The rankings and sensitivities on this page describe how the thirty catalog strategies interact with bull difficulty. The Shape Reader (shown in the results tables as X188) ranks first across the bull-difficulty sweep as well, confirming the qualitative story below — bull difficulty modulates rank inside the field without inverting it. The Shape Reader’s priority list rarely aims at bull in the scoring phase, which is why its ranking moves with the field rather than against it.
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 Shift
The Headline Effect
Bull difficulty modulates the ranking without inverting it. E12 still leads at nearly every skill level under every bull setting. E1 (Early Bull) is the most bull-sensitive strategy — it loses 1.67 percentage points on average under realistic bull (0.75×) and slips from the top spot to #7–#8. Chase strategies (S11–S17) move in the opposite direction, gaining 1.2–1.4pp. Under extreme bull difficulty (0.25×), the effect amplifies: E1 loses roughly 6pp and falls to #17–#20.
E1’s plan is to close bull early (after 17 instead of last) to unlock it as a 25-point scoring avenue. When bull accuracy is reduced, this approach pays less: E1 spends extra darts trying to close a harder target while opponents close easier numbered targets and pull ahead. The effect is present but modest under realistic bull, and pronounced only when bull difficulty is pushed to an extreme.
E1 Rank Trajectory
| MPR | Full-strength Bull | Realistic Bull (0.75×) | Extreme Bull (0.25×) |
|---|---|---|---|
| 0.8 | #5 (62.2%) | #8 (61.4%) | #9 (58.7%) |
| 1.0 | #5 (61.1%) | #8 (60.0%) | #9 (57.3%) |
| 1.2 | #4 (60.2%) | #8 (59.1%) | #8 (56.0%) |
| 1.5 | #3 (59.6%) | #8 (58.3%) | #9 (54.9%) |
| 2.0 | #3 (59.2%) | #8 (57.7%) | #16 (53.8%) |
| 2.5 | #3 (59.2%) | #9 (57.6%) | #17 (53.7%) |
| 3.0 | #3 (59.5%) | #9 (57.7%) | #17 (53.5%) |
| 3.6 | #3 (59.7%) | #9 (57.8%) | #17 (53.4%) |
| 4.0 | #3 (59.9%) | #9 (58.1%) | #17 (53.4%) |
| 4.9 | #7 (59.9%) | #8 (57.9%) | #19 (51.9%) |
| 5.6 | #5 (60.1%) | #8 (57.8%) | #19 (50.3%) |
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 | −1.3 | −1.4 | −1.4 | −1.5 | −1.6 | −1.5 | −1.8 | −1.8 | −1.8 | −2.0 | −2.3 | −1.67 |
| S17 | +1.2 | +1.3 | +1.3 | +1.3 | +1.2 | +1.3 | +1.4 | +1.4 | +1.6 | +1.8 | +2.1 | +1.44 |
| S16 | +1.1 | +1.1 | +1.4 | +1.2 | +1.2 | +1.2 | +1.3 | +1.3 | +1.4 | +1.8 | +2.0 | +1.37 |
| S12 | +1.2 | +1.1 | +1.2 | +0.9 | +1.1 | +1.3 | +1.3 | +1.4 | +1.3 | +1.9 | +2.0 | +1.34 |
| S13 | +1.1 | +1.2 | +1.0 | +1.2 | +1.2 | +1.2 | +1.1 | +1.5 | +1.3 | +1.8 | +2.0 | +1.34 |
| S10 | +0.4 | +0.4 | +0.6 | +0.7 | +0.9 | +0.9 | +0.8 | +0.9 | +0.9 | +1.3 | +1.6 | +0.85 |
| S7 | −0.4 | −0.5 | −0.4 | −0.5 | −0.6 | −0.6 | −0.6 | −0.7 | −0.8 | −1.1 | −1.3 | −0.68 |
| PS | −0.3 | −0.3 | −0.2 | −0.4 | −0.2 | −0.3 | −0.2 | −0.3 | −0.5 | −0.6 | −0.6 | −0.35 |
| S2 | −0.3 | −0.2 | −0.4 | −0.2 | −0.2 | −0.3 | −0.3 | −0.3 | −0.3 | −0.4 | −0.3 | −0.27 |
Showing the biggest movers and a few reference strategies. Full table includes all 30. Values are percentage point deltas (0.75× bull minus full-strength bull). 20,000 games per matchup at each of 11 MPR levels.
Key Patterns
- Biggest gainers: S17 (+1.44), S16 (+1.37), S12 (+1.34), S13 (+1.34) — all chase strategies with high lead thresholds. When bull is harder for everyone, their defer-bull-for-late approach pays relatively better.
- Biggest loser: E1 (−1.67) — the one strategy that leans on early bull closure.
- Near-invariant: S2 (−0.27), PS (−0.35) — neither leans on bull as a scoring lane, so their rankings barely shift between bull settings.
- 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 | Full-strength Bull Top 5 | Realistic Bull (0.75×) Top 5 | Change |
|---|---|---|---|
| 0.8 | E1, E10, E12, S2, E3 | E12, E10, E2, S2, E3 | E1 exits; E2 enters |
| 1.0 | E1, E10, E12, E2, E3 | E12, E3, E10, S2, E2 | E1 exits; S2 enters |
| 1.5 | E1, E12, E10, S2, E3 | E12, E10, S2, E2, E3 | E1 exits; E2 enters |
| 2.0 | E1, E12, E10, E2, S2 | E12, E10, S2, E2, E3 | E1 exits; E3 enters |
| 3.0 | E1, E12, E10, E3, S2 | E12, E10, E2, E3, S2 | E1 exits; E2 enters |
| 4.0 | E12, E1, E10, S2, E2 | E12, S2, E3, E10, E2 | E1 exits; E3 enters |
| 5.6 | E12, S2, E2, E3, E1 | E12, E3, S2, E2, E10 | E1 exits; E10 enters |
E12 and E10 are the stable leaders at casual-through-competitive skill under both bull settings. E1 is #1 at casual/amateur skill when bull is full-strength, but is displaced entirely from the top five once bull drops to 0.75×. At pro level (MPR 4.9–5.6), E12 remains #1 and S2 settles firmly into the top three.
The change is a reshuffling, not a reversal. Under full-strength bull E1 sits at #1–#2 from casual through competitive skill; under realistic bull it slips to #7–#8. The rest of the order is largely intact.
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 |
|---|---|---|---|
| Novice / Casual (MPR 0.8–1.5) |
E1 leads narrowly | E12 leads; E10, S2, E2, E3 close behind | E1’s early bull closure punished; top five compresses to within ~1pp |
| Amateur / league (MPR 2.0–3.6) |
E1 or E12 share #1 | E12 leads; E10, S2, E2, E3 compete closely | More competitive field; no single dominant strategy |
| Competitive / Pro (MPR 4.0–5.6) |
E12 leads; E1 top-5 | E12 leads; S2/E3/E2 top-4 | E1 exits the top five; E12’s closing focus is bull-robust |
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 #1 (59.4%) at MPR 3.0. At 0.75× it falls to #8 (57.6%). At 0.5× #9 (55.7%). At 0.25× it collapses to #17 (53.3%) — a total drop of 6.1pp. There is no sharp threshold; the decline is smooth and accelerating. E1 transitions from top-tier to bottom-tier 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.
PS is Moderately Bull-Resistant
Phase Switch loses only 1.3pp from 1.0× to 0.25× bull at MPR 3.0 — comparable to E12 (−1.3) and S2 (−1.2), and less than E10 (−1.6). PS hovers around rank #7–#9 across bull levels. Its phase-switching mechanism naturally defers bull (closing it only in the endgame S2 phase), insulating it from bull difficulty more than E1 but without separating it from the top tier.
Dose-Response Summary (MPR 3.0)
| Strategy | Bull 1.0× | Bull 0.75× | Bull 0.5× | Bull 0.25× | Delta |
|---|---|---|---|---|---|
| E12 | 59.4% | 59.2% | 58.9% | 58.1% | −1.3 |
| S2 | 58.8% | 58.6% | 58.2% | 57.6% | −1.2 |
| E10 | 59.2% | 58.8% | 58.2% | 57.6% | −1.6 |
| PS | 57.5% | 57.3% | 56.8% | 56.2% | −1.3 |
| E1 | 59.4% | 57.6% | 55.7% | 53.3% | −6.1 |
| S10 | 50.0% | 50.8% | 51.9% | 53.2% | +3.2 |
| S13 | 48.6% | 49.8% | 51.6% | 54.2% | +5.6 |
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.