Prediction Market Math Glossary
Authoritative mathematical definitions, closed-form equations, and worked case examples for 35 foundational concepts in binary event derivatives and market microstructures.
Contract Pricing & AMM Mechanics
(10)Implied Probability
#P_{implied} = P_m / \sum_{j=1}^n P_j The probability of an event occurrence discounted by the market clearing price of an Arrow-Debreu contract share that pays $1.00 upon favorable resolution.
Arrow-Debreu Security
#\Phi(S_i) = \mathbf{1}_{\{S = S_i\}} \cdot \$1.00 A state-contingent financial derivative that pays exactly one unit of currency if a specific state of the world occurs and zero otherwise.
Risk-Neutral Measure (Q-Measure)
#P_t = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}[\Phi(S) \mid \mathcal{F}_t] A mathematical probability measure under which the discounted asset price process behaves as a martingale, equating price to expected payoff.
Constant Product Market Maker (CPMM)
#x \cdot y = k, \quad p_{Yes} = \frac{y}{x+y} An automated market maker algorithm maintaining invariant collateral reserves x and y such that their product remains constant, setting price via reserve ratios.
Logarithmic Market Scoring Rule (LMSR)
#C(\mathbf{q}) = b \ln\left(\sum_{i=1}^n e^{q_i / b}\right), \quad p_i = \frac{e^{q_i / b}}{\sum e^{q_j / b}} Robin Hanson's bounded-loss scoring rule for combinatorial prediction markets where price is the softmax of outstanding contract shares scaled by liquidity b.
Overround (Vigorish / Vig)
#\text{Overround} = \sum_{i=1}^n \frac{1}{d_i} - 1.000 The theoretical percentage margin by which the sum of bookmaker implied probabilities exceeds 100%, representing operator profit extraction.
Shin's Information-Theoretic Model
#\pi_i = \frac{\sqrt{z^2 + 4(1-z) \frac{P_i^2}{\sum P_k}} - z}{2(1-z)} A mathematical normalization framework proving bookmakers skew margins toward longshots to protect against informed insider traders.
Bonding Curve
#P(x) = f(x), \quad \text{Cost} = \int_0^{\Delta x} f(x) \, dx A deterministic mathematical curve establishing the continuous functional relationship between contract token supply and marginal price.
Discrete Expected Value (EV)
#\text{EV} = S \cdot \left[ \frac{p_{true} - P_m}{P_m} \right] The probability-weighted mean payoff of an event contract minus initial capital commitment, measuring mathematical trading edge.
Price Impact & Execution Slippage
#P_{avg} = P_{spot} \cdot \left[ 1 + \frac{\Delta c}{2L} + \mathcal{O}\left(\frac{\Delta c^2}{L^2}\right) \right] The adverse shift in average execution price caused by consuming available liquidity depth along an AMM bonding curve or order book.
Cross-Venue Arbitrage & Spreads
(7)Synthetic Cross-Market Arbitrage
#P_A(Yes) + P_B(No) < 1.000 - \gamma_{total} A risk-free trading strategy executing simultaneous complementary positions across fragmented venues when the sum of purchase prices is strictly under $1.00.
Dual-Leg Capital Sizing
#S_A = C \cdot \frac{P_A}{P_A + P_B}, \quad S_B = C \cdot \frac{P_B}{P_A + P_B} Closed-form capital allocation formulas ensuring invariant dollar payout across both resolution states in a cross-market arbitrage trade.
Negative Synthetic Spread
#\text{Spread} = 1.000 - [ P_A(Yes) + P_B(No) ] > 0 A transient market inefficiency where buying the full set of exhaustive outcome shares costs less than the guaranteed $1.00 redemption settlement.
Legging-In Risk
#\Delta P_{\text{leg}} = P_B(t + \Delta t) - P_B(t) The operational vulnerability where price movements or latency delays occur between executing the first leg and confirming the second leg of an arbitrage trade.
Fee Hurdle Rate
#\gamma_{total} = \phi_{taker, A} + \phi_{taker, B} + \frac{\text{Gas}}{C} + \phi_{cashout} The minimum gross spread threshold required to cover all exchange commissions, network gas fees, and capital rebalancing costs.
Fill-or-Kill (FOK) Order
#\text{Order Condition} = \{ \text{Fill} \text{ if } V_{available} \ge V_{order} \text{ else } \text{Cancel} \} An order execution parameter requiring the immediate total fulfillment of an order or its immediate total cancellation to prevent partial execution.
Law of One Price
#P_A(E) = P_B(E) \quad \forall E \in \Omega An economic theorem stating that identical financial assets with identical future payoffs must clear at identical prices in efficient markets.
Kelly Sizing & Capital Preservation
(7)Binary Kelly Criterion
#f^* = \frac{p - P_m}{1 - P_m} The information-theoretic formula determining the optimal fraction of bankroll to wager on binary prediction contracts to maximize geometric wealth growth.
Fractional Kelly (Half / Quarter Sizing)
#f_{\text{frac}} = c \cdot f^*, \quad c \in [0.25, 0.50] A risk-mitigation practice scaling down full Kelly fractions by factor c to safeguard against model estimation errors and severe drawdowns.
Logarithmic Utility
#U(W) = \ln(W), \quad \mathbb{E}[U(W_T)] = \max_f \sum p_i \ln(1 + f \cdot R_i) A concave von Neumann-Morgenstern utility function reflecting constant relative risk aversion (CRRA = 1), preventing overbetting ruin.
Volatility Drag
#R_{\text{geom}} \approx R_{\text{arith}} - \frac{\sigma^2}{2} The mathematical reduction in compound geometric return caused by portfolio variance, proving that high arithmetic gains can yield negative compound growth.
The Longshot Fade Trap (Polymarket Fade Trap)
#\text{Lock} = \$0.95, \quad \text{Yield} = +5.26\%, \quad \text{Tail Loss} = -100\% The capital inefficiency trap of purchasing No shares on 3¢–5¢ contracts, locking up vast capital for meager yields while absorbing massive black swan risk.
Drawdown Probability Formulation
#P(\text{Drawdown} \ge D) = (1 - D)^{2c - 1} The probability that an investment account will experience a peak-to-trough decline of at least fraction D before doubling capital under fractional Kelly c.
Portfolio Covariance Sizing
#\max_{\mathbf{f}} \left[ \mathbf{f}^T \boldsymbol{\mu} - \frac{1}{2} \mathbf{f}^T \boldsymbol{\Sigma} \mathbf{f} \right] \quad \text{s.t.} \quad \sum f_i \le 1.0 Multivariate capital allocation optimization across correlated binary event markets incorporating covariance matrix Sigma to prevent simultaneous drawdowns.
Brier Scoring & Empirical Calibration
(8)Brier Score
#\text{BS} = \frac{1}{N} \sum_{t=1}^N (f_t - o_t)^2 A strictly proper quadratic scoring rule measuring the mean squared error between probabilistic forecasts f_t in [0, 1] and binary realizations o_t in {0, 1}.
Murphy Decomposition
#\text{BS} = \text{Reliability} - \text{Resolution} + \text{Uncertainty} An algebraic identity partitioning the aggregate Brier Score into three orthogonal components: Reliability (calibration error), Resolution, and Uncertainty.
Reliability (Calibration Error)
#\text{Reliability} = \frac{1}{N} \sum_{k=1}^K n_k (f_k - o_k)^2 The component of the Murphy decomposition measuring how closely observed frequencies match forecast probabilities across discrete probability bins.
Resolution (Discriminatory Power)
#\text{Resolution} = \frac{1}{N} \sum_{k=1}^K n_k (o_k - \bar{o})^2 The component measuring a forecasting mechanism's ability to sort outcomes into distinct subsets whose frequencies differ from the base rate.
Uncertainty (Base-Rate Entropy)
#\text{Uncertainty} = \bar{o}(1 - \bar{o}) The irreducible environmental variance of the event sample space, peaking at 0.250 when historical base occurrence rate is exactly 50%.
Favorite-Longshot Bias (FLB)
#P_m(E_{\text{rare}}) > p_{\text{true}}(E_{\text{rare}}), \quad P_m(E_{\text{fav}}) < p_{\text{true}}(E_{\text{fav}}) An empirical pricing distortion where low-probability outcomes trade at substantial premiums while high-probability favorites are underpriced.
Log-Loss (Cross-Entropy)
#\text{LogLoss} = -\frac{1}{N} \sum_{t=1}^N [ o_t \ln(f_t) + (1 - o_t) \ln(1 - f_t) ] An information-theoretic proper scoring rule that heavily penalizes overconfident incorrect predictions with asymptotic infinity.
Kullback-Leibler (KL) Divergence
#D_{KL}(P \parallel Q) = \sum_{x} P(x) \ln\left( \frac{P(x)}{Q(x)} \right) The measure of information entropy lost when approximating the true probability distribution P with market belief distribution Q.
Oracles, Settlement & Risk Mechanics
(3)Optimistic Oracle Architecture
#\text{State} = \{ \text{Proposed} \to \text{Dispute Window } \Delta t \to \text{Finalized} \} A game-theoretic mechanism where proposed event outcomes are assumed truthful unless challenged with a collateral bond during a challenge period.
UMA Tokenholder Dispute Voting
#\text{Voting Result} = \text{Mode}\left( \{ v_i \cdot w_i \} \right), \quad w_i = \text{Token Weight} The Schelling-point consensus protocol where tokenholders vote on disputed resolutions, rewarding voters aligned with the majority consensus.
Settlement Semantic Ambiguity
#\mathcal{S}(E) = \{ s \in \Omega \mid \text{Text}(E) \text{ is ungrounded} \} The systemic risk occurring when event resolution rules contain linguistic or boundary edge cases not anticipated in the market specification.
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