Price a coin-doubling game rationally
Coin-Doubling (St. Petersburg) Game: Expected Value, Log-Utility Pricing, Kelly Staking, and House Cap
Context and Assumptions
This problem explores a variant of the St. Petersburg game. You purchase a ticket to play a game that begins with a bankroll of $1. A fair coin is flipped repeatedly until the first Tail appears. With each Head, the current bankroll doubles. Upon the first Tail, the game stops, and you are paid the current bankroll.
- Notation: Let K be the number of Heads observed before the first Tail. K can be any non-negative integer (K ∈ {0, 1, 2, …}). The probability of observing exactly k Heads before the first Tail is P(K=k) = 2^-(k+1). The payout for a given K is X = 2^k.
- Repeated Play (Scalability Assumption): For repeated play scenarios, you can stake a fraction
fof your current wealth per round. The house charges a pricepfor every $1 staked. The random gross payoff per $1 staked is 2^K. If you stakes = f * W_tdollars, your wealthW_tupdates multiplicatively asW_{t+1} = W_t * [1 + f * (2^K − p)]. This formulation enables Kelly analysis.
Tasks
(a) Expected Payout
Compute the expected payout E[X] of the game and state whether it converges. Provide a clear derivation.
(b) One-Shot Reservation Price Under Log Utility
Given a utility function U(w) = ln(w) and an initial wealth W, determine the maximum ticket price p* you would be willing to pay to play the game once. Provide the equation that defines p* and discuss its existence and uniqueness.
(c) Repeated Play with Kelly Staking
Using the multiplicative wealth model W_{t+1} = W_t * [1 + f * (2^K − p)]:
- Derive the condition on
(p, f)for positive expected log-growthg(f) = E[ln(1 + f * (2^K − p))] > 0. - Provide the first-order condition that defines the optimal fraction
f*. - Discuss edge cases, specifically for
p ≤ 1versusp > 1.
(d) House Cap C on Payout
Consider a scenario where the house imposes a maximum payout cap C. This means the actual payoff becomes X_C = min(2^K, C).
- Compute the expected payout E[X_C] under this cap.
- Explain how the introduction of this cap changes the risk-neutral expected value and your reservation price, considering both the one-shot log-utility perspective and the repeated Kelly staking perspective.
AI interview prep
Core dilemma
The central engineering conflict is reconciling the theoretical infinite expected value of the St. Petersburg game with practical, finite pricing and risk management strategies for both players and the house. It's about moving from pure mathematical expectation to rational decision-making under uncertainty and risk aversion.
3 tradeoffs and 3 stress tests locked
Interview prep
Unlock the interviewer X-Ray rubric, tradeoff matrix, stress tests, clarifying questions, and key concepts.
Price a coin-doubling game rationally
Coin-Doubling (St. Petersburg) Game: Expected Value, Log-Utility Pricing, Kelly Staking, and House Cap
Context and Assumptions
This problem explores a variant of the St. Petersburg game. You purchase a ticket to play a game that begins with a bankroll of $1. A fair coin is flipped repeatedly until the first Tail appears. With each Head, the current bankroll doubles. Upon the first Tail, the game stops, and you are paid the current bankroll.
- Notation: Let K be the number of Heads observed before the first Tail. K can be any non-negative integer (K ∈ {0, 1, 2, …}). The probability of observing exactly k Heads before the first Tail is P(K=k) = 2^-(k+1). The payout for a given K is X = 2^k.
- Repeated Play (Scalability Assumption): For repeated play scenarios, you can stake a fraction
fof your current wealth per round. The house charges a pricepfor every $1 staked. The random gross payoff per $1 staked is 2^K. If you stakes = f * W_tdollars, your wealthW_tupdates multiplicatively asW_{t+1} = W_t * [1 + f * (2^K − p)]. This formulation enables Kelly analysis.
Tasks
(a) Expected Payout
Compute the expected payout E[X] of the game and state whether it converges. Provide a clear derivation.
(b) One-Shot Reservation Price Under Log Utility
Given a utility function U(w) = ln(w) and an initial wealth W, determine the maximum ticket price p* you would be willing to pay to play the game once. Provide the equation that defines p* and discuss its existence and uniqueness.
(c) Repeated Play with Kelly Staking
Using the multiplicative wealth model W_{t+1} = W_t * [1 + f * (2^K − p)]:
- Derive the condition on
(p, f)for positive expected log-growthg(f) = E[ln(1 + f * (2^K − p))] > 0. - Provide the first-order condition that defines the optimal fraction
f*. - Discuss edge cases, specifically for
p ≤ 1versusp > 1.
(d) House Cap C on Payout
Consider a scenario where the house imposes a maximum payout cap C. This means the actual payoff becomes X_C = min(2^K, C).
- Compute the expected payout E[X_C] under this cap.
- Explain how the introduction of this cap changes the risk-neutral expected value and your reservation price, considering both the one-shot log-utility perspective and the repeated Kelly staking perspective.