The Traveling AP Problem — break-even distance & optimal trip length

Correction: an earlier version of this page said no published formal treatment of this problem existed. That was wrong — Chase McCloskey's The Traveling AP Problem: Optimal Scheduling & Travel Decisions in Casino Gambling formalizes the same decision as a renewal-reward process nested inside a fixed-cost travel choice, and extends it with a stochastic casino-heat detection layer and a full multi-trip dynamic program well beyond what's modeled here. The deterministic base layer of that paper matches the structure below: break-even distance by bisection, optimal trip length by single-peaked scan. Adjust the inputs on the left; the two decision outputs update live. Model detail is in the panel at the bottom.

Game Economics

Game A — Local, hourly EV ($)

Game B — Travel, hourly EV ($)

Hours/day playable — Local

Hours/day playable — Travel destination

Travel & Trip Parameters

One-way distance (miles)

Planned trip length (days)

Driving speed (mph)

Fuel + wear cost ($/mile)

Hotel ($/night)

Meals ($/day)

Ground transport at destination ($/day)

Comps offset ($/day)

Risk & Fatigue

Mean playable days before backoff

Playable casinos at destination

Fatigue onset threshold (hrs/session)

Fatigue decay rate (k, per hr past threshold)

Net advantage at current inputs

Trip profit minus what Game A would have paid over the same days, net of all travel costs.

Optimal trip length @ this distance

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Break-even max distance @ this trip length

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Effective hourly EV, Game B (fatigue-adjusted)

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Averaged across the daily session length, given the fatigue decay curve.

Net advantage vs. trip length (at current distance)

Net advantage vs. distance (at current trip length)

Formal model

Decision variables D = one-way distance to Game B (miles) N = trip length (days) Per-day survival (backoff hazard) Let m = mean playable days before backoff (extended by rotating across c playable casinos at the destination, m_eff = m · c). Hazard h = 1 / m_eff S(d) = (1 − h)^(d−1) probability day d is still reached Fatigue-adjusted hourly EV Session length H, fatigue threshold τ, decay rate k. If H ≤ τ: EV_eff = EV_travel If H > τ: EV_eff = EV_travel · [ τ + (H−τ)·avg_decay ] / H avg_decay = (1 − e^(−k(H−τ))) / (k(H−τ)) (hourly EV holds until the fatigue threshold, then decays exponentially for the remaining hours of the session; avg_decay is the mean of that exponential tail) Travel cost & time (round trip) Driving: Cost(D) = 2·D·cost_per_mile Time(D) = 2·D / speed Flying: Cost(D) = 2·(base + D·$/mile) Time(D) = 2·(overhead + D/cruise) Net advantage of the trip vs. grinding Game A for N days NetAdv(D,N) = Σ[d=1..N] S(d) · EV_eff · H_travel gaming profit, backoff-discounted − EV_local · H_local · N forgone Game A profit, same N days − Cost(D) fuel/airfare − EV_local · Time(D) opportunity cost of travel hours − max(0, N−1) · (hotel − comps) lodging net of comps − N · (meals + ground) daily living cost at destination Two outputs solved from NetAdv(D,N) 1. Break-even distance D*: NetAdv(D,N) is strictly decreasing in D (only the cost terms depend on D), so D* is found by bisection on NetAdv(D*,N) = 0. 2. Optimal trip length N*: argmax over N of NetAdv(D,N), found by direct scan (1–60 days). The curve is generally single-peaked because each added day's marginal payoff — S(d)·EV_eff·H_travel − EV_local·H_local − (hotel−comps) − meals − ground — shrinks geometrically with backoff survival while its marginal cost stays roughly flat, guaranteeing the marginal contribution eventually turns negative. What this leaves out Bankroll/Kelly constraints and ruin risk, correlation between backoff probability and bet spread, non-linear comp accrual, and the fact that EV and variance both scale with bet size — this model treats EV_travel and EV_local as given hourly rates rather than deriving them from a Kelly-sized bankroll. Those would be the natural next layer.