Why Parrondo's paradox is irrelevant for utility theory, stock buying, and the emergence of life

Abstract

ince the initial publication of Parrondo’s paradox [1, 2], a number of articles onthe subject have appeared both in the academic literature and in the media.The paradox describes a reversal of fortune for a gambler who loses money byplaying either of two games of chance, but who makes money by randomly playingthe two games. The academic literature suggests that Parrondo’s paradox advancesvon Neumann and Morgenstern’s work on utility theory, that it offers money for free,that it has implications for investing strategies, and that it may hold the key forunderstanding certain biological processes, possibly even the emergence of life [3, 4].HarmerandAbbott[2]suggestthattheparadoxmightoperateineconomicsorsocialdynamics to extract benefits from ostensibly detrimental situations. For example, ifa society or an ecosystem suffers from declines in either the birth rate or the deathrate, declines in both together might combine with favorable consequences.Media reports compare Parrondo’s paradox with the sacrifice of chess pieces towin a game. There are suggestions that Parrondo’s paradox offers the possibility ofmaking money by investing in losing stocks. Paulos [5] observes that “standard stockmarket investments cannot be modeled by games of this type, but variations of thesegames might conceivably give rise to counterintuitive investment strategies. Al-though a much more complex phenomenon, the ever-increasing valuations of somedot-coms with continuous losses may not be as absurd as they seem. Perhaps they’llone day be referred to as Parrondo profits.” The same article quotes Derek Abbott assaying that “President Clinton, who at first denied having a sexual affair with MonicaS. Lewinsky saw his popularity rise when he admitted that he had lied. The addedscandal created more good for Mr. Clinton.” All these examples seem to suggestmiraculous possibilities for turning unfavorable situations into favorable ones, eitherby purposeful actors or by acts of nature. Parrondo’s paradox appears to be aremarkable finding with far-reaching consequences in wide ranging situations.Or is it? That’s the question we ask and hope to answer. To do so, we need to lookmore closely at the rationale behind Parrondo’s paradox, stripping it of all extrane-ous baggage and descriptive analogies as ratchets and pawls. We need to understand