Category Allocation Best-Response

Table of Contents

• Example
• Workflow
• Common Patterns
• Guardrails
• Quick Reference

Example

Scenario: Yahoo MLB 10-category H2H, Week 8. Threshold = 6 of 10. Upstream matchup-win-probability-sim returned per_cat_win_probability for the current week. Our resources: 5 open roster slots, $80 FAAB, 3 streamer starts.

Inputs:

Cat	our_capacity	opp_projection	per_cat_win_prob	Inverse?
R	42	38	0.64	no
HR	12	14	0.35	no
RBI	40	41	0.47	no
SB	6	4	0.72	no
OBP	0.335	0.328	0.64	no
K	55	50	0.64	no
ERA	3.85	4.10	0.65	yes
WHIP	1.22	1.28	0.68	yes
QS	4	3	0.69	no
SV	2	5	0.08	no

Classification by win probability:

• SV (0.08): conceded — dominated strategy. Opponent has 5 projected SV to our 2; no marginal roster slot flips this. Leverage = 0.
• HR (0.35): contested — borderline-losing but within reach. Leverage = 1.5.
• RBI (0.47): contested — coin-flip. Leverage = 1.5.
• R (0.64), OBP (0.64), K (0.64), ERA (0.65), WHIP (0.68), QS (0.69): pushed (lock-in needs attention, 0.25–0.85). Leverage = 1.2. (0.70 is the boundary — one of these would sit on the contested side at 1.5 if very close to the line.)
• SB (0.72): pushed. Leverage = 1.2.

No cat exceeds 0.85 on this week; no cats are locked.

Outputs:

• pushed_cats: [SB, QS, WHIP, ERA, K, R, OBP] (7 cats at 0.64–0.72, ordered by win prob descending)
• conceded_cats: [SV]
• contested_cats: [HR, RBI]
• leverage_weights: {R: 1.2, HR: 1.5, RBI: 1.5, SB: 1.2, OBP: 1.2, K: 1.2, ERA: 1.2, WHIP: 1.2, QS: 1.2, SV: 0.0}
• k_of_n_win_probability: 0.605 (Poisson-binomial over the 10 per-cat probabilities with threshold 6)
• rationale: "SV is a dominated strategy (8% flip probability) — do not spend a reliever slot. The 7 pushed cats cover the 6-cat threshold on median, so lock them in and invest marginal resources into HR and RBI (contested, 1.5x leverage) where one extra power bat can flip the matchup."

Interpretation: if we simply defend the 7 pushed cats we hit threshold in expectation. The contested cats (HR, RBI) are where the highest-leverage moves live — every marginal HR or RBI unit maps almost 1-for-1 to matchup win probability.

Workflow

Copy this checklist and track progress:

Category Allocation Best-Response Progress:
- [ ] Step 1: Validate inputs and confirm upstream signals
- [ ] Step 2: Classify each cat by per_cat_win_probability
- [ ] Step 3: Assign leverage_weights
- [ ] Step 4: Check threshold satisfaction (pushed + contested >= K?)
- [ ] Step 5: Apply borderline-upgrade logic if threshold not met
- [ ] Step 6: Compute k_of_n_win_probability via Poisson-binomial
- [ ] Step 7: Write rationale and emit outputs

Step 1: Validate inputs and confirm upstream signals

per_cat_win_probability is the critical upstream dependency and must come from matchup-win-probability-sim (or an equivalent simulator). Do not invent per-cat probabilities from raw capacity vs projection — matchup-win-probability-sim accounts for variance, and variance is what determines flip probability for contested cats. See resources/methodology.md.

• Every cat appears in our_per_cat_capacity, opp_per_cat_projection, and per_cat_win_probability
• All per_cat_win_probability values are in [0, 1]
• cat_win_threshold is in [1, N] where N = len(cats)
• inverse_cats is a subset of the cat list
• resources_available dict is present (even if empty — downstream skills may still consume leverage weights without a resource plan)
• Upstream random_seed from matchup-win-probability-sim is recorded for audit

Step 2: Classify each cat by per_cat_win_probability

Apply the four-tier classification in Quick Reference. The thresholds are deliberate — see resources/methodology.md for the Blotto-derived rationale.

• < 0.25 → conceded_cats (dominated strategy — any roster slot here is strictly worse than redeploying)
• 0.25–0.70 → contested_cats (highest marginal value of one more unit)
• 0.70–0.85 → pushed_cats (needs attention but should hold)
• > 0.85 → locked (do not waste marginal resources — returns are near-zero)
• Inverse cats (ERA, WHIP, TO, GAA, etc.) use per_cat_win_probability directly — the inverse handling has already been applied upstream by matchup-win-probability-sim. Do not re-invert.

Step 3: Assign leverage_weights

Leverage weights propagate to mlb-lineup-optimizer, mlb-streaming-strategist, mlb-waiver-analyst, and any downstream consumer that maximizes Σ daily_quality × leverage[cat]. See resources/methodology.md.

• leverage = 0.0 for conceded_cats (hard zero — not low, zero)
• leverage = 1.5 for contested_cats (high marginal value of one more unit)
• leverage = 1.2 for pushed_cats (above default but below contested)
• leverage = 1.0 for locked cats (default — marginal gains are redundant)

Step 4: Threshold satisfaction check

Verify len(pushed_cats) + len(contested_cats) >= cat_win_threshold. If not, we are mathematically unable to reach the win threshold even if we go 100% on our defensible cats; the matchup is presumptively losing and we must either upgrade a borderline-conceded cat or pivot to variance-seeking play. See principle #6 in frameworks/game-theory-principles.md.

• Count pushed_cats + contested_cats (these are the cats where we have nonzero flip probability)
• If count >= cat_win_threshold: threshold satisfied, proceed to Step 6
• If count < cat_win_threshold: apply Step 5 borderline-upgrade logic
• If still insufficient after Step 5: flag the matchup for variance-strategy-selector as a high-variance play candidate

Step 5: Borderline-upgrade logic (conditional)

If Step 4 fails, look for conceded_cats with per_cat_win_probability in the [0.20, 0.25) "upgrade band". These are just below the concede line — a moderate resource investment (one waiver add, one FAAB bid, one streamer start) can lift them over 0.25 and into the contested tier.

• Sort conceded_cats by per_cat_win_probability descending
• Pick the top candidate(s) with per_cat_win_probability >= 0.20
• Reclassify to contested and set leverage = 1.5
• Document the upgrade in the rationale with the specific resource cost (e.g., "upgrading HR from conceded: spend 1 roster slot + $15 FAAB on a power bat")
• Re-run Step 4 — if still short, this matchup is presumptively losing; set a flag and defer to the variance-strategy-selector skill

Step 6: Compute k_of_n_win_probability via Poisson-binomial

Treat per-cat wins as independent Bernoullis (the same approximation used by matchup-win-probability-sim in poisson_binomial mode). Compute P(sum >= cat_win_threshold) via the standard PB recurrence. See resources/methodology.md.

P_0(0) = 1
P_i(k) = P_{i-1}(k) * (1 - p_i) + P_{i-1}(k-1) * p_i   for i = 1..N, k = 0..N
k_of_n_win_probability = sum over k >= threshold of P_N(k)

• Use the post-upgrade per_cat_win_probability vector (Step 5 may have modified one entry)
• Return the overall k_of_n_win_probability
• If the value diverges from the matchup_win_probability returned by matchup-win-probability-sim by more than 0.03, investigate — the two should match within PB approximation error

Step 7: Write rationale and emit outputs

Rationale is a 2–4 sentence plain-English summary of the allocation logic. Name the conceded cats and why, name the contested cats and the highest-leverage resource move, call out any borderline upgrades, and state the computed k_of_n_win_probability. See resources/template.md for worked examples.

• All outputs present: pushed_cats, conceded_cats, contested_cats, leverage_weights, k_of_n_win_probability, rationale
• leverage_weights has one entry per cat (not one per push/concede/contest bucket)
• Rationale names at least one conceded cat and one contested cat explicitly
• Validate using resources/evaluators/rubric_category_allocation_best_response.json. Minimum standard: average score 3.5 or above.

Common Patterns

Pattern 1: MLB 10-cat (Yahoo 5x5)

• cats: [R, HR, RBI, SB, OBP, K, ERA, WHIP, QS, SV]
• cat_win_threshold: 6
• inverse_cats: [ERA, WHIP]
• Typical concedes: SV (vs closer-heavy opponents), SB (vs power-heavy rosters), QS (vs all-RP staffs)
• Resources: roster_slots (typically 4–6 bench), faab ($0–$100), streamer_starts (2–4 per week)
• Downstream consumers: mlb-lineup-optimizer (principle #5), mlb-streaming-strategist (principle #1), mlb-waiver-analyst (principle #2)

Pattern 2: NBA 9-cat

• cats: [PTS, REB, AST, STL, BLK, 3PM, FG%, FT%, TO]
• cat_win_threshold: 5
• inverse_cats: [TO]
• Typical concedes: FT% (vs Giannis/Simmons-type rosters), TO (vs low-usage rosters), 3PM (vs shooting-punt rosters)
• Resources: roster_slots, streamer_games (NBA has daily streaming via DTD lineups)
• Special: NBA has higher cat-to-cat correlation (PTS-AST-3PM from team usage); the pushed tier tends to move together, so a single roster move can lift multiple cats at once. This makes contested-cat leverage particularly high.

Pattern 3: NHL 10-cat

• cats: [G, A, +/-, PIM, PPP, SOG, W, GAA, SV%, SO]
• cat_win_threshold: 6
• inverse_cats: [GAA]
• Typical concedes: SO (shutouts are low-count lottery), PIM (vs goon-free rosters), +/- (high variance)
• Resources: roster_slots, streamer_games, goalie_starts
• Special: Goalie cats (W, GAA, SV%, SO) are deeply correlated through the same player — a single streamer start affects four cats at once. Treat goalie-cat leverage as a bundle rather than four independent decisions.

Pattern 4: Variance-seeking underdog (cross-domain)

• When k_of_n_win_probability < 0.40: we are the underdog. Raise leverage on contested cats to 1.5, and flag the matchup for variance-strategy-selector. High-variance lineup construction (boom-bust players, one-start studs) can lift our win probability from ~35% to ~45% even without roster improvements. See principle #6 in frameworks/game-theory-principles.md.

Guardrails

1. Do not invent per-cat probabilities. This skill is a downstream consumer of matchup-win-probability-sim. If you do not have per_cat_win_probability from a proper simulator, do not proceed — compute them first. Eyeballing probabilities from raw projections ignores variance, which is the whole point of the contested-cat classification.

2. Leverage weight 0.0 is a hard constraint, not a preference. When mlb-lineup-optimizer reads leverage[SV] = 0.0, it will refuse to start a pure-SV reliever even if the reliever has a high daily_quality. This is correct behavior (principle #1 in frameworks/game-theory-principles.md) — do not soften the zero to 0.1 or 0.2 to "keep options open". Zero means zero.

3. Contested-cat leverage stays 1.5 even if we are favored in the cat. The 1.5 multiplier captures the marginal value of one extra unit — which is high whenever the cat is close. A cat at per_cat_win_probability = 0.68 still has room for marginal gains to flip it to a near-certain win; leverage remains 1.5 (it is a pushed cat at 1.2, and moves to 1.5 if it drops into the contested band below 0.70).

4. Threshold and count must match the league format. cat_win_threshold = 6 for 10-cat MLB (strict majority). 5 for 9-cat NBA. 6 for 10-cat NHL. Passing the wrong threshold produces a silently wrong classification. Always confirm the league's tie-break rules — some leagues award ties for half-wins.

5. Dominated-strategy elimination is per-matchup, not per-season. A cat conceded this week vs a closer-heavy opponent may be a push cat next week vs a different opponent. Do not cache conceded_cats across weeks. Re-run the classification every matchup.

6. Inverse-cat handling is upstream's job. matchup-win-probability-sim already returns per_cat_win_probability with inverse handling applied (ERA 3.50 beats ERA 4.20 = win probability near 1.0). This skill consumes those probabilities directly. Do not re-invert, and do not treat inverse cats differently at the classification step.

7. Upgrade band is narrow (0.20–0.25). Only upgrade a conceded cat when its per_cat_win_probability is within the narrow [0.20, 0.25) band and we have resources available. Upgrading a 0.15-probability cat is throwing resources at a losing cause. Upgrading a 0.24-probability cat with a single waiver add may flip it to 0.30 — worth it.

8. Document resource cost when recommending upgrades. Abstract "upgrade HR" is unactionable. Say "upgrade HR with 1 roster slot + $15 FAAB on a power bat who adds ~3 HR/week", so mlb-waiver-analyst and mlb-faab-sizer have a concrete target.

9. When the matchup is presumptively losing, pivot to variance, not to pushing harder. If k_of_n_win_probability < 0.40 after classification and upgrades, the best move is to maximize variance (principle #6), not to redouble on pushed_cats. Flag the matchup in the rationale and call variance-strategy-selector downstream.

10. The skill is domain-neutral — resist MLB-specific assumptions. When called from NBA or NHL contexts, the thresholds (0.25 / 0.70 / 0.85), bands, leverage weights, and upgrade-band logic all apply unchanged. Only the cat list and win threshold change between sports. Keep the core logic sport-agnostic.

Quick Reference

Four-tier classification (from per_cat_win_probability):

Range	Bucket	Leverage	Rationale
[0.00, 0.25)	conceded	0.0	Dominated strategy. Marginal unit has near-zero flip impact.
[0.25, 0.70)	contested	1.5	Highest marginal value — one more unit often flips outcome.
[0.70, 0.85]	pushed	1.2	Should hold but not guaranteed; defend with attention.
(0.85, 1.00]	locked	1.0	Default weight; marginal gains are near-redundant.

Upgrade band (for borderline concedes):

Range	Action
[0.20, 0.25)	Candidate for upgrade if Step 4 threshold check fails.
[0.00, 0.20)	Do not upgrade. Resources are better spent on contested cats.

Poisson-binomial recurrence (for k_of_n_win_probability):

Given p = [p_1, p_2, ..., p_N] and threshold K:

P_0(0) = 1
P_0(k) = 0 for k >= 1

For i = 1..N:
  P_i(0) = P_{i-1}(0) * (1 - p_i)
  For k = 1..i:
    P_i(k) = P_{i-1}(k) * (1 - p_i) + P_{i-1}(k-1) * p_i

k_of_n_win_probability = sum_{k=K}^{N} P_N(k)

Inputs required:

• our_per_cat_capacity: dict[cat, number] — our projected per-cat output (remaining-week or full-week)
• opp_per_cat_projection: dict[cat, number] — opponent's projected per-cat output
• per_cat_win_probability: dict[cat, float in [0,1]] — FROM matchup-win-probability-sim
• cat_win_threshold: int — 6 for MLB 10-cat, 5 for NBA 9-cat, 6 for NHL 10-cat
• resources_available: dict[str, number] — e.g., {"roster_slots": 5, "faab": 80, "streamer_starts": 3}
• inverse_cats: list[str] — cats where lower is better (ERA, WHIP, TO, GAA, etc.)

Outputs produced:

• pushed_cats: list[cat] — ordered by per_cat_win_probability descending
• conceded_cats: list[cat] — dominated; leverage 0.0
• contested_cats: list[cat] — highest marginal leverage
• leverage_weights: dict[cat, float] — values in {0.0, 1.0, 1.2, 1.5}
• k_of_n_win_probability: float in [0,1] — overall matchup win prob under this allocation
• rationale: string — 2–4 sentences of plain-English allocation logic

Upstream dependencies:

• matchup-win-probability-sim (REQUIRED) — supplies per_cat_win_probability and overall matchup_win_probability for cross-validation
• *-category-state-analyzer (optional, MLB/NBA/NHL) — supplies the per-cat capacity and opponent projection

Downstream consumers:

• mlb-lineup-optimizer / NBA / NHL equivalents — consume leverage_weights (principle #5)
• mlb-streaming-strategist — reads conceded_cats as hard constraint (principle #1)
• mlb-waiver-analyst / mlb-faab-sizer — consume contested_cats + resources_available to target high-leverage adds
• variance-strategy-selector — consumes k_of_n_win_probability to decide favorite-vs-underdog play style (principle #6)

Key resources:

• resources/template.md: Input/output contract, MLB 10-cat worked example, NBA 9-cat worked example, rationale templates
• resources/methodology.md: Colonel Blotto origins and H2H-variant math, why K-of-N is not winner-takes-all, Nash equilibrium for symmetric cases, heuristic best-response for asymmetric cases, dominated-strategy elimination, upstream-signal derivation, variance-pivot logic
• resources/evaluators/rubric_category_allocation_best_response.json: 10 criteria — classification accuracy, leverage weight assignment, dominated-strategy identification, threshold satisfaction check, rationale quality, borderline-upgrade logic, generalization to non-10-cat leagues, output completeness, upstream-signal integration, citations