same game with a bankroll of 1024
The bank can pay at most 1024, which is 2^10, so tosses 1 to 10 pay in full and later tosses pay 1024.
tosses 1 to 10: 10 terms of 1 = 10
toss 11: 1024/2048 = 1/2
toss 12: 1024/4096 = 1/4
tail from toss 11: 1/2 + 1/4 + 1/8 + ... = 1
value of the game = 10 + 1 = 11
Twenty terms of an uncapped game sum to 20 and the sum keeps growing; the capped game sums to 11 and stops. A price of 11 or less is then a fair one against this bank, which is a very different answer to the one the infinite sum gave.
Common mistakes
1
Thinking an infinite expected value means you should pay any priceThe expected value is infinite only if the bank can pay any prize. Against a bank of 1024 dollars the value is 11 and against one of 1,048,576 it is 21. The infinity belongs to an idealised bank, and a rule applied to a game the bank could never run is not a rule applied to a game you could play.
2
Reading the cap as a refutation of expected valueThe cap removes the infinity but not the gap between the expected value and what most people would pay, which for a bank of a million dollars is 21. Whether expected value is the right price or whether the utility of late prizes should be discounted is still open.
3
Expecting the value to grow in step with the bankEach doubling of the bankroll adds exactly 1 to the value. A bank a thousand times larger adds about ten. The sum is dominated by the early terms, because the late prizes, though large, are very unlikely.
Check your understanding
✓
Finish when you can sum the game and price it against a given bank.
Given a bankroll, you should be able to write the term for each toss, show that without a cap each term is 1 and the sum has no ceiling, and compute the value of the capped game as the count of full terms plus the tail.
Note: The lab sums the expectation exactly, one toss at a time, and the bankroll must be a whole number of dollars. It scores dollars at face value. A bounded utility, the other standard reply, is described here and not computed.