Quantitative playbook · five sports

The maths of winning

Each sport below is reduced to the few equations that decide it: a logistic shot model and a Poisson scoring process for football, an impulse-response training model and an optimal-control pacing model for the Olympics, a Markov chain for tennis, an expected-value shot map and end-game decision tree for basketball, and an energy-budget pacing model with cubic drag for swimming. Every model runs live in the page, so you can move the parameters and watch the optimal strategy move with them.

Football

0.76

xG from 3 m, central

Olympics

14–21 d

optimal taper

Tennis

52 → 75%

point → match (best of 5)

Basketball

1.17

pts per corner three

Swimming

v³

power cost of speed

Association football · 11 v 11 · 105 × 68 m

Football: shoot close, score first, pass into space

Goals are rare Poisson events, so the whole game is about shifting a small rate. The best strategies raise your goals-per-shot by shooting from inside the box, defend a lead by cutting variance, kick penalties unpredictably, and move the ball into the space your team controls.

Penalty spot 11 m · goal 7.32 × 2.44 m

Logistic regression · shot quality

Expected goals: distance kills, angle wounds

xG = σ(β₀ + β₁·ln d + β₂·|θ| + β₃·header)
σ(z) = 1 / (1 + e^−z)d = distance to goal line centre (m), θ = angle off the goal's centre line, β = (3.70, −2.29, −1.03, −0.70). Calibrated so a central 11 m shot ≈ 0.14, 6 m ≈ 0.40, 20 m ≈ 0.04.

Every shot is a Bernoulli trial whose success probability depends almost entirely on where it is taken. Doubling the distance roughly cuts the odds by a factor of five (2^2.29); a 45° angle halves them again. Over a season the sum of xG predicts future goals far better than goals themselves do.

Strategy

Work the ball inside the box before shooting. Ten shots from 25 m are worth about the same as one from the penalty spot.
Cut-backs beat crosses. A pass into the central 6–12 m zone produces shots with 3–4× the xG of a header from a wide cross.
Defend by pushing shots outward. A defence that concedes 14 shots at 0.05 xG is better than one that concedes 8 at 0.12.

Shot map: xG field with a season of shots raining in

Colour = model xG at that spot · dots = sampled shots, filled when scored

xG against distance for three shooting angles

Model curve · foot shots

Poisson process · Dixon–Coles match model

Goals are Poisson arrivals, so leads are worth more than they look

X_home ~ Poisson(λ) , λ = α_h·β_a·γ
X_away ~ Poisson(μ) , μ = α_a·β_h
P(x,y) = τ_ρ(x,y)·e^−λλ^x/x!·e^−μμ^y/y!α = attack strength, β = defensive weakness, γ ≈ 1.3 home advantage, τ_ρ = Dixon–Coles correction for 0–0, 1–0, 0–1, 1–1 (ρ ≈ −0.13). In play, remaining goals are Poisson with rate λ·(90 − t)/90.

Because goal arrivals are memoryless, the probability of the current leader winning depends only on the lead and the time left. The simulation replays a match minute by minute and recomputes the three outcome probabilities after every event. Watch how a single early goal moves the favourite's chance by 25–30 points.

Strategy

Score first, then cut variance. Once ahead, a compact low block that lowers both teams' rates protects the lead; the same tactic when level mostly buys a draw.
Underdogs need variance. Weaker sides should play high-tempo, high-shot football: more goals in total means more ways the better team's edge fails to show.
Chase early. A goal down at 60' still leaves a rate of λ/3; at 85' it is λ/18. Substitutions and risk should front-load, not wait.

In-play win probability, one simulated match

Home · draw · away recomputed every minute

Pre-match scoreline probabilities

Dixon–Coles grid · darker = more likely · rows home goals, columns away goals

Zero-sum game · mixed-strategy equilibrium

Penalties: be unpredictable in exactly the right proportion

Kicker maximises min_keeper Σ_i,j p_i q_j S_ij
p* = (S_RR − S_RL) / (S_LL − S_LR − S_RL + S_RR)S_ij = scoring probability when the kicker chooses i and the keeper dives j. Empirical values (Palacios-Huerta, 1,417 kicks): natural side vs same dive 0.58, natural vs wrong dive 0.95, other side vs wrong dive 0.93, other side vs same dive 0.70.

There is no best side. Any predictable pattern is punished, so the only stable strategy is a random mix in which the keeper is indifferent between diving either way. Solving the 2 × 2 game gives the kicker's natural side about 38% of the time and the keeper diving there about 42%, for an 80% expected conversion rate. Real professionals mix within a few points of this.

Strategy

Randomise, don't alternate. Sequences like L-R-L-R are as exploitable as always-left; draw each kick independently.
Mix toward your weaker side more than feels natural. The equilibrium puts 62% of kicks on the "other" side because the keeper leans natural.
Keepers: guess by the kicker's history, not the last kick. A kicker at 70% natural is worth diving natural on every time.

Shootout under equilibrium mixing

Each kick is drawn from p* and q*; the conversion rate converges to the game value

Kicker's guaranteed scoring rate against a best-responding keeper

x = share of kicks to the natural side · peak is the minimax strategy

Spatial model · pitch control field

Pass to the space you own, not to the player you see

PC_team(x) = σ( (T_opp(x) − T_team(x)) / s )
T_team(x) = min_i∈team ‖x − r_i‖ / v_iT = time for the fastest player of each team to reach point x; s ≈ 0.45 s sets how sharply control flips. Fernández & Bornn (2018) and Spearman (2018) formulations, simplified.

Every point on the pitch is "owned" by whichever team can reach it first. Colouring the field by that probability shows where a pass will actually be received. In the animation the ball carrier always releases to the teammate standing in the highest-control zone ahead of the ball; watch how the defence's shifts open and close channels.

Strategy

Progress through control, not through bodies. The valuable pass is the one that lands where control is above 0.7 and closer to goal.
Runs create control before they receive. A sprint into a channel flips ownership of the space in front of it, even if the ball never comes.
Press to shrink the opponent's high-control area, especially around the ball carrier's forward options.

Live pitch control, attack left to right

Blue field = attacking team controls · red = defending team · ball follows the best-controlled receiver

Olympic Games · peaking, pacing, portfolio

Olympics: peak on the day, pace by physics, fund by marginal medals

An Olympic medal is won twice: once in the four-year plan and once in the race. The plan is an impulse-response problem (arrive at the start line with fitness high and fatigue gone), the race is an optimal-control problem (spend a fixed energy store over a fixed distance), and a nation's medal table is a resource-allocation problem.

400 m track · 8 lanes · 1.22 m each

Impulse-response ODE · Banister fitness-fatigue

Peaking: fatigue decays three times faster than fitness, so taper

p(t) = p₀ + k₁ Σ_s<t w(s)e^{−(t−s)/τ₁} − k₂ Σ_s<t w(s)e^{−(t−s)/τ₂}w(s) = daily training load, τ₁ ≈ 45 d (fitness), τ₂ ≈ 15 d (fatigue), k₂/k₁ ≈ 2. Performance is fitness minus fatigue; the taper trades a little fitness for a lot of fatigue relief.

Training deposits both fitness and fatigue. Fatigue is larger but decays faster, so after a heavy block performance is suppressed even though fitness is at its highest. The taper exploits the time-constant gap: cut load and fatigue drains within two weeks while fitness barely moves. The optimisation chart solves for the taper length that maximises performance on competition day.

Strategy

Taper 2–3 weeks, cut volume 40–60%, keep intensity and frequency. This matches the Bosquet meta-analysis of tapering studies.
Overreach first. A deliberate heavy block 3–4 weeks out raises the fitness ceiling the taper then reveals.
Too long a taper costs more than too short. Past 3 weeks the fitness curve falls faster than the remaining fatigue relief.

120-day build with a taper into competition day

Bars = load · lines = fitness, fatigue, modelled performance

Competition-day performance against taper length

All other parameters as set above · marker = optimum

Optimal control · Keller's running model

Pacing: get to cruise speed fast, then hold it until the tank hits zero

dv/dt = f − v/τ , 0 ≤ f ≤ F
dE/dt = σ − f·v , E(0) = E₀ , E ≥ 0f = propulsive force per unit mass, τ = 0.89 s internal resistance, F = 12.2 m/s², σ = 41.6 W/kg aerobic supply, E₀ = 2,409 J/kg anaerobic store (Keller 1973, fitted to world records).

For anything longer than about 300 m the optimal control is bang-bang then singular: full force until a cruise speed, then exactly the force that holds it, so that the energy store reaches zero at the line. Going out faster spends the store on a v·f cost that grows with speed and forces a slower finish. The race animation puts an optimally paced runner in lane 1 against a "sprint-and-hang-on" runner in lane 2 with identical physiology.

Strategy

Even effort, not even splits. The first 30–50 m are always maximal; after that the target is constant speed.
The penalty for a fast start is convex. Five percent too fast costs a little; ten percent costs a lot. Err slow rather than fast.
Sprints are the exception: under ~290 m the store cannot be exhausted, so the answer is maximal force throughout.

Optimal pacing versus a fast start

Same runner, same energy store · lane 1 optimal, lane 2 goes out hard

Velocity profiles

Speed over distance for both strategies

Concave returns · Lagrangian allocation

The medal table: spend where the next pound buys the most medals

M_s(b) = M̄_s·(1 − e^−b/c_s)
maximise Σ_s M_s(b_s) s.t. Σ_s b_s = B
optimum: dM_s/db_s = λ for every funded sportM̄_s = medals realistically on offer (events × podium depth reachable), c_s = cost scale (how expensive it is to become competitive). The optimum equalises marginal medals per unit budget; sports whose marginal return never reaches λ get nothing.

Nations do not win Olympics by being good at everything. Medal counts scale with GDP and population (Bernard & Busse), but the lever a federation actually controls is allocation. Since returns to funding are concave, the rule is simple: keep pouring money into a sport until its marginal medal rate equals that of the next-best sport. The greedy animation allocates a budget in small increments to whichever sport has the highest marginal return at that moment, which is exactly the Lagrangian solution.

Strategy

Target medal-dense, shallow-field sports. Cycling, rowing, sailing and canoeing offer many events with few genuinely competitive nations; this was the UK Sport "no compromise" model from 2004 to 2016.
Cut sports whose marginal return is below λ, however popular; funding them is buying medals at the highest price.
Host once. The host effect alone adds roughly 50% to a nation's medal count.

Greedy budget allocation across eight sports

Each step funds the sport with the highest marginal medals · bars fill as money arrives

Medal return curves, with the funded point marked

Expected medals against budget for each sport · same tangent slope λ at every funded point

Tennis · best of 3 / 5 sets · 23.77 × 8.23 m singles

Tennis: a tiny edge per point becomes a huge edge per match

Tennis scoring is a Markov chain nested three deep: points make games, games make sets, sets make matches. Each level amplifies whatever edge exists below it, which is why the player who wins 52% of points wins three-quarters of five-set matches. Strategy is about buying that 2% where it is cheapest: on serve, and on the points that matter most.

Net 0.914 m · service box 6.40 × 4.115 m

Absorbing Markov chain · point → game → set → match

The amplifier: point probability to match probability

P_game(p) = p⁴(1 + 4q + 10q²) + 20p³q³·p²/(1 − 2pq)
P_set, P_match by recursion over game and set statesp = probability the server wins a point, q = 1 − p. Sets alternate serve and go to a tiebreak at 6–6; matches are best of 3 or 5. The recursion tracks whose serve it is, so the two players can have different serve strengths.

From deuce the server wins with probability p²/(1 − 2pq): at p = 0.6 that is 0.69, at p = 0.7 it is 0.84. Each level applies the same convex map, so the match curve is a steep sigmoid around 0.5. The live simulation plays a match point by point and re-solves the chain after every point to give a running win probability.

Strategy

Every point of serve percentage is worth about 5 points of match probability near parity. Fitness and focus that keep p from sagging late are the best investment.
Longer formats favour the better player. Underdogs should want best-of-3 and tiebreaks; favourites should welcome five sets.
The chain is memoryless: streaks have no effect on the next point. Momentum is a story, not a parameter.

Point-by-point match with live win probability

Player A vs Player B · probability re-solved after every point

Amplification curve: point win rate to game, set and match win rate

Both players serve equally well; x = the better player's point win rate on every point

Point importance · Morris (1977)

Not all points are equal: spend effort where the chain is steepest

I(score) = P(win game | win this point) − P(win game | lose this point)Computed from the same game chain. The importance of a point within the match multiplies game importance by the importance of the game within the set and of the set within the match.

At 30–40 or ad-out the next point swings the game by 40–70 points of probability; at 40–0 it swings it by under 10. Since a player cannot hold peak intensity on every point, and a server cannot hit maximum-risk first serves forever, the rational allocation puts risk, effort and best patterns on the high-importance cells. Studies of elite players show they do exactly this: first-serve speed and ace rates rise on break points.

Strategy

30–30, 30–40, deuce and ad-out are the match. Use your best serve pattern and highest-percentage return there.
Coast on 40–0 and 40–15: take a risk-free look at a new pattern; the cost of losing that point is tiny.
The weaker player gains more from importance-weighting because their lower p makes the cells more unequal.

Importance of every point in a game

Rows = server's points, columns = receiver's points · darker = the game turns more on this point

Risk optimisation · serve aggressiveness

How hard to hit the first serve, and whether to hit two of them

V(a) = p_in(a)·p_win(a) + (1 − p_in(a))·V₂
p_in(a) = 0.95 − 0.55a , p_win(a) = 0.52 + 0.33a
a* = argmax V(a)a ∈ [0,1] is first-serve aggressiveness (0 = safe kick serve, 1 = flat maximum). V₂ = expected point win when a second serve is needed = p_in,2·p_win,2. The sliders set the player's serve profile.

A harder serve is more likely to miss but wins more of the points it starts, and a miss is not a lost point, only a second serve. The optimum is interior and usually more aggressive than players choose, because the downside of a miss is cushioned by V₂. The chart also plots the "two first serves" policy: aggressive on both deliveries. It wins when the safe second serve is weak (V₂ low) or the player's aggressive serve is very reliable.

Strategy

Most players under-hit the first serve. Move a toward the marked optimum; expected point gain is typically 1–3 points of p, which the amplifier turns into 5–15 points of match probability.
Two first serves is right for big servers with poor kick serves, and on high-importance points where the upside is worth the variance.
Improve V₂ before a. A better second serve lifts the whole curve and lets the first serve go bigger.

Expected point win against first-serve aggressiveness

Solid = normal second serve · dashed = two first serves · marker = optimum

Basketball · 5 v 5 · 28 × 15 m · 24 s shot clock

Basketball: maximise points per possession, then win the last 30 seconds

A game is about 100 possessions per side, so the only currency is expected points per possession. Shot selection decides most of it (rim and corner threes dominate), the four factors explain the rest, and a handful of end-game decisions turn on explicit probability trees.

Three-point arc 7.24 m (NBA), corner 6.71 m · rim 3.05 m

Expected value · shot-zone economics

Shot selection: the mid-range is a tax you pay for nothing

EPA_z = FG%_z × pts_z + FTR_z × 1.5
Points per 100 = 100 × Σ_z f_z·EPA_zEPA = expected points per attempt in zone z, FTR = free-throw trips drawn per attempt (rim attempts draw fouls), f_z = share of attempts from zone z. Zone rates are league-typical: rim 0.64, short mid 0.41, long mid 0.40, corner 3 0.39, above-break 3 0.36.

A 40% mid-range jumper is worth 0.80 points; a 36% above-the-break three is worth 1.08 and a 39% corner three 1.17. The rim is worth 1.28 before free throws and about 1.45 with them. That 0.3–0.6 gap per attempt, over 90 attempts, is a 15–25 point swing per game. The animation fires a season of shots under a chosen "shot diet" and shows the points-per-100 that diet earns.

Strategy

Rim, corner three, free throw line: everything else is a fallback. Modern offences keep long twos under 10% of attempts.
Space the floor to make the drive-and-kick geometry work: the corner three exists because the defence must collapse on the drive.
Defend the inverse: take away rim and corners, and force the shots that pay least.

Expected points per attempt by zone, with shots raining in

Colour = EPA · dots = simulated attempts, filled when made

Points per 100 attempts for four shot diets

Same zone efficiencies, different attempt mix

Pythagorean expectation · four factors

Net rating to wins: one point per 100 possessions is 2.7 wins

Win% = ORtg^k / (ORtg^k + DRtg^k) , k ≈ 14
ORtg = 100 × Pts / Poss
Poss ≈ FGA − ORB + TOV + 0.44·FTAMorey's exponent 13.91 for the NBA. Dean Oliver's four factors decompose ORtg: effective FG% (weight 0.40), turnover rate (0.25), offensive rebounding (0.20), free-throw rate (0.15).

Because possessions are the denominator on both sides, the model strips out pace and reveals the true exchange rate between efficiency and wins. Around .500 the curve is nearly linear: each point of net rating is worth roughly 2.7 wins over 82 games. The four-factor weights say where a point of rating is cheapest to find.

Strategy

Shooting quality (eFG%) is 40% of the game. Roster and scheme should buy it first.
Turnovers are the cheapest fix: each one is a possession worth ~1.1 points thrown away, and reducing them needs no talent, only discipline.
Do not chase offensive rebounds at the cost of transition defence: the 0.20 factor is small and the counter-cost is large.

Win percentage against net rating

Pythagorean curve · marker = the team set by the sliders

Decision tree · end-game probability

Up three with seconds left: foul or defend?

P(lose | defend) = p₃ · P(lose OT) + ε
P(lose | foul) = p_FT1 · p_ORB · (p₃ + p₂·P(lose OT)) + p_3FTp₃ = chance the opponent hits a tying three against your defence, p_ORB = chance they rebound a deliberately missed second free throw, p_3FT = risk you foul a shooter and give three free throws. P(lose OT) = 0.5 by default.

Defending gives the opponent one clean shot at a three worth a coin-flip overtime. Fouling forces them through a chain of unlikely events: make the first free throw, miss the second on purpose, win the rebound, then score. The chain multiplies to a few percent, so fouling is right unless your fouling is clumsy or their three-point look is very poor. The second chart shows the end-of-quarter "two-for-one": a slightly worse quick shot is worth it whenever it guarantees an extra possession.

Strategy

Foul when up three with under ~6 seconds, before the shooter starts a shooting motion. The model favours it for any realistic p₃ above ~0.2.
Two-for-one is worth it whenever the quick shot loses less than about 40% of its normal efficiency.
Down three at the end, take the three. A 33% three to tie beats a 55% two followed by a coin-flip on getting the ball back.

Probability of losing: foul vs defend, across the opponent's three-point threat

Lines from the tree above · vertical marker = the p₃ set below

Two-for-one at the end of a quarter

Expected points for the rest of the quarter against how much efficiency the quick shot gives up

Expected possession value · drive-and-kick

Why the play ends in the corner

EPV(t) = E[points | state of all ten players at t]
≈ Σ_options P(option) × value(option)Cervone, D'Amour, Bornn & Goldsberry (2016) framework, illustrated here with a simplified value: the zone EPA of the ball's location scaled by how open the ball-handler is, blended with the value of continuing the possession.

A possession has a value at every instant, before any shot is taken. The pick-and-roll animation follows one possession: the screen frees the handler, the drive pulls the help defender off the corner, and the kick-out lands the ball on the highest-value spot on the floor with nobody near it. The EPV trace under the court shows the value rising with each decision.

Strategy

The drive is a threat, not a shot. Its job is to move the help defender; the pass that follows is the payoff.
Park your best shooters in the corners where the arc is shortest and the help has the furthest to recover.
Judge plays by EPV gain, not by outcome. A good open corner three that misses was still the right decision.

Pick-and-roll into a corner three, with live EPV

Blue = offence, red = defence · trace below tracks the possession's expected points

Swimming · 50 m pool · 8 lanes · 15 m underwater limit

Swimming: fight the cube, spend the store evenly, stay under the water

Water resistance grows with the square of speed and power with its cube, so every strategy in the pool is about buying speed cheaply: streamlining, even pacing that never spends energy at high speed twice, and staying in the fast, low-drag underwater phase as long as the rules and lungs allow.

15 m marks · lanes 2.5 m · depth 2–3 m

Energy budget · critical power with cubic cost

Pacing: the cube makes even speed the cheapest way to cover the distance

P(v) = k·v³ , F_drag = k·v²
dW'/dt = −(k·v³ − CP) while k·v³ > CP
v ≤ (CP/k)^1/3 once W' = 0k ≈ 30 kg/m for an elite freestyler, CP = aerobic power the swimmer can sustain indefinitely, W' = finite anaerobic work capacity. Any pacing that spends W' faster than needed forces the swimmer down to critical speed for the rest of the race.

With a linear cost model every pacing plan that empties the store at the wall finishes in the same time. The cube breaks the tie: by Jensen's inequality, speed variation raises the average power for the same average speed, so the plan with the least variation wins. The race puts three swimmers with identical CP, W' and drag in three lanes: even pace, a fast first 100 m, and a held-back negative split.

Strategy

Even splits from the 200 m up, allowing for the faster first length that the dive provides for free.
Going out 4% too fast costs more than going out 4% too slow, and the cost grows with the cube of the excess.
Train CP and drag before W'. Raising CP or lowering k moves the whole race; W' only moves the finish.

400 m freestyle, three pacing plans, same physiology

Bars beside each lane show remaining W' · lane 1 even, lane 2 fast start, lane 3 negative split

Speed by 50 m split

The fast starter is forced to critical speed once W' is spent

Exponential decay · breakout optimisation

Underwater: surface at the moment the kick stops beating the stroke

v_uw(t) = v_kick + (v_push − v_kick)·e^−t/τ
t₂₅(d) = t_uw(d) + t_break + (25 − d)/v_surf + c·t_uw²v_push ≈ 3.0 m/s off the wall, decaying with τ ≈ 1.6 s toward the sustainable dolphin-kick speed v_kick; v_surf = surface stroke speed; t_break ≈ 0.3 s lost in the breakout; c·t² is the oxygen-debt penalty paid later in the race. d ≤ 15 m by rule.

Off the wall a swimmer is briefly faster than they can ever be on the surface, and a strong dolphin kick keeps them above surface speed for several metres. The optimal breakout is the distance where the decaying underwater speed crosses the surface speed, pulled back slightly by the oxygen cost. For an elite kicker that lands at or beyond the 15 m mark, which is why the rule exists.

Strategy

Kick to 15 m if your v_kick exceeds your surface speed; otherwise break out where the curves cross, not earlier.
The push-off matters more than the kick: v_push sets the whole decay curve, and it is free speed from the wall.
In a 400 m race, shorten the underwaters: the c·t² term grows with every turn.

Underwater phase, side view

The swimmer surfaces at the model's optimal distance · 15 m rule marked

Time to 25 m against breakout distance

Marker = optimum · shaded region is illegal beyond 15 m

Hydrodynamics · stroke mechanics

Drag, power and the stroke-rate trade-off

P = k·v³ ⇒ Δv/v ≈ ⅓·ΔP/P − ⅓·Δk/k
v = SR × SL , SL(SR) = SL₀ − c·SR
SR* = SL₀ / (2c)A 10% drag reduction is worth about 3.4% speed at the same power; 10% more power buys only 3.2%. Stroke length shortens as stroke rate rises, so speed is a parabola in stroke rate with an interior optimum.

The cube is the whole story of swimming technique: drag reductions are worth three times their size in speed, and power increases are worth a third of theirs. That is why body position, streamlining and the underwater phase dominate coaching, and why the fastest swimmers are rarely the strongest. The second chart shows why simply stroking faster stops working: each extra stroke per minute shortens the stroke until the product peaks.

Strategy

Buy drag reduction first: head position, hip height, tight streamlines off every wall.
Find your stroke-rate optimum and race at it; the sprint rate is 10–20% above the distance rate, not double.
Power training pays only through the cube root; use it to hold stroke length at the optimum rate, not to raise rate.

Power required against speed for two drag coefficients

Horizontal line = the swimmer's sustainable power

Speed against stroke rate

v = SR × SL with stroke length falling as rate rises · marker = optimum rate

Models and sources

Dixon & Coles (1997), Modelling association football scores; Spearman (2018), Beyond expected goals; Fernández & Bornn (2018), Wide open spaces; Palacios-Huerta (2003), Professionals play minimax.
Banister et al. (1975) impulse-response model; Bosquet et al. (2007) taper meta-analysis; Keller (1973), A theory of competitive running; Bernard & Busse (2004), Who wins the Olympic Games.
Morris (1977), The most important points in tennis; Klaassen & Magnus (2001) on point independence; Walker & Wooders (2001) on serve mixing.
Oliver (2004), Basketball on Paper; Morey's Pythagorean exponent; Cervone, D'Amour, Bornn & Goldsberry (2016), A multiresolution stochastic process model for predicting basketball possession outcomes.
Toussaint & Beek (1992) on active drag; Monod & Scherrer critical power; Skiba et al. (2012) W' balance; Veiga & Roig (2016) on underwater phases.

Zone efficiencies, physiological constants and penalty statistics are representative values from the literature above, not a specific team or athlete. Every number on the page is recomputed from the parameters you set.