Frame the problem as maximizing expected score under uncertainty, where each round's optimal interval balances width (probability of containment) and ratio (score if correct). Derive the optimal interval for a single round given a belief distribution, then adapt across rounds based on remaining score needed and observed outcomes.
Pro tip: In market-making interviews, emphasize that you'd quote tighter intervals when you have higher confidence and wider when uncertain, but always consider the score function L/U—it rewards narrower intervals, so there's a trade-off. Also, mention that you'd update your beliefs after each round based on whether you hit or missed.
Clarify that score per round is L/U if true value in [L,U], else 0. Total score must be ≥2.0 over 5 rounds, so average 0.4 per round. Note that L/U is maximized when L is as large as possible and U as small as possible, but that reduces hit probability.
Assume a prior distribution for the unknown quantity (e.g., log-normal or based on market context). For a given distribution, compute the optimal [L,U] that maximizes expected score E[L/U * I(true in [L,U])]. This is a constrained optimization: maximize (L/U) * P(L ≤ X ≤ U).
After each round, update your belief about the distribution based on the outcome (hit or miss) and any new information. Also, adjust your strategy based on the cumulative score: if behind, you may need to take more risk (wider intervals or higher L/U trade-off) to catch up.
Since you need a total of 2.0, you might need to balance expected score with variance. If you're close to the target, you can play safer; if far, you might need to gamble with narrower intervals to get higher L/U.
Simulate the game with your strategy to see if it reliably reaches 2.0. Tune parameters (e.g., how aggressive to be) based on simulation results.
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The decomposition into placement and width is the key insight.
Frame the problem as maximizing expected score under uncertainty, where the optimal interval balances the width of the interval against the probability of containing the true answer. Use a probabilistic model of your belief (e.g., a distribution) and choose L and U to maximize the expected payoff function (e.g., score = 1 if correct, 0 otherwise, or a continuous scoring rule).
Pro tip: In trading interviews, they care about your thought process more than the exact answer. Explicitly state your assumptions about the scoring rule and your prior, and show how you would adapt if the scoring rule changes (e.g., from binary to linear).
Ask or state what the payoff is for a correct answer and what happens if incorrect. Is it binary (all-or-nothing) or does it depend on the interval width? The optimal strategy depends heavily on this.
Represent your belief about the true answer as a probability distribution (e.g., normal, uniform, or a set of scenarios). This captures your uncertainty quantitatively.
Write the expected score as a function of L and U. For a binary payoff, it's the probability mass between L and U; for a continuous payoff, it may involve the width or other factors.
Maximize the expected payoff. For a binary payoff, the optimal interval is the highest-density region containing a certain probability mass (e.g., the 90% credible interval). For other payoffs, solve the optimization problem.
Account for risk aversion, market conditions, or other constraints. In a trading context, you might widen the interval to avoid ruin or narrow it to maximize expected value.
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Per-hit score is 1/r, so you need ceil(2 * r) hits out of 5.
First, clarify the problem setup: define U and L as the upper and lower bounds of a random walk, and r = U/L as the fixed ratio. Then, model the pass probability as a function of r, likely using a random walk or Brownian motion framework, and analyze how it changes as r varies from tight (r close to 1) to loose (r large). Compare different tightness levels by computing or simulating the probability for specific r values, and discuss the trade-offs between pass probability and expected time or risk.
Pro tip: Emphasize that in trading interviews, the goal is often to balance risk and reward; a tighter ratio (r close to 1) may yield higher pass probability but lower profit per trade, while a looser ratio (r large) may have lower pass probability but higher potential profit. Show you understand the business context.
Clearly state that U and L are the upper and lower bounds of a random walk, and r = U/L is the fixed ratio. Assume a symmetric random walk or Brownian motion with drift, and define pass probability as the probability of hitting U before L.
For a symmetric random walk, the probability of hitting U before L is L/(U+L) = 1/(1+r). For a biased random walk with drift, use the exponential martingale or Gambler's Ruin formula.
Show that for a symmetric walk, pass probability decreases as r increases (since P = 1/(1+r)). For a biased walk, the dependence may be more complex; discuss how drift affects the relationship.
Choose specific r values (e.g., r=1.1, r=2, r=10) and compute pass probabilities. Discuss how tighter ratios (r close to 1) yield higher pass probability but smaller profit per trade, while looser ratios yield lower pass probability but larger profit.
Relate to trading: a tighter ratio may be safer but less profitable, while a looser ratio is riskier but more rewarding. Mention that optimal r depends on risk tolerance and edge.
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The dynamic programming framing is V(k, t) = best pass probability with k rounds left and t score still needed.
Frame the problem as a sequential decision under uncertainty where you balance exploration and exploitation. Explain that you would start with wider intervals to gather information, then tighten as your score stabilizes or you gain confidence, while always considering the cost of being wrong. Emphasize that the optimal strategy depends on the scoring rule and risk tolerance.
Pro tip: Mention that you would set a minimum interval width to avoid overfitting to noise, and that you would use a Bayesian approach to update your beliefs about your performance after each round.
Clarify how intervals are scored (e.g., penalty for width vs. coverage) and what the goal is (e.g., maximize expected score, minimize regret). This determines the trade-off between tightness and accuracy.
In early rounds, use wider intervals to gather information about your performance and the difficulty of the questions. This reduces the risk of large penalties from being overconfident.
After each round, update your estimate of your ability and the variance of your performance. If your score is consistently high, you can tighten intervals; if it's volatile, keep them wider.
As you gain confidence (e.g., through more data or stable performance), gradually tighten intervals. But always consider the downside risk: if a tight interval is wrong, the penalty may be severe.
Avoid making intervals too narrow, as this can lead to overfitting to recent performance. Set a minimum width based on the inherent uncertainty of the task.
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If you genuinely don't know whether the answer is 1,000 or 1,000,000, any interval wide enough to cover that range has a tiny L/U ratio.
First, clarify the scoring rule: if it's based on relative error or a fixed tolerance, a wide uncertainty range makes a high score unattainable. Then, explain that the optimal strategy shifts from precision to risk management—you should choose an estimate that maximizes expected score given the distribution, not the midpoint. Finally, discuss how to communicate this trade-off and adapt your approach to still extract value from the round.
Pro tip: In estimation rounds with huge uncertainty, the scoring rule often rewards being within an order of magnitude rather than exact accuracy. So, anchor on a plausible order of magnitude and state your confidence interval explicitly—this shows you understand the scoring mechanics and can manage ambiguity like a trader.
Ask or infer whether the score is based on absolute error, relative error, or a threshold (e.g., within 10x). This determines the maximum achievable score.
Identify the range and shape of the uncertainty. If it spans orders of magnitude, a point estimate is unlikely to be close; consider a log-normal or uniform distribution.
Choose an estimate that maximizes the expected score under the scoring rule, which may be the median or a value that hedges against extreme errors.
Explain that with such uncertainty, the attainable score is capped, and focus on demonstrating sound reasoning and calibration rather than chasing an unattainable perfect score.
Reflect on how to improve future estimates by gathering more information or using better priors, showing adaptability and a growth mindset.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Short answer: you'd concentrate tighter intervals on questions where your uncertainty is lowest and deliberately sacrifice the hardest ones.
Start by clarifying the scenario: quoting intervals likely refers to providing confidence intervals or price ranges for a set of problems, and risk allocation means how you distribute your confidence or capital across them. Then contrast sequential quoting (where you must commit to each interval before seeing the next question, so you manage risk adaptively based on remaining questions and accumulated performance) with simultaneous quoting (where you can optimize the entire portfolio of intervals, balancing risk across all five at once). Emphasize that with full information you can diversify risk more efficiently, but you must still account for correlation and uncertainty in your own knowledge.
Pro tip: Tie your answer to Optiver's trading context: they care about probabilistic thinking, calibration, and dynamic decision-making under uncertainty. Mention that sequential quoting introduces path dependency and the option value of waiting, while simultaneous quoting allows for global optimization but may reduce the value of information gained from earlier questions.
Restate the question in your own words to ensure you understand what 'quoting intervals' and 'risk allocation' mean. Assume each question requires a confidence interval (e.g., 90% CI) and that risk is the width of the interval or the amount of capital wagered.
Explain that when answering sequentially, you must quote each interval without knowing future questions. You would likely be more conservative early on, preserving capital or confidence for later, and adjust based on how well you did on previous questions (e.g., if you nailed the first two, you might take more risk on the third).
With all five questions visible, you can assess the overall difficulty and correlation between questions. You can allocate risk more efficiently by diversifying: for example, if two questions are on similar topics, you might widen intervals on both to avoid correlated losses, or if one is very easy, you can tighten its interval and use the saved risk budget on harder ones.
Highlight that simultaneous quoting allows for global optimization (e.g., minimizing total expected interval width subject to a overall confidence level), while sequential quoting is a dynamic programming problem where you update your strategy after each question. The key difference is the ability to hedge across questions.
Relate this to A/B testing, model deployment, or trading: sequential decisions are common in online learning and multi-armed bandits, while batch decisions occur in portfolio optimization. Emphasize that Optiver values the ability to think in terms of expected value, risk, and adaptation.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.