# Quant Trading Game: Expected-Value Betting and Market Making on Coins, Dice, and Cards You are playing a live, 30-minute trading game against the interviewer. You start with a bankroll of $1,000 and play an unlimited number of rounds until time runs out. Each round you decide whether to bet, and how much, using only your current bankroll. The game moves through three stages: 1. **Coin flips.** The interviewer quotes you odds on the outcome of a fair coin flip, and you choose whether to bet and how much. 2. **Sum of two dice.** The interviewer quotes you odds on events defined over the sum of two fair six-sided dice (for example, hitting a specific total), and you again choose whether and how much to bet. 3. **Playing cards.** Cards are drawn from a standard, well-shuffled 52-card deck. This stage has two flavors: bets quoted on the **product** of the drawn cards' values, and a final segment on the **sum of three drawn cards**, where instead of quoting odds the interviewer makes you a two-sided market — a bid price and an ask price — and you may **buy** or **sell** the sum at those prices, or pass. Partway through, the interviewer introduces additional constraints on how you may bet, and at the end asks follow-up questions about the reasoning behind your betting decisions. You are expected to act quickly — a couple of minutes per round at most. Work through how you would play this game. ### Constraints & Assumptions - 30-minute session; unlimited rounds until time expires. - Starting bankroll is $1,000; a bet may not exceed your current bankroll, and a busted bankroll ends the game. - The coin is fair, the dice are fair six-sided dice, and the deck is a standard 52-card deck. Assume card values are the ranks: Ace = 1, number cards at face value, Jack = 11, Queen = 12, King = 13. - For odds-based bets, "$b:1$ odds" means a $1 stake returns $b$ of profit if you win and loses the $1 stake if you lose. - In the market segment you may buy at the ask or sell at the bid (settling against the realized sum), or decline to trade. - The interviewer may change the rules or add constraints mid-game; you are expected to adapt your strategy on the spot. ### Clarifying Questions to Ask - Are card values the standard ranks (Ace = 1 through King = 13), and are multi-card draws made without replacement? - Am I allowed to pass on any round, or is there a minimum forced bet? - In the market segment, is there a limit on the size I can trade at the quoted prices? - Am I evaluated on final profit and loss, or on the quality and speed of my decision process? - When cards are drawn one at a time, can I trade again after seeing partial information? ### Part 1 For the odds-based stages, build the machinery you will reuse every round: derive the fair value and the bet/pass rule for (a) a coin flip at quoted odds, (b) events on the sum of two dice, and (c) the product of dice or card values. What quantities should you precompute so that each round takes seconds, not minutes? ```hint Precompute, don't re-derive Fair values never change between rounds of the same stage. Compute the key expectations **once** (coin, the two-dice sum distribution, the mean card value) and reduce every quote to a single comparison: bet only when $p \cdot b > 1 - p$. ``` ```hint Two dice and products The sum of two dice has a triangular distribution with mean $7$. For a **product** of independent quantities, use $E[XY] = E[X]\,E[Y]$ — no need to enumerate outcomes. ``` #### What This Part Should Cover - The correct expected-value condition for taking a bet at quoted odds. - The distribution and moments of the two-dice sum, and the mean card value. - Recognizing which numbers can be precomputed once per stage and reused. - A stated pass rule for zero- or negative-edge quotes. ### Part 2 In the final segment the interviewer quotes a two-sided market on the sum of three cards drawn from the deck — a bid and an ask around some level. Determine the fair value of the three-card sum, decide when you should buy, sell, or pass, and explain how your fair value updates if the cards are revealed one at a time. ```hint Fair value of the sum The mean rank of a card is $7$, so linearity of expectation gives the fair value of the three-card sum directly — sampling without replacement does not change the mean. ``` ```hint After a card is revealed Condition on what you have seen: once a card is removed, the remaining 51 cards still sum to a known total, so their mean is easy to recompute. Update your fair value round by round rather than re-deriving it from scratch. ``` #### What This Part Should Cover - The fair value of the three-card sum and the buy-below-fair / sell-above-fair decision rule. - The effect of drawing without replacement on variance (and why the mean is unaffected). - Updating the fair value as cards are revealed sequentially. - Weighing the edge against the spread and the variance of the payoff. ### Part 3 The interviewer hints that the Kelly criterion is not the right sizing rule for this game. Explain what Kelly sizing is, why its assumptions break down here, and give a practical bet-sizing policy you would actually use across all three stages. ```hint Kelly's assumptions Kelly sizing maximizes long-run log growth, but that guarantee only holds when the edge is known exactly and the same bet repeats many times. Ask which of those premises a 30-minute, finite-round game with quoted (sometimes estimated) edges actually satisfies. ``` #### What This Part Should Cover - A correct statement of the Kelly fraction and what it optimizes. - At least two concrete reasons Kelly's assumptions fail in this setting. - A usable alternative policy (for example fractional Kelly with a hard cap) and how size should scale with edge and variance. - Bankroll-preservation reasoning: why risking ruin is unacceptable even when positive-edge bets keep coming. ### What a Strong Answer Covers Across all parts, the interviewer is evaluating you as a trader, not just as a probabilist: - **Speed with correctness** — fair values precomputed and reused, decisions made in seconds, arithmetic verbalized cleanly under time pressure. - **One consistent framework** — every quote reduced to fair value versus price, so the coin, dice, and card stages become the same decision in different clothes. - **Risk discipline** — bet sizing that scales with edge and variance, never risks ruin, and adjusts when constraints are injected mid-game. - **Communication under pressure** — stating fair value, edge, and size out loud so the interviewer can follow the reasoning behind every action. ### Follow-up Questions - The first card revealed is a King. How does your fair value for the three-card sum move, and would you trade against the original market? - Mid-game the interviewer imposes a rule that you must bet at least 10% of your bankroll every round. How does your strategy change? - If the roles were reversed and you had to quote the market on the three-card sum yourself, how would you choose the width of your bid-ask spread? - You have found one clearly mispriced market with a large edge. Why might betting your entire bankroll on it still be wrong?