§ VII · Reference

last revised 2026-04-22snapshot 2026-06-15T03:47Z

Notation

The symbol table used in the model specification, evaluation metrics, and the Phase 1 framework paper; with canonical values from the pre-registration where applicable.

By The 45% Problem project

Contents

This table mirrors the notation in the Phase 1 framework paper. Every symbol listed here appears in the model specification, the evaluation metrics, or the pre-registration constants. The canonical value column shows the pre-registered value where the symbol has one; "n/a" denotes a variable or a value estimated from data.

Symbols are grouped thematically: model parameters first, then market and evaluation quantities, then meta and infrastructure notation.

Match and score variables

Symbol	Definition	Canonical value	Notes
$X$	Home team goals (random variable)	n/a	Poisson-distributed with rate $\lambda_h$
$Y$	Away team goals (random variable)	n/a	Poisson-distributed with rate $\lambda_a$
$x, y$	Realized home / away goal counts	n/a	Lowercase = realization of $X$ , $Y$
$N$	Number of settled matches	n/a	Grows monotonically during the tournament
$i$	Match index	n/a	Ranges from 1 to $N$

Poisson rate parameters

Symbol	Definition	Canonical value	Notes
$\lambda_h$	Expected home team goals	n/a	$> 0$ ; parameterizes the home Poisson
$\lambda_a$	Expected away team goals	n/a	$> 0$ ; parameterizes the away Poisson
$\mu$	Mean log-goals across calibration corpus	n/a	Baseline intercept in the log-linear model
$\alpha_i$	Attack strength for team $i$	n/a	Estimated from historical match data
$\beta_i$	Defence strength for team $i$	n/a	Estimated from historical match data
$\delta_\text{HFA}$	Home-field advantage offset	n/a	Log-scale; estimated from historical data
$\lambda_3$	Bivariate Poisson shared-shock parameter	0.10	Pre-registered; common-shock term in Karlis-Ntzoufras decomposition
$c^{*}$	Elo-to-log-rate slope	0.580386	Pre-registered; calibrated on 2010 to 2021 corpus
$\mu^{*}$	Elo-to-log-rate intercept (mean goals at parity)	1.715874	Pre-registered; calibrated on 2010 to 2021 corpus

The log-linear parameterization of expected goals:

\log \lambda_h = \mu + \alpha_h - \beta_a + \delta_\text{HFA}

\log \lambda_a = \mu + \alpha_a - \beta_h

The simulation engine's calibrated Elo-to-rate mapping (used at run time):

\lambda_h = \mu^{*} \cdot e^{\,c^{*} (R_h + H - R_a)/400}

\lambda_a = \mu^{*} \cdot e^{-c^{*} (R_h + H - R_a)/400}

Dixon-Coles correction

Symbol	Definition	Canonical value	Notes
$\rho$	Dixon-Coles low-score correlation parameter	n/a	Small negative value, $\rho \in (-1/(\lambda_h \lambda_a), 0)$
$\tau(x,y)$	Correction factor for score $(x,y)$	n/a	Equals 1 for $\max(x,y) \geq 2$

The correction factor in full:

\tau(x,y,\lambda_h,\lambda_a,\rho) = \begin{cases} 1 - \lambda_h\lambda_a\rho & x=0,\; y=0 \\ 1 + \lambda_h\rho & x=0,\; y=1 \\ 1 + \lambda_a\rho & x=1,\; y=0 \\ 1 - \rho & x=1,\; y=1 \\ 1 & \text{otherwise} \end{cases}

Elo and strength parameters

Symbol	Definition	Canonical value	Notes
$R_i$	Current Elo rating for team $i$	n/a	Updates after each settled match
$R'_i$	Updated Elo rating	n/a	Post-match value
$E_i$	Elo expected score for team $i$	n/a	$E_i = 1/(1 + 10^{(R_j - R_i)/400})$
$S_i$	Actual match score for team $i$	n/a	1 = win, 0.5 = draw, 0 = loss
$K$	Elo update magnitude	20 (group) / 32 (KO)	Pre-registered; higher $K$ for knockout rounds
$w_\text{Elo}$	FIFA-blend weight on Elo ratings (M2)	n/a	Estimated on 2018 to 2022 hold-out corpus
$s_i$	Blended strength for team $i$ (M2)	n/a	$s_i = w_\text{Elo}\,\tilde{R}^\text{Elo}_i + (1-w_\text{Elo})\,\tilde{R}^\text{FIFA}_i$
$f_{i,t}$	Recency-weighted form factor (M3)	n/a	Rolling exponential decay over recent matches
$\gamma$	Form factor decay constant (M3)	0.30	Pre-registered

Market and edge quantities

Symbol	Definition	Canonical value	Notes
$p_m$	Model-implied probability for outcome	n/a	$p_m \in [0,1]$
$q_\text{raw}$	Raw bookmaker decimal odds (inverted)	n/a	Before margin removal
$q_d$	De-vigged market probability	n/a	Power-method de-vigging; $\sum_j q_{d,j} = 1$
$E$	Edge	n/a	$E = p_m - q_d$ ; positive = model higher than market
$\varepsilon$	Edge threshold	0.03 (mainline) / 0.05 (derivative)	Pre-registered; governs gate status
$k$	Power-method exponent for de-vigging	n/a	Estimated to zero overround
$z$	Power-method exponent (unnormalized form)	n/a	$\sum_i r_i^{z} = 1$ ; $z > 1$ for overround Pinnacle, $z < 1$ for underround exchanges
$\Delta_{\text{draw}}$	Pinnacle knockout-draw bias correction	+0.014	Pre-registered; additive on draw leg in knockout markets
$\Delta_{\text{host}}$	Pinnacle group-stage host-win bias correction	−0.006	Pre-registered; additive on host-win leg in group-stage markets
$\sigma_p$	Monte Carlo standard error on $p_{\text{model}}$	n/a	$\sqrt{p(1-p)/N_{\text{MC}}}$
$\sigma_q$	De-vigging uncertainty (parametric bootstrap)	n/a	50 resamples, tick noise $\pm 0.005$ on decimal odds
$E^{*}$	Variance-adjusted standardized edge	n/a	$E / \sqrt{\sigma_p^{2} + \sigma_q^{2}}$ ; logged but not flag-decisive

The power-method de-vigging formula:

q_{d,i} = \frac{o_i^{-1/k}}{\displaystyle\sum_j o_j^{-1/k}}

where $o_i$ is the decimal odds for outcome $i$ . The implementation uses the unnormalized form $q_i = r_i^{z}$ with $z$ chosen so that $\sum_i r_i^{z} = 1$ , which produces an identical $q$ vector under a different parameterization of the exponent.

Evaluation and scoring metrics

Symbol	Definition	Canonical value	Notes
$p_i$	Model probability on the realized outcome of match $i$	n/a	The "score" that feeds all loss functions
$o_i$	Indicator: 1 if outcome $i$ realized, 0 otherwise	n/a	In practice always 1 for the event that occurred
$\text{BS}$	Brier score	n/a	$\text{BS} = \frac{1}{N}\sum_i (p_i - o_i)^2$
$\mathcal{L}$	Log-loss (cross-entropy loss)	n/a	$\mathcal{L} = -\frac{1}{N}\sum_i \log p_i$
$\text{RPS}$	Ranked probability score	n/a	Cumulative distribution squared error
$\text{CLV}_i$	Closing-line value for forecast $i$ (M★ only)	n/a	$\log(p_m/q_\text{close}) \times 10{,}000$ bps
$p_k$	Mean model probability in calibration bin $k$	n/a	For reliability diagram
$\hat{f}_k$	Empirical frequency in calibration bin $k$	n/a	Fraction of events that occurred

Monte Carlo and simulation

Symbol	Definition	Canonical value	Notes
$n_\text{MC}$	Number of Monte Carlo simulation runs	10,000	Pre-registered; fixed for all snapshots
$s$	Random seed	42	Pre-registered; reproducibility anchor
$[p_\text{lo}, p_\text{hi}]$	95% Monte Carlo confidence interval	n/a	2.5th and 97.5th percentile across simulations

Bet sizing and trading policy

Symbol	Definition	Canonical value	Notes
$f_{\text{full}}$	Full Kelly stake fraction	n/a	$(p \cdot o - 1)/(o - 1)$
$\phi$	Fractional Kelly multiplier	1/4 (mainline) / 1/8 (longshot)	Pre-registered; class set by $q_d$
$f_{\text{cap}}$	Per-market hard cap	0.05 (mainline) / 0.025 (longshot)	Pre-registered; applied after fractional multiplier
$f_{\text{recommended}}$	Recommended stake fraction	n/a	$\mathrm{clip}(\phi \cdot f_{\text{full}},\, 0,\, f_{\text{cap}})$
$f_{\text{event}}$	Per-event cap (sum across legs of one match)	0.08	Pre-registered; orchestrator-level enforcement
$f_{\text{day}}$	Per-day cap (sum across all events on one UTC day)	0.15	Pre-registered; orchestrator-level enforcement

Hierarchical prior (M3)

Symbol	Definition	Canonical value	Notes
$c$	Confederation index	n/a	One of: CONMEBOL, UEFA, CONCACAF, AFC, CAF, OFC
$\mu_c$	Confederation-level mean attack strength	n/a	Hyper-parameter, estimated jointly
$\sigma_c$	Confederation-level attack strength spread	n/a	Hyper-parameter, estimated jointly

The partial-pooling prior:

\alpha_i \sim \mathcal{N}(\mu_c,\; \sigma_c^2), \quad c = \text{confederation}(i)

Infrastructure and audit notation

Symbol	Definition	Example
`snapshot_id`	ISO-8601 timestamp identifying a data bundle	`2026-06-11T04:00Z`
`code_sha`	Short git commit SHA of the build that generated the snapshot	`a1b2c3d`
`data_sha`	SHA-256 of the data bundle, for tamper detection	`sha256:7f2e…`
`forecast_id`	ULID uniquely identifying a single forecast row	`01HX…`
`match_id`	Unique match identifier following the canonical schema	`FIFA2026-GRP-…`
`pre_reg_tag`	Git tag for the pre-registration lockdown commit	`v1.0.0-mstar-lock`
`model_id`	Model identifier	`M0`, `M1`, `M2`, `M3`, `M_STAR`