27_bonus_value.md

27 — Bonus Value: Are Knowledgeable-but-Slow Players Undervalued?

27 — Bonus Value: Are Knowledgeable-but-Slow Players Undervalued?

Question (GM critique): Bonuses are worth 10 pts vs 4 for toss-ups, so bonuses are where games are won. Our player value model (PPTF — toss-up points per toss-up faced) excludes bonuses because they are team-answered. Does this badly UNDERVALUE a knowledgeable-but-slow player — strong combine, poor buzzer speed — who would raise their team's bonus conversion?

Data: clean games only (reconciles==True, 181 games), all three seasons (nsba1_2022, nsba2_2023, nsba3_2025). Code: analysis_bonus_value.py. Artifacts in outputs/. Combine theta is a PREDICTOR (not outcome); nsba3 PPTF uses estimated-TUH proxy.

CRITICAL DATA FACT (changes the whole question)

bonuses_long.csv logs only bonuses that were converted. Every row is worth 10 pts (99.6%). There are no "failed bonus" rows. I verified this by reconciling scores: toss_up_pts + logged_bonus_pts equals the reported game score exactly (corr = 1.000, mean abs diff = 0.0 across all 181 clean games).

Therefore a won toss-up that has no matching bonus row earned 0 bonus points — it is a genuinely failed bonus. This lets us recover the true bonus conversion rate:

bonus conversion = (logged bonus opportunities) / (toss-ups won)

This rate is real and varies across teams: mean 0.488, std 0.196, range 0.20–1.00 at the team-season level. So "bonus conversion" IS a separable, measurable skill here, and the critique is genuinely testable (not a degenerate constant). Good.

Task 1 — Bonus share of scoring: bonuses DO dominate

Per-season bonus share (mean over games): nsba1 0.54, nsba2 0.51, nsba3 0.57. Confirmed: bonuses are ~55–61% of all points scored — the majority of the scoreboard. The GM's framing is correct on the scoreboard arithmetic.

Task 2 — Do bonuses decide games? Bonus points matter; conversion rate less so

Team-game level (n = 359), correlation with winning:

Joint logit win ~ pptf_z + conv_z: pptf coef +2.69, conv coef +0.64 (p=0.0003), pseudo-R² 0.455. So conversion adds some independent signal, but it is ~4× weaker than toss-up production.

Win decomposition (177 head-to-head games): the winner's average margin splits +18.9 toss-up / +25.1 bonus → bonuses are 57% of the winning margin. 26 games (14.7%) were won despite losing or tying the toss-up battle — won purely on bonuses. So bonuses can and do flip a meaningful minority of games.

But the key nuance: bonus points track wins because winning toss-ups is the prerequisite for getting bonus opportunities at all — bonus pts corr with toss-up pts is +0.68. The conversion rate (the part not mechanically tied to toss-up wins) is only weakly tied to winning (+0.16).

Task 3 — Player bonus impact: knowledge does NOT predict conversion

Three independent estimators, all pointing the same way:

(a) Trigger attribution (who buzzed the toss-up that earned each bonus — 99.6% of bonuses linkable to a buzzer): per-player conversion mean 0.489, std 0.253 (249 player- seasons). Player-level conversion has essentially zero correlation with combine theta: +0.016 (n=73).

(b) ON/OFF (lineup recoverable via tossups_present — nsba1 full, nsba2 partial; 78 games, 659 rows): corr(player presence-share, team bonus conversion) = −0.034. No detectable lineup effect on conversion.

(c) Roster-knowledge regression (35 team-seasons), bonus_conv ~ pptf_z + knowledge_z:

The knowledge coefficient is negative and non-significant. Roster book-knowledge does not predict bonus conversion beyond buzzer skill. corr(conversion, mean theta) = −0.15.

Conclusion: there is no evidence in this data that book-knowledge raises bonus conversion. Conversion is roughly a constant-plus-noise (~49%) that does not load on combine theta, on specific players, or on lineup composition.

Task 4 — The key test: knowledgeable-but-slow players are NOT hiding bonus value

Knowledge and speed ARE partly distinct: corr(theta_z, pptf_z) = +0.56 (so combine and buzzer skill are correlated but far from identical — slow-but-smart players exist).

Eight "knowledgeable-but-slow" players (theta_z > 0.5, pptf_z < 0):

Their bonus trigger-conversion (0.542) is barely above league (0.489) on n=8 — well within noise, and below league for half of them. They do not convert bonuses at an elevated rate. The mechanism the critique posits — "smart player quietly carries bonuses" — does not show up.

Why: in NSBA bonuses are team-answered and conversion is nearly knowledge-agnostic at the margins we can measure. The thing that actually generates bonus points is winning the toss-up to earn the bonus — and that is exactly what PPTF already captures. A slow player who can't buzz never gets the bonus to the table, regardless of how much they know.

Task 5 — Combined value board: re-ranks almost not at all

Built nsba3 board (≥4 games, n=50): value = toss-up VORP + estimated bonus value, where each won toss-up generates a 10-pt opportunity at conversion conv_hat = 0.489 + β·theta_z (β = −0.013, from Task 3 — i.e. knowledge barely moves it).

• Full-bonus credit (gives every toss-up win its bonus value): Spearman vs PPTF rank = 0.945, mean |rank change| = 3.7. The reshuffle is driven almost entirely by toss-up volume (n_correct), not by knowledge — because bonus opportunities are won by buzzing.
• Knowledge-premium-only model (credits only the conversion lift above replacement that knowledge buys): Spearman vs PPTF = 0.996, mean |rank change| = 0.9. Essentially no reordering.

Biggest movers are high-volume toss-up scorers (Edwin He, Theenash Sengupta rise on toss-up count), not knowledgeable-but-slow players. Under the premium model the slow-smart guys (Advaith M, Aldric Benalan) actually drift down slightly, since their few toss-up wins cap their bonus exposure.

VERDICT & RECOMMENDATION

The critique is right about the scoreboard (bonuses = ~55–61% of points, decide ~15% of games outright) but wrong about the player-valuation implication. In this data:

1. Bonus conversion is a real, varying team skill (~49%, std 0.20) — but it is not predicted by combine knowledge (β ≈ 0, p = 0.37), by individual players, or by lineup.
2. Bonus points are overwhelmingly a function of winning the toss-up to earn the bonus, which PPTF already captures. Crediting bonuses re-ranks the board by ≤1 spot once you strip out the toss-up-volume component.
3. "Knowledgeable-but-slow" players add no measurable hidden bonus value. A player who can't buzz never brings the bonus to the table.

Fix (recommended): Do not add a speculative per-player bonus-conversion term to the value model — the data does not support attributing conversion to individuals, and doing so would mostly re-launder toss-up volume. Instead:

• Keep PPTF as the core toss-up metric (it already absorbs the bonus-generation pathway).
• Optionally add a team-level bonus-conversion adjustment (a roster trait, ~±0.2 of value swing), NOT a per-player one, and flag it as low-confidence.
• If we want to reward subject knowledge for roster construction, do it through coverage / breadth (which Task-2 north-star already shows: n_subjects corr +0.53 with win%), not through a phantom individual bonus credit.

Caveats (be explicit): per-player bonus attribution is inherently speculative — bonuses are team-answered, lineup data exists only for nsba1 + partial nsba2 (none for the live nsba3 draft season), and the KBS sample is n=8. The team-knowledge regression is n=35 team-seasons. The negative knowledge→conversion coefficient should be read as "no positive effect," not as a precise estimate. But across three independent methods the sign is consistently non-positive, so the conclusion (no exploitable individual bonus value) is robust to the noise.

Artifacts

• outputs/bonus_value_summary.csv — all headline numbers
• outputs/value_board_with_bonus.csv — nsba3 board: pptf-only vs combined ranks
• outputs/team_knowledge_bonus.csv — team-season conversion vs combine knowledge
• outputs/player_knowledge_speed_gap.csv — every player's theta_z, pptf_z, gap, trigger conv
• outputs/bonus_trigger_by_player.csv — per-player bonus trigger conversion

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