The Pasadena Game

The setup

A fair coin is flipped until it lands heads. On flip n, the payoff depends on whether n is odd or even:

• If n is odd: you win $2^n / n
• If n is even: you lose $2^n / n

So: heads on flip 1 wins $2. Heads on flip 2 loses $2. Heads on flip 3 wins $8/3. Heads on flip 4 loses $4. And so on.

To find the expected value, you sum each outcome multiplied by its probability. But this series does not converge. It oscillates without settling on any limit. Depending on the order in which you add the terms, you can make the expected value equal any real number you like, including positive infinity, negative infinity, and zero. Mathematicians call a series like this conditionally convergent: it has no single well-defined sum.

The game is worse than having an infinite expected value. It has no expected value at all.

What this reveals about expected value theory

The St. Petersburg Paradox showed that infinite expected monetary value is not a reliable guide to rational choice. The natural response was to move to expected utility: the value function people should maximize has diminishing returns, so it stays finite. The St. Petersburg Paradox then becomes a problem only for unbounded utility functions.

The Pasadena Game attacks at a deeper level. It is not that the expected utility is inconveniently large. It is that standard expected utility theory cannot even assign a number. Any decision procedure that tells you to choose the option with the highest expected utility is silent when expected utility is undefined.

This is not just a problem for exotic pathological cases. It exposes something about the foundations: expected utility theory assumes that utilities can be summed in a well-defined way. The Pasadena Game is constructed to violate exactly that assumption. If you accept the mathematics, the game has no rational answer.

What it suggests about the limits of formal decision theory

The standard responses to the Pasadena Game are either to reject it as too pathological to take seriously, or to modify expected utility theory to handle undefined expectations.

Neither response is fully satisfying. Rejecting it as pathological requires explaining which games count as legitimate and which don't, and that line is hard to draw without already assuming the theory you're defending. Modifying the theory to handle undefined expectations opens questions about what alternative framework you're substituting.

Nover and Hájek use the game to press a broader point: formal decision theory has been developed around cases it can handle well. When you push outside that range, you find that the framework rests on assumptions that look necessary but aren't. A complete theory of rational choice under uncertainty needs to say what to do when expected values don't exist. Expected utility theory does not say this. It assumes the problem away.

The Pasadena Game is less a puzzle to be solved than a pressure test. It reveals what current decision theory is actually doing, and what it has quietly assumed.