Bolt v. Chestnut — a Monte Carlo race

Model

Index athletes \(i \in \{B, C\}\), legs \(\ell \in \{\text{eat}, \text{run}\}\), rounds \(r = 1, \dots, R\). Leg durations are independent draws

\[ X_{i\ell r} \sim \mathcal N\!\big(\mu_{i\ell}\,\varphi_{i\ell}^{\,r-1},\; \sigma_{i\ell}^{2}\,\varphi_{i\ell}^{\,2(r-1)}\big), \quad T_{i\ell r} = \max\{\varepsilon,\, X_{i\ell r}\} \]

with \(\varepsilon = 0.05\) s. The fatigue factor \(\varphi_{i\ell}\) compounds per round and scales mean and dispersion alike: \(\varphi = 1\) is tireless, \(\varphi = 1.05\) is 5% slower every round. The variability toggle \(v\) switches the noise off — at \(v = 0\), \(T_{i\ell r} = \mu_{i\ell}\,\varphi_{i\ell}^{\,r-1}\) exactly and all trials collapse onto the deterministic mean race. One trial = one complete race = one lane.

Totals & the analytic benchmark

\[ S_i = \sum_{r=1}^{R} \big(T_{i,\text{eat},r} + T_{i,\text{run},r}\big), \qquad \Delta = S_B - S_C \]

Ignoring the clamp, sums of normals are normal,

\[ S_i \sim \mathcal N\!\Big(\textstyle\sum_{\ell} \mu_{i\ell}\, g_R(\varphi_{i\ell}),\ \sum_{\ell} \sigma_{i\ell}^{2}\, g_R(\varphi_{i\ell}^{2})\Big), \quad g_R(\varphi) = \frac{\varphi^{R} - 1}{\varphi - 1} \]

with \(g_R(1) = R\). The summary therefore quotes the closed form \(P(B \text{ wins}) = \Phi\big({-\mathbb{E}[\Delta]}/\!\sqrt{\operatorname{Var}\Delta}\big)\) as Φ, next to its Monte Carlo estimate — watching the two agree is half the point of the exercise.

Estimators in the race summary

Across the \(N\) trials the card reports each athlete's sample mean and standard deviation \(\bar S_i \pm s_i\), the mean margin \(\bar\Delta\) with its large-sample interval \(\bar\Delta \pm 1.96\, s_\Delta / \sqrt{N}\), the win tally, and the Monte Carlo win probability

\[ \hat p_B = \frac{1}{N} \sum_{n=1}^{N} \mathbf 1\big\{S_B^{(n)} < S_C^{(n)}\big\} \]

shown as MC beside the \(\Phi\)-benchmark.

Reading the picture

Each track lane is one trial with both athletes in it; 🌭 marks where the eating happens and a depleting ring shows bite progress. Below, every path unrolls its legs \(T_{i\ell r}\) as a bar alternating light (eat) and solid (run) segments on a common time axis — the sweep line is the race clock — and the totals strip places a tick at every \(S_i^{(n)}\), with μ at the mean. Superimposed on the rows, per athlete, is a kernel density estimate of the totals,

\[ \hat f_h(t) = \frac{1}{Nh} \sum_{n=1}^{N} K\!\left(\frac{t - S_i^{(n)}}{h}\right), \]

with Gaussian kernel \(K\) and Silverman's bandwidth \(h = 0.9\,\min\{s,\ \mathrm{IQR}/1.34\}\, N^{-1/5}\), sharing the time axis with the bars and the totals strip. Both curves use a common density scale, and at \(v = 0\) the estimate degenerates and is omitted.

Why Chestnut wins

At the defaults, \(\mathbb E[S_B] = 10\,(45 + 9.6) = 546\) s against \(\mathbb E[S_C] = 10\,(6 + 22) = 280\) s, while \(\operatorname{sd}(\Delta) = \sqrt{10\,(8^2 + 0.3^2 + 1.5^2 + 2.5^2)} \approx 27\) s — Bolt's deficit is a \({\approx}10\)-sigma event. Break-even requires \(\mu_{B,\text{eat}} = 18.4\) s per hotdog.

Controls

Presets load scenarios — Chestnut's default cruise, coin-flip photo finishes, one-in-five upsets, wildcard chaos, clockwork races won by a hair, and "the wall", where fatigue decides — and switch variability on; editing any number returns you to custom. The round-order toggle opens each round with the hotdog or the sprint (totals unaffected; the mid-race picture is not). Space pause · R re-run · S skip to finish · F full-screen · H help.