Five-Round Interval-Estimation Game: Optimal Intervals and Risk Allocation

You play five independent rounds. In round i, an unknown numeric answer A is drawn from a continuous distribution. You announce a closed interval [L, U]. The round score is:

• 0 if A ∉ [L, U]
• L/U if A ∈ [L, U]

Total score is the sum across five rounds. The target is total ≥ 2.0. Assume that for each round you have a subjective continuous pdf f(a) (cdf F) for A.

Answer the following:

(a) Formulate the optimization for choosing L and U that maximizes the expected per-round score E[(L/U)·1{A ∈ [L, U]}] given a known continuous pdf.

(b) Derive the first-order optimality conditions (FOCs). Discuss how the optimal coverage probability compares to conventional confidence levels under light- vs heavy-tailed beliefs.

(c) Describe a strategy to allocate risk across five rounds (e.g., dynamic programming or heuristics) to maximize the probability of achieving total ≥ 2.0, including how to adjust interval tightness after early wins/losses.

(d) Provide a quick mental method for approximately optimal [L, U] under log-normal beliefs (A > 0), and explain how you’d adapt for heavy tails.

(e) Outline a simple simulation to validate your strategy and estimate the probability of meeting the ≥ 2.0 target.

Assume U > 0 and typically A > 0 (most market-making questions are positive). If A could be negative, you may work on a transformed scale (e.g., log of absolute value) or restrict to positive-support questions.