Expected Value on kenji.blog

St. Petersburg Paradox: How Much Would You Pay for a Gamble with "Infinite" Expected Value?

Thu, 10 Sep 2026 05:00:00 +0900

1. The Dream Gamble with “Infinite” Expected Value

As you walk through a casino, a dealer invites you to play a new coin toss game.

[Game Rules]

You pay an entry fee to start the game.
You toss a coin. If it lands on Heads, your prize money doubles, and you get to toss again.
The game ends as soon as it lands on Tails. You receive the prize money accumulated up to that point.

The initial prize starts at $2.

If it lands on Tails on the 1st toss, you get $2 and the game ends.
If it’s Heads on the 1st and Tails on the 2nd, you get $4 and the game ends.
If it’s Heads on the 1st and 2nd, and Tails on the 3rd, you get $8 and the game ends.
…From then on, as long as Heads keep appearing, the prize doubles to $16, $32, $64… and so on.

graph TD Start["Game Start"] --> Toss1{"1st Coin Toss"} Toss1 -->|Tails (1/2)| End1["End: Win $2"] Toss1 -->|Heads (1/2)| Toss2{"2nd Coin Toss"} Toss2 -->|Tails (1/2)| End2["End: Win $4"] Toss2 -->|Heads (1/2)| Toss3{"3rd Coin Toss"} Toss3 -->|Tails (1/2)| End3["End: Win $8"] Toss3 -->|Heads (1/2)| Toss4{"..."} Toss4 -.->|The longer the streak| Infinite["Prize doubles infinitely!"]

Now, here is a question for you. If the entry fee for this game were “$10,000 (about 1.5 million yen)”, would you participate?

Most people would probably say “I won’t participate”. Because there is a 50% chance of getting Tails on the first toss, meaning you would only get $2 and suffer a massive loss.

However, if you calculate this strictly according to mathematical probability theory (expected value), a surprising fact emerges. Mathematically, whether the entry fee is $10,000 or $100 million, you should participate in this game even if you have to borrow your entire net worth.

Why on earth is that?

2. Let’s Calculate the Expected Value

To determine whether a gamble is “profitable or not”, we use a mathematical metric called “expected value”. The expected value is a number that represents “how much you will make on average per game if you repeat the game many times”. The formula is the sum of all “(prize money) × (probability of getting it)”.

Let’s calculate the expected value for this game.

Probability of getting Tails on the 1st toss: $\frac{1}{2}$ Prize is $2$. Contribution to expected value = $2 \times \frac{1}{2} = 1$ dollar
Probability of getting Tails on the 2nd toss: Getting Heads then Tails, so $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ Prize is $4$. Contribution to expected value = $4 \times \frac{1}{4} = 1$ dollar
Probability of getting Tails on the 3rd toss: Getting Heads, Heads, then Tails, so $(\frac{1}{2})^3 = \frac{1}{8}$ Prize is $8$. Contribution to expected value = $8 \times \frac{1}{8} = 1$ dollar
Probability of getting Tails on the $n$-th toss: $(\frac{1}{2})^n$ Prize is $2^n$ dollars. Contribution to expected value = $2^n \times (\frac{1}{2})^n = 1$ dollar

In other words, no matter what toss the game ends on, the expected value for that pattern is always “$1”. Since the game can potentially continue infinitely, adding all these expected values together results in the following:

$$ \text{Total Expected Value} = 1 + 1 + 1 + 1 + \dots = \infty \text{ (Infinity)} $$

The answer derived by mathematics is “the expected value of this game is infinite”. Since the expected value is infinite, no matter how high the entry fee is, theoretically it is an absolutely “profitable gamble”.

This is the “St. Petersburg Paradox”, proposed by Nicolaus Bernoulli in 1713. There is a fierce contradiction between the correct mathematical calculation result (having infinite value) and human realistic sensation (only wanting to pay a few dollars).

3. The Discovery of “Utility” that Resolves the Gap between Mathematics and Humans

The one who solved this paradox was Daniel Bernoulli, a genius mathematician and cousin of Nicolaus. (It got this name because he presented this paper at the Academy of Sciences in St. Petersburg.)

Daniel delved into human psychology. He thought, “Humans do not judge things by the ‘absolute monetary amount’, but by the ‘satisfaction (utility)’ that the money brings.”

This is called the “Law of Diminishing Marginal Utility”.

The value of money decreases depending on the amount you hold

For example, when you are extremely thirsty in a desert, the first glass of water has enough value (satisfaction) that you would “want to drink it even if you had to pay 10,000 yen”. However, as you drink the second and third glasses, the value of a single glass of water rapidly drops. By the 10th glass, you would likely say “I wouldn’t want it even if it were free”.

The same goes for money.

“1 million yen” given to someone with zero savings has immense, life-saving value.
However, “1 million yen” given to Elon Musk, who has a net worth of tens of billions, only holds about as much value (satisfaction) as a 1-yen coin found on the street.

In other words, even if the prize money doubles infinitely like $2 \rightarrow $4 \rightarrow $8 \rightarrow $16…, the “happiness (utility)” a human feels does not increase infinitely in proportion to the amount.

4. Recalculating the Expected Value using “Utility”

Daniel Bernoulli assumed that “the value (utility) of money felt by humans is proportional to the logarithm ($\log$) of the amount”.

Let the amount be $x$, and let’s express the value (utility) felt by humans $u(x)$ as a logarithmic function (here we consider a simple model with a base of 2).

Utility of $2 prize: $\log_2(2) = 1$
Utility of $4 prize: $\log_2(4) = 2$
Utility of $8 prize: $\log_2(8) = 3$
Utility of $2^n$ prize: $\log_2(2^n) = n$

The amount doubles each time, but human “happiness” only increases little by little like 1, 2, 3… Using this “utility”, let’s calculate the expected value (expected utility) again.

$$ \text{Expected Utility} = \sum_{n=1}^{\infty} \left( n \times \left(\frac{1}{2}\right)^n \right) $$ $$ = 1 \cdot \frac{1}{2} + 2 \cdot \frac{1}{4} + 3 \cdot \frac{1}{8} + 4 \cdot \frac{1}{16} + \dots $$

When you calculate the sum of this infinite series, the result does not become “infinite”, but converges to “2”. If we reverse calculate the amount for which the utility is “2”, it becomes $2^2 = 4$ dollars.

In other words, when recalculated by incorporating human psychology (utility), a very common-sense and realistic answer is derived: **“The value of this game is about ‘$4' according to human sensation."** That is precisely why we do not feel like paying $10,000 for this game.

5. Conclusion: The Paradox that Opened the Door to Economics

The St. Petersburg Paradox was a groundbreaking paradox that mathematically proved that the objective number of “monetary amount” and the subjective value of “human satisfaction” do not match.

The concept of “Utility” proposed by Daniel Bernoulli, 200 years later, became the most important foundation of modern microeconomics and financial engineering (such as portfolio theory). The behavior of us buying insurance or diversifying investments can all be explained by this human psychological mechanism of “diminishing marginal utility (the pain of a huge loss is far greater than the joy of a huge gain)”.

A simple calculation problem of gambling ended up deciphering the human mind and triggering the birth of the massive academic discipline of economics.

The Two Envelopes Paradox: The Collapse of Logic and Decision-Making Traps Caused by Infinite Expected Values

Thu, 10 Sep 2026 00:00:00 +0900

1. The Ultimate Choice: To Switch or Not to Switch?

You are standing on the final stage of a game show. On the table in front of you are two identical-looking envelopes (A and B). The host says to you:

“One envelope contains twice as much money as the other. Please choose one.”

After some hesitation, you choose Envelope A. Just as you are about to look inside, the host whispers the devil’s temptation:

“You can exchange your Envelope A with the remaining Envelope B right now if you want. Would you like to switch?”

Now, should you switch your envelope?

2. The “Infinite Loop” Derived from Expected Value Calculations

Let’s exercise some mathematical thinking here. Suppose the amount in your Envelope A is $X$ yen. According to the rules, the amount in Envelope B is either “half of $X$ yen ($\frac{X}{2}$)” or “twice $X$ yen ($2X$)”. The probability for each is $\frac{1}{2}$ (50%).

Now, let’s calculate the expected value (the estimated average amount) if you switch envelopes.

$$ E = \frac{1}{2} \times \left(\frac{X}{2}\right) + \frac{1}{2} \times (2X) $$ $$ E = \frac{X}{4} + X = \frac{5}{4}X = 1.25X $$

A surprising result emerges. By simply switching envelopes, the expected value jumps to $1.25$ times (a 25% increase) the original $X$ yen. The conclusion becomes, “If you think mathematically, it’s definitely better to switch!”

However, a collapse of logic occurs here. Suppose you switched to Envelope B. What happens if the host asks again right after, “Do you want to switch back to A after all?” The exact same calculation formula applies, and this time it means “Switching from B to A will increase the expected value by 1.25 times.”

In other words, just by continuously switching “from A to B” and “from B to A”, the theoretical expected value will keep increasing infinitely. This clearly contradicts reality (the contents of the envelopes are fixed from the start and do not increase just because you switch them).

graph TD Start["You choose Envelope A (contains X yen)"] --> Think["Calculate if it's profitable to switch"] Think --> Case1["Envelope B has half (X/2 yen) : 50% probability"] Think --> Case2["Envelope B has double (2X yen) : 50% probability"] Case1 --> Calc["Expected Value = (X/4) + X = 1.25X"] Case2 --> Calc Calc --> SwitchToB["Switch to Envelope B! (contains Y yen)"] SwitchToB --> ThinkAgain["Calculate again"] ThinkAgain --> Case3["Envelope A has half (Y/2 yen) : 50% probability"] ThinkAgain --> Case4["Envelope A has double (2Y yen) : 50% probability"] Case3 --> Calc2["Expected Value = 1.25Y"] Case4 --> Calc2 Calc2 --> SwitchToA["Switch back to Envelope A!"] SwitchToA --> Start style Calc fill:#ff9999,stroke:#333,stroke-width:2px style Calc2 fill:#ff9999,stroke:#333,stroke-width:2px style SwitchToA fill:#ff4444,color:#fff,stroke:#333,stroke-width:4px

Why did a seemingly perfect expected value calculation produce such a strange paradox?

3. Demystifying the Mathematical Trick: The Swap of Variables

The trap of this paradox lies in “how the random variable $X$ is used”.

In the previous calculation formula, we treated the amount $X$ in Envelope A as a fixed constant, and assumed Envelope B is either “$\frac{X}{2}$ or $2X$”. However, what is actually fixed is the “total amount of money in the two envelopes”, or the “smaller amount”.

Let $S$ be the amount in the envelope with less money. Then, the envelope with more money contains $2S$. There are only two possible scenarios for the entire game (the probability of each is $\frac{1}{2}$).

Pattern 1: Envelope A you chose has the smaller amount ($S$), and Envelope B has the larger amount ($2S$)
Pattern 2: Envelope A you chose has the larger amount ($2S$), and Envelope B has the smaller amount ($S$)

Now, let’s correctly calculate the expected values for “not switching” and “switching” the envelopes.

Expected value when not switching $E_{stay}$:

$$ E_{stay} = \frac{1}{2} \times S + \frac{1}{2} \times 2S = \frac{3}{2}S = 1.5S $$

Expected value when switching $E_{switch}$: You get $2S$ in Pattern 1, and $S$ in Pattern 2.

$$ E_{switch} = \frac{1}{2} \times 2S + \frac{1}{2} \times S = \frac{3}{2}S = 1.5S $$ $$ E_{stay} = E_{switch} $$

The expected values match perfectly! In the first incorrect calculation, we treated the $X$ in Pattern 1 (which is actually $S$) and the $X$ in Pattern 2 (which is actually $2S$) as different values using the same variable $X$, which created the illusion that “switching increases the expected value.”

pie title "The Truth of Expected Values (assuming the smaller amount is S)" "Expected value of not switching (1.5S)" : 50 "Expected value of switching (1.5S)" : 50

4. What If You Open the Envelope?

The paradox seems to be resolved. However, a deeper problem awaits.

What if you looked inside your Envelope A before exchanging envelopes? When you open Envelope A, you find “10,000 yen” inside.

At this moment, $X = 10000$ becomes a fixed value. Envelope B contains either “5,000 yen” or “20,000 yen”. What happens if we apply the very first calculation formula here?

$$ E_{switch} = \frac{1}{2} \times 5000 + \frac{1}{2} \times 20000 = 2500 + 10000 = 12500 $$

The expected value is 12,500 yen. It is certainly higher than the current 10,000 yen. Moreover, since $X$ is now a “specific constant” of 10,000 yen, the previous counterargument of the “swap of variables” no longer applies. In this case, is it absolutely better to switch?

The Disproof by Bayesian Inference: The Missing “Prior Distribution”

In response to this, mathematicians introduced the concept of the “prior distribution of amounts (prior probability)”. The question is whether we can truly say that 5,000 yen and 20,000 yen are each inside with a $\frac{1}{2}$ probability.

For example, suppose the maximum budget for the show is 100 million yen. If you open Envelope A and find “60 million yen”, the probability that Envelope B contains “120 million yen” is zero (because it’s over budget). In other words, as the amount in Envelope A gets larger, the probability that Envelope B is “double” must decrease, and the probability that it is “half” must increase.

When calculating the expected value using Bayes’ theorem assuming an arbitrary prior distribution $P(x)$, it has been mathematically proven that under any realistic probability distribution (where the sum is 1), there is no magical distribution that makes it “better to switch” for all amounts of $X$.

5. The Infinite Trap: Connection to the St. Petersburg Paradox

There is only one case where it is “better to switch for all $X$”. That is only if we assume the show’s budget is infinite and all amounts (1 yen, 2 yen, 4 yen, 8 yen… up to infinity) appear uniformly—an “improper prior distribution” (a distribution whose sum is infinity).

However, in the real world, no television station has infinite assets. The bug caused by this “infinite expected value” shares deep roots with the St. Petersburg paradox (the problem of how much a person would be willing to pay for a gamble with an infinite expected value).

6. Conclusion: The Terrors of Probability and Expected Value

Even though the “Two Envelopes Paradox” consists only of simple multiplication and addition, it teaches us the following lessons:

Errors caused by ambiguity in definitions: If you do not clarify what a variable refers to (whether $X$ always refers to the same amount), logic can easily collapse.
The illusion of “no information = 50% probability”: The assumption that “because we don’t know, it must be fifty-fifty” (the principle of insufficient reason) can sometimes lead to fatal miscalculations.
The difficulty of handling infinity: Introducing the concept of “infinity,” which cannot be applied to the real world, into calculation formulas produces results that defy common sense.

The next time in life you think, “The grass is greener on the other side, so it’s better to switch,” remember this paradox. In your calculation formula, the variables might just be getting swapped.