Coin-Doubling (St. Petersburg) Game: EV, Log-Utility Pricing, Kelly Staking, and House Cap

Context and assumptions

• Single play: You buy one ticket to a game that starts at $1. You flip a fair coin until the first Tail. With each Head, the bankroll doubles; at the first Tail you stop and are paid the current bankroll. If the flip sequence is H,H,T you receive$ 4.
• Notation: Let K be the number of Heads before the first Tail. Then K ∈ {0,1,2,…} with P(K=k) = 2^{-(k+1)} and the payout is X = 2^k.
• Part (c) repeated play (scalability assumption): You can stake a fraction f of wealth per round. The house charges a price p per $1 staked; the random gross payoff per$ 1 staked is 2^K. If you stake s = f W_t dollars, your wealth updates multiplicatively as W_{t+1} = W_t [1 + f (2^K − p)]. This makes Kelly analysis well-defined.

Tasks (a) Compute the expected payout E[X] and state whether it converges; show your derivation.

(b) One-shot reservation price under log utility. With U(w) = ln w and initial wealth W, what maximum ticket price p* would you pay to play once? Give the equation that determines p* and discuss existence/uniqueness.

(c) Repeated play with Kelly staking. Using the multiplicative model W_{t+1} = W_t [1 + f (2^K − p)]:

• Derive the condition on (p, f) for positive expected log-growth g(f) = E[ln(1 + f (2^K − p))] > 0.
• Give the first-order condition that defines the optimal fraction f*.
• Discuss edge cases (e.g., p ≤ 1 vs p > 1).

(d) House cap C on payout. If the house caps the maximum payout at C (i.e., payoff becomes X_C = min(2^K, C)):

• Compute E[X_C].
• Explain how the cap changes the risk-neutral expected value and your reservation price (one-shot log-utility and repeated Kelly perspectives).