A Study of Proxies for Shapley Allocations of Transport Costs

Abstract

We survey existing rules of thumb, propose novel methods, and comprehensively evaluate a number of solutions to the problem of calculating the cost to serve each location in a single-vehicle transport setting. Cost to serve analysis has applications both strategically and operationally in transportation settings. The problem is formally modeled as the traveling salesperson game (TSG), a cooperative transferable utility game in which agents correspond to locations in a traveling salesperson problem (TSP). The total cost to serve all locations in the TSP is the length of an optimal tour. An allocation divides the total cost among individual locations, thus providing the cost to serve each of them. As one of the most important normative division schemes in cooperative games, the Shapley value gives a principled and fair allocation for a broad variety of games including the TSG. We consider a number of direct and sampling-based procedures for calculating the Shapley value, and prove that approximating the Shapley value of the TSG within a constant factor is NP-hard. Treating the Shapley value as an ideal baseline allocation, we survey six proxies for it that are each relatively easy to compute. Some of these proxies are rules of thumb and some are procedures international delivery companies use(d) as cost allocation methods. We perform an experimental evaluation using synthetic Euclidean games as well as games derived from real-world tours calculated for scenarios involving fast-moving goods; where deliveries are made on a road network every day. We explore several computationally tractable allocation techniques that are good proxies for the Shapley value in problem instances of a size and complexity that is commercially relevant.

The Shapley value for a fair division of group discounts for coordinating cooling loads

“…A common assumption in cooperative game theory literature is that the characteristic function has a negligible computational cost, e.g., it can be done in constant time. However, in many real world problems, such as our discount scheme, and those considered in [33,34], computing the value of a coalition C involves solving a hard optimisation problem. In our setting, v(C) is hard to compute optimally.…”

Stratied Finite Empirical Bernstein Sampling

“…with characteristic functions given by the solution to non-trivial optimization problems). Two examples of this are the problems of pricing (i) logistics, involving solutions to the travelling salesman problem [33], and (ii) electricity networks, which requires solving optimzation problems that incorporate the power flow equations [16,34,41,42]. Focusing on electricity networks in particular: these are complicated technical system used to transport electrical power from generators to loads, subject to the non-linear physical and operational constraints of the system's components.…”

“…It is an axiomatic approach to allocating a divisible reward or cost between participants where there is a clearly defined notion of how much surplus or profit a group or "coalition" of participants could achieve by themselves [31]. It has many applications, including analyzing the power of voting blocks in weighted voting games [32], in cost and surplus division problems [33,34], and as a measure of network centrality [35]. Formally, a cooperative game, N, v ∈ G N , comprises a set of n players, N = {1, 2, .…”

Stratied Finite Empirical Bernstein Sampling

“…It is an axiomatic approach to allocating a divisible reward or cost between participants, where there is a clearly defined notion of how much reward any group (or 'coalition') of participants could achieve by themselves (Chalkiadakis, Elkind and Wooldridge, 2012). The Shapley value has many applications, including analysing the power of voting blocks in weighted voting games (Bachrach et al, 2009), in cooperative cost and surplus division problems (Aziz et al, 2016;Chapman, Mhanna and Verbič, 2017), and as a measure of network centrality (Michalak et al, 2013). Formally, a cooperative game, (N, v) ∈ G N , comprises a set of n players, N = {1, 2, .…”