Paradox

232 A BRIEF HISTORY OF THE PARADOXImam could reason just as well this way.” Pascal has left outthe possibility that there is a Baal or a Moloch or a Zeus orsome other claimant of faith. In the absence of doctrinalconstraints, we must also include inversions such as thepossibility that God rewards the doubters and punishesbelievers. To narrow the field of contenders, Pascal mustrevert to the theologizing that initially seemed bypassed bythe wager.THE ST. PETERSBURG PARADOXPascal’s wager is also challenged by economists who reject thepossibility of infinite utility. Their finitism emerged indiscussions of a problem that was first stated in a letter writtenby Nicholas Bernoulli and published in 1713. A fair coin willbe tossed until a head results. You will then be paid $2 n-1where n equals the number of tosses. So the expected returnis: (½ × $1) + (¼ × $2) + (8 × $4) + . . . :+ (½n × $2 n-1 +. .. . Since each addend equals a half dollar, and there areinfinitely many of them, the sum is infinite. Thus, someonewho maximized expected money should be willing to pay anyamount of money for this bet. Yet few people would pay $100for the deal.Jean d’Alembert named this puzzle the St. Petersburgparadox because the first article on the paradox was publishedby the Imperial Academy of Sciences at St. Petersburg. Theauthor was Nicholas Bernoulli’s cousin Daniel Bernoulli. Hepointed out that doubling one’s cash holdings from a millionto two million does not really double its value to you. Eachnew dollar tends to have less influence on your welfare thanthe preceding dollar. Daniel Bernoulli’s insight is enshrined