Generating functions and Owen value in cooperative network cover game
Introduction
There exists a wide range of problems regarding mutual relations between pairs of agents within a network. Real life gives us many examples of such relations: supplier–customer, male–female, predator–prey, criminal–policeman, source–receiver, to name a few. In addition, agents may communicate with each other using transport, mobile or social networks.
This paper studies a cooperative game based on a graph in which vertices represent players and the characteristic function is defined using a maximal covering by the pairs of connected vertices.
For example, in a mobile network, the vertices of the corresponding graph represent mobile devices and connections between them appear within a range of visibility. In practice, it is important to find the maximal load on a mobile network under which any two devices can communicate with each other. Such problems arise in many source–receiver networks, e.g., in electrical power and radio networks.
Consider another example. Assume a graph represents a transport network in which each vertex is a departure or destination point. If two vertices are connected by an edge, then the corresponding points are connected by a route. The problem is to cover the transport network using non-intersecting edges in order to maximize the number of vertices covered.
One very useful method of analyzing social networks [1] is to split members into pairs in a maximal way. This is also important for network security in patrolling problems.
A network covering defines a partition of the set of players into coalitions. After determining the characteristic function, the Owen value is calculated. The Owen value [2], [3] can be regarded as an adaptation of the Shapley value in the situation where a coalition structure is pre-defined. This value describes the significance of each player in the network and can be treated as the reward of the players from communicating via the network. We will demonstrate that the method of generating functions [4], [5], [6] is useful for calculating the Owen value.
The method of generating functions has low space complexity. The method is suitable for symbolic computer packages such as MAPLE, MATHEMATICA, etc. The method is versatile, it can be used for computing different power indices, Owen and Aumann–Dreze values, etc.
A set of axioms defining the Owen value was introduced in [7]. Note that the linearity property of the payoffs plays a key role in this axiomatic characterization. A number of publications have been dedicated to the properties of the Owen value, e.g. [8], [9]. When calculating the payoffs of the players, it is necessary to consider the permutations corresponding to a given partition into coalitions. The characteristic function is defined for coalitions and hence calculation of the Owen value generally has exponential computational complexity. Therefore, the development of methods for calculating the Owen value is topical. Among the research works in this field, we should mention the paper [10]. The authors of [11] compared several concepts of solutions to a cooperative patrolling game and showed that, in practice, the Owen value is applicable to a larger extent than the Shapley or Aumann–Dreze values.
Just as in work [12], we will assume below that all the players act benevolently towards each other. Formally speaking, this means that considerations can be limited to coalitional partitions with a maximal number of winning coalitions. The Owen value is calculated by expanding the characteristic function via a basis of unanimity games. In [13], the Owen value was expressed in analytic form for such characteristic functions , and we will employ this approach for more general cases.
The expansion of characteristic functions mentioned above involves the method of generating functions. As a matter of fact, generating functions have demonstrated high efficiency for calculating the solutions of cooperative games. They have been used for the algorithmic determination of the Shapley–Shubik and Banzhaf values in voting games [4], [5]. Note that the papers, [6], [14], [15], calculated the Myerson vector for network games based on generating functions. More specifically, the cited authors developed a modification of the original algorithm from [16], in which generating functions were used to calculate the number of subtrees with vertices in a tree .
In that paper, the method of generating functions yielded a simple representation of characteristic functions. We find the explicit formula for the generating function for the number of connected coalitions in a tree. Moreover, it was adapted to develop a network covering algorithm with non-intersecting sets. We show that this algorithm is very useful for calculating the Owen value in a cooperative network cover game and illustrate this approach for various network structures.
This paper is organized as follows. Section 2 states the problem, introduces the necessary definitions, as well as proving the monotonicity and super-additivity of the characteristic function. Using the method of generating functions, in Section 3 we demonstrate that the expansion of the basis of the characteristic function for tree networks is of a special form. Similar results are obtained for cyclic network structures. Also, Section 3 shows how the maximal number of minimal winning coalitions can be found. Section 4 is dedicated to calculating the Owen value for various network structures (linear, star, and some others).
Section snippets
Cooperative game-theoretic model
Let be the set of players. A subset is called a coalition. We consider the cooperative game . The following definitions will be useful in the sequel.
Definition 1 A coalition is called winning if .
Definition 2 A coalition is called minimal winning if and .
Definition 3 A coalitional partition of the set is a set that satisfies the following properties: The element is defined to be the element of a partition that
Generating functions and basis expansion of characteristic function
If the characteristic function of a cooperative game is represented as a linear combination of unanimity games, then the Owen value can be calculated without considering all the permutations of players. As is well-known, any characteristic function has the basis expansion
Note that expansion may contain null terms. It turns out that for all unconnected coalitions.
Denote by the set of all subsets of vertices for the
Owen value calculation
Here we find the Owen value for different network structures.
Conclusions
In this paper, we have analyzed the paired interactions in networks using cooperative game theory methods. In accordance with this approach, network vertices are players and their payoffs depend on their arrangement in a given network. The characteristic function is defined as the maximal number of paired interactions in a given network. The associated cooperative game is solved using the Owen value. This game-theoretic interpretation of network structures yields new interesting conclusions
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
Supported by the Shandong Province, China “Double Hundred Talent Plan” (No. WST2017009). Support from Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged.
Vladimir Mazalov is Research Director at the Institute of Applied Mathematical Research, Karelian Research Center, Russian Academy of Sciences. He is also Professor of the Probability Theory Department at Petrozavodsk State University. He finished his Ph.D. studies at the Faculty of Applied Mathematics, Leningrad University in 1979. After that he has mainly worked in research projects funded by the Russian Academy of Sciences, in 1980-1998 at the Chita Institute of Natural Resources, East
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Vladimir Mazalov is Research Director at the Institute of Applied Mathematical Research, Karelian Research Center, Russian Academy of Sciences. He is also Professor of the Probability Theory Department at Petrozavodsk State University. He finished his Ph.D. studies at the Faculty of Applied Mathematics, Leningrad University in 1979. After that he has mainly worked in research projects funded by the Russian Academy of Sciences, in 1980-1998 at the Chita Institute of Natural Resources, East Siberia and, currently, at the Institute of Applied Mathematical Research, Karelian Research Center. His research interests are related with game theory and stochastic analysis and applications in behavioral biology, networking and economical systems. He was a supervisor of 20 Ph.D. Thesis. He is an editorial board member of International Game Theory Review, International Journal of Mathematics, Game Theory and Algebra, Scientiae Mathematicae Japonicae. He has published more than 100 publications in international conferences, journals and books and been awarded many competitive grants such as SNSF (Switzerland), JSPS (Japan), DAAD (Germany), Swedish Institute and Russian Fund for Basic Research.
Vasily Gusev was born in 1991 in Chita city, Russia. In period from 2013 to 2017 he was a Ph.D. student of Institute of Applied Mathematical Research, Karelian Research Center of Russian Academy of Science. In 2017 he defended a Ph.D. thesis. Now he work at National Research University Higher School of Economics as a research fellow. His research interests include mathematical programming, game theory and stability theory.