The Mean, the Median, and the St. Petersburg paradox
Abstract (Via Sjdm)
The St. Petersburg Paradox is a famous economic and philosophical puzzle that has generated numerous conflicting explanations. To shed empirical light on this phenomenon, we examined subjects’ bids for one St. Petersburg gamble with a real monetary payment. We found that bids were typically lower than twice the smallest payoff, and thus much lower than is generally supposed. We also examined bids offered for several hypothetical variants of the St. Petersburg Paradox. We found that bids were weakly affected by truncating the gamble, were strongly affected by repeats of the gamble, and depended linearly on the initial “seed” value of the gamble. One explanation, which we call the median heuristic, strongly predicts these data. Subjects following this strategy evaluate a gamble as if they were taking the median rather than the mean of the payoff distribution. Finally, we argue that the distribution of outcomes embodied in the St. Petersburg paradox is so divergent from the Gaussian form that the statistical mean is a poor estimator of expected value, so that the expected value of the St. Petersburg gamble is undefined. These results suggest that this classic paradox has a straightforward explanation rooted in the use of a statistical heuristic.
Introduction (Via SJDM)
In the St. Petersburg paradox, originally proposed in 1738, the house offers to flip a coin until it comes up heads. The house pays $1 if heads appears on the first trial; otherwise the payoff doubles each time tails appears, with this compounding stopping and payment being given at the first heads (Bernoulli, 1738; shown in Figure 1). By conventional definitions, the St. Petersburg gamble has an infinite expected value; nonetheless, most people share the intuition that they should not offer more than a few dollars to play. Explaining why people offer such small sums to play a gamble with infinite expected value is an important question in economics and philosophy (Datson, 1988; Samuelson, 1977; Martin, 2008; Gigerenzer & Selten, 2002).