Does Batting Variance Matter in Test Cricket?

A Monte Carlo simulation study exploring whether "boom-or-bust" or consistent batsmen are more valuable in Test cricket.

Research Question

If two batsmen have the same batting average, but one is high-variance (more ducks AND more centuries) and one is consistent (steady scores), which is more valuable to a Test cricket team?

Key Findings

Strategy	Win Rate	Draw Rate	Loss Rate
All Consistent	53.8%	~8%	38.2%
High Variance Openers Only	~52%	~8%	~40%
All High Variance	44.8%	~7%	48.2%

Bottom line: Consistency wins. High-variance batting creates collapse risk that isn't compensated by occasional big scores.

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How the Simulation Works

Ball-by-Ball Engine

Every delivery is simulated individually. The base scoring distribution (before normalization):

Outcome	Base Weight
Dot ball	0.650
Single	0.220
Two	0.070
Four	0.040
Six	0.020
Wicket	Dynamic

The wicket probability changes based on balls faced - this is the core innovation.

After calculating the dynamic wicket probability, all outcomes are normalized to sum to 100%. This means when wicket probability is higher (early in an innings for high-variance batsmen), scoring shot probabilities are slightly reduced proportionally.

The Variance Model

Both batsman types produce identical expected runs per dismissal (~38 runs for openers/middle order):

LOW VARIANCE (Consistent):
  p(wicket) = 1.68% constant per ball

  Result: Steady scores, few ducks, few centuries

HIGH VARIANCE (Boom-or-Bust):
  p(wicket) = 0.75% + (3.4% - 0.75%) * e^(-0.025 * balls_faced)

  At ball 0:   3.4% (very risky - often out early)
  At ball 50:  1.8% (average risk)
  At ball 100: 1.0% (safe when "set")

  Result: More ducks AND more centuries

Both are calibrated using survival analysis to have the same expected batting average.

Example normalization:

• High-variance batsman at ball 0: wicket = 3.4%, so scoring outcomes scaled to 96.6%
• High-variance batsman at ball 100: wicket = 1.0%, so scoring outcomes scaled to 99.0%
• Low-variance batsman (any ball): wicket = 1.68%, so scoring outcomes scaled to 98.3%

Match Structure

A full Test match simulation includes:

1. Four innings - Team 1 bats, Team 2 bats, Team 1 bats again, Team 2 chases

2. Weather variation - 60% good (400-450 overs), 25% moderate (320-400), 15% poor (250-320)

3. Pitch deterioration - Later innings are harder to bat:

   Innings	Wicket Multiplier	Effect
   1st	1.00x	Fresh pitch
   2nd	1.10x	Slightly worn
   3rd	1.22x	Worn
   4th	1.45x	Difficult chase

4. Declarations - Teams can declare when:

   • 1st innings: 350+ runs with 6+ wickets down, or 450+ runs
   • 3rd innings: Lead of 250+ with 120+ overs remaining

How Matches End

Outcome	Condition
Team 1 wins	Team 2 all out in 4th innings before reaching target
Team 2 wins	Team 2 reaches target in 4th innings
Draw	Time runs out (max overs reached) before result

Draw rate is calibrated to ~8-10%, matching modern Test cricket.

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Team Composition

Each team has 11 players with different base averages:

Position	Players	Expected Average	Role
Openers	2	~38 runs	Face new ball
Middle Order	4	~38 runs	Main run scorers
All-rounders	2	~25 runs	Balance bat/bowl
Tail	3	~13 runs	Primarily bowlers

Team Strategies Tested

We tested 25 different variance allocation strategies:

Uniform strategies:

• All Very Low - Everyone consistent (decay=0.0)
• All High - Everyone boom-or-bust (decay=0.85)

Opener-focused:

• Both Openers High - Aggressive openers, consistent rest
• Aggro + Anchor Open - One aggressive, one steady opener

Position-based:

• Solid Middle - Consistent middle order, aggressive ends
• Swinging Tail - Aggressive tail-enders
• Top Heavy Gradient - Variance decreases down the order

Mixed:

• Alternating Pair - Alternate high/low through lineup
• Explosive Start - Top 3 aggressive, rest consistent

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Experimental Results

Round-Robin Tournament (25 strategies, 50 matches each)

Rank	Strategy	Win Rate	Key Insight
1	All Very Low	51.7%	Consistency is optimal
2	Explosive Start	50.5%	Top 3 aggressive works
3	Mixed Middle	50.3%	Alternating in middle
...	...	...	...
21	All High	45.0%	Too much variance hurts
25	Both Openers Low	43.4%	Worst strategy

Head-to-Head Statistical Tests (300 matches each)

Matchup	Result	Z-score	Significant?
All Very Low vs Both Openers High	142-143	0.12	NO (tied)
All Very Low vs Explosive Start	161-120	4.73	YES
All Very Low vs All High	157-132	2.89	YES
Both Openers High vs All High	154-125	3.35	YES

Collapse Analysis

High variance teams experience significantly more collapses:

Team Type	Collapse Rate (<150 all out)
All Low Variance	~7%
All High Variance	~18%

The centuries from high-variance batsmen don't compensate for the collapses.

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Why Does Consistency Win?

1. Collapse Risk: High variance means more early dismissals. When multiple batsmen fail early in the same innings, you get a collapse that's very hard to recover from.

2. Test Cricket is Long: Unlike T20/ODI, there's no "required rate" pressure in most situations. Steady accumulation works.

3. Compounding Effect: One collapse in a 4-innings match can lose you the game, even if other innings went well.

4. "Getting Set" Comes Too Late: By the time a high-variance batsman becomes safe (50+ balls), a consistent batsman has scored similar runs with less risk.

But High Variance Openers Are "Free"

Interestingly, having just the two openers be high-variance (with a consistent middle/lower order) performs identically to all-consistent teams. This is because:

• Openers face the new ball anyway (inherently risky)
• If they survive, they score big; if not, consistent middle order rescues
• The consistent tail prevents collapses

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Quick Start

# Clone the repository
git clone https://github.com/rbpilgrim/cricket_variance_simulation.git
cd cricket_variance_simulation

# Run the main experiment
python team_composition_experiment.py

# Run head-to-head analysis
python analyze_matchups.py

# Or open the Jupyter notebook
jupyter notebook cricket_variance_analysis.ipynb

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Project Structure

File	Description
ball_by_ball_simulation.py	Core simulation engine - ball outcomes, innings, matches
team_composition_experiment.py	25-strategy round-robin tournament
analyze_matchups.py	Head-to-head statistical analysis
analyze_match_details.py	Detailed match statistics and collapse analysis
variance_exchange_rate.py	Quantifies variance value in runs
cricket_variance_analysis.ipynb	Self-contained Colab notebook
RESEARCH_NOTE.md	Full research writeup with methodology

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Limitations

• Simplified bowling: All bowlers treated equally (no swing, spin, pace variation)
• No player matchups: Real cricket has bowler-batsman specific interactions
• Fixed conditions: Same pitch model for all matches (no spinning tracks, green seamers)
• Basic declarations: Real captains use more nuanced decision-making
• No follow-on: The follow-on rule is not implemented

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Future Work

• Model different pitch types (spinning, seaming, flat)
• Add bowler skill variation
• Implement follow-on rule
• Model batting partnerships (not just individuals)
• Analyze specific match situations (chasing 300+ in 4th innings)

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License

MIT

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