Methods

This page describes how Fourth & Value models NFL player props and calculates "value." It's intentionally general (no week-specific numbers) so it stays accurate as we update models.

1) What we model

We focus on common player-prop markets, normalized to canonical keys (market_std):

• Normal/Gaussian: rush_yds, recv_yds, pass_yds, receptions, rush_attempts, pass_attempts, pass_completions.
• Poisson: anytime_td, pass_tds, pass_interceptions.

We map sportsbook labels (e.g., "player_rushing_yards") to these keys so we can merge and compare across books.

2) Data sources (high level)

• Odds (per book): Player props and prices from multiple sportsbooks via The Odds API.
• Player game logs: Weekly player stats from nflverse (official NFL data).
• Schedule & kickoff: Used for display (ET) and filtering upcoming games.
• Defensive stats: Computed from opponent performance (yards allowed per game).

3) Modeling approach

Family-based multi-market modeling (Oct 2024):

Instead of modeling each market independently, we now model linked market families through shared latent factors:

• Rush family: rush_attempts and rush_yds are linked through:
  • Volume latent: expected carries (μ, σ)
  • Efficiency latent: yards per carry (μ, σ) bounded [2.5, 6.5]
  • Derived relationship: rush_yds = rush_attempts × YPC
• Receive family: receptions and recv_yds linked through:
  • Volume latent: expected targets
  • Catch rate: receptions/targets bounded [0.50, 0.95]
  • Yards per reception: bounded [6.0, 18.0]
  • Derived: recv_yds = targets × catch_rate × YPR
• Pass family: pass_attempts, pass_completions, and pass_yds linked through:
  • Volume latent: expected attempts
  • Completion percentage: bounded [0.50, 0.75]
  • Yards per completion: bounded [5.0, 12.0]
  • Derived: pass_yds = attempts × comp% × Y/C

Why this matters: When a sportsbook moves the rush attempts line but not rush yards, our model detects the implied efficiency change. If a book prices 18 carries at 120 yards (6.67 YPC—above our max bound), we know they're mispricing at least one market. This surfaces arbitrage opportunities where both linked markets show value.

Core methodology:

• Recency-weighted L4: Per player × market, we use only the last 4 games (L4) with exponential weighting (alpha=0.4). Most recent game receives ~64% of the total weight.
• Normal markets: Estimate mean μ and standard deviation σ → assume Normal(μ,σ).
• Poisson markets: Estimate rate λ for counts (TDs, INTs) → assume Poisson(λ).
• Sanity guards: Efficiency metrics are bounded to realistic ranges; floor σ to avoid overconfidence.

Exponential weighting:

Recent games receive exponentially more weight than older games. For a 4-game window:

• Oldest game (Week N-3): 10.3% weight
• 2nd oldest (Week N-2): 25.6% weight
• Most recent (Week N-1): 64.1% weight

This captures form changes, hot streaks, and role adjustments much faster than simple averages.

Matchup adjustments:

We apply two layers of matchup adjustments to account for both opponent strength and game location:

A. Defensive adjustments

Player projections are adjusted based on opponent defensive strength.

Defensive rating calculation:

• Computed from current season yards allowed per game (weeks 1 to N-1)
• Normalized using z-scores to a 0.5–2.0 scale:
  • 0.5 = Easiest defense (most yards allowed)
  • 1.0 = League average defense
  • 2.0 = Toughest defense (fewest yards allowed)
• Separate ratings for pass defense and rush defense
• Updated weekly as more games are played

Adjustment formula:

• adjustment_factor = 1.0 - (def_rating - 1.0) × 0.3
• Applied to μ after deriving from latent factors: μ_adjusted = μ × adjustment_factor
• Market mapping:
  • Receptions, receiving yards: use pass_def_rating
  • Rush yards, rush attempts: use rush_def_rating
  • QB passing markets: use pass_def_rating

Examples:

• Easiest defense (0.5 rating) → 1.0 - (0.5 - 1.0) × 0.3 = 1.15 → +15% boost
• Average defense (1.0 rating) → 1.0 - (1.0 - 1.0) × 0.3 = 1.00 → No change
• Tough defense (1.5 rating) → 1.0 - (1.5 - 1.0) × 0.3 = 0.85 → -15% penalty

Note: In practice, defensive ratings typically range from 0.5 to ~1.5 early in the season, so actual adjustments are usually in the ±7% to ±15% range.

B. Home/Away field advantage adjustments

After defensive adjustments, we apply home field advantage multipliers to account for location-based performance differences.

Why this matters: NFL data shows consistent home field advantage across all positions. Home teams score 2-4 more points per game on average. This stems from:

• Crowd noise disrupting opponent communication (helps your offense)
• Familiarity with field conditions, locker rooms, and routine
• Reduced travel fatigue and timezone changes
• Referee bias (subtle but measurable)

Implementation approach: We use evidence-based multipliers applied to player expectations (μ) after defensive adjustments:

4) Edge calculation & consensus

Converting probabilities to odds:

For each prop, we:

1. Compute model probability of Over (e.g., P(rush_yds ≥ 50.5)).
2. Convert to fair odds via fair_odds = round(100/p - 100) for p ≥ 50%, or 100×(1-p)/p for p < 50%.
3. Compare to sportsbook's American odds to compute basis-point edge.

Edge (basis points):

edge_bps = 10000 × (model_prob - market_prob)

A positive edge means our model thinks the Over is more likely than the book's implied probability. Negative edge suggests the Under is favorable.

Consensus view:

When 3+ books offer the same player × market × side × line:

• We show the median price across books.
• Calculate consensus market probability from median odds.
• Highlight these props with a green background in the UI.

Consensus props provide confidence that the line is well-established across the market, reducing the risk of stale or outlier pricing.

5) Known limitations

• Sample size: Early in the season (Weeks 1-4), we have limited data. Model confidence is lower.
• Team context: We don't yet adjust for pace, game script, or team totals/spreads.
• Weather: No weather adjustments (particularly important for passing games).
• Venue effects: Dome vs outdoor conditions not yet factored.

6) Roadmap

• ✅ Exponential weighting for recent form (Oct 2024)
• ✅ L4 rolling window (Oct 2024)
• ✅ Injury/role adjustment framework (Oct 2024)
• ✅ Defensive matchup framework (Oct 2024)
• ✅ Family-based multi-market modeling (Oct 2024)
• ✅ Home/away field advantage adjustments (Oct 2024)
• 🔄 Team context (pace, pass rate, game totals)
• 🔜 Market fusion (weak Bayesian updates from consensus lines)
• 🔜 Monte Carlo joint simulation with correlated shocks
• 🔜 Weather and dome/outdoor venue adjustments

Why transparency matters: By documenting our methods, we hold ourselves accountable to users and enable informed decision-making. If you find issues with our projections or have suggestions for improvements, please reach out via @fourthandvalue on X.