Blue-Zone Analytics: Methodology, Prior Evidence, and Design Results

1. Executive summary

The brief asks for a black-box reverse-engineering programme. That framing should be revised before any data is collected, because the box is not black.

Tencent — the developer and publisher of PUBG Mobile — holds a published patent family describing a safe-region narrowing algorithm in a document that names PUBG explicitly. Patents are public documents; reading one is neither reverse engineering nor a terms-of-service concern. The patent gives a specific, implementable candidate for the exact routine this project set out to infer.

That changes the shape of the work in three ways.

The research question sharpens. Instead of "what distribution generates the next circle", the question becomes "does the observed data match this distribution", which is a goodness-of-fit problem with an exact null.

The data requirement collapses. The computed power analysis in §7 shows that the structural questions — is the next circle contained in the previous one, are bearing and displacement coupled, are bearings confined to a half plane — are settled at n ≈ 10–30 annotated transitions. Subtler questions about radial bias need 100–200. Nobody needs thousands of matches to start.

A specific, high-value ambiguity appears. The patent uses the same symbol for the angle draw and the radius draw. Read literally, bearing and displacement are deterministically coupled and the next centre lies on a one-dimensional spiral — which would make the zone far more predictable than anyone believes. It is very probably a drafting artefact. It is also falsifiable with about ten observations, at power 1.00. Test it first.

Separately, §4.1 records an exact result that needs no data whatsoever: whenever the shrink ratio exceeds 0.5, a disc of radius 2r_{t+1} − r_t around the current centre cannot be caught outside the next zone, and below 0.5 that guarantee disappears entirely. Locating the phase where the schedule crosses 0.5 is the cheapest actionable output in this whole document.

The most important negative result is in §8: at broadcast video resolution, measurement error in the key statistic grows from ±0.012 at phase 1 to ±0.79 at phase 8. The late circles that matter most tactically are the ones that cannot be measured from VODs. Any study that pools all phases will be dominated by noise from exactly the phases it most wants to understand.

2. What the prior evidence actually says

2.1 The patent family

US 11,529,559 B2, assigned to Tencent Technology (Shenzhen), inventor Xin Sun, from Chinese priority application CN 201811369245.6 (filed 16 Nov 2018), with continuations US 12,017,144 B2 and US 12,515,133 B2. The description defines PUBG by name and describes, for a large-map shooter, the following:

1. Zone radii per phase are fixed constants, decreasing in order, and circle N is contained within circle N−1. Choosing the next zone therefore reduces entirely to choosing a centre.
2. The admissible centre region is the disc of radius R = r_prev − r_next, concentric with the current circle.
3. A random number r ∈ [0,1] is drawn. An angle of r·360 (the text adds "or r·π") is selected, and a radial offset of √r · R.
4. The candidate centre is checked against a whitelist of pre-authored map regions that exclude terrain unsuitable for combat — the text names sea, mountains and cliffs. A candidate outside the whitelist is redrawn, capped at about 100 iterations.
5. A distribution map from tens of thousands of internal trials shows: zero probability in a border region, low probability in an edge transition band, and equal probability between the map's central and peripheral regions.

Point 5 is worth dwelling on. It is a direct statement that there is no radial pull toward the map centre — a belief widespread in player communities. It also means the observable non-uniformity, if any, comes from the whitelist and the containment geometry, not from a centre-seeking term.

2.2 The three readings of step 3

Reading A is the textbook way to sample a disc uniformly and is almost certainly what is implemented. Readings B and C are cheap to eliminate and expensive to have missed.

2.3 What the patent does not establish

A patent describes an embodiment, not shipped code, and the priority date is 2018. PUBG Mobile has changed repeatedly since. Publicly documented changes to the PC title over the same period include a reduced probability of "extreme" zones far from the previous centre after the first phase, a final circle that closes to the centre rather than to a random point, and an increasing preference for land over water toward the end of a match — all of which are phase-dependent departures from the plain patent model. Whether any of them apply to the Mobile codebase, which is a separate implementation, is unknown and is precisely what the data collection is for.

Treat the patent as a sharp prior. Do not cite it as a finding.

2.4 Source quality warning

Searches on this topic return a large volume of SEO content carrying confident, mutually contradictory numbers for zone damage and timings, including figures that contradict each other within the same site. Several such pages were encountered while preparing this document. No zone timing or damage value has been entered into the phase table in §3 from those sources. The table is mostly empty on purpose.

3. Phase model

The brief asks for a phase-by-phase table and instructs that values must not be invented. Accordingly:

The only entries that can be filled without measurement are the maximum centre displacements, which follow from the containment constraint and are exact if containment holds — itself the first thing to test.

Radii are constants per phase and per map, so this table is cheap to complete properly: RadiusSchedule.from_observations() recovers the whole schedule from a handful of annotated matches, and the result is reusable indefinitely. Timings and damage are best measured directly in custom rooms rather than scraped.

Everything downstream — every probability the dashboard reports — is a function of this table. Until it is filled with measured values, the placeholder shrink ratio propagates into every number the system produces. This is the single highest-leverage piece of data collection in the project.

4. Coordinate system

Normalised map frame: x, y ∈ [0,1] with the origin at the north-west corner, r as a fraction of map edge length. Maps of different physical sizes then share a frame, which is what makes cross-map comparison meaningful.

Each transition reduces to two scalars:

R  = r_t − r_{t+1}                  admissible displacement bound
d  = |c_{t+1} − c_t|                observed displacement
s  = (d / R)²                       normalised squared displacement
θ  = atan2(Δy, Δx) mod 2π           bearing

The squaring is what makes s uniform under Reading A: it is the area-measure transform for a uniform disc. Comparing raw displacement d across phases — which is what most community analyses do — mixes the radius schedule into the statistic and destroys the comparison.

4.1 An exact consequence: the guaranteed-safe disc

One useful result falls out of the containment constraint alone, with no distributional assumption and no data at all.

The next centre can move at most R = r_t − r_{t+1} from the current one. So a point at distance ρ from the current centre is inside the next circle for every admissible draw when

ρ + (r_t − r_{t+1}) ≤ r_{t+1}      ⟺      ρ ≤ 2·r_{t+1} − r_t

Two things follow.

Above a shrink ratio of 0.5, there is a disc of ground that cannot be caught outside. Its radius is 2r_{t+1} − r_t, and the current circle's centre is always inside it. Dead centre is not merely a good bet at those ratios; it is risk-free with respect to the zone.

Below 0.5 the guarantee vanishes completely. The circle can shrink past its own centre, and no ground is safe. The phase at which the schedule crosses 0.5 is therefore the phase at which centre play stops being free — and locating that crossover is a pure measurement of the radius schedule, needing no zone observations at all beyond the radii themselves.

Because total centre drift telescopes, the same formula extends to a k-phase horizon by substituting r_{t+k}. The disc shrinks quickly with k, which is the exact sense in which a long-horizon guarantee is unavailable.

This is implemented as geometry.guaranteed_safe_radius and verified in the test suite by checking that no point of the disc ever falls outside 20,000 sampled circles. It is drawn as the dotted inner ring on the interactive chart.

5. Hypothesis set

The brief's hypotheses A–H map onto this frame as follows.

Hypothesis D deserves a note on how it is tested, because the usual version of this test is wrong. Asking "do zone centres land on land more often than water?" compares against the wrong baseline — the admissible disc may barely contain any water in the first place. terrain_association_test compares the observed rate against the disc-restricted expected rate, transition by transition, using a Poisson-binomial null. That is the difference between measuring the algorithm and measuring the map.

5.1 A confound found by running the battery against itself

Running the pipeline on synthetic data generated from a known rule caught a false-positive mode that would have produced a spectacular wrong answer on real data.

The coupling test asks whether the residual φ = θ − span·s concentrates. Under genuine shared-draw coupling φ ≡ 0 and the circular resultant is 1. But any non-uniform bearing marginal also concentrates φ: with bearings confined to a half plane and s drawn independently, θ − π·s is triangular and peaked at zero, and Rayleigh reports it as highly significant at a resultant near 0.33.

So a significant coupling p-value is not evidence of coupling. The resultant has to be near 1. verdict() requires ≥ 0.70 for that reason, and the half-plane case now falls through to the bearing tests where it belongs.

The general lesson applies beyond this one test: several tests in the battery share the same inputs, and a departure in one marginal leaks into tests aimed at something else. Read the whole battery before drawing a conclusion, never one line of it.

5.2 A second one: testing several things at once

Rendering the simulator during packaging surfaced a related problem. For data generated by the uniform-disc rule, the verdict reported "departs from uniform" in about one batch in ten.

No single test was at fault. Each marginal test is calibrated on its own — on uniform data the radial KS test fired 5.4% of the time and Kuiper 5.0%, against a nominal 5%. But the verdict flagged a departure when any of them fired, and the chance that at least one of several independent 5% tests fires is well above 5%. Measured on uniform data:

The verdict now judges each of the m marginal tests at 0.05 / m, which caps its false-alarm rate at 5% whatever the dependence between the tests. The cost in power is small: at 25 matches the simulator still identifies every non-uniform rule 100% of the time, and the Python verdict still detects a mild pull toward the previous centre (Beta(0.7, 1)) in 88.7% of 175-transition batches.

The power table in §7 is unaffected — it describes each test on its own at 0.05. A conclusion drawn from the verdict needs slightly more data than that table implies, because the verdict is deliberately stricter than any one test.

6. Whitelist cartography

The whitelist is the one component of the model with genuine reconstruction value, because it is map-specific authored content rather than a formula.

Its signature is regions where centres never land despite the geometry permitting it. With enough observed centres and their admissible discs, the whitelist can be estimated: for each map cell, compare the number of times a centre landed there against the number of times a centre could have landed there. Cells with high opportunity and zero landings are excluded terrain.

This is a coverage-limited estimate — cells that rarely fall inside any admissible disc will stay unknown for a long time — and the estimate should carry per-cell confidence rather than being rendered as a hard boundary. The rejection cap noted in §2.1 also means the exclusion is not absolute: when the admissible disc barely intersects the whitelist, a small amount of probability leaks into excluded terrain. That leak is itself a testable signature.

Not implemented yet. It needs data first, and its structure depends on what §7's tests find.

7. Computed result: how much data is needed

Simulated power of the battery at α = 0.05, 400 trials per cell. Data are generated from each scenario; the figure is the fraction of trials where the test correctly rejects.

The false-positive row confirms the battery is calibrated: 0.03–0.05 against a nominal 0.05.

Reading this as an annotation budget:

• 10 transitions settle containment, coupling, and the half-plane reading. That is roughly two matches. Do this before anything else.
• 30 transitions detect a strong radial bias at 94% power.
• 100–200 transitions are needed for a mild bias, and this is the regime any realistic departure from the patent model probably lives in.
• Beyond 400, the marginal return on more of the same measurement is small. Effort is better spent on phase-stratified sampling and on cross-map coverage.

Because a mild bias needs ~200 transitions and each match yields at most 8, a per-map, per-phase analysis needs on the order of 25–30 matches per map for the pooled tests, and considerably more to stratify by phase. Plan for that shape, not for "as many as possible".

8. Computed result: the measurement precision ceiling

The RANSAC + algebraic circle fitter was characterised against rendered circles with known ground truth, 150 trials per condition. Errors are fractions of map edge length.

On an 8×8 km map, 0.0011 map units is about 9 metres. The fitter holds up well on clipped arcs down to 90°, which matters because minimap views rarely show a complete circle.

Now propagate that into s. Since s = (d/R)² divides by a difference of two fitted radii, the error is amplified as the circles get small:

(Using the placeholder radius schedule; the qualitative pattern is schedule-independent, since R shrinks geometrically whatever the ratios are.)

s lives on [0,1]. By phase 7 the measurement uncertainty covers nearly half that range, and by phase 8 it covers the whole of it. Phases 1–4 carry essentially all of the usable signal available from broadcast footage.

Three consequences:

1. Do not pool phases. A pooled test is dominated by noise contributed by the phases with the least information.
2. Community claims about late-circle patterns are, at the precision anyone outside the studio actually has, unfalsifiable. Treat them accordingly.
3. If late phases must be studied, the measurement method has to change: full-screen map captures at native resolution, custom rooms where the observer controls the camera, or frame-averaging across many frames of a static circle. Not minimap fits from a 1080p stream.

9. What the system will and will not tell an analyst

The forecast engine produces two distinct fields, and conflating them is the most common error in this genre:

• Centre density — where the next circle's centre will be. This is what every community heatmap shows.
• Survival field — the probability that a team standing at a point is still inside the zone. This is what positioning decisions actually depend on.

They differ substantially. A point well off-centre can have high survival probability because it falls inside many sampled circles, while a point at the mode of the centre density gains little if the circle is large. The strategy layer scores positions on the survival field, penalised by rotation cost from a Dijkstra pass over a traversability grid, and by a contention term that reflects the obvious fact that every other team is looking at the same map.

Multi-phase lookahead compounds uncertainty at the rate the game does. The three-step field is close to flat, which is the correct answer, and the system reports it as such rather than manufacturing a confident-looking mode.

The honest form of an output is: "Region A holds an estimated 34% of the next-centre probability mass and covers 9% of the map; region B holds 21% over 6%." An estimate that spreads 90% of the mass over 60% of the map is not a prediction, and highest_density_regions reports region area alongside every probability so that this cannot be hidden.

10. Player distribution

The brief asks whether player positions influence future zones. Two things must be separated.

Zone centre selection. The patent's routine takes no player-position input. If the shipped implementation matches, player distribution cannot influence where the circle goes, and any apparent correlation is either coincidence, selection effect, or reverse causation — players rotate toward where they expect the zone, so surviving players cluster near zone centres by their own choice. This confound is severe and is the reason essentially every community claim on this topic is uninterpretable.

Zone timing and damage. Separately, the PC title has documented "dynamic blue zone" behaviour keyed to survivor count on some maps. That is a real player-count dependency, but it acts on timing and damage, not on location.

Conflating the two produces the widespread belief that "the circle follows players". Test them as separate claims with separate data.

The only clean test of the location claim available without studio access is a custom room with players deliberately clustered in a known region, repeated enough times to compare the centre distribution against the disc-uniform null. That is feasible and cheap, and it is the recommended way to settle this.

11. Validation plan

Split by match, never by transition — transitions within a match share a first circle and a radius schedule, so a random split leaks. schema.match_level_split enforces this.

Because the model is a probability distribution rather than a point predictor, evaluate it as one:

• Log score of the observed centre under the predicted density, against the disc-uniform baseline. Positive mean improvement is the only meaningful sense in which the model "beats chance".
• PIT calibration on s: transform each observed value through the predicted CDF and test the result for uniformity. A model can rank well and still be badly calibrated, and calibration is what makes a stated 34% mean 34%.
• Coverage of the stated 50/75/90/95% regions on held-out matches, reported with region area.
• Cross-map and cross-tournament held-out sets, since whitelist geometry is map-specific by construction and patch-specific in practice.

A model that improves the log score on training matches but not on unseen ones has fitted the annotation, not the algorithm.

12. Limitations

• No match data has been collected. Nothing here is a measurement of live behaviour.
• The radius schedule is a placeholder and propagates into every probability.
• The whitelist is unknown; the coast-band mask is a crude stand-in, labelled as such in code.
• Late-phase measurement is precision-limited as quantified in §8.
• The patent may not describe shipped code, and may be out of date.
• Esports rooms may use different zone parameters from public matches; this is untested and would invalidate pooling the two.
• The contention term in the strategy layer is a heuristic proxy, not a game-theoretic equilibrium.

13. Recommended sequence

1. Fill the radius schedule from ~5 annotated matches per map. Everything else depends on it.
2. Annotate 10 transitions. Run containment, coupling, and Kuiper. This is a day's work and eliminates or confirms three hypotheses outright.
3. Characterise your own annotation error on your own footage before trusting any tolerance.
4. Extend to ~200 transitions on one map, phases 1–4 only, and test for radial and phase structure.
5. Only then begin whitelist estimation, and only on the map with the best coverage.
6. Custom-room experiments for the player-distribution question and for late phases, where VOD precision runs out.

14. Ethics and scope

Everything in this project uses published patent documents, publicly broadcast tournament footage, and observations made by playing the game normally. No component touches game memory, network traffic, the client binary, or anti-cheat systems, and none should be added. If a tournament organiser restricts analytical use of their VODs, that restriction governs regardless of what is technically possible.

Sources

• US 11,529,559 B2, "Method and device for adjusting region, storage medium, and electronic device", Tencent Technology (Shenzhen), inventor Xin Sun; priority CN 201811369245.6, 16 Nov 2018. Continuations US 12,017,144 B2 and US 12,515,133 B2.
• PUBG Corp. patch notes describing blue-zone changes to the PC title (reduced probability of extreme safe zones after the first, final circle closing to the centre, increased land preference late in a match).
• PUBG community wiki, on dynamic blue-zone behaviour keyed to survivor count.

Power and precision figures in §7 and §8 were computed by bluezone.power and bluezone.cv.synthetic_error_report in this repository and are reproducible with python -m bluezone.power.