Chapter 15: Key Takeaways
1. Multi-Outcome Markets Offer Richer Opportunities Than Binary Markets
Multi-outcome markets with $n$ candidates, teams, or brackets have $n$ prices that must be internally consistent. More dimensions mean more ways the market can be wrong. The structural constraints --- probabilities must sum to 1, outcomes are mutually exclusive --- create mathematical relationships that can be exploited when violated.
2. Raw Market Prices Are Not Probabilities
Market prices in multi-outcome markets almost always sum to more than 1 due to the overround (vig, margin). You must remove the overround before using prices as probability estimates. Four methods exist:
- Multiplicative: Simple but slightly biased for extreme outcomes.
- Additive: Can produce invalid (negative) probabilities.
- Shin's method: Theoretically motivated, best for markets with informed trading.
- Power method: Empirically reliable, good default choice.
For serious quantitative work, use Shin's or the power method.
3. Dutch Books Are Theoretically Important but Practically Rare
A Dutch book is a set of bets that guarantees profit regardless of outcome. They exist when prices are inconsistent (sum too high or too low, or cross-market discrepancies). In practice, transaction costs, execution risk, and capital requirements eliminate most Dutch book opportunities. The concept remains important as a check on market efficiency.
4. Kelly Criterion Extends to Multi-Outcome Markets
The Kelly criterion for mutually exclusive outcomes maximizes expected log-wealth growth:
$$\max \sum_{j=1}^n q_j \ln(1 - F + f_j/p_j)$$
Key practical considerations: - Use numerical optimization (not closed-form solutions). - Apply fractional Kelly (0.25 to 0.50 of full Kelly) due to higher model risk. - Budget constraints limit total exposure and improve robustness.
5. Relative Value Beats Absolute Value
Trading relative mispricings ("Candidate A is overpriced versus Candidate B") is often more reliable than absolute probability bets. Reasons: - Comparative judgments are easier and more accurate than absolute probability estimation. - Pair trades have natural hedging properties (profit in most states). - You do not need a complete probability model to identify relative value.
The odds ratio $\text{OR}_{ij} = (q_i/q_j)/(p_i/p_j)$ quantifies relative mispricing.
6. Scalar Markets Reveal Distributional Disagreements
Scalar bracket markets imply a probability distribution over a continuous variable. By fitting parametric distributions (normal, log-normal, t) to the bracket probabilities, you can: - Identify brackets that are mispriced relative to a smooth distribution. - Trade mean shifts, variance changes, or specific shape disagreements. - Use the smoothness constraint as an edge: discrete bracket markets frequently violate distributional smoothness.
7. Hedging Is Essential in Multi-Outcome Markets
With $n$ possible outcomes, your portfolio can have wildly different values across states. Partial hedging reduces the worst-case loss while preserving most expected value. Key hedging approaches: - Buy "the field" to protect against your primary bet losing. - Use cross-outcome positions to flatten the payoff profile. - Dynamic hedging adjusts positions as new information arrives.
8. Market Making in Multi-Outcome Markets Requires Multidimensional Inventory Management
Market makers face $n$-dimensional inventory risk. Effective approaches include: - Inventory-dependent pricing (skew quotes to reduce imbalanced positions). - The LMSR automated market maker (bounded loss of $b \ln n$). - Dynamic spread adjustment based on volatility and information flow.
9. Behavioral Biases Are Amplified in Multi-Outcome Markets
Several documented biases create systematic mispricings: - Favorite-longshot bias: Longshots tend to be overpriced, favorites underpriced. - Salience bias: Well-known outcomes attract excess buying. - Narrative bias: Outcomes with compelling stories are overpriced. - Anchoring: Prices are sticky to initial levels, slow to update.
These biases are more pronounced with many outcomes because attention and analysis are spread thin.
10. Model Risk Is the Dominant Risk
In multi-outcome and scalar markets, the primary risk is not market risk --- it is the risk that your probability model is wrong. Mitigations include: - Fractional Kelly sizing (reduces the cost of being wrong). - Diversification across markets (not just across outcomes within a market). - Sensitivity analysis (how much do allocations change if probabilities shift?). - Robustness checks (do conclusions hold under alternative distributional assumptions?). - Continuous model validation (track calibration and adjust).
Core Formulas to Remember
| What | Formula |
|---|---|
| Overround | $\omega = \sum p_i - 1$ |
| Multiplicative removal | $\hat{p}_i = p_i / \sum p_j$ |
| Expected portfolio return | $E[R] = \sum f_j(q_j/p_j - 1)$ |
| Odds ratio | $\text{OR}_{ij} = (q_i/q_j)/(p_i/p_j)$ |
| Implied mean (scalar) | $\hat{\mu} = \sum q_k m_k$ |
| LMSR max loss | $b \ln(n)$ |
Decision Checklist
Before trading a multi-outcome or scalar market:
- Have you removed the overround using an appropriate method?
- Do you have an independent probability model, not just intuition?
- Have you identified specific mispricings, not just "I like this outcome"?
- Have you used fractional Kelly (not full Kelly) for sizing?
- Have you analyzed the payoff in every possible state?
- Have you considered partial hedging for your worst-case scenario?
- Have you stress-tested your model (what if your probabilities are off by 5%)?
- Do you have a plan for monitoring and exit?