Methodology

Game Engine

The simulation is built around a DartsCricketGame class that enforces all standard Cricket rules:

• 7 targets: 15, 16, 17, 18, 19, 20, Bullseye
• 3 marks to close each target (single = 1 mark, double = 2, triple = 3)
• 3 darts per turn
• Alternating turns, with first-player advantage mitigated by alternating who goes first each game
• Points scored at face value (15–20, Bull = 25) after closing, only while the opponent hasn’t closed the same target
• Win condition: all 7 targets closed and score ≥ opponent’s score
• Maximum 200 turns per game to prevent infinite loops

Skill Model

Frongello’s original study tested strategies at equal skill (all players hit targets with identical accuracy) and at a 95% relative skill difference. We extend this with probabilistic skill profiles that model how darts actually land. When a strategy aims for a specific hit type (e.g., triple 20), a random draw against the player’s profile determines the actual outcome — triple, double, single, or complete miss.

What Is MPR?

MPR (Marks Per Round) is the standard metric for darts player skill, measuring marks earned per 3-dart round. In real-world darts leagues, MPR counts all marks thrown — including excess marks on already-closed targets. Our simulation measures empirical MPR the same way. Typical real-world ranges:

• Beginner: MPR ~1.0–1.5
• League player: MPR ~2.0–3.0
• Competitive: MPR ~3.0–4.0
• Elite / Pro: MPR ~4.0–5.5+

How We Build Skill Profiles

We don’t have access to raw professional throw data. Instead, we hand-calibrated three base profiles — Amateur, Good, and Pro — informed by published research on darts accuracy (Tibshirani et al., 2011; Haugh & Wang, 2022). Each base defines a probability distribution over outcomes when aiming for triples: what fraction hit the triple, downgrade to a double, downgrade to a single, or miss entirely.

The key difference between base profiles isn’t just accuracy — it’s the shape of the outcome distribution. An amateur who misses a triple is more likely to hit a single (the dart lands in the wider single segment); a pro who misses is relatively more likely to hit a double (they’re close to the target but just off).

Three Base Profiles

Outcomes shown are for aiming at triples. Each base profile also defines distributions for aiming at doubles and singles (used when strategies target those hit types). MPR values are theoretical: 3 × (T×3 + D×2 + S×1).

Scaling to 11 Skill Levels

To cover the full range of real-world ability, we generate 11 skill levels from these three bases. For each target MPR, the system:

1. Picks the closest base profile by MPR distance
2. Computes a scale factor: scale = target_MPR / base_MPR
3. Multiplies all non-miss probabilities (triple, double, single) by that factor uniformly
4. Assigns the freed probability mass to “miss”

This preserves each base profile’s characteristic outcome shape (the ratio of triple:double:single downgrades) while adjusting overall accuracy. Three levels (MPR 3.6, 4.9, 5.6) use base profiles directly; the other eight are scaled versions.

All 11 Skill Levels

The complete set of outcome distributions when aiming for triples at each skill level. Note the shape change between MPR 4.0 and 4.9 where the base profile switches from Amateur to Good:

Highlighted rows use base profiles directly. The MPR values shown on the Results page are empirical (measured from tournament games), which are slightly lower than the theoretical values here.

Outcome Distribution by Skill Level

This chart visualizes the outcome breakdown when aiming for triples at each of the 11 skill levels. At the lowest levels, misses dominate (>80%); at pro level, triples are the most common outcome. Notice the shape shift between MPR 4.0 and 4.9 where the base switches from Amateur to Good — the triple percentage jumps while singles drop.

The Shape Discontinuity

The chart reveals an important artifact of this approach. From MPR 0.8 through 4.0, all profiles are scaled from Amateur — so they share the same outcome shape where singles are always more common than triples (a missed triple is more likely to land in the wider single ring). At MPR 4.9, the base switches to Good, and at 5.6 to Pro. These profiles have a fundamentally different shape: triples are as common as or more common than singles, reflecting that skilled players miss by smaller margins.

This means the jump from MPR 4.0 to 4.9 isn’t just an accuracy increase — it’s a qualitative change in how darts miss. In reality, this transition is gradual. Our three-base system approximates it as a step function. This is a known limitation, but the alternative (fitting continuous models to professional data we don’t have access to) would introduce its own assumptions.

Empirical vs. Theoretical MPR

The MPR values in the table above are theoretical — the expected marks per round if every dart aimed at triples:

Theoretical MPR = 3 × (T×3 + D×2 + S×1)

The empirical MPR values shown on the Results page are measured from actual tournament games, where strategies don’t always aim for triples. Strategies sometimes aim for singles or doubles depending on game state, which slightly lowers the observed MPR. The empirical values are typically 1–5% lower than theoretical.

Tournament Design

• Format: Full round-robin — every strategy plays every other strategy
• 22 strategies: S1–S17 (Frongello), E1–E4 (experimental), PS (Phase Switch)
• Matchups per level: 231 (upper triangle of 22×22, mirrored for symmetry)
• Games per matchup: 20,000 (with alternating first player)
• Total games: ~50 million across all 11 MPR levels
• Symmetry: A vs B and B vs A from the same 20,000 games. If A wins 55%, B wins 45%.
• Diagonal: Always 50% (a strategy playing itself)

Statistical Considerations

Sample size: 20,000 games per matchup gives a standard error of ~0.35% for a 50% win rate (√(0.5×0.5/20000) ≈ 0.0035). A 2% difference is statistically significant at p < 0.001.

First-player advantage: Mitigated by alternating who throws first. In even-numbered games, player B goes first. This ensures neither strategy has a systematic advantage from turn order.

Maximum turns: Games are capped at 200 turns to prevent infinite loops with degenerate strategy pairs. In practice, pro-level games average ~18 turns and amateur-level games ~22 turns. Even at the lowest skill level, games average ~68–82 turns — well below the limit.

Deterministic strategies: All strategy bots are deterministic given the game state. The only randomness comes from the skill profile’s throw resolution. This means results are fully reproducible given the same random seed.

Comparison with Frongello

Limitations

• Strategy space: We tested 22 strategies but the space of possible strategies is infinite. There may be better strategies we haven’t explored.
• No learning: All strategies are fixed rules. A reinforcement learning agent might discover different principles.
• Simplified skill model: Real darts have target adjacency effects (missing 20 might hit 1 or 5). Our model treats misses as complete misses.
• No psychological factors: Real games involve pressure, fatigue, and opponent observation. Our simulation is purely mathematical.

References

1. Frongello, A. (2018). “Optimal Strategy in Darts Cricket.” UNLV Theses, Dissertations, Professional Papers, and Capstones. 3464.
2. Tibshirani, R. J., Price, A., & Taylor, J. (2011). “A statistician plays darts.” Journal of the Royal Statistical Society: Series A, 174(1), 213–226.
3. Haugh, M. B., & Wang, C. (2022). “Play Like the Pros? Solving the Game of Darts as a Dynamic Zero-Sum Game.” INFORMS Journal on Computing.