Case Study 2: Conditional Markets for Policy Analysis

Overview

Can prediction markets help us understand the causal effects of policy decisions? This case study explores the design and implementation of conditional prediction markets for policy analysis, focusing on the question: "If a carbon tax is enacted, what happens to GDP growth, employment, and carbon emissions?"

We implement a multi-outcome conditional market using the conditional token framework, simulate trading by participants with diverse beliefs, and analyze the resulting conditional probability estimates. We also critically examine the extent to which these market prices can be interpreted as causal effects rather than mere correlations.

Motivation

Policy makers face a fundamental challenge: they need to predict the consequences of actions they have not yet taken. Traditional approaches include:

• Economic models: Structural models that simulate policy effects. Precise but dependent on assumptions.
• Expert surveys: Polls of economists and domain experts. Quick but subject to groupthink and anchoring.
• Natural experiments: Study similar policies in other jurisdictions. Rigorous but slow and limited.

Conditional prediction markets offer a complementary approach: aggregate the beliefs of many informed participants about what would happen under each policy scenario. If the market functions well, it provides a continuously updated, incentive-compatible estimate of policy effects.

Market Design

Events and Structure

We define the following market structure:

Condition Event (C): "Will the US Congress enact a carbon tax of at least $50/ton by end of 2027?"

Outcome Events (each conditional on C): 1. GDP: "Will US GDP growth exceed 2% in the year following enactment/non-enactment?" 2. Employment: "Will the unemployment rate remain below 5% in the year following?" 3. Emissions: "Will US carbon emissions decrease by at least 10% within two years?" 4. Innovation: "Will clean energy patent filings increase by at least 20% within two years?"

For each outcome, we create two conditional sub-markets: - "Outcome given Carbon Tax passes" (8 tokens per outcome) - "Outcome given Carbon Tax does not pass" (8 tokens per outcome)

Token Structure

Following the conditional token framework (Section 30.8):

$1 Collateral
  |
  +-- Carbon Tax Passes ($P_C)
  |     |
  |     +-- GDP > 2% ($P_C * P(GDP|C))
  |     +-- GDP <= 2% ($P_C * P(not GDP|C))
  |     +-- Employment < 5% ($P_C * P(Emp|C))
  |     +-- Employment >= 5% ($P_C * P(not Emp|C))
  |     +-- Emissions down 10% ($P_C * P(Em|C))
  |     +-- Emissions not down ($P_C * P(not Em|C))
  |     +-- Innovation up 20% ($P_C * P(Inn|C))
  |     +-- Innovation not up ($P_C * P(not Inn|C))
  |
  +-- Carbon Tax Does Not Pass ($1 - P_C)
        |
        +-- [Same structure as above, conditional on NOT C]

Total: 1 condition event + 4 outcome events x 2 condition values = 9 independent probability parameters. This is far more tractable than the full combinatorial space of 2^5 = 32 states, which would require 31 parameters.

Note that this structure assumes conditional independence between outcomes given the condition. This is a simplifying assumption --- in reality, GDP and employment are correlated even after conditioning on the carbon tax.

LMSR Configuration

We use separate LMSR market makers for: - The condition event (b = 200, providing deep liquidity since this is the most-traded market) - Each conditional outcome market (b = 100)

The higher b for the condition event reflects its role as the "backbone" of the market structure --- all other prices depend on the condition probability.

Participant Model

We simulate five types of participants:

1. Economists (15% of traders)

Beliefs based on mainstream economic consensus: - Carbon tax slightly reduces GDP growth (P(GDP>2% | Tax) = 0.35, P(GDP>2% | No Tax) = 0.50) - Small increase in unemployment (P(Emp<5% | Tax) = 0.55, P(Emp<5% | No Tax) = 0.70) - Significant emissions reduction (P(Em down | Tax) = 0.75, P(Em down | No Tax) = 0.15) - Moderate innovation boost (P(Inn up | Tax) = 0.65, P(Inn up | No Tax) = 0.30)

2. Industry Representatives (20% of traders)

More pessimistic about economic effects: - P(GDP>2% | Tax) = 0.20, very negative on GDP - P(Emp<5% | Tax) = 0.35, concerned about job losses - P(Em down | Tax) = 0.60, somewhat skeptical of emissions reduction - P(Inn up | Tax) = 0.40, skeptical of innovation benefits

3. Environmental Advocates (15% of traders)

More optimistic about economic effects and very positive on environmental outcomes: - P(GDP>2% | Tax) = 0.45, believe green transition creates growth - P(Emp<5% | Tax) = 0.60, believe green jobs offset losses - P(Em down | Tax) = 0.90, highly confident in emissions reduction - P(Inn up | Tax) = 0.85, very confident in innovation stimulus

4. Political Traders (25% of traders)

Primarily trade on the condition event (whether the tax passes), with mild views on outcomes: - Focus on P(Carbon Tax) based on political analysis - Weak beliefs on outcomes (close to market consensus)

5. Arbitrageurs (25% of traders)

Look for inconsistencies between conditional and marginal markets. If the conditional prices imply a marginal price that differs from the directly-traded marginal, they exploit the difference.

Implementation Walkthrough

The full code is in code/case-study-code.py. We highlight the key steps:

Step 1: Market Initialization

# Create the multi-conditional market
market = MultiConditionalMarket(
    condition_name="CarbonTax",
    outcome_names=["GDP_Growth", "Low_Unemployment",
                   "Emissions_Down", "Innovation_Up"],
    b=100.0
)

# Initialize condition probability
# Political analysis suggests 30% chance of passage
condition_market = ConditionalMarket("CarbonTax", "dummy", b=200.0)

Step 2: Trading Simulation

Each round: 1. A trader is selected (weighted by population proportion). 2. The trader evaluates the current market prices. 3. If any price differs from their belief by more than their "trading threshold" (how much mispricing they need to see before trading), they submit a trade. 4. The LMSR updates prices.

Step 3: Analyzing Market Consensus

After 1,000 trades, we extract the market's implied beliefs:

# P(Carbon Tax passes)
p_tax = market.condition_probability()

# P(GDP > 2% | Tax)
p_gdp_given_tax = market.conditional_probability("GDP_Growth", True)

# P(GDP > 2% | No Tax)
p_gdp_given_no_tax = market.conditional_probability("GDP_Growth", False)

# Implied causal effect
causal_effect_gdp = p_gdp_given_tax - p_gdp_given_no_tax

Results

Market Consensus After 1,000 Trades

P(Carbon Tax passes) = 0.28

Interpretation

The market consensus tells a coherent story:

1. GDP: The carbon tax is expected to reduce GDP growth slightly. The -0.18 difference suggests a mild negative economic impact, closer to the economists' view than the industry representatives' dire predictions.

2. Employment: A similar mild negative effect on employment (-0.16). Again, the market balances between the industry pessimists and the environmental optimists.

3. Emissions: Strong positive effect on emissions reduction (+0.56). This is the area of greatest agreement across trader types, with even industry representatives acknowledging significant impact.

4. Innovation: Moderate positive effect on clean energy innovation (+0.30). The market believes a carbon tax would meaningfully stimulate innovation, though not as dramatically as environmental advocates predict.

Price Convergence

Prices converge after approximately 300 trades. The first 100 trades show high volatility as different trader types push prices in different directions. Between trades 100-300, arbitrageurs smooth out inconsistencies. After trade 300, prices stabilize around the consensus values with small fluctuations.

Trader Profitability

Economists and political traders are the most profitable, suggesting their beliefs are closest to the evolving consensus. Industry representatives lose the most, suggesting their views are too extreme. Note that actual profitability can only be determined after events resolve.

Causal Inference Analysis

The Fundamental Problem

The conditional prices P(Outcome | Tax) and P(Outcome | No Tax) are not necessarily causal effects. They could reflect confounding: perhaps traders believe that a carbon tax is more likely to pass during periods of strong economic growth, making the "Tax world" systematically different from the "No Tax world" in ways beyond the tax itself.

Testing for Confounding

We can partially test for confounding by examining how the condition probability P(Carbon Tax) varies with exogenous shocks. In our simulation:

1. Shock 1: A report indicates stronger-than-expected GDP growth. If P(Carbon Tax) increases (because the tax is more politically feasible during good times), this suggests confounding --- the economic environment affects both the tax and the outcomes.

2. Shock 2: A key senator changes their vote for purely ideological reasons (uncorrelated with economic conditions). If prices for P(GDP | Tax) do not change in response, this suggests the conditional prices are closer to causal estimates.

In our simulation, Shock 1 produces a small increase in P(Carbon Tax) (from 0.28 to 0.32), and P(GDP | Tax) remains stable. This is mildly encouraging for a causal interpretation but not conclusive.

The Decision Market Solution

To obtain true causal estimates, we would need a decision market structure (Chapter 31) where the decision to implement the carbon tax is partially randomized based on market prices. This breaks the confounding pathway by making the "treatment assignment" independent of background conditions.

In practice, few real policies can be randomized this way. But the conditional market still provides valuable information: it reveals the market's best estimate of what would happen under each scenario, incorporating the beliefs and private information of all participants. Even if the causal interpretation is imperfect, this is often the best available evidence for policy decisions.

Design Recommendations

Based on this case study, we recommend the following for real-world policy conditional markets:

1. Separate the condition and outcome markets clearly. Traders should understand that conditional shares are voided if the condition does not occur, so they are only expressing beliefs about outcomes in specific scenarios.

2. Include multiple outcomes. A single conditional market (e.g., only GDP) is less informative than a multi-outcome market that reveals the full scenario picture.

3. Subsidize liquidity generously. Policy questions attract fewer traders than sports or election markets. Higher b values ensure prices are responsive even with lower trading volume.

4. Recruit diverse trader populations. The market's accuracy depends on having traders with different perspectives (economists, industry, advocates). A market dominated by one group will reflect that group's biases.

5. Be transparent about the correlation-causation gap. Market prices are conditional probabilities, not causal effects. Policy makers should use them as one input among many, not as definitive causal estimates.

6. Run multiple conditional markets for robustness. If a market for P(GDP | Carbon Tax = $50/ton) gives different results than P(GDP | Carbon Tax = $25/ton), this reveals information about the dose-response relationship.

Conclusion

Conditional prediction markets are a powerful tool for policy analysis. They aggregate diverse beliefs about policy consequences into clear, quantitative scenarios. While the causal interpretation requires caution, the information provided by well-designed conditional markets surpasses what most alternative approaches can offer. The conditional token framework provides a practical implementation path, and our simulation demonstrates that markets can converge to sensible consensus values even with highly diverse participant beliefs.

The next frontier --- decision markets that can support genuine causal inference --- is the subject of Chapter 31.