Chapter 30 Quiz

Instructions: Select the best answer for each question. Answers are provided at the end.

Question 1

For n binary events, the full combinatorial state space has how many states?

A) n B) 2n C) n^2 D) 2^n

Question 2

A full combinatorial LMSR with 10 binary events requires how many outcome slots?

A) 10 B) 20 C) 100 D) 1,024

Question 3

In a combinatorial market, the price of the Boolean expression "A OR B" equals:

A) P(A) + P(B) B) P(A) + P(B) - P(A AND B) C) P(A) * P(B) D) 1 - P(NOT A AND NOT B)

Question 4

The maximum loss for an LMSR-C with n binary events and liquidity parameter b is:

A) b * ln(n) B) b * n C) b * n * ln(2) D) b * 2^n

Question 5

Which property makes the LMSR-C maximum loss manageable despite exponential state spaces?

A) The loss grows exponentially in n B) The loss grows linearly in n C) The loss grows logarithmically in n D) The loss is constant regardless of n

Question 6

In a partition-based combinatorial market, events in different partitions are assumed to be:

A) Perfectly correlated B) Negatively correlated C) Independent D) Conditionally independent given a common cause

Question 7

A partition market with three groups of sizes 5, 4, and 3 requires how many total states?

A) 12 B) 60 C) 56 D) 4,096

Question 8

The mean-field approximation for combinatorial markets assumes:

A) All events are perfectly correlated B) All events are independent C) Events form a tree structure D) Events follow a Bayesian network

Question 9

In importance sampling for combinatorial market pricing, the proposal distribution should ideally:

A) Be uniform over all states B) Be close to the target distribution C) Only cover states where the Boolean expression is true D) Be the exact posterior distribution

Question 10

Lazy evaluation in LMSR-C works by:

A) Pre-computing all state prices at initialization B) Only instantiating states for events that have been traded C) Randomly sampling states to evaluate D) Using a neural network to approximate prices

Question 11

In a conditional prediction market for P(Y | X), what happens to "Y given X=True" shares if X resolves to False?

A) They pay out $1 B) They pay out $0 C) They are voided and the purchase price is refunded D) They convert to unconditional Y shares

Question 12

The Gnosis Conditional Token Framework implements conditional markets through:

A) Order book matching B) Token splitting and merging C) Dutch auctions D) Continuous double auctions

Question 13

A bundle order that buys event A and sells event B simultaneously is called:

A) A parlay bet B) A spread bet C) A hedging order D) A conditional order

Question 14

Atomicity in bundle trading means:

A) The bundle is very small B) The entire bundle executes or none of it does C) The bundle cannot be divided D) The bundle uses atomic operations on the blockchain

Question 15

For which of the following scenarios is a full combinatorial LMSR most practical?

A) 50 binary events about world politics B) 5 binary events about related economic indicators C) 30 binary events about sports outcomes D) 100 binary events about technology trends

Question 16

The correlation between two events in a combinatorial market is defined as:

A) P(A AND B) B) P(A AND B) - P(A) * P(B) C) [P(A AND B) - P(A)P(B)] / sqrt[P(A)(1-P(A))P(B)(1-P(B))] D) P(A | B) - P(A)

Question 17

Belief propagation provides exact marginal probabilities when the graphical model is:

A) A complete graph B) A tree C) A cycle D) Any graph with fewer than 10 nodes

Question 18

If a combinatorial market reports P(A) = 0.6, P(B) = 0.7, and P(A AND B) = 0.75, what can we conclude?

A) The prices are consistent B) There is an arbitrage opportunity because P(A AND B) > P(A) C) There is an arbitrage opportunity because P(A AND B) > P(B) D) Both B and C

Question 19

The "causal effect" estimate from a conditional market P(Y|X) - P(Y|NOT X) measures:

A) The true causal effect of X on Y B) The associational difference, which may include confounding C) The average treatment effect D) The counterfactual probability

Question 20

In a tree-structured market, the total number of parameters is proportional to:

A) 2^n where n is the number of events B) n^2 C) The number of edges in the tree D) The depth of the tree

Question 21

A combinatorial market with 20 events requires approximately how much memory for the full share vector (at 8 bytes per state)?

A) 160 bytes B) 8 KB C) 8 MB D) 8 GB

Question 22

Which approximation method is best suited for a market where events have a known causal structure with sparse dependencies?

A) Mean-field approximation B) Uniform sampling C) Bayesian network approximation D) Full enumeration

Question 23

The key advantage of LMSR-C over running separate LMSR markets for each event is:

A) Lower computational cost B) Ability to capture and price correlations between events C) Higher liquidity per market D) Simpler implementation

Question 24

In the context of decision markets, a conditional market P(Outcome | Action) is used to:

A) Predict which action will be taken B) Estimate the causal effect of the action on the outcome C) Calculate the probability of the outcome regardless of action D) Determine the market maker's profit

Question 25

When is an approximation "good enough" for a combinatorial market?

A) When it gives the exact answer B) When the approximation error is smaller than the market's bid-ask spread and trading noise C) When it runs in under 1 millisecond D) When it uses less than 1 MB of memory

Answer Key

  1. D --- 2^n. Each of n binary events can be True or False, giving 2^n combinations.

  2. D --- 1,024. 2^10 = 1,024 states for 10 binary events.

  3. B --- P(A) + P(B) - P(A AND B). This is the inclusion-exclusion principle. Note that D is also correct (they are equivalent), but B is the standard form. Both B and D are acceptable.

  4. C --- b * n * ln(2). The maximum loss equals b * ln(2^n) = b * n * ln(2).

  5. B --- The loss grows linearly in n. Despite 2^n states, the LMSR-C maximum loss is b * n * ln(2), which is linear in n.

  6. C --- Independent. The partition assumption is that events in different partitions are statistically independent.

  7. C --- 56. 2^5 + 2^4 + 2^3 = 32 + 16 + 8 = 56. The full state space would be 2^12 = 4,096.

  8. B --- All events are independent. Mean-field approximation finds the best product (independent) distribution.

  9. B --- Be close to the target distribution. A proposal close to the target minimizes the variance of importance weights.

  10. B --- Only instantiating states for events that have been traded. Unactivated events do not affect relative prices.

  11. C --- They are voided and the purchase price is refunded. This is the defining feature of conditional shares.

  12. B --- Token splitting and merging. The CTF uses a split/merge mechanism to create conditional tokens from collateral.

  13. B --- A spread bet. Going long on one event and short on another is a spread position.

  14. B --- The entire bundle executes or none of it does. This prevents partial execution risk.

  15. B --- 5 binary events about related economic indicators. 2^5 = 32 states is very manageable.

  16. C --- The Pearson correlation formula adapted for binary variables.

  17. B --- A tree. Belief propagation gives exact results on trees; on general graphs it is approximate ("loopy BP").

  18. D --- Both B and C. P(A AND B) cannot exceed either P(A) or P(B). Here 0.75 > 0.7 > 0.6, violating both constraints.

  19. B --- The associational difference, which may include confounding. Without randomization or proper adjustment, the conditional difference includes confounding.

  20. C --- The number of edges in the tree. Each edge requires a conditional probability table, and the total parameters are proportional to the number of edges.

  21. C --- 8 MB. 2^20 = 1,048,576 states * 8 bytes = 8,388,608 bytes, which is approximately 8 MB.

  22. C --- Bayesian network approximation. It directly exploits the known causal/dependency structure.

  23. B --- Ability to capture and price correlations between events. Separate markets cannot express joint distributions.

  24. B --- Estimate the causal effect of the action on the outcome. Decision markets use conditional prices to evaluate the likely effect of different actions.

  25. B --- When the approximation error is smaller than the market's bid-ask spread and trading noise. The approximation need not be more precise than the inherent noise in the market.