← Jane Street Interview Insights
First, reframe the game as a payoff matrix and identify dominated strategies. Then, look for a mixed-strategy Nash equilibrium, likely with support on a small set of numbers, and prove optimality by showing indifference and no profitable deviations.
Pro tip: In zero-sum games with a penalty for being too high, the equilibrium often involves a geometric distribution; test small cases to spot the pattern before proving it.
Clearly state the rules: each player picks an integer 1-100; the larger pick is reduced by 10, then compared to the smaller pick; the larger resulting value wins. Define the payoff as +1 for a win, -1 for a loss, 0 for a tie.
Show that picking 1 is weakly dominated (e.g., picking 2 does at least as well against all opponent choices). Similarly, argue that very high numbers are risky because subtracting 10 may make them lose to slightly smaller numbers.
Assume a mixed strategy with support on a set of numbers. Set up indifference conditions: the expected payoff of each pure strategy in the support must be equal. Solve for probabilities, often leading to a geometric distribution.
Verify that no pure strategy outside the support yields a higher expected payoff against the proposed mixed strategy. Also, show that the opponent cannot exploit the strategy by deviating.
Mention how the solution changes with different bounds or penalties, and relate to real-world applications like bidding or pricing strategies.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the rules of the number-picking game, including the range of numbers and the winning condition. Then, derive the outcome as a function of the gap by considering the parity of the gap and the positions of the numbers relative to the midpoint. Finally, express the conditions for win, lose, or draw in terms of the gap and the smaller number.
Pro tip: Demonstrate structured thinking by breaking the problem into cases based on the parity of the gap and the position of the smaller number relative to the midpoint. This shows analytical rigor and the ability to handle edge cases, which is highly valued at Jane Street.
Confirm the range of numbers (e.g., 1 to N), the objective (e.g., to have the larger number), and any constraints. This ensures you and the interviewer are aligned before diving into analysis.
Let the smaller pick be x and the larger be y, with gap d = y - x. Define the winning condition for the player who chose y in terms of x and d, considering the game's mechanics.
Examine how the parity of d and the position of x relative to the midpoint of the range affect the outcome. Use symmetry arguments to simplify the analysis.
Based on the analysis, express the exact conditions for the larger-number player to win, lose, or draw as a function of d and x. Consider edge cases where x is near the boundaries.
Test the derived conditions with small numerical examples to ensure correctness and to illustrate the logic clearly.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
The dominance argument for the lower bound is actually clean once you see it: any pick below some threshold is strictly dominated by shifting up by the penalty amount, because you never do worse against any opponent pick and sometimes do strictly better.
First, clarify the game rules and the definition of optimality. Then, use a proof by contradiction: assume a pure strategy is optimal, and show that an opponent can exploit it by choosing a number that beats it. Finally, derive the lowest rational number by iterated elimination of dominated strategies, typically leading to a threshold like 0 or 1 depending on the game.
Pro tip: Connect the game to real-world strategic interactions, such as bidding in auctions or pricing decisions, to show practical relevance. Emphasize that mixed strategies are essential in competitive settings where predictability is a disadvantage.
Restate the rules, including the number range, winning condition, and what 'optimal' means (e.g., maximizing expected payoff). Confirm assumptions like rationality and common knowledge.
Assume a pure strategy (a fixed number) is optimal. Show that if everyone else plays it, an opponent can choose a number that wins more often, contradicting optimality. Use a symmetry or best-response argument.
Apply iterated elimination of dominated strategies: start from the highest possible number and reason downwards. Show that any number above a certain threshold is dominated, leading to the lowest rational choice.
Explain that the optimal strategy is mixed, often uniform over a range. Mention Nash equilibrium and how it ensures no pure strategy can be exploited.
Conclude with the lowest number and relate the reasoning to data science concepts like game theory, decision theory, and strategic thinking in modeling.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
This is where I basically ran out of steam.
First, clarify the dominance elimination process and identify the surviving numbers. Then, analyze the win/loss structure by constructing a tournament matrix and checking for a Condorcet winner. If none exists, formulate the problem as a zero-sum game and solve for the optimal mixed strategy using linear programming or iterative methods.
Pro tip: In game theory interviews, always connect the solution to practical implications, such as market making or resource allocation, to show you understand the business context.
Ask clarifying questions to understand how dominance elimination works and what 'survive' means. Confirm the set of surviving numbers.
For each pair of surviving numbers, determine which one beats the other based on the given rules. Create a matrix where entry (i,j) is 1 if i beats j, -1 if j beats i, and 0 if tie.
Look for a number that beats all others (a Condorcet winner). If found, that number is the pure optimal strategy. If not, proceed to mixed strategies.
Set up the problem as a two-player zero-sum game where the payoff matrix is the win/loss matrix. The optimal mixed strategy is the Nash equilibrium of this game.
Use linear programming (e.g., simplex method) or iterative methods (e.g., fictitious play) to find the mixed strategy that maximizes the minimum expected payoff. Interpret the probabilities.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the underlying game: two players alternately pick numbers from 1 to 100 without replacement, and the player who picks the larger number pays a penalty k (originally 10) to the other. Then analyze how the optimal strategy depends on k and N by considering the trade-off between picking a high number to win the round and the penalty cost. Finally, discuss how the strategy changes as k varies (e.g., for small k, picking high is more attractive; for large k, picking low to avoid penalty may be better) and as N increases (e.g., the game becomes more about relative ranks and the penalty becomes less significant relative to the range).
Pro tip: Emphasize that the optimal strategy is not just about maximizing the chance of picking the larger number, but about maximizing expected net payoff, which depends on both the probability of winning and the penalty size. Also, note that as N grows, the penalty k becomes relatively smaller, so the strategy converges to simply picking the highest available number.
Restate the rules: two players alternately pick numbers from 1 to N without replacement; the player who picks the larger number pays a penalty k to the other. Confirm that the payoff for a round is +1 (or +k?) for the winner and -k for the loser, or similar.
Explain the known optimal strategy for the original parameters: likely a threshold strategy where you pick the highest number if it's above some threshold, otherwise pick the lowest to minimize penalty.
Discuss how changing k affects the threshold: as k increases, the penalty for losing becomes larger, so you become more conservative and may pick lower numbers to avoid being the larger pick. As k decreases, you become more aggressive.
Explain that as N increases, the relative impact of k diminishes, so the optimal strategy approaches simply picking the highest available number. Conversely, for small N, the penalty is more significant.
Conclude that the optimal strategy balances the expected gain from winning against the expected penalty from losing, and that the balance depends on the ratio k/N or similar.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Interesting pivot because now it's not adversarial, it's just optimization against a fixed distribution.
First, clarify the game's payoff structure and rules, as the question is underspecified. Then, assuming a common setup like a symmetric zero-sum game (e.g., higher number wins), compute the expected payoff for each pure strategy against the uniform distribution and identify the best response. Finally, discuss mixed strategies if needed and verify the solution.
Pro tip: Show that you recognize the ambiguity and ask clarifying questions before diving into calculations. This demonstrates careful thinking and prevents solving the wrong problem.
Ask about the payoff structure: Is it a zero-sum game? What happens in ties? Are players choosing numbers simultaneously? This ensures you understand the problem correctly.
Assume a common payoff: you win if your number is higher than your opponent's, lose if lower, and tie if equal. Write the expected payoff for choosing a number x against a uniform opponent.
For a given x, the probability of winning is (x-1)/100, losing is (100-x)/100, and tying is 1/100. If win payoff is +1, lose is -1, tie is 0, then expected payoff = (2x-101)/100.
The expected payoff increases with x, so the best pure strategy is x=100, yielding expected payoff 99/100 = 0.99. If ties are resolved differently, adjust accordingly.
If the game allows mixed strategies, note that against a uniform opponent, any pure strategy is a best response if the opponent is fixed. But in equilibrium, the opponent would not play uniformly if it's not optimal.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.