Chapter 31 Exercises: Decision Markets and Futarchy

Conceptual Exercises

Exercise 1: Futarchy Basics

Explain in your own words the difference between a standard prediction market and a decision market. Why is the distinction important? What new incentive issues arise when a market's prices directly influence a decision?

Exercise 2: Vote on Values, Bet on Beliefs

Hanson's slogan is "vote on values, bet on beliefs." Identify three specific examples where the boundary between "values" and "beliefs" is blurred. For each example, explain why the clean separation is difficult.

Exercise 3: Conditional Interpretation

A company opens two conditional markets: - Market A: "Expected revenue if we launch Product A" -- current price: $45M - Market B: "Expected revenue if we launch Product B" -- current price: $52M

A manager concludes that Product B is unambiguously better. Identify at least three reasons this conclusion might be wrong, drawing on the concepts of causal inference, selection effects, and market quality.

Exercise 4: Selection Bias Scenario

Consider a government deciding whether to implement a stimulus package. The conditional market "GDP if stimulus is implemented" shows 2.1% growth, while "GDP if no stimulus" shows 2.8% growth. A naive interpretation says the stimulus hurts GDP. Construct a plausible scenario where the stimulus actually helps GDP but the conditional markets still show this pattern due to selection effects.

Exercise 5: Welfare Metric Design

Design a welfare metric for a city government using futarchy to decide on infrastructure investments. Your metric should: - Include at least four components - Specify weights and justify them - Address potential gaming/Goodhart's Law concerns - Specify the measurement timeline

Exercise 6: Manipulation Incentives

A pharmaceutical company has a drug worth $10 billion if approved. A decision market is being used to advise the FDA on approval, with total market liquidity of $5 million. Analyze: (a) What is the maximum the company would be willing to spend on manipulation? (b) Is this amount sufficient to move the market? (c) What defenses could be put in place?

Exercise 7: Thin Market Analysis

A DAO is using futarchy to decide among 8 possible treasury allocation strategies. The total liquidity budget is $200,000. Calculate the approximate cost of moving a single conditional market's price by 15 percentage points under LMSR. Is this market likely to be robust? What would you recommend?

Exercise 8: Democratic Legitimacy

Write a 500-word argument in favor of futarchy from a democratic legitimacy perspective, addressing the common criticism that markets are undemocratic. Then write a 500-word argument against futarchy on the same grounds. Which argument do you find more compelling, and why?

Exercise 9: Corporate vs. Government Futarchy

List five reasons why futarchy might work better in corporate settings than in government settings, and five reasons why it might work worse. For each reason, explain the underlying mechanism.

Exercise 10: Minority Rights Under Futarchy

A futarchy system uses "average citizen wellbeing" as its welfare metric. A proposed policy would significantly benefit 90% of citizens while severely harming 10%. Analyze: (a) What would the conditional market predict? (b) Is this outcome ethically acceptable? (c) How could the welfare metric be modified to protect minority interests?

Mathematical Exercises

Exercise 11: Conditional Expectation Calculation

In a decision market with two decisions (A, B) and an outcome $Y \in \{0, 1, 2, 3\}$, the joint probability distribution believed by the market is:

(a) Compute $P(D=A)$ and $P(D=B)$. (b) Compute $\mathbb{E}[Y \mid D=A]$ and $\mathbb{E}[Y \mid D=B]$. (c) Which decision does the market recommend under the argmax rule? (d) Can you determine the causal effect of A vs. B from this table? Why or why not?

Exercise 12: LMSR Cost of Manipulation

An LMSR market has liquidity parameter $b = 1000$ and 5 outcome buckets with current shares $q = [100, 200, 300, 200, 100]$. (a) Compute the current price vector. (b) What is the cost of buying 50 additional shares of bucket 5 (the highest outcome)? (c) How does the expected value change after this purchase? (d) If the manipulator wants to raise the expected value by 10%, approximately how many shares of the highest bucket must they buy?

Exercise 13: Selection Bias Formula

Derive the selection bias formula for a decision market. Specifically, show that:

$$\mathbb{E}[Y \mid D=A] - \mathbb{E}[Y \mid D=B] = \text{ATE}_{A \text{ vs } B} + \text{Selection Bias}$$

where:

$$\text{Selection Bias} = \left(\mathbb{E}[Y(B) \mid D=A] - \mathbb{E}[Y(B) \mid D=B]\right)$$

Interpret this formula: under what conditions is the selection bias zero? Give an example of positive selection bias and negative selection bias.

Exercise 14: Manipulation Budget

A decision market uses LMSR with parameter $b$. There are two decisions (A, B) and the current expected values are $\mathbb{E}[Y|A] = 50$ and $\mathbb{E}[Y|B] = 52$ (so B is currently recommended). A manipulator wants to switch the recommendation to A.

(a) Approximately what change in expected value does the manipulator need to achieve in Market A (or reduce in Market B)? (b) Express the approximate cost of this manipulation as a function of $b$. (c) If the manipulator gains $V$ from decision A being chosen, derive the condition on $b$ that makes manipulation unprofitable.

Exercise 15: Combinatorial Decision Space

A company must decide on three binary dimensions: pricing (high/low), marketing channel (online/offline), and target market (enterprise/consumer). This creates $2^3 = 8$ possible strategies.

(a) How many conditional markets are needed under the "separate markets" approach? (b) If total liquidity is $L$, what is the per-market liquidity? (c) Compare this to a combinatorial market approach. What are the advantages and disadvantages? (d) Propose a tournament structure that reduces the number of simultaneous markets to at most 3.

Programming Exercises

Exercise 16: Basic Conditional Market

Implement a conditional prediction market with two decisions (A, B) using a simple order-book mechanism (not LMSR). Your market should: - Accept limit orders conditioned on each decision - Match orders when possible - Report the current best bid/ask for each conditional market - Void orders in the non-chosen market upon decision

Test your implementation with at least 20 orders.

Exercise 17: Selection Bias Simulation

Write a Python simulation that: (a) Generates 10,000 "worlds" with a confounder U, potential outcomes Y(A) and Y(B), and an endogenous decision D. (b) Computes the true ATE and the naive conditional difference. (c) Varies the strength of the confounder's effect on the decision from 0 to 2. (d) Plots the true ATE vs. naive estimate as a function of confounder strength. (e) Implements an instrumental variable estimator and shows that it can recover the true ATE.

Exercise 18: Decision Market with Multiple Metrics

Extend the DecisionMarket class from Section 31.6 to support multiple outcome metrics simultaneously. For example, a decision might affect both revenue and customer satisfaction. Implement: - Separate LMSR markets for each (decision, metric) pair - A weighted combination rule that aggregates across metrics - A demonstration with 3 decisions and 2 metrics

Exercise 19: Manipulation Detection

Write a Python function that takes a sequence of trades in a decision market and detects potential manipulation. Your detector should flag: - Single traders making unusually large trades - Trades that consistently push the expected value in one direction - Trading patterns that are inconsistent with information-motivated trading - Sudden price movements followed by reversal

Test your detector on simulated legitimate and manipulated trading sequences.

Exercise 20: Futarchy DAO Simulation

Implement a complete futarchy simulation for a DAO: - The DAO has a treasury and a governance token - Proposals are submitted and evaluated via conditional markets - Traders have different information and beliefs - The token price serves as the welfare metric - Simulate 20 proposals, some good and some bad - Track how often the market makes the correct decision - Compare to a simple majority vote among the same agents

Exercise 21: Scoring Rule Decision Market

Implement a decision market based on proper scoring rules (Section 31.11.1): - Use the logarithmic scoring rule - Each of N traders reports a probability distribution over outcomes for each decision - Aggregate reports using linear opinion pooling - Score traders based on the actual outcome - Compare the aggregate prediction quality to the LMSR-based approach

Exercise 22: Dynamic Decision Market

Implement a decision market where: - Information arrives over time (new signals about the true effect of each decision) - Traders update their beliefs and trade accordingly - The market price evolves over time - At any point, the market has a current recommendation - After a fixed number of periods, the decision is made - Track how the recommendation evolves and whether it converges to the correct decision

Exercise 23: Multi-Round Futarchy

Implement a multi-round futarchy system where: - In each round, a new proposal is evaluated - The welfare metric incorporates outcomes from previous rounds - Trader reputations are updated based on past accuracy - Better traders get more influence (weighted by past performance) - Simulate 50 rounds and analyze how the system's accuracy changes over time

Exercise 24: Privacy-Preserving Decision Market

Implement a simplified privacy-preserving decision market where: - Traders submit encrypted reports (simulated encryption) - Reports are aggregated without revealing individual values - Add calibrated noise (differential privacy) to the aggregate - Compare the information quality (MSE of the aggregate prediction) with and without privacy

Exercise 25: Conditional Token Implementation

Implement the Gnosis-style conditional token framework: - Create tokens for condition-outcome pairs - Implement splitting (splitting a base token into conditional tokens) and merging - Build an AMM that prices conditional tokens - Demonstrate a complete futarchy workflow: create proposal, open markets, trade, resolve

Applied Exercises

Exercise 26: Real-World Futarchy Design

Choose a real policy debate (e.g., universal basic income, carbon tax, immigration reform) and design a complete futarchy mechanism for it: - Define the welfare metric (at least 3 components with weights) - Specify the conditional market structure - Address the resolution timeline - Analyze potential manipulation risks and defenses - Discuss ethical concerns specific to this policy area

Write a 1,000-word proposal.

Exercise 27: Corporate Decision Market Pilot

You are a consultant hired to design a prediction market pilot for a 500-person technology company. The CEO wants to use markets to decide between two product roadmaps for the next year.

Design the complete system: - Market structure (conditional vs. advisory) - Participation incentives (real money, play money, bonuses) - Question design - Resolution criteria - Timeline - Risk management - Success metrics for the pilot

Exercise 28: Meta-DAO Analysis

Research Meta-DAO's futarchy implementation (or use the description in Section 31.9.3). Analyze: (a) What is the welfare metric and what are its limitations? (b) How does liquidity provision work? (c) What types of proposals have been evaluated? (d) What evidence exists about decision quality? (e) What improvements would you suggest?

Write a 1,500-word analysis.

Exercise 29: Futarchy vs. Democracy Debate

Organize a structured debate (written or with classmates) on the proposition: "Futarchy would produce better governance outcomes than representative democracy for a medium-sized city." Present arguments for and against, using specific evidence from this chapter and external sources. Conclude with a judgment and justification.

Exercise 30: Decision Market for Climate Policy

Design a decision market for international climate policy. The decision is between three carbon reduction strategies: - Strategy A: Carbon tax at $100/ton - Strategy B: Cap-and-trade with declining caps - Strategy C: Direct subsidies for renewable energy

Your design should address: - The welfare metric (balancing economic and environmental outcomes) - How to handle the 30-year resolution timeline - Cross-border participation and incentives - The role of corporate actors with financial interests - Why this might (or might not) be better than the current UNFCCC process