Approximation of the Shapley value for the Euclidean travelling salesman game

Abstract

The travelling salesman problem (TSP) consists of finding a minimal route to serve a set customers using one vehicle. This naturally leads to the problem of finding a fair way to subdivide the overall cost of a trip between all participating customers. The Shapley value associated with the Euclidean travelling salesman game (TSG) has been proven to provide a solution to the fair cost allocation problem that satisfies several very important axiomatic properties. Unfortunately the calculation of the Shapley value involves high computational complexity, which makes it impractical for many real applications. This has led to substantial research effort dedicated to finding approximations with lower computational complexity. We develop a novel methodology of approximating the Shapley value of the Euclidean TSG, which is inspired by an extension of the 1D case. From this we derive two approximation methods having polynomial computational complexity, and also indicate how they could, in principle, be further refined. We provide experimental results which show that our proposed approximations compare favorably with the state-of-the-art approximations of the Euclidean TSG found in the literature.

Appendix

In this “Appendix” we take a closer look at the errors introduced by Appro-O(1). Since the Shapley values scale up linearly with the distances, it is normal to ask what is the error as a relative fraction of the true Shapley value. With reference to Fig. 5 and Eq. (20) let us assume we use Appro-O(1) for the unfavourable case when $s_{23} <$ min{$s_{12},s_{13}$}. To fix notations let us assume that we have:

$$\begin{aligned} s_{23} < s_{13} \le s_{12} \end{aligned}$$

(33)

or equivalenty by (12):

$$\begin{aligned} d_{03} + d_{12} \le d_{02} + d_{13} < d_{01} + d_{23} \end{aligned}$$

(34)

The relative error on the value $\varPhi _{1}^{(1)}$ by using Appro-O(1) according to (21), instead of the correct value $\varPhi _{1}$ given by (20) is:

$$\begin{aligned} \epsilon = \frac{\varPhi _{1}^{(1)} - \varPhi _{1} }{\varPhi _{1}} = \frac{\frac{1}{3}(s_{13} - s_{23})}{ s_{11} - \frac{1}{2}(s_{12} + s_{13}) + \frac{1}{3}s_{23}} \end{aligned}$$

(35)

We have that $\epsilon \ge 0$, since both the numerator and the denominator in (35) are positive, by the assumptions of (33) and the fact that all Shapley values are positive, as averages of positive marginal costs. This means that the error will lead to an overestimation of the true value. We next show that as an upper bound for this relative error we have:

$$\begin{aligned} \epsilon \le \frac{1}{3} \end{aligned}$$

(36)

To prove the inequality of (36) we can substitute the $s_{ij}$ values according to (12) to get the equivalent inequality:

$$\begin{aligned} d_{03} + 8 d_{23} \le 9 d_{13} + 3 d_{12} + 5 d_{02} \end{aligned}$$

(37)

We can rewrite (37) as:

$$\begin{aligned} (d_{03} + d_{12} - d_{13} - d_{02}) + 8 d_{23} \le 8 d_{13} + 4 d_{12} + 4 d_{02} \end{aligned}$$

(38)

and notice that the expression in the brackets is negative according to (34). It is therefore sufficient to prove that

$$\begin{aligned} 2 d_{13} + d_{12} + d_{02} \ge 2 d_{23} \end{aligned}$$

(39)

We prove (39) by analysing separately the two complementary cases (a):$d_{02} \ge d_{12}$ and (b):$d_{02} < d_{12}$. For case (a) we have that $d_{02} + d_{12} + 2 d_{13} \ge d_{12} + d_{12} + 2 d_{13} = 2(d_{12} + d_{13}) \ge 2 d_{23}$, with the last inequality following from the triangle inequality. For the case (b) we have that from $d_{12} > d_{02}$ and $d_{02} + d_{13} \le d_{03} + d_{12}$ according to (34) it follows that $d_{12} + d_{13} > d_{03} + d_{12}$ hence $d_{13} > d_{03}$. From that we have that $2 d_{13} + d_{12} + d_{02} > 2 d_{03} + d_{02} + d_{02}$ = $2(d_{03} + d_{02})$$\ge$$2 d_{23}$ with the last inequality again following from the triangle inequality. We conclude that (39) holds in both situations (a) and (b) and therefore (37) are (36) always true.

The error limit of (36) is tight, however situations where this limit can be approached are statistically rare. A typical example leading to this high relative error is shown in Fig. 9.

Fig. 9

Diagram for highlighting case for extreme relative error on the approximation of the Shapley value at location A1. The distance from the depot D to A1 is 1, while the distances from D to both A2 and A3 are a, with a very large

Here A1 is at distance 1 from the depot D, while A2 and A3 are both at distance a from D, with a having a large value. Simple calculations show that in this case at location A1 we have the true Shapley value and its approximation using Appro-O(1) respectively as:

$$\begin{aligned} \varPhi _{1}= & {} 1 + \frac{1}{a + \sqrt{1 + a^2}} \nonumber \\ \varPhi _{1}^{(1)}= & {} \frac{4}{3} + \frac{2}{3(a + \sqrt{1 + a^2})} \end{aligned}$$

(40)

while for locations A2 and A3 both the true Shapley values and their values given by Appro-O(1) are:

$$\begin{aligned} \varPhi _{2} = \varPhi _{2}^{(1)} = \varPhi _{3} = \varPhi _{3}^{(1)} = 2a - \frac{a}{1+a+\sqrt{1+a^2}} \end{aligned}$$

(41)

The relative error $\epsilon$ of (35) is:

$$\begin{aligned} \epsilon = \frac{1}{3} \cdot \frac{\sqrt{1+a^2}+a-1}{\sqrt{1+a^2}+a+1} \end{aligned}$$

(42)

and we have that $\epsilon \rightarrow \frac{1}{3}$ as $a \rightarrow \infty$. Therefore the limit of (35) is indeed tight, however some additional comments are needed. To come close to this extreme relative error, the value of a has to be very large. Say locations A2 and A3 are at 1000 km from D, with A1 just 1 km from D, and suppose 1 dollar would be the price charged per km. Then, by using the approximation of Appro-O(1) the customer at location A1 would be charged about 33 cents more over the “fair” value of just over 1 dollar. While this 33% overcharge may look large in relative terms, it is to be noted that it occurs in the context of a pricing scheme where the other prices are around 2000 dollars each. Also to be noted is that even with this extreme relative error, the customer at location A1 still benefits from the sharing scheme, by paying 1.33 dollars instead of the 2 dollars that would be charged if location A1 were to be served independently.

Fig. 10

Diagram of relative error to the true Shapley value introduced by Appro-O(1) for the worst case scenario on 3 locations. The distribution is obtained for measurements of the worst case scenario over 100,000 randomly generated 2D networks, with uniform distribution of the locations into a square of fixed size

The situation of having a 33% relative error is very unlikely, and it is more insightful to assess the expected value of this error. To that end, we have run an experiment where we have generated 100,000 random 2D configurations, consisting of a depot D and three locations A1, A2, A3 to be served. All coordinates have been generated with a random uniform distribution, within a square of fixed size. For each generated network, the indices of A1, A2, A3 have been permuted such as to ensure that we have $s_{23} \le$ min{$s_{12}, s_{13}$}. This would lead to the least favourable case when $\varPhi _{1}^{(1)}$ would display an error compared to $\varPhi _{1}$. The frequency graph on the relative error, expressed as a percentage of the true Shapley value at location A1 over these 100,000 random networks is shown in Fig. 10. The graph shows that getting relative errors above 15% is very unlikely, with the average value of the relative error being around 5.4%. As mentioned before, these non-zero errors on $\varPhi ^{(1)}$ only occur when $s_{23}$ = min{$s_{12},s_{13},s_{23}$} which, assuming no restrictions on the coordinates of the locations, should happen in one third of the situations. In the other two thirds of the situations, the value $\varPhi ^{(1)}$ will have no error. Therefore, in general, we conclude that the average relative error of using Appro-O(1) on the scenario in Fig. 5 will be about 1.8%.