Chapter 15: Quiz

Instructions

Select the best answer for each question. Some questions may have multiple correct interpretations; choose the most complete or precise answer.

Question 1

In a multi-outcome prediction market with 5 candidates, the prices are [0.30, 0.25, 0.20, 0.15, 0.12]. What is the overround?

(A) 2%
(B) 5%
(C) 8%
(D) 12%

Answer: (A) The overround is $\sum p_i - 1 = 0.30 + 0.25 + 0.20 + 0.15 + 0.12 - 1 = 0.02 = 2\%$.

Question 2

Which overround removal method is most likely to produce negative probabilities for low-probability outcomes?

(A) Multiplicative method
(B) Additive method
(C) Shin's method
(D) Power method

Answer: (B) The additive method subtracts $\omega/n$ from each price. If any price is less than $\omega/n$, the result is negative, which is not a valid probability.

Question 3

In Shin's model of overround, the parameter $z$ represents:

(A) The total overround percentage
(B) The proportion of informed traders in the market
(C) The number of outcomes in the market
(D) The liquidity parameter of the market maker

Answer: (B) Shin's model explicitly accounts for a fraction $z$ of market participants who have inside information about the true outcome.

Question 4

The power method for overround removal finds exponent $k$ such that $\sum p_i^k = 1$. For a market with overround, this exponent satisfies:

(A) $k < 0$
(B) $0 < k < 1$
(C) $k = 1$
(D) $k > 1$

Answer: (D) Since $\sum p_i > 1$ and each $p_i < 1$, raising each to a power $k > 1$ shrinks them, reducing the sum to 1.

Question 5

A Dutch book of Type 1 (buyer's Dutch book) exists when:

(A) Prices sum to more than 1
(B) Prices sum to less than 1
(C) Prices sum to exactly 1
(D) Any price exceeds 0.50

Answer: (B) When prices sum to less than 1, buying one contract of each outcome costs less than 1 but guarantees a payout of exactly 1, yielding a risk-free profit.

Question 6

In a 4-outcome market, you construct a pair trade: long Outcome B, short Outcome A. In how many of the 4 possible states do you at least break even?

(A) 1
(B) 2
(C) 3
(D) 4

Answer: (C) You profit if B wins (your long pays out) and you break even or profit if C or D wins (neither position pays out, but you collected net credit from selling A at higher price). You lose only if A wins (your short obligation pays out). So you are profitable in 3 of 4 states (assuming A is priced higher than B and neither wins gives the spread credit).

Question 7

The expected return of a multi-outcome portfolio with fractions $f_j$ at prices $p_j$ with true probabilities $q_j$ is:

(A) $\sum f_j (p_j / q_j - 1)$
(B) $\sum f_j (q_j / p_j - 1)$
(C) $\sum f_j (q_j - p_j)$
(D) $\sum q_j / \sum p_j - 1$

Answer: (B) Each bet on outcome $j$ contributes $f_j(q_j/p_j - 1)$ to expected return. The ratio $q_j/p_j$ compares the true probability to the price; subtracting 1 gives the per-unit expected edge.

Question 8

Why is fractional Kelly (using $\alpha < 1$ times the full Kelly bet) especially recommended for multi-outcome markets?

(A) Multi-outcome markets have lower expected returns
(B) The Kelly criterion does not apply to mutually exclusive outcomes
(C) Probability estimation across many outcomes has higher model risk
(D) Transaction costs are always higher in multi-outcome markets

Answer: (C) Estimating a complete probability distribution over many outcomes is inherently more uncertain than estimating a single binary probability. This higher model risk means full Kelly is too aggressive, making fractional Kelly prudent.

Question 9

The odds ratio $\text{OR}_{ij} = (q_i/q_j)/(p_i/p_j)$ is used to measure:

(A) The absolute mispricing of outcome $i$
(B) The relative mispricing of outcome $i$ versus outcome $j$
(C) The total overround in the market
(D) The liquidity of outcome $i$ compared to outcome $j$

Answer: (B) The odds ratio compares the true probability ratio to the market price ratio. An OR above 1 indicates $i$ is underpriced relative to $j$; below 1 indicates overpriced.

Question 10

In a scalar bracket market, the "implied mean" is calculated using:

(A) The highest-probability bracket midpoint
(B) The median bracket price
(C) The probability-weighted average of bracket midpoints
(D) The bracket with the lowest overround

Answer: (C) The implied mean $\hat{\mu} = \sum q_k m_k$ is the expected value computed from the bracket probabilities and bracket midpoints.

Question 11

You fit a normal distribution to a scalar bracket market and find that one bracket's market probability is significantly lower than the fitted probability. This suggests:

(A) The bracket is overpriced and you should sell
(B) The bracket is underpriced and you should buy
(C) The normal distribution is a poor fit
(D) Either (B) or (C), and further analysis is needed

Answer: (D) The discrepancy could indicate a genuine mispricing (buy opportunity) or could indicate that the normal distribution does not adequately capture the true distribution. Further analysis with alternative distributions or domain knowledge is needed.

Question 12

In the LMSR automated market maker, the price of outcome $i$ is given by $p_i = e^{q_i/b}/\sum_j e^{q_j/b}$. What happens to all prices as $b \to \infty$?

(A) All prices approach 0
(B) All prices approach 1
(C) All prices approach $1/n$ (equal probability)
(D) Prices become more extreme (closer to 0 or 1)

Answer: (C) As $b \to \infty$, each $e^{q_i/b} \to e^0 = 1$, so all prices approach $1/n$. A very large $b$ means trades have almost no price impact, and the market remains near its initial uniform prior.

Question 13

The maximum loss for an LMSR market maker with liquidity parameter $b$ and $n$ outcomes is:

(A) $b$
(B) $b \cdot n$
(C) $b \cdot \ln(n)$
(D) $b \cdot n \cdot \ln(n)$

Answer: (C) The maximum loss is $b \ln(n)$, which occurs when all trading concentrates on a single outcome that eventually wins.

Question 14

A market maker in a multi-outcome market adjusts midpoint prices based on inventory. If the market maker is long Outcome A, they should:

(A) Increase the ask on A and decrease the bid on A
(B) Decrease both the bid and ask on A
(C) Increase the midpoint on A (raising both bid and ask)
(D) Widen the spread on A only

Answer: (C) By increasing the midpoint (and thus both bid and ask), the market maker makes it more likely that the next trade on A is a sale (at the higher ask), reducing the long inventory position.

Question 15

Which of the following is NOT a reason why Dutch books are rare in practice?

(A) Transaction costs consume the arbitrage profit
(B) Prices are always perfectly efficient
(C) Execution risk means prices can move during multi-leg execution
(D) Capital must be available on multiple platforms for cross-market arbitrage

Answer: (B) Prices are not always perfectly efficient --- that is precisely why Dutch books theoretically exist. However, transaction costs (A), execution risk (C), and capital requirements (D) make them difficult to exploit in practice.

Question 16

In a 10-candidate election market, the favorite-longshot bias predicts:

(A) The leading candidate is underpriced and longshots are overpriced
(B) The leading candidate is overpriced and longshots are underpriced
(C) All candidates are correctly priced
(D) The leading candidate is overpriced and longshots are overpriced

Answer: (A) The classic favorite-longshot bias in multi-outcome markets suggests that favorites tend to be slightly underpriced (offering positive expected value) while longshots tend to be overpriced (their prices exceed their true probability of winning).

Question 17

When constructing a pair trade in a multi-outcome market (long $i$, short $j$), the trade profits from:

(A) Only outcome $i$ winning
(B) Outcome $i$ winning or any outcome other than $j$ winning
(C) Any outcome winning
(D) Outcome $i$ winning, with partial profit if neither $i$ nor $j$ wins

Answer: (D) The trade profits most if $i$ wins (long position pays out), profits partially if neither $i$ nor $j$ wins (net credit from the spread is retained), and loses if $j$ wins (short obligation triggers).

Question 18

For a scalar bracket market, which strategy would you employ if your model predicts a higher variance than the market implies?

(A) Buy center brackets, sell tail brackets
(B) Buy tail brackets, sell center brackets
(C) Buy all brackets
(D) Sell all brackets

Answer: (B) If the true distribution has higher variance, the tails have more probability than the market implies (underpriced) and the center has less probability (overpriced). Buy the underpriced tails and sell the overpriced center.

Question 19

The covariance between the indicator random variables $X_i$ and $X_j$ (where $X_i = 1$ if outcome $i$ wins) in a mutually exclusive market is:

(A) $q_i q_j$
(B) $-q_i q_j$
(C) $q_i(1 - q_j)$
(D) $0$

Answer: (B) For mutually exclusive outcomes, $\text{Cov}(X_i, X_j) = E[X_i X_j] - E[X_i]E[X_j] = 0 - q_i q_j = -q_i q_j$, since $X_i$ and $X_j$ cannot both be 1.

Question 20

A combinatorial prediction market that combines a 3-outcome election market with a 4-outcome economic market has how many joint outcomes?

(A) 7
(B) 12
(C) 24
(D) 81

Answer: (B) The number of joint outcomes is the product of the individual outcome counts: $3 \times 4 = 12$.

Question 21

When fitting a parametric distribution to a scalar bracket market, maximum likelihood estimation maximizes:

(A) $\sum_k q_k \cdot [F(a_k;\theta) - F(a_{k-1};\theta)]$
(B) $\sum_k q_k \cdot \ln[F(a_k;\theta) - F(a_{k-1};\theta)]$
(C) $\sum_k [q_k - (F(a_k;\theta) - F(a_{k-1};\theta))]^2$
(D) $\max_k |q_k - (F(a_k;\theta) - F(a_{k-1};\theta))|$

Answer: (B) The MLE treats the bracket probabilities as multinomial weights and maximizes the log-likelihood, which is $\sum_k q_k \ln[F(a_k;\theta) - F(a_{k-1};\theta)]$.

Question 22

Which of the following is the best reason to use relative value strategies rather than absolute probability betting?

(A) Relative value trades are always more profitable
(B) Relative value trades have zero risk
(C) Comparative judgments between outcomes are often more reliable than absolute probability estimates
(D) Relative value trades require less capital

Answer: (C) The key advantage of relative value is that traders can often make more reliable assessments about which of two outcomes is more likely than about the exact probability of either outcome individually.

Question 23

In a multi-outcome market, "partial hedging" refers to:

(A) Hedging only some of the outcomes in your portfolio
(B) Reducing (but not eliminating) the worst-case loss while preserving most expected value
(C) Hedging only during part of the market's lifetime
(D) Using partial Kelly instead of full Kelly

Answer: (B) Partial hedging is a deliberate choice to reduce the worst-case outcome to an acceptable level without fully eliminating risk, which would also eliminate most of the expected profit.

Question 24

A market has 8 outcomes. Using the LMSR with $b = 200$, the maximum loss for the market maker is approximately:

(A) $200$
(B) $277$
(C) $416$
(D) $1,600$

Answer: (C) Maximum loss = $b \ln(n) = 200 \ln(8) = 200 \times 2.079 \approx 416$.

Question 25

You observe a 6-outcome prediction market where prices after removing overround are [0.05, 0.10, 0.15, 0.30, 0.25, 0.15]. Your model estimates [0.03, 0.08, 0.20, 0.32, 0.22, 0.15]. Which outcome has the highest relative edge?

(A) Outcome 3 (market: 0.15, model: 0.20)
(B) Outcome 4 (market: 0.30, model: 0.32)
(C) Outcome 1 (market: 0.05, model: 0.03)
(D) Outcome 5 (market: 0.25, model: 0.22)

Answer: (A) Relative edge = $(q_i - \hat{p}_i)/\hat{p}_i$. For Outcome 3: $(0.20 - 0.15)/0.15 = 33.3\%$. For Outcome 4: $(0.32 - 0.30)/0.30 = 6.7\%$. Outcome 3 has the highest positive relative edge. (Note: Outcomes 1 and 5 have negative edges from the model's perspective, so they would be sell candidates, not buy candidates.)