The Bull Problem
In our initial simulations — and in Frongello's original study — the bullseye was treated identically to numbered targets. A player who hit triples 30% of the time had the same accuracy on bull as on 20. But the real dartboard tells a different story: the bullseye is geometrically much smaller than numbered segments.
We introduced a bull difficulty multiplier of 0.75×, reducing all non-miss probabilities when aiming at bull by 25%. This single change produced the most dramatic strategy shift in our entire study.
The E1 Collapse
The Single Biggest Effect
E1 (Early Bull) drops from the undisputed #1 strategy at 9 of 11 skill levels to dead last at MPR 3.6 and above. Its average win rate falls by 5.67 percentage points — no other strategy moves more than ±1.31pp.
E1’s entire value proposition is closing bull early (after 17 instead of last) to unlock it as a 25-point scoring avenue. When bull accuracy is reduced by 25%, this approach becomes a liability: E1 spends extra darts trying to close a harder target while opponents close easier numbered targets and pull ahead.
E1 Rank Trajectory
| MPR | Standard Bull Rank | Realistic Bull Rank | Drop |
|---|---|---|---|
| 0.8 | #1 | #4 | −3 |
| 1.0 | #1 | #5 | −4 |
| 1.2 | #1 | #6 | −5 |
| 1.5 | #1 | #7 | −6 |
| 2.0 | #1 | #8 | −7 |
| 2.5 | #1 | #11 | −10 |
| 3.0 | #1 | #12 | −11 |
| 3.6 | #1 | #22 (last) | −21 |
| 4.0 | #1 | #22 (last) | −21 |
| 4.9 | #7 | #22 (last) | −15 |
| 5.6 | #6 | #22 (last) | −16 |
Bull Sensitivity Per Strategy
For each strategy, the change in average win rate (realistic bull minus standard bull) at each MPR level. Sorted by magnitude of impact.
| Strategy | 0.8 | 1.0 | 1.2 | 1.5 | 2.0 | 2.5 | 3.0 | 3.6 | 4.0 | 4.9 | 5.6 | Mean |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| E1 | −3.3 | −3.6 | −4.0 | −4.5 | −5.0 | −5.5 | −6.1 | −7.1 | −7.5 | −7.4 | −8.4 | −5.67 |
| S9 | +0.5 | +0.8 | +1.0 | +1.0 | +1.2 | +1.3 | +1.5 | +1.7 | +1.7 | +1.7 | +2.0 | +1.31 |
| S8 | +0.3 | +0.4 | +0.5 | +0.6 | +0.8 | +0.9 | +1.0 | +1.1 | +1.1 | +1.5 | +1.6 | +0.89 |
| S5 | +0.5 | +0.4 | +0.4 | +0.5 | +0.7 | +0.6 | +0.8 | +1.0 | +1.0 | +0.9 | +0.9 | +0.70 |
| S1 | −0.7 | −0.5 | −0.6 | −0.6 | −0.5 | −0.5 | −0.6 | −0.5 | −0.4 | −0.8 | −0.7 | −0.58 |
| E4 | +0.2 | +0.1 | +0.1 | +0.1 | −0.1 | −0.1 | +0.1 | −0.1 | +0.0 | −0.2 | −0.2 | −0.01 |
| PS | +0.0 | +0.0 | +0.1 | +0.0 | +0.1 | −0.1 | +0.0 | +0.1 | −0.1 | −0.5 | −0.7 | −0.10 |
| E2 | +0.1 | +0.0 | +0.1 | +0.1 | +0.0 | +0.0 | −0.2 | −0.3 | −0.3 | −0.4 | −0.6 | −0.14 |
| S2 | −0.3 | −0.4 | −0.5 | −0.2 | +0.0 | −0.1 | −0.3 | +0.0 | +0.1 | −0.4 | −0.6 | −0.25 |
Showing selected strategies. Full table includes all 30. Values are percentage point deltas (realistic bull minus standard bull). 20,000 games per matchup at each of 11 MPR levels.
Key Patterns
- Biggest winners: S9 (+1.31), S8 (+0.89), S5 (+0.70) — all high-threshold strategies that defer bull closure the longest. When bull is harder for everyone, delaying bull becomes relatively more valuable.
- Bull-insensitive: E4 (−0.01), PS (−0.10), E2 (−0.14) — their win rates come from number-targeting rather than bull play.
- The effect scales with MPR. Nearly every strategy shows larger deltas at higher skill levels, because bull-closure situations occur more often and matter more.
Ranking Stability
Top-5 strategies at each MPR level, before and after bull difficulty adjustment.
| MPR | Standard Bull Top 5 | Realistic Bull Top 5 | Change |
|---|---|---|---|
| 0.8 | E1, E2, E3, E4, PS | E2, E3, E4, E1, PS | E1 drops to #4 |
| 1.0 | E1, E2, E3, PS, E4 | E2, E3, PS, E4, E1 | E1 drops to #5 |
| 1.2 | E1, E2, PS, E3, E4 | E2, PS, E3, E4, S2 | E1 exits; S2 enters |
| 2.0 | E1, E2, S2, PS, E4 | E2, S2, PS, E4, S10 | E1 exits; S10 enters |
| 3.0 | E1, S2, E2, PS, E4 | E2, S2, PS, E4, S10 | E1 exits; S10 enters |
| 4.0 | E1, PS, E2, S2, E4 | PS, S2, E2, E4, S10 | E1 exits; S10 enters |
| 5.6 | PS, S2, E2, E4, S10 | PS, S2, E2, E4, S9 | S10 exits; S9 enters |
The remaining top-5 is remarkably stable: E2, S2, PS, and E4 form a consistent top tier across all conditions. When E1 falls out, its slot is typically filled by S10 (chase-and-cover) at mid MPR or S9 (high-threshold extra-darts) at very high MPR.
Pre-bull, E1 dominated so heavily it masked the true competitive landscape. With E1’s artificial advantage removed, the post-bull rankings reveal a tighter, more competitive field where E2, S2, and PS trade the #1 spot.
Game Length Impact
Harder bull means more darts thrown to close it, extending game length uniformly across all skill levels.
| MPR | Std Turns | Real Turns | Turns +% | Std Darts | Real Darts | Darts +% |
|---|---|---|---|---|---|---|
| 0.8 | 82.3 | 84.3 | +2.4% | 245.9 | 252.1 | +2.5% |
| 1.5 | 46.9 | 48.4 | +3.2% | 139.8 | 144.3 | +3.2% |
| 2.5 | 29.5 | 30.5 | +3.4% | 87.5 | 90.7 | +3.7% |
| 3.6 | 21.6 | 22.4 | +3.7% | 63.9 | 66.4 | +3.9% |
| 4.0 | 19.9 | 20.7 | +4.0% | 59.0 | 61.3 | +3.9% |
| 5.6 | 17.5 | 17.8 | +1.7% | 51.5 | 52.7 | +2.3% |
Showing selected MPR levels. Peak effect at MPR 4.0 (+4.0% turns).
The effect peaks at MPR 4.0 and decreases at very high MPR. At mid-high skill, players reach the bull-closure phase quickly but still struggle with the 0.75× penalty. At very high skill, even the penalty is overcome relatively fast.
The Equalizing Effect
In unequal-skill matchups, harder bull consistently acts as a mild equalizer, slightly boosting the weaker player’s win rate.
| Matchup | Std P1 WR | Real P1 WR | Delta | Effect |
|---|---|---|---|---|
| 2.0 vs 1.5 (strong first) | 83.10% | 82.94% | −0.16 | Slight equalizer |
| 2.5 vs 2.0 (strong first) | 78.56% | 78.34% | −0.22 | Slight equalizer |
| 3.6 vs 3.0 (strong first) | 76.88% | 76.49% | −0.39 | Slight equalizer |
| 4.0 vs 3.6 (strong first) | 67.81% | 67.17% | −0.64 | Mild equalizer |
| 4.9 vs 4.0 (strong first) | 79.06% | 76.87% | −2.20 | Strong equalizer |
Showing strong-first matchups. Negative delta = strong player’s win rate decreases = weaker player benefits.
The equalizing effect grows with skill level. At MPR 1.5 vs 2.0, the shift is negligible. By MPR 4.0 vs 4.9, it reaches +2.15pp for the underdog — the largest equalizing effect observed.
Strategy recommendations also change in 83% of unequal matchup slots. The dominant pattern: E1 is replaced as the recommended strategy for the stronger side, typically by S6 or S2. For the weaker side, S1 is often replaced by S10 (chase-and-cover).
Strategy Regimes
The bull difficulty multiplier creates three distinct regimes in the equal-skill tournament:
| Regime | Standard Bull | Realistic Bull | What Changed |
|---|---|---|---|
| Low MPR (0.8–1.5) |
E1 dominates | E2 (Honeypot) leads | E1’s early bull closure punished; E2’s trap-and-score unaffected |
| Mid MPR (2.0–3.6) |
E1 dominates | E2/S2/PS compete closely | More competitive field; no single dominant strategy |
| High MPR (4.0–5.6) |
E1 or PS | PS solidifies dominance | PS’s pattern-based approach avoids bull-dependent decisions |
A Healthier Competitive Landscape
The post-bull landscape is more strategically interesting. Without E1’s outsized dominance, the spread between #1 and #5 narrows from ~4pp to ~2pp, creating a genuinely competitive field where multiple viable approaches exist at each skill level.
Full Sensitivity Sweep
Tournaments at all four bull multiplier levels (1.0×, 0.75×, 0.5×, 0.25×) are now complete. 20,000 games per matchup at each of 11 MPR levels. Here are the answers.
E1’s Collapse is Gradual, Not Threshold
E1 (Early Bull) drops steadily as bull gets harder. At 1.0× it ranks #2 (59.8%). At 0.75× it falls to #8 (58.2%). At 0.5× still #8 (56.2%). At 0.25× it collapses to #16 (53.9%) — a total drop of 5.9pp. There is no sharp threshold; the decline is smooth and accelerating. E1 transitions from top-tier to mid-pack across the range.
Chase Strategies Gain at Extreme Bull Difficulty
The biggest winners from harder bull are the chase strategies (S10–S17). At 0.25× bull difficulty, S11–S17 all gain +5.0 to +5.7pp compared to standard bull. S10 gains +3.2pp. The reason: when bull is nearly impossible, strategies that defer bull closure (by chasing opponent-closed targets first) avoid wasting darts on it. S9’s advantage does not compound — S9 actually loses 2.3pp at 0.25× because its high scoring threshold delays covering.
PS is Remarkably Bull-Resistant
Phase Switch loses only 0.7pp from 1.0× to 0.25× bull — the smallest decline of any competitive strategy. It remains #1 at every bull difficulty level at MPR 3.0. Its phase-switching mechanism naturally defers bull (closing it only in the endgame S2 phase), insulating it from bull difficulty changes.
Dose-Response Summary (MPR 3.0)
| Strategy | Bull 1.0× | Bull 0.75× | Bull 0.5× | Bull 0.25× | Delta |
|---|---|---|---|---|---|
| PS | 61.1% | 60.9% | 60.8% | 60.4% | −0.7 |
| E10 | 59.6% | 59.2% | 58.8% | 58.0% | −1.6 |
| S2 | 59.1% | 58.7% | 58.2% | 57.5% | −1.6 |
| E1 | 59.8% | 58.2% | 56.2% | 53.9% | −5.9 |
| S10 | 50.4% | 51.1% | 52.3% | 53.6% | +3.2 |
| S13 | 48.9% | 50.1% | 52.1% | 54.6% | +5.7 |
Selected strategies at MPR 3.0. Delta is percentage point change from 1.0× to 0.25× bull. Full data for all 30 strategies is available in the interactive results page.