# GTO Strategy Analysis: Lead vs NoLead on 9s6d5c

**Date:** 2025-12-04
**Project:** LLM Range Tool - PLO4 GTO Analysis
**Board:** 9s6d5c (connected, two-tone)

---

## Experiment Overview

This analysis compares Hero's optimal continuation betting (c-bet) strategies in two distinct scenarios on the flop 9s6d5c in PLO4 (4-card Omaha). Both scenarios involve Hero being in position after raising preflop and facing Villain's check, but differ critically in whether Villain had the option to donk bet.

### Game Context

- **Hero:** In position, made preflop raise
- **Villain:** Out of position, checks on flop 9s6d5c
- **Hero's Decision:** Should I c-bet? How often? With what hands?

### Scenario Definitions

**Lead Scenario:**
- Villain CAN make donk bet (lead out) but CHOOSES to check
- Villain's check is VOLUNTARY → provides information about range
- Villain's range is FILTERED (removed hands that would donk)

**NoLead Scenario:**
- Villain CANNOT make donk bet (option disabled in game rules)
- Villain's check is FORCED → provides NO information
- Villain's range is UNFILTERED (contains entire range)

---

## Methodology

### 1. Data Sources

Two sets of GTO solutions from PLO4 solver for board 9s6d5c:

**Lead Range Files:**
- `1_2 POT.csv` - 66,550 combos (Hero bets 1/2 pot after Villain's voluntary check)
- `CHECK.csv` - 62,788 combos (Hero checks back after Villain's voluntary check)
- **Total:** 129,338 combos

**NoLead Range Files:**
- `NoLead_1_2 POT.csv` - 49,746 combos (Hero bets 1/2 pot after Villain's forced check)
- `NoLead_CHECK.csv` - 79,510 combos (Hero checks back after Villain's forced check)
- **Total:** 129,256 combos

### 2. Bucketing System

Each combo analyzed through **85 binary features** (buckets):

**Categories:**
- Made hands: flush, sets, trips, pairs, straights, full houses
- Draws: flush draws, straight draws, wraps, gutshots, backdoor draws
- Blockers: straight blockers, flush blockers, draw blockers (at various levels)
- Hand strength: top pair, middle pair, pocket pairs, kicker strength

**Function:** `get_all_buckets(combo, board)` returns [0,1,0,...,1] array of 85 features

### 3. Matrix Building

For each scenario:
1. Load GTO ranges from CSV files
2. Bucket each combo (85 features)
3. Group by unique feature combinations
4. Sum weights (frequencies) by action
5. Normalize to percentages
6. **Output:** Bucket matrix (1,368 unique combinations × 2 actions)

### 4. Decision Tree Training

Used scikit-learn DecisionTreeClassifier with:
- **Algorithm:** CART (Gini impurity)
- **Sample weights:** Combo frequencies from GTO solver
- **Parameter:** `min_samples_leaf` ∈ {1, 10, 50}
- **Features:** 85 bucket features (readable names)
- **Target:** Action with max percentage (bet_1/2pot vs check)

### 5. Exports Generated

For each tree (6 total: Lead/NoLead × min1/min10/min50):

**Mermaid Diagram (.mmd):**
- Flowchart format with readable bucket names
- Node decisions and leaf percentages
- Viewable on GitHub, mermaid.live, VS Code

**Metadata JSON:**
- Feature importance (sorted)
- Node structure (splits, thresholds, samples)
- Leaf statistics (decisions, percentages)
- Tree statistics (depth, leaves, nodes)

**Feature Importance CSV:**
- Ranked list of features by importance
- Decimal values (0-1 scale)

### 6. Analysis Focus

This analysis uses **min_samples_leaf=50** trees for:
- Balance between accuracy and interpretability
- Lead: 18 leaves, depth 7, 75.56% accuracy
- NoLead: 20 leaves, depth 6, 79.61% accuracy

---

## Deep Analysis: C-betting with vs without Villain's Donk Option

### Tree Statistics Comparison

| Metric | NoLead (forced check) | Lead (voluntary check) |
|--------|----------------------|------------------------|
| Max depth | 6 | 7 |
| Leaves | 20 | 18 |
| Nodes | 39 | 35 |
| Features used | 12 | 14 |
| Accuracy | 79.61% | 75.56% |

**Key insight:** Lead tree is more complex (deeper, more features) because Villain's voluntary check creates informational advantage for Hero.

---

### Feature Importance Comparison

#### TOP-5 NoLead (vs forced check):
1. **straight_blocker2** (46.40%)
2. **straight_draw_blocker_nut** (16.63%)
3. **pair** (10.42%) - ⚠️ showdown value important
4. **bdfd2** (7.63%) - equity for multi-street
5. **straight_draw_blocker2** (6.13%)

#### TOP-5 Lead (vs voluntary check):
1. **straight_blocker** (37.27%)
2. **no_draw** (12.51%) - ⚠️ KEY: blocker bluffing signal
3. **straight_draw_blocker_nut** (6.54%)
4. **straight_draw_blocker1** (6.48%)
5. **top_pair** (5.88%)

---

### Strategic Interpretation

#### 1. NoLead: Villain Forced to Check (Wide Range)

**What it means:**
- Villain's range is UNFILTERED (contains everything)
- His check tells us NOTHING about hand strength
- Could have monsters, air, draws - entire range

**GTO Response:**
- More CONSERVATIVE c-betting
- Need real equity (pairs, draws) to bet
- Can't over-bluff against wide range
- Pairs important for showdown value
- Backdoor draws justify multi-street plan

**Most passive: node_id 6 (check 99.8%):**

No straight_blocker2, no straight_draw_blocker_nut,
no set, no top_pair, no bdfd2, no straight_draw_blocker2

**Interpretation:** Complete air vs Villain's wide range → give up

---

#### 2. Lead: Villain Voluntarily Checked (Filtered Range)

**What it means:**
- Villain COULD have donk bet but didn't
- His check FILTERS his range (removed donk-betting hands)
- Likely capped/weakened range (no monsters that would donk for protection)

**GTO Response:**
- More AGGRESSIVE c-betting
- Can bluff-bet with just blockers
- `no_draw` importance = blocker bluffing logic
- When Hero has no draws, Villain's "honest check" less likely to be draw
- Top pair more valuable (Villain capped)

**Most aggressive: node_id 33 (bet 98.7%):**

Path: straight_blocker=YES → no_draw=YES → pocket_pair=NO

**Interpretation:** Pure blocker bluff!
- Block Villain's value hands
- No draws in our hand = Villain less likely has draws
- Villain's voluntary check = capped range
- **Bet aggressively!**

---

### Decision Tree Visualizations

#### NoLead Tree (min_samples_leaf=50)

flowchart TD
    node0{"straight_blocker2"}
    node0 -->|No| node1
    node1{"straight_draw_blocker_nut"}
    node1 -->|No| node2
    node2{"set"}
    node2 -->|No| node3
    node3{"top_pair"}
    node3 -->|No| node4
    node4{"bdfd2"}
    node4 -->|No| node5
    node5{"straight_draw_blocker2"}
    node5 -->|No| node6
    node6["check<br/>bet_1/2pot: 0.2%<br/>check: 99.8%"]
    node5 -->|Yes| node7
    node7["check<br/>bet_1/2pot: 2.1%<br/>check: 97.9%"]
    node4 -->|Yes| node8
    node8["check<br/>bet_1/2pot: 17.7%<br/>check: 82.3%"]
    node3 -->|Yes| node9
    node9{"tp_tk"}
    node9 -->|No| node10
    node10["check<br/>bet_1/2pot: 39.4%<br/>check: 60.6%"]
    node9 -->|Yes| node11
    node11["check<br/>bet_1/2pot: 14.7%<br/>check: 85.3%"]
    node2 -->|Yes| node12
    node12{"pair"}
    node12 -->|No| node13
    node13["bet_1/2pot<br/>bet_1/2pot: 57.6%<br/>check: 42.4%"]
    node12 -->|Yes| node14
    node14["check<br/>bet_1/2pot: 11.6%<br/>check: 88.4%"]
    node1 -->|Yes| node15
    node15{"bdfd2"}
    node15 -->|No| node16
    node16{"straight_blocker_nut1"}
    node16 -->|No| node17
    node17{"straight_draw_blocker2"}
    node17 -->|No| node18
    node18{"pair"}
    node18 -->|No| node19
    node19["check<br/>bet_1/2pot: 13.4%<br/>check: 86.6%"]
    node18 -->|Yes| node20
    node20["check<br/>bet_1/2pot: 11.2%<br/>check: 88.8%"]
    node17 -->|Yes| node21
    node21["check<br/>bet_1/2pot: 36.4%<br/>check: 63.6%"]
    node16 -->|Yes| node22
    node22{"pair"}
    node22 -->|No| node23
    node23{"straight_draw_blocker2"}
    node23 -->|No| node24
    node24["bet_1/2pot<br/>bet_1/2pot: 60.7%<br/>check: 39.3%"]
    node23 -->|Yes| node25
    node25["check<br/>bet_1/2pot: 39.2%<br/>check: 60.8%"]
    node22 -->|Yes| node26
    node26["check<br/>bet_1/2pot: 20.2%<br/>check: 79.8%"]
    node15 -->|Yes| node27
    node27{"pair"}
    node27 -->|No| node28
    node28{"straight_blocker_nut"}
    node28 -->|No| node29
    node29["bet_1/2pot<br/>bet_1/2pot: 54.6%<br/>check: 45.4%"]
    node28 -->|Yes| node30
    node30["bet_1/2pot<br/>bet_1/2pot: 87.3%<br/>check: 12.7%"]
    node27 -->|Yes| node31
    node31{"pocket_pair"}
    node31 -->|No| node32
    node32["check<br/>bet_1/2pot: 44.3%<br/>check: 55.7%"]
    node31 -->|Yes| node33
    node33["bet_1/2pot<br/>bet_1/2pot: 70.6%<br/>check: 29.4%"]
    node0 -->|Yes| node34
    node34{"bdfd"}
    node34 -->|No| node35
    node35["bet_1/2pot<br/>bet_1/2pot: 75.0%<br/>check: 25.0%"]
    node34 -->|Yes| node36
    node36{"straight_draw_blocker2"}
    node36 -->|No| node37
    node37["bet_1/2pot<br/>bet_1/2pot: 97.9%<br/>check: 2.1%"]
    node36 -->|Yes| node38
    node38["bet_1/2pot<br/>bet_1/2pot: 89.5%<br/>check: 10.5%"]

#### Lead Tree (min_samples_leaf=50)

flowchart TD
    node0{"straight_blocker"}
    node0 -->|No| node1
    node1{"straight_draw_blocker_nut"}
    node1 -->|No| node2
    node2{"backdoor_straight_draw4"}
    node2 -->|No| node3
    node3{"set"}
    node3 -->|No| node4
    node4{"top_pair"}
    node4 -->|No| node5
    node5{"pair"}
    node5 -->|No| node6
    node6["check<br/>bet_1/2pot: 3.6%<br/>check: 96.4%"]
    node5 -->|Yes| node7
    node7["check<br/>bet_1/2pot: 27.0%<br/>check: 73.0%"]
    node4 -->|Yes| node8
    node8["check<br/>bet_1/2pot: 42.9%<br/>check: 57.1%"]
    node3 -->|Yes| node9
    node9{"bdfd1"}
    node9 -->|No| node10
    node10["check<br/>bet_1/2pot: 47.1%<br/>check: 52.9%"]
    node9 -->|Yes| node11
    node11["bet_1/2pot<br/>bet_1/2pot: 75.5%<br/>check: 24.5%"]
    node2 -->|Yes| node12
    node12["bet_1/2pot<br/>bet_1/2pot: 58.3%<br/>check: 41.7%"]
    node1 -->|Yes| node13
    node13{"straight_draw_blocker1"}
    node13 -->|No| node14
    node14{"pair"}
    node14 -->|No| node15
    node15["bet_1/2pot<br/>bet_1/2pot: 51.2%<br/>check: 48.8%"]
    node14 -->|Yes| node16
    node16["bet_1/2pot<br/>bet_1/2pot: 81.8%<br/>check: 18.2%"]
    node13 -->|Yes| node17
    node17{"top_pair"}
    node17 -->|No| node18
    node18["check<br/>bet_1/2pot: 19.3%<br/>check: 80.7%"]
    node17 -->|Yes| node19
    node19["check<br/>bet_1/2pot: 39.9%<br/>check: 60.1%"]
    node0 -->|Yes| node20
    node20{"no_draw"}
    node20 -->|No| node21
    node21{"straight_blocker2"}
    node21 -->|No| node22
    node22{"straight_draw_blocker3"}
    node22 -->|No| node23
    node23{"two_pairs"}
    node23 -->|No| node24
    node24{"bdfd2"}
    node24 -->|No| node25
    node25{"pocket_pair"}
    node25 -->|No| node26
    node26["check<br/>bet_1/2pot: 38.7%<br/>check: 61.3%"]
    node25 -->|Yes| node27
    node27["bet_1/2pot<br/>bet_1/2pot: 58.1%<br/>check: 41.9%"]
    node24 -->|Yes| node28
    node28["bet_1/2pot<br/>bet_1/2pot: 78.4%<br/>check: 21.6%"]
    node23 -->|Yes| node29
    node29["bet_1/2pot<br/>bet_1/2pot: 82.6%<br/>check: 17.4%"]
    node22 -->|Yes| node30
    node30["bet_1/2pot<br/>bet_1/2pot: 80.0%<br/>check: 20.0%"]
    node21 -->|Yes| node31
    node31["bet_1/2pot<br/>bet_1/2pot: 96.6%<br/>check: 3.4%"]
    node20 -->|Yes| node32
    node32{"pocket_pair"}
    node32 -->|No| node33
    node33["bet_1/2pot<br/>bet_1/2pot: 98.7%<br/>check: 1.3%"]
    node32 -->|Yes| node34
    node34["bet_1/2pot<br/>bet_1/2pot: 84.9%<br/>check: 15.1%"]

---

### Core Strategic Difference

#### The Information Gap:

**NoLead (no info from check):**

Villain's Range = [nuts, strong hands, medium, weak, air] - FULL SPECTRUM
Hero Strategy = Conservative, need equity, respect the range

**Lead (check = information):**

Villain's Range = [nuts, strong hands, medium, weak, air] - DONK HANDS
                = [medium hands, weak hands, some air] - CAPPED
Hero Strategy = Aggressive, exploit the cap, bluff with blockers

---

### Practical Application

#### Against Forced Check (NoLead):

**C-bet when:**
- ✅ You have pair + blocker (thin value)
- ✅ You have backdoor draws + blockers (equity + fold equity)
- ✅ You have sets/strong hands (value)

**Check back when:**
- ❌ Complete air without blockers
- ❌ Villain's range too wide to profitably bluff
- ❌ No equity for multi-street plan

**Philosophy:** "Respect Villain's uncapped range, build pot with equity"

---

#### Against Voluntary Check (Lead):

**C-bet when:**
- ✅ You have straight blockers (even without pair!)
- ✅ You have `no_draw` in your hand (blocker logic)
- ✅ Top pair or better (Villain capped)
- ✅ Exploiting Villain's range weakness

**Check back when:**
- ❌ No blockers and no equity
- ❌ Villain could check-raise with traps

**Philosophy:** "Punish Villain's capped range with aggressive bluffing"

---

### Advanced Insight: The `no_draw` Paradox

**Why is `no_draw` 12.51% important vs voluntary check?**

Standard poker logic: "If I have draws, I should bet them"
**But here:** "If I have NO draws, I should bet MORE"

**Explanation:**
1. Villain voluntarily checked (filtered out strong hands + draws that would donk)
2. If Hero has `no_draw` = fewer draws in deck = Villain even less likely has draws
3. Villain's "honest check" more likely to be weak pair/air
4. Hero's blockers become powerful bluffing tools
5. **Conclusion:** Bet aggressively with blockers even without draws!

---

### Comparison Table

| Scenario | Villain's Range | Hero's Approach | Key Features |
|----------|----------------|-----------------|--------------|
| **NoLead** | Wide/uncapped (forced check) | Conservative value-betting | Pairs, bdfd equity |
| **Lead** | Narrow/capped (voluntary check) | Aggressive blocker-bluffing | Blockers, no_draw |

---

### Example Hands

**Board: 9s6d5c**

**Scenario 1 (NoLead - forced check):**
- Hero holds: **Ah Kh Qd Jd** (no pair, backdoor flush draw, straight blockers)
- GTO: Check back (~80%+) - need more equity vs wide range

**Scenario 2 (Lead - voluntary check):**
- Hero holds: **Ah Kh Qd Jd** (same hand!)
- GTO: Bet (~70%+) - exploit capped range with blockers
- Villain checked when he could donk → likely doesn't have 78, doesn't have strong draws

**Same hand, different action based on information!**

---

## Key Takeaway

**The voluntary check is INFORMATION:**
- In **Lead** scenarios, Villain's check tells us his range is capped
- We can exploit this with aggressive blocker bluffing
- `no_draw` becomes a WEAPON, not a weakness

**The forced check is NO INFORMATION:**
- In **NoLead** scenarios, Villain's check means nothing
- We must play carefully against uncapped range
- Need real equity and showdown value

**This is why the same Hero hand plays differently in these two trees!**

---

## Technical Details

### Files Generated

output/
├── bucket_matrix_lead_9s6d5c.csv           # Lead bucket matrix
├── bucket_matrix_nolead_9s6d5c.csv         # NoLead bucket matrix
├── tree_lead_9s6d5c_min50.mmd              # Lead tree diagram
├── tree_lead_9s6d5c_min50_metadata.json    # Lead tree metadata
├── tree_lead_9s6d5c_min50_importance.csv   # Lead feature importance
├── tree_nolead_9s6d5c_min50.mmd            # NoLead tree diagram
├── tree_nolead_9s6d5c_min50_metadata.json  # NoLead tree metadata
└── tree_nolead_9s6d5c_min50_importance.csv # NoLead feature importance

### Code Repository

- `build_bucket_matrix.py` - Main script for matrix building and tree training
- `lib/buckets.py` - 85 bucket functions for hand analysis
- `lib/card.py` - Card representation classes
- `README_BUCKETS.md` - Full documentation of bucket system

### Reproducibility

# Install dependencies
pip install pandas scikit-learn tqdm

# Run analysis
python3 build_bucket_matrix.py

# Processing time: ~6 minutes
# Output: 2 matrices + 6 trees with metadata

---

**Generated:** 2025-12-04
**Project:** LLM Range Tool - PLO4 GTO Analysis
**Board:** 9s6d5c