How Game Theory Solves the Hardest Problem in Delivery Logistics

Every delivery company faces the same problem: a driver picks up from 12 stops on a single route. The route costs $180 in fuel, labor, and vehicle time. How do you allocate that cost across the 12 customers?

The naive answer — split it evenly at $15 each — is wrong. The customer 2 miles from your facility shouldn’t subsidize the customer 25 miles away. But the customer 25 miles away shouldn’t bear the full marginal cost of the detour either, because the driver was already heading in that direction.

This is a cooperative game theory problem, and the solutions are surprisingly elegant.

The Shapley value: fair allocation from first principles

The Shapley value, developed by Lloyd Shapley in 1953, answers a deceptively simple question: given a group of players cooperating to create value (or share costs), what is each player’s fair share?

The key insight is marginal contribution. For each possible ordering of customers on a route, you calculate what each customer adds to the total cost when they join. Average across all possible orderings, and you get the Shapley value — each customer’s fair allocation.

For delivery logistics, this means:

A customer close to the depot who’s always “on the way” pays less, because their marginal contribution to route cost is low regardless of ordering.
A customer in an isolated area pays more, because adding them to any route consistently increases total distance.
Customers clustered together share the cost of reaching their area, because once one is served, the others add minimal marginal cost.

The Shapley value satisfies four properties that make it uniquely suited to cost allocation:

Efficiency — the allocations sum to the total cost. No money left on the table, no overcharging.
Symmetry — customers with identical cost impacts pay the same amount.
Null player — a customer who adds zero marginal cost to every route pays nothing.
Additivity — if costs have multiple components (fuel, labor, time), you can allocate each independently and sum the results.

No other allocation method satisfies all four simultaneously.

The problem with pure Shapley: not all coalitions are feasible

In theory, the Shapley value considers every possible coalition of customers. In practice, not every group of customers can be efficiently served together. Geographic constraints, time windows, vehicle capacity, and road networks all limit which customers can realistically share a route.

A customer in Bergen County and a customer in Queens might both be in your service area, but serving them on the same route is operationally absurd. The Shapley value, applied naively, would still consider coalitions containing both — distorting the allocation.

This is where Roger Myerson’s extension becomes essential.

The Myerson value: network-constrained cooperation

The Myerson value (1977) modifies the Shapley value for graph-restricted games. Instead of assuming any subset of customers can cooperate, you define a network — a graph where edges represent feasible pairings. Only connected subgroups within this network form valid coalitions.

In delivery logistics, the network encodes operational reality:

Two customers within the same delivery zone share an edge — they can feasibly be on the same route.
Customers separated by a river, a toll bridge, or 40 miles of highway have no edge — they’ll never share a route efficiently.
The network can also encode time compatibility. A customer who needs morning pickup and one who needs evening delivery may be geographically close but operationally disconnected.

The Myerson value computes fair allocations using only the coalitions that the network permits. The result: customers in dense, well-connected zones pay less per stop (because they enable efficient routing), while isolated customers see their true cost reflected.

What makes the Myerson value powerful for logistics is that the network isn’t hypothetical — it’s directly derivable from your routing data. If your routing engine has ever paired two customers on the same route, there’s an edge. The graph builds itself from operational history.

The Owen value: hierarchical cost structures

Real delivery operations have structure beyond individual customers. Routes are organized into zones. Zones roll up into regions. Regions are assigned to depots. This hierarchy matters for cost allocation, because costs accumulate at every level.

The Owen value (1977) extends the Shapley value for games with a priori unions — pre-defined groups within the larger coalition. It computes allocation in two stages:

Between groups: how should total cost be divided among zones?
Within groups: given a zone’s share, how should it be divided among that zone’s customers?

For a delivery operation serving 7 counties across two states, the Owen value first allocates cost between the NJ and NY service regions, then between counties within each state, then between customers within each county. At each level, the allocation follows Shapley principles — but respects the operational hierarchy.

This is particularly valuable for commercial accounts with multiple locations. A hotel chain with 4 properties across 3 zones should see cost allocation that reflects the structure of the service — not a flat average that ignores the fact that 3 of their properties cluster together while one is isolated.

Combining the three: a complete allocation framework

In practice, these three methods aren’t alternatives — they’re layers.

Owen provides the hierarchical structure. Your service areas, zones, and route clusters define the coalition structure. Cost flows down from total to region to zone to route.

Myerson constrains which customers can share routes within each zone. The feasibility network eliminates impossible coalitions and ensures allocations reflect operational reality, not theoretical combinations.

Shapley provides the foundational fairness guarantee at every level. Each customer’s allocation reflects their true marginal impact on the system — not an average, not a simple distance-based prorate, but a game-theoretically optimal fair share.

The result is a cost allocation system where:

Customers in dense urban zones pay less per stop (more route sharing, lower marginal cost)
Isolated customers pay proportionally more (their routes are less shareable)
Multi-location accounts see per-location pricing that reflects geographic reality
Volume discounts emerge naturally from the math — more stops in an area reduce everyone’s marginal cost
New customers can be priced accurately before their first pickup, based on how they’d integrate into existing route structures

Why this matters for commercial laundry

Most commercial laundry services price on weight and frequency alone. A hotel in Jersey City and a hotel in rural Morris County pay the same per-pound rate, even though the logistics cost of serving them differs by 3-4x.

This creates two problems:

Urban customers subsidize rural ones — which means you’re either overcharging your easiest accounts or undercharging your hardest ones.
You can’t profitably expand — because your pricing doesn’t distinguish between a new customer that slots into an existing route and one that requires a dedicated detour.

Game-theoretic cost allocation solves both. It gives you a defensible, mathematically fair basis for location-specific pricing. And it gives you a tool for evaluating new customers before you commit — does this account make the network more efficient, or less?

The math is well-established. The data is already in your routing system. The gap is implementation — and that’s a solvable engineering problem, not a research problem.

This article reflects our approach to operations at We Deliver. If you’re a business evaluating commercial laundry partners, get in touch — we’d be happy to walk through how our pricing works.