Undefined Expected Value
A Game Worth
Nothing and Everything
The St. Petersburg game has infinite expected value. The Pasadena Game is worse: its expected value does not exist at all.
You know the St. Petersburg Paradox: a coin-flip game with infinite expected value. It challenged decision theory for 300 years. But there is a more troubling variant.
The Pasadena Game, introduced by Nover and Hájek in 2004. The rules:
Tails on flip 1
+$2
Tails on flip 2
-$4
Tails on flip 3
+$8
Tails on flip 4
-$16
The formula: if tails appears on flip n, you receive (-1)^(n+1) * 2^n dollars. Odd flips win, even flips lose.
Question: How much should you pay to play?
For St. Petersburg, the answer was "infinitely much" (a paradox, but at least a number). For Pasadena, the answer is there is no answer. The expected value is undefined.
Not infinite. Not zero. Not any number.
The expected value simply does not exist.
Play the Pasadena Game
Experience the game firsthand. Watch how wins and losses alternate, with magnitudes doubling each flip.
The Pasadena Game Rules
- 1.A fair coin is flipped until it lands TAILS
- 2.If tails appears on flip n, you get (-1)^(n+1) * 2^n dollars
- 3.Odd flips = WIN, Even flips = LOSE
Flip 1 = +$2, Flip 2 = -$4, Flip 3 = +$8, Flip 4 = -$16...
Play several times. You will see a pattern: 50% chance of +$2 (tails on flip 1), 25% chance of -$4 (tails on flip 2), 12.5% chance of +$8, and so on.
The wins feel good, the losses sting. But notice: there is no clear sense of whether you are "ahead" or "behind" relative to some expected baseline - because there is no expected baseline.
An Uncomfortable Feeling
Without an expected value, you cannot say whether the game is "good" or "bad" for you. It exists in a strange limbo outside the usual framework of rational betting.
The Alternating Series
Let us calculate the expected value. For each flip number n:
- Probability of tails on exactly flip n: (1/2)^n
- Payout if tails on flip n: (-1)^(n+1) * 2^n
- Contribution to EV: (1/2)^n * (-1)^(n+1) * 2^n = (-1)^(n+1)
Each term contributes either +1 or -1. The expected value is:
The Pasadena Expected Value Series
Every term alternates between +1 and -1
The probability halves each time (1/2, 1/4, 1/8...) but the payout doubles (2, -4, 8, -16...). The magnitudes always multiply to 1, but the signs keep flipping.
The sum 1 - 1 + 1 - 1 + ... has no value.
This is Grandi's Series, known since 1703. It does not converge. The partial sums alternate between 1 and 0 forever.
Unlike St. Petersburg where EV diverges to infinity, here the series oscillates without settling anywhere.
Watch the Partial Sums
For a series to converge, its partial sums must approach a single limit. Watch what happens with Grandi's series:
Current partial sum after 0 terms:
--
The partial sums oscillate forever
S_1 = 1, S_2 = 0, S_3 = 1, S_4 = 0, ... The sequence (1, 0, 1, 0, ...) never converges.
There is no limit. The expected value does not exist.
The partial sums bounce between 1 and 0 forever. They never settle. No matter how many terms you add, you cannot determine what the sum "really" is.
Contrast with St. Petersburg
In St. Petersburg, each partial sum is larger than the last: 1, 2, 3, 4, 5... The limit is infinity. Here, the partial sums are 1, 0, 1, 0, 1, 0... There is no limit at all.
A sum without a limit is not a number. The expected value of the Pasadena Game is not just hard to calculate - it is mathematically undefined.
Pasadena vs. St. Petersburg
The St. Petersburg Paradox is famous for its infinite expected value. The Pasadena Game represents an even deeper challenge to decision theory.
St. Petersburg Game
Payout: 2^n on flip n
EV Series: 1 + 1 + 1 + 1 + ...
Expected Value: INFINITY
The sum diverges to positive infinity
Pasadena Game
Payout: (-1)^(n+1) * 2^n on flip n
EV Series: 1 - 1 + 1 - 1 + ...
Expected Value: UNDEFINED
The sum does not converge to any value
| Property | St. Petersburg | Pasadena |
|---|---|---|
| Expected Value | +inf | undefined |
| Series Type | Divergent (to +inf) | Oscillating (no limit) |
| Utility Fix Works? | Yes | No |
| Bounded Wealth Fix? | Yes | Partially |
| Philosophical Trouble | High | Extreme |
St. Petersburg asks: "What if EV is infinite?"
Pasadena asks: "What if EV does not exist at all?"
The standard fixes for St. Petersburg - logarithmic utility, bounded wealth, probability weighting - do not help here. Even with diminishing marginal utility, the signs still alternate. Even with bounded wealth, the oscillation remains.
The Core Difference
St. Petersburg has a divergent but positive series. You can say it is "worth infinity." Pasadena has an oscillating series. You cannot assign it any value - positive, negative, or otherwise.
Rearranging to Get Any Value
Here is the most disturbing property of conditionally convergent (or non-convergent) series: by rearranging the terms, you can make the partial sums approach any value you want.
This is the Riemann Rearrangement Theorem. Applied to the Pasadena Game, it means: there is no "true" expected value waiting to be discovered. The answer depends on how you choose to sum.
The Riemann Rearrangement Theorem
For any conditionally convergent series (a series that converges, but not absolutely), the terms can be rearranged to sum to any value you want - including positive infinity, negative infinity, or any real number.
Grandi's series (1 - 1 + 1 - 1 + ...) does not converge at all, but the same principle applies: by choosing which +1s and -1s to sum first, we can make the partial sums approach any target.
The expected value of the Pasadena Game is whatever you want it to be
Depending on how you sum the contributions (which flip you count first), you can make the "expected value" equal any number. This is not a number that can be calculated - it is fundamentally undefined.
Why This Matters for Decision Theory
If a game's expected value depends on the arbitrary order you sum probabilities, then expected value cannot guide rational decision-making for that game. The Pasadena Game shows that some gambles are not just risky - they are fundamentally incomparable to anything else.
Want the expected value to be 5? Rearrange the terms. Want it to be -3? Rearrange differently. Want infinity? Sure. Negative infinity? Also possible.
The Implication for Decision Theory
If the expected value of a gamble depends on arbitrary choices about how to sum probabilities, then expected value theory cannot tell you whether to take that gamble. The framework simply breaks down.
Philosophical Implications
The Pasadena Game forces us to confront deep questions about rationality, probability, and decision-making. Expand each section to explore:
The Pasadena Game is not just a mathematical curiosity.
It is a limit case showing where our foundations of rational choice run out.
What Do We Do Now?
Several responses have been proposed to the Pasadena Game:
Use Weak Expectations
Define expected value only when the series converges absolutely. The Pasadena Game would simply have no expected value, and we accept that.
Regularization Methods
Use Cesaro summation or similar techniques to assign 1/2 to Grandi's series. But this feels like imposing structure that isn't really there.
Dominance Reasoning
Compare gambles by outcomes rather than expected values. The Pasadena Game would be compared case-by-case against alternatives.
Accept Incomparability
Some gambles simply cannot be compared. The Pasadena Game exists outside the space where expected value theory applies.
There is no consensus. The Pasadena Game remains an open problem in philosophy, showing that even our most foundational tools for reasoning about uncertainty have sharp limits.
Some questions are not puzzles to be solved,
but boundaries to be understood.
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Reference: Nover & Hajek (2004)