Efficiently computing the Shapley value of connectivity games in low-treewidth graphs (original) (raw)

Algorithms for computing the Shapley value of cooperative games on lattices

Discrete Applied Mathematics

We study algorithms to compute the Shapley value for a cooperative game on a lattice L Σ = (F Σ , ⊆) where F Σ is the family of closed sets given by an implicational system Σ on a set N of players. The first algorithm is based on the generation of the maximal chains of the lattice L Σ and computes the Shapley value in O(|N | 3 .|Σ|.|Ch|) time complexity using polynomial space, where Ch is the set of maximal chains of L Σ. The second algorithm proceeds by building the lattice L Σ and computes the Shapley value in O(|N | 3 .|Σ|.|F Σ |) time and space complexity. Our main contribution is to show that the Shapley value of weighted graph games on a product of chains with the same fixed length is computable in polynomial time. We do this by partitioning the set of feasible coalitions relevant to the computation of the Shapley value into equivalence classes in such a way that we need to consider only one element of each class in the computation.

Algorithms for the Shapley and Myerson values in graph-restricted games

Graph-restricted games, first introduced by Myerson [20], model naturally-occurring scenarios where coordination between any two agents within a coalition is only possible if there is a communication channel(a path) between them. Two fundamental solution concepts that were proposed for such a game are the Shapley value and the Myerson value. While an algorithm has been proposed to compute the Shapley value in arbitrary graph-restricted games, no such general-purpose algorithm has yet been developed for the Myerson value.

Spanning connectivity games

2009

The Banzhaf index, Shapley-Shubik index and other voting power indices measure the importance of a player in a coalitional game. We consider a simple coalitional game called the spanning connectivity game (SCG) based on an undirected, unweighted multigraph, where edges are players. We examine the computational complexity of computing the voting power indices of edges in the SCG. It is shown that computing Banzhaf values and Shapley-Shubik indices is #P-complete for SCGs. Interestingly, Holler indices and Deegan-Packel indices can be computed in polynomial time. Among other results, it is proved that Banzhaf indices can be computed in polynomial time for graphs with bounded treewidth. It is also shown that for any reasonable representation of a simple game, a polynomial time algorithm to compute the Shapley-Shubik indices implies a polynomial time algorithm to compute the Banzhaf indices. As a corollary, computing the Shapley value is #P-complete for simple games represented by the set of minimal winning coalitions, Threshold Network Flow Games, Vertex Connectivity Games and Coalitional Skill Games.

A randomized method for the Shapley value for the voting game

2007

The Shapley value is one of the key solution concepts for coalition games. Its main advantage is that it provides a unique and fair solution, but its main problem is that, for many coalition games, the Shapley value cannot be determined in polynomial time. In particular, the problem of finding this value for the voting game is known to be #P-complete in the general case. However, in this paper, we show that there are some specific voting games for which the problem is computationally tractable. For other general voting games, we overcome the problem of computational complexity by presenting a new randomized method for determining the approximate Shapley value. The time complexity of this method is linear in the number of players. We also show, through empirical studies, that the percentage error for the proposed method is always less than 20% and, in most cases, less than 5%.

Compensations in the Shapley value and the compensation solutions for graph games

International Journal of Game Theory, 2012

We consider an alternative expression of the Shapley value that reveals a system of compensations: each player receives an equal share of the worth of each coalition he belongs to, and has to compensate an equal share of the worth of any coalition he does not belong to. We give an interpretation in terms of formation of the grand coalition according to an ordering of the players and define the corresponding compensation vector. Then, we generalize this idea to cooperative games with a communication graph. Firstly, we consider cooperative games with a forest (cycle-free graph). We extend the compensation vector by considering all rooted spanning trees of the forest (see Demange ) instead of orderings of the players. The associated allocation rule, called the compensation solution, is characterized by component efficiency and relative fairness. The latter axiom takes into account the relative position of a player with respect to his component. Secondly, we consider cooperative games with arbitrary graphs and construct rooted spanning trees by using the classical algorithms DFS and BFS. If the graph is complete, we show that the compensation solutions associated with DFS and BFS coincide with the Shapley value and the equal surplus division respectively.

The V L Value for Network Games

SSRN Electronic Journal, 2000

In this paper we consider a proper Shapley value (the V L value) for cooperative network games. This value turns out to have a nice interpretation. We compute the V L value for various kinds of networks and relate this value to optimal strategies in an associated matrix game.

Shapley Value for Shortest Path Routing in Dynamic Networks

Transportation is a crucial component of supply chain management, responsible for delivering goods and services to customers. This paper explores the application of game theory concepts, precisely the Shapley value, to cost allocation in transportation operations involving drones and trucks. Our focus is on shortest-path games in which agents own nodes in a network and seek to transport items between nodes at the lowest possible cost. We provide a comprehensive literature review of the Shapley value and its use in shortest-path games, with particular emphasis on transportation networks. Our proposed model includes sets of customers, drones, and trucks and uses binary decision variables to indicate whether a drone or truck serves a given customer. The objective is to minimize the total cost of serving all customers while adhering to capacity and synchronization constraints. We use the Shapley value to determine the contribution and cost-sharing of each drone and truck in serving the ...

Power Indices in Spanning Connectivity Games

Lecture Notes in Computer Science, 2009

The Banzhaf index, Shapley-Shubik index and other voting power indices measure the importance of a player in a coalitional game. We consider a simple coalitional game called the spanning connectivity game (SCG) based on an undirected, unweighted multigraph, where edges are players. We examine the computational complexity of computing the voting power indices of edges in the SCG. It is shown that computing Banzhaf values is #P-complete and computing Shapley-Shubik indices or values is NP-hard for SCGs. Interestingly, Holler indices and Deegan-Packel indices can be computed in polynomial time. Among other results, it is proved that Banzhaf indices can be computed in polynomial time for graphs with bounded treewidth. It is also shown that for any reasonable representation of a simple game, a polynomial time algorithm to compute the Shapley-Shubik indices implies a polynomial time algorithm to compute the Banzhaf indices. This answers (positively) an open question of whether computing Shapley-Shubik indices for a simple game represented by the set of minimal winning coalitions is NP-hard.

An analysis of the shapley value and its uncertainty for the voting game

2006

The Shapley value provides a unique solution to coalition games and is used to evaluate a player's prospects of playing a game. Although it provides a unique solution, there is an element of uncertainty associated with this value. This uncertainty in the solution of a game provides an additional dimension for evaluating a player's prospects of playing the game. Thus, players want to know not only their Shapley value for a game, but also the associated uncertainty. Given this, our objective is to determine the Shapley value and its uncertainty and study the relationship between them for the voting game. But since the problem of determining the Shapley value for this game is #P-complete, we first present a new polynomial time randomized method for determining the approximate Shapley value. Using this method, we compute the Shapley value and correlate it with its uncertainty so as to allow agents to compare games on the basis of both their Shapley values and the associated uncertainties. Our study shows that, a player's uncertainty first increases with its Shapley value and then decreases. This implies that the uncertainty is at its minimum when the value is at its maximum, and that agents do not always have to compromise value in order to reduce uncertainty.

Computational aspects of extending the Shapley Value to coalitional games with externalities

2010

Until recently, computational aspects of the Shapley value were only studied under the assumption that there are no externalities from coalition formation, i.e., that the value of any coalition is independent of other coalitions in the system. However, externalities play a key role in many real-life situations and have been extensively studied in the game-theoretic and economic literature. In this paper, we consider the issue of computing extensions of the Shapley value to coalitional games with externalities proposed by Myerson [21], Pham Do and Norde and McQuillin [17]. To facilitate efficient computation of these extensions, we propose a new representation for coalitional games with externalities, which is based on weighted logical expressions. We demonstrate that this representation is fully expressive and, sometimes, exponentially more concise than the conventional partition function game model. Furthermore, it allows us to compute the aforementioned extensions of the Shapley value in time linear in the size of the input.