Vexing Expectations

We introduce a St. Petersburg-like game, which we call the ‘Pasadena game’, in which we toss a coin until it lands heads for the first time. Your pay-offs grow without bound, and alternate in sign (rewards alternate with penalties). The expectation of the game is a conditionally convergent series. As such, its terms can be rearranged to yield any sum whatsoever, including positive infinity and negative infinity. Thus, we can apparently make the game seem as desirable or undesirable as we want, simply by reordering the pay-off table, yet the game remains unchanged throughout. Formally speaking, the expectation does not exist; but we contend that this presents a serious problem for decision theory, since it goes silent when we want it to speak. We argue that the Pasadena game is more paradoxical than the St. Petersburg game in several respects. We give a brief review of the relevant mathematics of infinite series. We then consider and rebut a number of replies to our paradox: that there is a privileged ordering to the expectation series; that decision theory should be restricted to finite state spaces; and that it should be restricted to bounded utility functions. We conclude that the paradox remains live. 1. The Pasadena paradox It’s your lucky day. We offer you at no charge the following game, which with winking homage to a more famous game that inspired it, we will call the Pasadena game. We toss a fair coin until it lands heads for the first time. We have written on consecutive cards your pay-off for each possible outcome. The cards read as follows: (Top card) If the first heads is on toss #1, we pay you $2. (2nd top card) If the first heads is on toss #2, you pay us $2. (3rd top card) If the first heads is on toss #3, we pay you $8/3. (4th top card) If the first heads is on toss #4, you pay us $4.