Case Study 1: Optimizing a Multi-Candidate Election Portfolio

Overview

This case study walks through the complete process of analyzing and trading a multi-candidate election market. We will take a realistic 8-candidate primary election market, extract true probabilities from overround-laden prices, identify mispricings using an independent polling model, construct a Kelly-optimal portfolio, and simulate the outcomes over the life of the market.

Scenario

It is early 2028, and the Democratic primary for the 2028 presidential nomination is underway. A major prediction market lists 8 candidates with the following prices:

The overround is 3%, which is typical for a prediction market with moderate liquidity.

Part 1: Extracting True Probabilities

Step 1: Apply Multiple Overround Removal Methods

We apply all four methods from Section 15.2 to understand the range of possible true probabilities.

Multiplicative Removal:

Divide each price by 1.03:

Additive Removal:

Subtract $0.03/8 = 0.00375$ from each:

Power Method:

We find $k$ such that $\sum p_i^k = 1$. Numerically, $k \approx 1.0384$.

Shin's Method:

With numerical optimization, we find $z \approx 0.0089$.

Step 2: Analysis

With only 3% overround, the methods produce similar results. The differences are largest for the longshots: - Klobuchar ranges from 3.63% (additive) to 3.88% (multiplicative) - Harris ranges from 26.96% (power) to 27.63% (additive)

For this case study, we will use Shin's method as our baseline market-implied probabilities.

Part 2: Independent Probability Estimation

Our Polling-Based Model

We have developed a polling aggregation model that produces the following probability estimates. Our model uses a weighted combination of national polls, state polls, endorsement scores, and fundraising metrics:

Key Observations

1. Whitmer is significantly underpriced: Our model gives her 22% versus the market's implied 15.5%. This is the strongest absolute edge.

2. Warnock is moderately underpriced: 8% model vs 5.8% market. The relative edge (2.24/5.76 = 38.9%) is actually larger than Whitmer's relative edge (6.54/15.46 = 42.3%), but the absolute edge is smaller.

3. Newsom and Harris are overpriced: Our model assigns lower probabilities to both leading candidates. This may reflect name recognition bias in the market.

4. The "Field" is overpriced: The market gives 7.7% to "Other" while we estimate 5%. This is common --- the "field" category often attracts speculative buying.

Part 3: Constructing the Kelly-Optimal Portfolio

Optimization Setup

We use the Kelly criterion for mutually exclusive outcomes as described in Section 15.4:

$$\max_{f_1, \ldots, f_8} \sum_{j=1}^{8} q_j \ln\left(1 - F + \frac{f_j}{p_j}\right)$$

where $q_j$ are our model probabilities, $p_j$ are market prices, and $F = \sum f_j$.

Constraints

• $f_j \geq 0$ for all $j$ (no short selling in this market)
• $\sum f_j \leq 0.30$ (budget constraint: risk at most 30% of bankroll)

Full Kelly Solution

Running the optimizer (see code/case-study-code.py):

Half-Kelly Solution (Recommended)

Given the uncertainty inherent in polling models, we use half-Kelly:

Expected Growth Rate

• Full Kelly expected log growth: 0.82% per resolution
• Half Kelly expected log growth: 0.62% per resolution
• Half Kelly reduces growth by 24% but reduces variance by approximately 50%

Part 4: Risk Analysis

Payoff Scenarios

With the half-Kelly portfolio on a $10,000 bankroll:

Risk Metrics

• Expected return: 22% x 38.2% + 12% x 4.0% + 8% x 29.3% + 53% x (-12.4%) = +2.20%
• Probability of profit: 42% (if C, D, or F wins)
• Probability of loss: 58% (if anyone else wins)
• Maximum loss: -12.4% (bounded and manageable)
• Maximum gain: +38.2% (if Whitmer wins)
• Sharpe-like ratio: 2.20% / 19.3% = 0.114

Interpretation

This is a positive-expected-value portfolio but with the characteristic of prediction market bets: negative outcome in the majority of scenarios, large positive outcome in fewer scenarios. The 42% probability of profit is typical for a well-constructed multi-outcome bet. The key is that when we win, we win substantially more than when we lose.

Part 5: Monte Carlo Simulation

Simulation Design

We simulate 10,000 independent elections drawn from our probability model and track the bankroll evolution if we repeatedly face similar markets.

Key Results (from simulation)

Over 10,000 simulated resolutions: - Median final bankroll (starting from $10,000): $12,445 (24.5% cumulative growth) - Mean final bankroll: $13,210 (32.1% cumulative growth) - 5th percentile: $7,890 (21.1% drawdown) - 95th percentile: $20,340 (103% cumulative growth) - Probability of being ahead after 10 resolutions: 67% - Probability of being ahead after 50 resolutions: 89%

Lessons

1. Individual bets often lose: In any single market, the most likely outcome is a loss (58% chance). This is psychologically challenging.

2. The edge compounds: Over many similar bets, the positive expected value accumulates and the probability of being ahead increases steadily.

3. Bankroll management is critical: Even with half-Kelly sizing, drawdowns of 20% or more are not uncommon. Without proper bankroll management, a trader might abandon a winning strategy during a normal drawdown.

4. Model accuracy matters enormously: If our model's probabilities are wrong by even 5 percentage points systematically, the expected value can flip to negative. This underscores the importance of model validation.

Part 6: Dynamic Strategy

As the Race Evolves

In practice, this market would be open for months, with prices changing as polls, debates, and events occur. A complete strategy includes:

1. Weekly rebalancing: Update model probabilities with new polling data. Reoptimize the portfolio. Add or trim positions.

2. Event hedging: Before major debates, consider reducing exposure. The market may move sharply, and your model's edge may be temporarily reduced by high volatility.

3. Profit taking: If Whitmer's price rises from 0.16 to 0.22 (approaching your model estimate), sell to lock in profit. The edge has diminished.

4. Dropout adjustments: When candidates drop out, their probability mass redistributes. Your model should anticipate which candidates benefit, potentially creating new mispricings.

5. Monitoring model performance: Track your model's calibration over time. If your predictions are consistently wrong, reduce bet sizes.

Conclusion

This case study demonstrates the full workflow for multi-candidate election trading:

1. Extract true probabilities from market prices (Shin's method).
2. Compare to independent model estimates.
3. Identify edges (Whitmer, Warnock, Pritzker).
4. Size positions with fractional Kelly (half-Kelly for safety).
5. Understand the risk profile (58% chance of loss, but positive expected value).
6. Simulate to build confidence in the strategy.
7. Plan for dynamic management over the market's lifetime.

The code for all calculations is available in code/case-study-code.py.