Case Study 2: Scalar Markets for Economic Indicators

Overview

This case study examines how prediction markets handle continuous numeric outcomes — specifically, economic indicators like GDP growth and inflation. We compare two market designs (bracket contracts and linear scalar contracts), analyze their payoff structures, calculate expected payoffs under different distributional assumptions, and show how traders can extract and use the market's implied distribution.

The Setup

Two platforms are running markets on the same economic question:

Question: "What will the annualized U.S. real GDP growth rate be for Q3 2026?"

Resolution source: The Bureau of Economic Analysis (BEA) advance estimate, published approximately 30 days after the end of the quarter.

Resolution date: Late October 2026.

Platform A: Bracket Contracts

Platform A offers six mutually exclusive brackets, each paying $1 if GDP growth falls within that range:

Note: The price sum is $0.98, slightly below $1.00. This means either (a) there is a small arbitrage opportunity, or (b) these are midpoint prices and the asks would sum above $1.00.

Platform B: Linear Scalar Contract

Platform B offers a single linear scalar contract:

• Floor: 0%
• Ceiling: 5%
• Current price: $0.46
• Volume: 85,000 contracts

Payoff formula: $\text{Payoff} = \text{clamp}\left(\frac{\text{GDP Growth} - 0\%}{5\% - 0\%}, 0, 1\right)$

Analysis of the Bracket Market (Platform A)

Implied Probability Distribution

Normalizing the bracket prices to sum to 1:

The probability density (probability divided by bracket width) tells us how concentrated the probability is within each range. B4 (2-3%) has the highest density at 36.7% per percentage point, making it the market's most likely range for GDP growth.

Implied Expected Value

Using bracket midpoints (and assuming -1% for B1 and 5% for B6 as representative values):

$$E[\text{GDP}] = (-1\%) \times 0.051 + 0.5\% \times 0.092 + 1.5\% \times 0.235 + 2.5\% \times 0.367 + 3.5\% \times 0.173 + 5.0\% \times 0.082$$

$$= -0.051 + 0.046 + 0.353 + 0.918 + 0.606 + 0.410 = 2.28\%$$

The market's implied expected GDP growth is approximately 2.28%.

Implied Variance and Standard Deviation

$$E[\text{GDP}^2] = (-1)^2 \times 0.051 + (0.5)^2 \times 0.092 + (1.5)^2 \times 0.235 + (2.5)^2 \times 0.367 + (3.5)^2 \times 0.173 + (5.0)^2 \times 0.082$$

$$= 0.051 + 0.023 + 0.529 + 2.294 + 2.119 + 2.050 = 7.066$$

$$\text{Var}[\text{GDP}] = E[\text{GDP}^2] - (E[\text{GDP}])^2 = 7.066 - (2.28)^2 = 7.066 - 5.198 = 1.868$$

$$\text{SD}[\text{GDP}] = \sqrt{1.868} \approx 1.37\%$$

The market implies GDP growth has a standard deviation of about 1.37 percentage points.

Distribution Shape

The implied distribution is: - Unimodal: Peak density in the 2-3% range. - Slightly right-skewed: The right tail (4%+) has more probability mass (8.2%) than the left tail (below 0%, 5.1%), relative to the distance from the mode. But the left tail has more extreme potential (recession), so the distribution accommodates both upside and downside risk. - Roughly consistent with a normal distribution centered at 2.28% with SD of 1.37%, but with slightly heavier tails.

Analysis of the Linear Scalar Contract (Platform B)

Implied Expected Value

For a linear scalar contract:

$$\text{Price} = E\left[\frac{\text{GDP} - \text{Floor}}{\text{Ceiling} - \text{Floor}}\right] = \frac{E[\text{GDP}] - \text{Floor}}{\text{Ceiling} - \text{Floor}}$$

(This holds when the distribution has negligible mass outside the [floor, ceiling] range. With clipping at the boundaries, the relationship is approximate.)

$$E[\text{GDP}] \approx \text{Price} \times (\text{Ceiling} - \text{Floor}) + \text{Floor} = 0.46 \times (5\% - 0\%) + 0\% = 2.30\%$$

Platform B's implied expected GDP growth is 2.30%, very close to Platform A's 2.28%. The small difference could be due to the clipping effect (probability mass below 0% and above 5% is truncated).

Payoff Profile for a Long Position

Buying the linear scalar at $0.46:

Break-even point: GDP growth = $0.46 \times 5\% + 0\% = 2.3\%$

Maximum profit: $1.00 - $0.46 = $0.54 (if GDP >= 5%)

Maximum loss: $0.46 (if GDP <= 0%)

Comparing Bracket vs. Linear Designs

Advantages of Bracket Contracts

1. Full distribution information: Each bracket reveals the market's implied probability for that range, giving you a complete picture of the probability distribution.

2. Targeted bets: You can bet on a specific range. If you think GDP will be 1-2% (a minority opinion), you can buy that bracket cheaply and earn a high return if correct.

3. Portfolio flexibility: You can combine brackets to create custom payoff profiles (e.g., "GDP above 2%" by buying B4 + B5 + B6).

Advantages of Linear Scalar Contracts

1. Simplicity: One contract, one price, one payoff formula. Easy to understand and trade.

2. Liquidity concentration: All trading activity is in one contract rather than spread across six brackets, resulting in tighter spreads and deeper order books.

3. Proportional payoff: Your reward scales linearly with how right you are. If GDP is 4%, you earn more than if GDP is 3%. Bracket contracts give you the same payoff anywhere within the winning bracket.

4. No bracket-boundary effects: Bracket contracts create discontinuities at bracket boundaries. GDP of 1.99% pays the "1-2%" bracket, while GDP of 2.01% pays the "2-3%" bracket — a tiny difference in outcome, a totally different payout.

Summary Comparison

Trading Strategies

Strategy 1: Distribution Disagreement (Bracket Market)

Scenario: You are an economist who believes GDP growth will come in below market expectations. Specifically, you think there is a 35% chance GDP will be between 1% and 2%, compared to the market's implied 23.5%.

Trade: Buy 200 contracts of B3 (1% to 2%) at $0.23.

• Cost: 200 x $0.23 = $46.00
• If GDP is between 1% and 2%: Payout = 200 x $1.00 = $200.00, P&L = +$154.00
• If GDP is outside that range: P&L = -$46.00

Expected value (using your estimates): - P(B3 wins) = 35%: EV = 0.35 x $200 + 0.65 x $0 - $46 = $70 - $46 = +$24.00

This trade has a positive expected value of $24 under your beliefs.

Strategy 2: Tail Risk Bet (Bracket Market)

Scenario: You believe the probability of a recession (GDP below 0%) is underestimated. The market prices this at 5.1%, but you estimate 12% based on leading indicators.

Trade: Buy 500 contracts of B1 (Below 0%) at $0.05.

• Cost: 500 x $0.05 = $25.00
• If GDP < 0%: Payout = 500 x $1.00 = $500.00, P&L = +$475.00
• If GDP >= 0%: P&L = -$25.00

Expected value (using your estimates): - EV = 0.12 x $500 + 0.88 x $0 - $25 = $60 - $25 = +$35.00

This tail risk bet risks only $25 but could pay $475 — a 19:1 payoff ratio if correct.

Strategy 3: Direction Bet (Linear Scalar)

Scenario: You believe GDP will be stronger than the market expects — you estimate 3.2% vs. the market's implied 2.3%.

Trade: Buy 1,000 contracts of the linear scalar at $0.46.

• Cost: 1,000 x $0.46 = $460.00
• If GDP = 3.2%: Payoff = (3.2 - 0) / 5 = $0.64 per contract
• Revenue = 1,000 x $0.64 = $640.00
• P&L = $640 - $460 = +$180.00
• If GDP = 2.3% (market consensus): Payoff = $0.46, P&L = $0.00 (break-even)
• If GDP = 1.0%: Payoff = $0.20, P&L = 1,000 x ($0.20 - $0.46) = -$260.00

Strategy 4: Cross-Platform Comparison

Both platforms should give consistent pricing. If the linear scalar on Platform B implies 2.30% expected GDP and the bracket market on Platform A implies 2.28%, the slight difference is within noise.

But suppose Platform B's linear scalar was priced at $0.52, implying E[GDP] = 2.60%, while Platform A's brackets still imply 2.28%. This divergence suggests a trading opportunity:

• Sell the linear scalar on Platform B (you think it is overpriced).
• Buy brackets on Platform A that correspond to lower GDP outcomes.

This is a form of relative-value trading across platforms and contract types.

Expected Payoffs Under Different Distributions

Let us calculate expected payoffs for the linear scalar contract under three different assumptions about the GDP distribution.

Assumption 1: Normal Distribution (mean = 2.3%, SD = 1.3%)

We can calculate the expected payoff analytically. The payoff is $\text{clamp}((x - 0) / 5, 0, 1)$ where $x \sim N(2.3, 1.3^2)$.

For a normal distribution clipped to [0, 5]:

$$E[\text{Payoff}] = \frac{1}{5} \left[ \mu \cdot (\Phi(b) - \Phi(a)) + \sigma^2 (\phi(a) - \phi(b)) + \mu(1 - \Phi(b)) \cdot 0 \right]$$

More practically, we can compute this numerically (see code/case-study-code.py). The approximate answer:

$$E[\text{Payoff}] \approx 0.459$$

Fair price: approximately $0.46 — consistent with the market.

Assumption 2: Pessimistic (mean = 1.5%, SD = 1.5%)

Under a more pessimistic view:

$$E[\text{Payoff}] \approx 0.310$$

If the contract is priced at $0.46 and you believe this distribution, you should sell: the contract is overpriced by about $0.15.

Assumption 3: Optimistic (mean = 3.5%, SD = 1.0%)

Under an optimistic view with tight distribution:

$$E[\text{Payoff}] \approx 0.696$$

If you believe this, the contract at $0.46 is a strong buy: you expect to receive $0.70 per contract on average.

Summary

Bracket Design Considerations

Granularity Trade-offs

The choice of bracket boundaries significantly affects market quality:

Too few brackets (e.g., just "Below 2%" and "2% and above"): - Very coarse — you cannot express nuanced views. - High liquidity per bracket (good). - Low information content.

Too many brackets (e.g., 20 brackets of 0.5% each): - Very fine-grained — captures detailed distribution information. - Liquidity is split across many brackets (bad — wide spreads, thin books). - Some brackets in the tails will have near-zero trading volume.

Good practice: Use 5-8 brackets with narrower intervals around the expected value and wider intervals in the tails. The market in our example (6 brackets) is a reasonable balance.

Non-Uniform Bracket Widths

The brackets in our example use uniform 1% widths (except for the open tails). An alternative design uses non-uniform widths:

This design provides higher resolution around the most probable range (1.5% to 3.0%) and coarser resolution in the tails. The narrower brackets near the center allow traders to express more precise views where it matters most.

Boundary Effects

Bracket contracts create a "boundary problem" — outcomes near a boundary have very different payoffs despite being almost identical in substance:

• GDP of 1.99%: B3 (1-2%) pays $1, B4 (2-3%) pays $0.
• GDP of 2.01%: B3 pays $0, B4 pays $1.

This matters more for economic indicators where rounding and revisions are common. The BEA might initially report 2.0% and later revise to 1.9%, completely flipping which bracket wins.

Mitigation strategies: - Use resolution criteria that specify "the bracket containing the first decimal place" to reduce ambiguity. - Some platforms offer "overlapping" brackets or "graduated" payoffs to smooth boundaries.

Real-World Examples

Federal Funds Rate Markets

The CME's Fed Funds futures market effectively functions as a bracket-style scalar market for the Federal Reserve's target rate. Each contract corresponds to a specific rate outcome, and prices imply the market's probability distribution over possible rate decisions.

Inflation Markets (CPI)

Platforms like Kalshi have offered bracket contracts on CPI readings: - "Will CPI be below 2.5%?" - "Will CPI be 2.5% to 3.0%?" - "Will CPI be above 3.0%?"

These are coarse (3 brackets), but they capture the key question: is inflation running hot, moderate, or cool?

TIPS Breakeven Inflation

Treasury Inflation-Protected Securities (TIPS) implicitly create a linear scalar market on future inflation. The "breakeven inflation rate" derived from TIPS prices is analogous to our linear scalar contract price — it represents the market's expected inflation rate.

Code

Full Python code for this case study is available in code/case-study-code.py, including: - Bracket market analysis and implied distribution calculation - Linear scalar contract payoff simulation - Expected payoff computation under different distributional assumptions - Monte Carlo simulation for strategy evaluation - Bracket design comparison

Discussion Questions

1. If you could only trade on one platform, would you choose the bracket market (Platform A) or the linear scalar market (Platform B)? What factors would influence your decision?

2. The bracket market implied E[GDP] = 2.28% while the linear scalar implied 2.30%. Under what conditions would you trade the difference between these platforms?

3. How would you design a bracket market for a quantity with a very wide range of possible outcomes, like "What will the price of Bitcoin be on December 31, 2026?" (range: $0 to potentially $500,000+)?

4. The tail risk bet (Strategy 2) has a very high potential payoff but low probability of winning. What position sizing rule would you use for such a trade?

5. If the BEA revises its GDP estimate after the initial release, but the contract resolves on the initial release, how might this create a disconnect between the "market truth" (revised figure) and the "contract truth" (initial figure)?

6. Consider a scenario where a major bank publishes a GDP forecast of 1.2% the day before the bracket market closes. How would you expect each bracket's price to change? Sketch the before-and-after probability distributions.