088概率中等derivationmedium

The St. Petersburg Paradox

题目

A casino offers the following game. A fair coin is flipped repeatedly until the first Tails appears. If the first Tails occurs on flip $n$ , you win $2^n$ dollars. (a) Compute the expected payoff of the game. (b) Despite the answer to (a), most people would pay no more than about $20 to play. Resolve this apparent paradox using Daniel Bernoulli's approach: assume the player has log utility $u(x) = \ln(x)$ and initial wealth $W$ . Compute the expected utility of the game and find the certainty equivalent (the sure amount that gives the same expected utility) for $W = 1{,}000{,}000$ . (c) A more practical resolution: suppose the casino has finite total capital $C$ . If the payout is capped at $C = 2^{40}$ (about $1 trillion), what is the expected payoff?

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Suppose a risk-neutral hedge fund with access to unlimited leverage is offered the St. Petersburg game at a price of $25 per play. They can play $N = 1{,}000$ independent rounds. What is the probability that their total payout exceeds $25,000 (i.e., they profit)? Use the structure of the distribution — not the Central Limit Theorem — to argue why the CLT fails here.

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