From Paradox to Praxis
Utility Function through the Lens of Saint Petersburg
In the heart of the 18th-century intellectual milieu emerged a paradox, subtle in its formulation yet profound in its implications, known as the Saint Petersburg Paradox. This enigma, nestled within the framework of a simple gambling game, unfurls a narrative that transcends the realm of chance, venturing into the domain of human rationality and the valuation of wealth. The scene is set with a fair coin tossed repeatedly until a heads appears, marking the game’s closure. The payoff for the player is orchestrated as 2n, where n is the count of tosses. As the coin spirals in the air, so does the mathematical expectancy of the game, soaring towards infinity. Yet, when the coin lands, the expectancy seems to plummet in the face of human valuation, with individuals unwilling to pay a significantly large sum to partake in this gamble. This discord between the cold calculus of expectancy and the warm, often whimsical, human valuation of wealth sets the stage for a deeper inquiry.
It was Daniel Bernoulli who, with a stroke of insight, proposed a resolution that would not only untangle the paradox but also lay the cornerstone for modern utility theory. He introduced the notion that humans do not perceive wealth in a linear fashion; rather, the relationship is logarithmic. The utility derived from…
