On the Shapley Value of Unrooted Phylogenetic Trees

Abstract

The Shapley value, a solution concept from cooperative game theory, has recently been considered for both unrooted and rooted phylogenetic trees. Here, we focus on the Shapley value of unrooted trees and first revisit the so-called split counts of a phylogenetic tree and the Shapley transformation matrix that allows for the calculation of the Shapley value from the edge lengths of a tree. We show that non-isomorphic trees may have permutation-equivalent Shapley transformation matrices and permutation-equivalent null spaces. This implies that estimating the split counts associated with a tree or the Shapley values of its leaves does not suffice to reconstruct the correct tree topology. We then turn to the use of the Shapley value as a prioritization criterion in biodiversity conservation and compare it to a greedy solution concept. Here, we show that for certain phylogenetic trees, the Shapley value may fail as a prioritization criterion, meaning that the diversity spanned by the top k species (ranked by their Shapley values) cannot approximate the total diversity of all n species.

A Appendix

In order to find non-isomorphic trees with permutation-equivalent Shapley transformation matrices, we have exhaustively analyzed all tree topologies up to 17 taxa and their split counts. To be precise, we have considered different necessary (but not sufficient) conditions for two non-isomorphic trees to have permutation-equivalent Shapley transformation matrices, the details of which will be explained in the following. Note that we have considered these necessary (but not sufficient) conditions as a first step, because they can quickly be checked, while directly examining whether two matrices a permutation-equivalent is time-consuming and not feasible for large matrices. Using these necessary conditions we have performed a candidate search for trees inducing permutation-equivalent Shapley transformation matrices, where the candidates were then further analyzed. We now describe the necessary conditions we used.

Fig. 6

Two non-isomorphic trees $\mathcal {T}_1'$ and $\mathcal {T}_2'$ on 17 leaves that are found by Algorithm 1 but do not induce permutation-equivalent Shapley transformation matrices and permutation-equivalent null spaces

• 1. Split size sequence:

  Recall that the Shapley transformation matrix of a tree $\mathcal {T}$ solely depends on the splits counts associated with its edges (cf. Theorem 1). In order for two tree topologies to have permutation-equivalent Shapley transformation matrices, they must exhibit the same split counts, in particular they must exhibit the same split sizes, where for a split $\sigma = A \vert B$ with $A, B \subset X, \, A \cap B = \emptyset ,$ and $A \cup B$ we let $\Vert \sigma \Vert = \Vert A \vert B \Vert = \min \{\vert A \vert , \vert B \vert \}$ denote its size. Note that any binary tree $\mathcal {T}$ on n leaves induces n trivial splits (where either $\vert A \vert = 1$ or $\vert B \vert = 1$) and $n-3$ non-trivial splits. Following Fischer and Liebscher (2015), we assume an arbitrary ordering of these splits $\sigma _1, \ldots , \sigma _{n-3}$ and define the $(n-3)$ tuple $\tilde{s}(\mathcal {T})$ as follows:

  $$\begin{aligned} \tilde{s}(\mathcal {T})_i = \Vert \sigma _i \Vert \text { for all } i=1, \ldots , n-3. \end{aligned}$$

  We now order the $n-3$ entries of $\tilde{s}(\mathcal {T})$ increasingly and call the resulting ordered sequence the split size sequence$s(\mathcal {T})$. Now, in order for two trees to have permutation-equivalent Shapley transformation matrices, their split size sequences must be identical, which gives us a first necessary condition. For $\mathcal {T}_1$ and $\mathcal {T}_2$ depicted in Fig. 3, we for example have

  $s(\mathcal {T}_1) = s(\mathcal {T}_2) = (2,2,2,2,2,2,3,3,4,4,4,4,8,8)$.

• 2. Matrix entries:

  If two trees exhibit the same split size sequence, we compute their Shapley transformation matrices and analyze them:

  • (a) For two matrices $\mathbf {M}_1$ and $\mathbf {M}_2$ to be permutation-equivalent, they must contain the same entries. To check if this is the case, we “flatten” both matrices and define $s(\mathbf {M}_1)$ to be the sequence containing all matrix elements of $\mathbf {M}_1$ in an increasing order and analogously we define $s(\mathbf {M}_2)$ to be the sequence containing all entries of $\mathbf {M}_2$ ordered increasingly. If $s(\mathbf {M}_1) = s(\mathbf {M}_2)$, the two matrices share the same entries and we proceed with a subsequent analysis of rows and columns.

  • (b) Recall that two matrices are permutation-equivalent if they are identical up to a permutation of rows and columns. Thus, we derive two additional necessary conditions for two matrices to be permutation-equivalent.

    • For all rows $r_i^1$ of $\mathbf {M}_1$, we define $s(r_i^1)$ to be the sequence containing the elements of $r_i^1$ in an increasing order. Analogously we define $s(r_j^2)$ to be the sequence containing the elements of a row $r_j^2$ of matrix $\mathbf {M}_2$. Now for all rows $r_i^1$ of $\mathbf {M}$, we check if $s(r_i^1) = s(r_j^2)$ for some row $r_j^2$ of $\mathbf {M}_2$.

    • Similarly, we compare the columns of $\mathbf {M}_1$ and $\mathbf {M}_2$. For any column $c_i^1$ of $\mathbf {M}_1$ or $c_j^2$ of $\mathbf {M}_2$, we define $s(c_i^1)$ and $s(c_j^2)$ to be the sequence containing the elements of the corresponding column in an increasing order. Now for all columns $c_i^1$ of $\mathbf {M}$, we check if $s(c_i^1) = s(c_j^2)$ for some column $c_j^2$ of $\mathbf {M}_2$.

We now summarize the above conditions in the following algorithm (Algorithm 1) that checks whether two non-isomorphic trees $\mathcal {T}_1$ and $\mathcal {T}_2$ are candidates for trees inducing permutation-equivalent Shapley transformation matrices.

Table 1 Split counts induced by edge $I_{13}$

Note that the algorithm returns TRUE, if the input trees possibly induce permutation-equivalent Shapley transformation matrices and FALSE if this can be ruled out (i.e., any of the necessary conditions introduced above is violated). However, if the algorithm returns TRUE the possible candidates have to be further analyzed, as all conditions mentioned above are necessary for two trees to have permutation-equivalent Shapley transformation matrices, but not sufficient (cf. Example 3). However, we have conducted this candidate search in Mathematica Wolfram Research Inc. (2017) and have analyzed all tree topologies up to 16 leaves. The only pair of candidates that we found is the pair $(\mathcal {T}_1, \mathcal {T}_2)$ depicted in Fig. 3 and used in the proof of Theorem 4. Thus, this pair is the smallest example for a pair of non-isomorphic trees inducing permutation-equivalent Shapley matrices (and thus permutation-equivalent null spaces). Subsequently, we have looked at the case of 17 taxa, where again only one pair of candidate trees was found (trees $\mathcal {T}_1'$ and $\mathcal {T}_2'$ depicted in Fig. 6). However, as we will explain below, $\mathcal {T}_1'$ and $\mathcal {T}_2'$ do not induce permutation-equivalent Shapley transformation matrices, which illustrates the fact that the conditions described above and used in Algorithm 1 are only necessary, but not sufficient conditions.

Example 3

Consider the pair of trees $(\mathcal {T}_1', \mathcal {T}_2')$ on 17 leaves depicted in Fig. 6. Algorithm 1 returns TRUE for this pair of trees, i.e., $\mathcal {T}_1'$ and $\mathcal {T}_2'$ are possible candidates for two non-isomorphic trees inducing permutation-equivalent Shapley transformation matrices. However, their Shapley transformation matrices are not permutation-equivalent. To see this, consider the split counts associated with edge $I_{13}$ and compare them for $\mathcal {T}_1'$ and $\mathcal {T}_2'$ (cf. Table 1). For leaves 1, 2, 3, 4, 9, 10, 11, 12 and 17 edge $I_{13}$ induces the same split counts in both $\mathcal {T}_1'$ and $\mathcal {T}_2'$. However, for leaves 5, 6, 7, 8 and leaves 13, 14, 15, 16 the split counts differ. To be precise, we have $f_{\mathcal {T}_1'}(i,I_{13}) = 9$ and $f_{\mathcal {T}_2'}(i,I_{13}) = 8$ for $i = 5,6,7,8$ and $f_{\mathcal {T}_1'}(j,I_{13}) = 8$ and $f_{\mathcal {T}_2'}(j,I_{13}) = 9$ for $j = 13,14,15,16$. At first glance, we can make the split counts associated with edge $I_{13}$ coincide for $\mathcal {T}_1'$ and $\mathcal {T}_2'$ by swapping leaves 5, 6, 7, 8 with leaves 13, 14, 15, 16 in $\mathcal {T}_2'$ (i.e., by permuting the rows associated with these leaves in the Shapley transformation matrix). However, then the split counts induced by for example edge $I_5$ will differ between $\mathcal {T}_1'$ and $\mathcal {T}_2'$. It can be checked that no permutation of rows or columns of the Shapley transformation matrix $\mathbf {M}_2'$ of $\mathcal {T}_2'$ exists such that it coincides with the Shapley transformation matrix $\mathbf {M}_1'$ of $\mathcal {T}_1'$. Thus, the Shapley transformation matrices of $\mathcal {T}_1'$ and $\mathcal {T}_2'$ are not permutation-equivalent even though Algorithm 1 suggests them as candidates. This shows that the criteria used in Algorithm 1 are necessary but not sufficient conditions for two non-isomorphic trees to have permutation-equivalent Shapley transformation matrices.