The Carousel Drop Problem — Mock Experiment Report

Companion experiments for _posts/2026-08-19-carousel-problem.md. Model: belt occupancy is i.i.d. Bernoulli(p) per cell, so the true free run A ∼ Geom(p); the detector reveals only the censored run G = min(A, ℓ); bag width W is drawn from a known distribution and revealed only on collision. A policy is a stopping time τ adapted to the observed history, bounded by a hard deadline T. The optimal policy solves the Bellman recursion V₀(g) = c(g), V_k(g) = min{ c(g), E[V_k-1(G_t+1) | G_t=g] }. All figures below are pure black/white/gray — line style and marker shape (not hue) distinguish series; heatmap cells use grayscale shading only.

Experiment 0 — Validate the analytic model against raw-bit simulation

Parameters: ℓ=3, p=0.25, n_trials=400,000. The analytic transition kernel and cost function are derived from the memoryless (geometric) belt assumption; here they are checked against a simulator that generates actual i.i.d. bits and reads off G_t directly, with no shortcuts.

g	analytic_pmf	empirical_pmf	abs_diff
0.0	0.2500	0.2509	0.00089
1.0	0.1875	0.1872	0.00030
2.0	0.1406	0.1408	0.00013
3.0	0.4219	0.4212	0.00072

Max abs. difference 0.00089, below the ~1/√n ≈ 0.00158 Monte Carlo noise floor.

Transition kernel P(G_t+1=g' | G_t=g)

Analytic

	g'=0	g'=1	g'=2	g'=3
g=0	0.25	0.1875	0.140625	0.421875
g=1	1.0	0.0	0.0	0.0
g=2	0.0	1.0	0.0	0.0
g=3	0.0	0.0	0.25	0.75

Empirical (raw-bit MC)

	g'=0	g'=1	g'=2	g'=3
g=0	0.2509	0.1854	0.1400	0.4238
g=1	1.0000	0.0000	0.0000	0.0000
g=2	0.0000	1.0000	0.0000	0.0000
g=3	0.0000	0.0000	0.2499	0.7501

Result: PASSED — analytic kernel matches simulation within Monte Carlo noise.

Experiment 1 — Bellman-optimal policy vs. baseline policies

Parameters: ℓ=2, p=0.2, T=5, W ∼ Geometric(q=0.4, mean width 2.5), 300,000 Monte Carlo trials per policy. Baselines: drop immediately, wait for the hard deadline, the naive “drop when the detector reports a fully clear view” heuristic (the airport rule at general ℓ), and swept fixed thresholds θ.

policy	analytic_collision_prob	mc_collision_prob	mc_mean_stop_delay	n_trials
bellman_optimal	0.1470	0.1482	0.773	300,000
naive_greedy (g==ell)	0.1500	0.1489	0.779	300,000
threshold_theta=2	0.1500	0.1505	0.782	300,000
threshold_theta=1	0.2308	0.2314	0.251	300,000
threshold_theta=0	0.3846	0.3846	0.000	300,000
immediate_drop	0.3846	0.3847	0.000	300,000
wait_to_deadline	0.3846	0.3880	5.000	300,000
threshold_theta=3	0.3846	0.3840	5.000	300,000

The Bellman-optimal policy is the empirical best (confirmed by both the exact analytic evaluation and independent 300k-trial Monte Carlo), beating the naive greedy rule by 2.0% (0.1500 → 0.1470).

Experiment 2 — threshold structure of the optimal policy

Parameters: ℓ=4, p=0.2, T=10, W ∼ Geometric(q=0.35, mean 2.86). Tests the post's open conjecture: is the optimal policy of the form “drop iff G_t ≥ θ_k”, with θ_k non-increasing as the deadline approaches (k → 0)?

k_remaining	theta_k	is_threshold_form	V_k(fresh_g=0)
0	0	True	1.0000
1	1	True	0.4167
2	2	True	0.3000
3	3	True	0.2393
4	3	True	0.2078
5	3	True	0.1768
6	4	True	0.1578
7	4	True	0.1414
8	4	True	0.1279
9	4	True	0.1170
10	4	True	0.1086

Every row is exactly a threshold rule in g: True. θ_k is monotone non-decreasing in k, i.e. non-increasing as k → 0 (the policy gets less picky as the deadline nears): True. Both parts of the conjecture hold in this instance.

Experiment 3 — what one more cell of view is worth

Parameters: p=0.2, T=6, W ∼ Geometric(q=0.3, mean 3.33), ℓ swept 0–12. Tests the post's second conjecture: does V_T*(ℓ) saturate, and near what statistic of W?

ell	V_star	marginal_gain_over_ell_minus_1
0.0	0.45455	—
1.0	0.31819	0.13635
2.0	0.22700	0.09119
3.0	0.18701	0.03999
4.0	0.18100	0.00600
5.0	0.18100	-0.00000
6.0	0.18100	0.00000
7.0	0.18100	-0.00000
8.0	0.18100	0.00000
9.0	0.18100	0.00000
10.0	0.18100	0.00000
11.0	0.18100	0.00000
12.0	0.18100	0.00000

Marginal gain from one more cell of view falls below 1% of the total observed gain at ℓ* = 5 (E[W]=3.33, median(W)=2) — saturation sits a little past the mean width, consistent with the conjecture that extra view beyond the bulk of the width distribution buys almost nothing.

Experiment 4 — sensitivity to belt congestion p and deadline slack T

Parameters: ℓ=2, W ∼ Geometric(q=0.4, mean 2.5); p ∈ {0.1,0.2,0.3,0.4}, T ∈ {0,1,2,3,5,8}. Grayscale only encodes magnitude (darker = higher value); no color is used.

Optimal collision probability V*(p, T) — darker = worse

p \ T	0	1	2	3	5	8
0.1	0.2174	0.1391	0.0969	0.0856	0.0792	0.0783
0.2	0.3846	0.2615	0.2025	0.1710	0.1470	0.1396
0.3	0.5172	0.3724	0.3116	0.2629	0.2171	0.1940
0.4	0.6250	0.4750	0.4150	0.3610	0.2976	0.2532

Relative reduction vs. immediate drop (%) — darker = bigger win

p \ T	1	2	3	5	8
0.1	36.0	55.4	60.6	63.6	64.0
0.2	32.0	47.4	55.6	61.8	63.7
0.3	28.0	39.8	49.2	58.0	62.5
0.4	24.0	33.6	42.2	52.4	59.5

Waiting helps most exactly where you'd expect: low congestion (small p) with enough slack (large T) gets close to a 64% relative cut in collisions versus always dropping immediately; even at the most congested setting tested (p=0.4) the optimal policy still nearly halves the collision rate once T ≥ 3.

Summary

Exp 0: analytic Markov kernel and cost function match raw-bit Monte Carlo — model validated.
Exp 1: the Bellman-optimal policy is the best of all candidates tested, beating the naive "wait for a fully clear view" heuristic.
Exp 2: optimal policy is threshold-form in g at every k, with the threshold non-increasing as the deadline approaches — conjecture 1 confirmed in this instance.
Exp 3: the value of field of view saturates a little past E[W] — conjecture 2 confirmed in this instance.
Exp 4: gains from optimal waiting are largest under light congestion and ample slack, but remain substantial even under heavy congestion.

Generated by make_report.py from the CSVs in results/, which were produced by run_experiments.py (seed 20260829).