Partition decision trees: representation for efficient computation of the Shapley value extended to games with externalities

Abstract

While coalitional games with externalities model a variety of real-life scenarios of interest to computer science, they pose significant game-theoretic and computational challenges. Specifically, key game-theoretic solution concepts—and the Shapley value in particular—can be extended to games with externalities in multiple, often orthogonal, ways. As for the computational challenges, while there exist two concise representations for coalitional games with externalities—called embedded MC-Nets and weighted MC-Nets—they allow the polynomial-time computation of only two of the six existing direct extensions of the Shapley values to games with externalities. In this article, inspired by the literature on endogenous coalition formation protocols, we propose to represent games with externalities in a way that mimic an intuitive process in which coalitions might form. To this end, we utilize Partition Decision Trees—rooted directed trees, where non-leaf nodes are labelled with agents’ names, leaf nodes are labelled with payoff vectors, and edges indicate membership of agents in coalitions. Interestingly, despite their apparent differences, the representation based on partition decision trees can be considered a subclass of embedded MC-Nets and weighted MC-Nets. The key advantage of this new representation is that it allows the polynomial-time computation of five out of six direct extensions of the Shapley value to games with externalities. In other words, by focusing on narrower Partition Decision Trees instead of wider embedded or weighted MC-Nets, a user is guaranteed to compute most extensions of the Shapley value in polynomial time.

Appendix: Summary of the main notation

N :

The set of agents. Also, the grand coalition

$i, a,b,c, \ldots$ :

The agents from N

v :

A game in the partition function form

$\hat{v}$ :

A game in the characteristic function form

S :

A coalition, $S \subseteq N$

$\mathscr {P}(N)$ :

The set of all partitions of the players in N

P :

A partition from $\mathscr {P}(N)$

(S, P):

A coalition S embedded in partition P

EC(N):

The set of all embedded coalitions

$(\varphi _i(v))_{i \in N}$ :

A value of game v

$\varOmega (N)$ :

The set of all permutations of N

$\omega$ :

A permutation from $\varOmega (N)$

$S_i^{\omega }$ :

The set of agents that appear in permutation $\omega$ after i

$v_e^{(S, P)}$ :

An elementary game in which only (S, P) has non-zero value

$\varPsi (N)$ :

The set of all games in which only coalitions in a single partition have non-zero values

$\varphi ^{EF}_i(v)$ :

The externality-free value, EF-value [17, 47]

$\varphi ^{MQ}_i(v)$ :

The McQuillin value, MQ-value [38, 56]

$\varphi ^{SS}_i(v)$ :

The Stochastic Shapley value, SS-value [35, 60]

$\varphi ^{HY}_i(v)$ :

The Hu-Yang value, HY-value  [27]

$\varphi ^{MY}_i(v)$ :

The Myerson value, MY-value  [44]

T :

A PDT rule

$\mathscr {T}$ :

A set of PDT rules

V :

A set of nodes

u :

A node from V

E :

A set of edges

$f_V$ :

A label function for internal nodes

$f_E$ :

A label function for edges

$f_L$ :

A label function for leaf nodes

$\varPi (T)$ :

The set of paths from the root to any leaf in T

$\pi$ :

A path from $\varPi (T)$

$v^{\pi }$ :

The game described by the PDT path $\pi$

$v^{T}$ :

The game described by the PDT rule T

$v^{\mathscr {T}}$ :

The game described by the set of PDT rules $\mathscr {T}$

x, y:

Real numbers.