Consider this game: start with $1. Flip a fair coin repeatedly. On each Head, your current bankroll doubles; on the first Tail, the game ends and you take your current bankroll. Example: H,H,T → you take $4. a) Compute the expected payout and state whether it converges; show your derivation. b) If you must state a maximum price you would pay to play once given risk‑averse logarithmic utility U(w)=ln(w) and initial wealth W, what price do you choose? c) If you can play repeatedly with Kelly staking fraction f of wealth against a fixed ticket price p per play, derive the condition on p and f for positive expected log‑growth and give the optimal f*. d) Discuss how a house cap (maximum payout C) changes the expected value and your reservation price.

This question evaluates understanding of probability distributions, divergent expected values, utility theory, and optimal betting strategies (Kelly staking) within the Statistics & Math domain for a Data Scientist role.

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This is a hard difficulty Statistics & Math question, commonly asked during Take-home Project rounds at Point72.

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This question is commonly asked for Data Scientist candidates at Point72 during technical interviews.

Price a coin-doubling game rationally | Point72 Interview Question

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.

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).

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

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.

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).

Price a coin-doubling game rationally

Quick Overview

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

Solution

Submit Your Answer

Price a coin-doubling game rationally

Quick Overview

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

Solution

Submit Your Answer