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Vexing expectations

Nover, Harris and Hájek, Alan (2004) Vexing expectations. Mind, 113 (450). pp. 237-249. ISSN 0026-4423. http://resolver.caltech.edu/CaltechAUTHORS:NOVmind04

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Abstract

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.


Item Type:Article
Additional Information:© 2004 by Mind Association. We thank John Broome, Antony Eagle, Terrence Fine, Matthias Hild, Bradley Monton, Andrew Reisner, Roy Sorensen, Manuel Vargas, Peter Vranas, and especially Mark Colyvan, Adam Elga, Braden Fitelson, Ned Hall, Chris Hitchcock, and Daniel Nolan for very helpful comments.
Record Number:CaltechAUTHORS:NOVmind04
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:NOVmind04
Alternative URL:http://dx.doi.org/10.1093/mind/113.450.237
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7496
Collection:CaltechAUTHORS
Deposited By: Lindsay Cleary
Deposited On:28 Feb 2007
Last Modified:26 Dec 2012 09:32

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