Oskar Skibski

A shapley value-based approach to determine gatekeepers in social networks with applications

Inspired by emerging applications of social networks, we introduce in this paper a new centrality measure termed gatekeeper centrality. The new centrality is based on the well-known game-theoretic concept of Shapley value and, as we demonstrate, possesses unique qualities compared to the existing metrics. Furthermore, we present a dedicated approximate algorithm, based on the Monte Carlo sampling method, to compute the gatekeeper centrality. We also consider two well known applications in social network analysis, namely community detection and limiting the spread of mis-information; and show the merit of using the proposed framework to solve these two problems in comparison with the respective benchmark algorithms.

Defeating Terrorist Networks with Game Theory

This column discusses the problem of identifying key members of a terrorist network. Game-theoretic centrality measures offer solutions but also raise computational challenges. The authors present a survey of this work and show how some of the computational challenges can be overcome.

Steady marginality: a uniform approach to shapley value for games with externalities

The Shapley value is one of the most important solution concepts in cooperative game theory. In coalitional games without externalities, it allows to compute a unique payoff division that meets certain desirable fairness axioms. However, in many realistic applications where externalities are present, Shapleys axioms fail to indicate such a unique division. Consequently, there are many extensions of Shapley value to the environment with externalities proposed in the literature built upon additional axioms. Two important such extensions are externality-free value by Pham Do and Norde and value that absorbed all externalities by McQuillin. They are good reference points in a space of potential payoff divisions for coalitional games with externalities as they limit the space at two opposite extremes. In a recent, important publication, De Clippel and Serrano presented a marginality-based axiomatization of the value by Pham Do Norde. In this paper, we propose a dual approach to marginality which allows us to derive the value of McQuillin. Thus, we close the picture outlined by De Clippel and Serrano.

The Stochastic Shapley Value for coalitional games with externalities

A long debated but still open question in the game theory literature is that of how to extend the Shapley Value to coalitional games with externalities. While previous work predominantly focused on developing alternative axiomatizations, in this article we propose a novel approach which centers around the coalition formation process and the underlying probability distribution from which a suitable axiomatization naturally follows. Specifically, we view coalition formation in games with externalities as a discrete-time stochastic process. We focus, in particular, on the Chinese Restaurant Process – a well-known stochastic process from probability theory. We show that reformulating Shapleys coalition formation process as the Chinese Restaurant Process yields in games with externalities a unique value with various desirable properties. We then generalize this result by proving that all values that satisfy the direct translation of Shapleys axioms to games with externalities can be obtained using our approach based on stochastic processes.

Marginality Approach to Shapley Value in Games with Externalities

One of the long-debated issues in coalitional game theory is how to extend the Shapley value to games with externalities. In particular, when externalities occur, a direct translation of Shapleys axioms does not imply a unique value. In this paper we study the marginality approach to this problem, based on the idea of an alpha-parametrized definition of the marginal contribution, where alpha is a vector of weights associated with an agent joining/leaving a coalition. We prove that all values that satisfy the direct translation of Shapleys axioms can be obtained using the marginality approach. Moreover, we show that every such value can be uniquely derived using marginality approach by choosing appropriate weights alpha. Next, we analyze how properties of a value translate to the requirements on the definition of the marginal contribution (i.e. weights alpha). Building upon this analysis, we show that under certain conditions, two other axiomatizations of the Shapley value (i.e., Youngs marginality axiomatization and Myersons axiomatization based on the concept of balanced contributions), translated to games with externalities using the proper definition of the alpha-parametrized marginal contribution, are equivalent to Shapleys axiomatization.

Axiomatic Characterization of Game-Theoretic Centrality

One of the fundamental research challenges in network science is centrality analysis, i.e., identifying the nodes that play the most important roles in the network. In this article, we focus on the game-theoretic approach to centrality analysis. While various centrality indices have been recently proposed based on this approach, it is still unknown how general is the game-theoretic approach to centrality and what distinguishes some game-theoretic centralities from others. In this article, we attempt to answer this question by providing the first axiomatic characterization of game-theoretic centralities. Specifically, we show that every possible centrality measure can be obtained following the game-theoretic approach. Furthermore, we study three natural classes of game-theoretic centrality, and prove that they can be characterized by certain intuitive properties pertaining to the well-known notion of Fairness due to Myerson.

Attachment Centrality for Weighted Graphs

Measuring how central nodes are in terms of connecting a network has recently received increasing attention in the literature. While a few dedicated centrality measures have been proposed, Skibski et al. [2016] showed that the Attachment Centrality is the only one that satisfies certain natural axioms desirable for connectivity. Unfortunately, the Attachment Centrality is defined only for unweighted graphs which makes this measure ill-fitted for various applications. For instance, covert networks are typically weighted, where the weights carry additional intelligence available about criminals or terrorists and the links between them. To analyse such settings, in this paper we extend the Attachment Centrality to node-weighted and edgeweighted graphs. By an axiomatic analysis, we show that the Attachment Centrality is closely related to the Degree Centrality in weighted graphs.

An Algorithm for the Myerson Value in Probabilistic Graphs with an Application to Weighted Voting

Myersons graph-restricted games are a well-known formalism for modeling cooperation thats subject to restrictions. In particular, Myerson considered a coalitional game in which cooperation is possible only through an underlying network of links between agents. A unique fair solution concept for graph-restricted games is called the Myerson value. One study generalized these results by considering probabilistic graphs in which agents can cooperate via links only to some extent, that is, with some probability. The authors algorithm is based on the enumeration of all connected subgraphs in the graph. As a sample application of the new algorithm, they consider a probabilistic graph that represents likelihood of pairwise collaboration between political parties before the 2015 general elections in the UK.

Additivity is (Almost) Sufficient - A New Axiomatization of the Shapley Value

The principle of additivity states the sum of payoffs in two separate games should equal the payoff in the combination of both games. We show that the Shapley value is the only value that is additive for arbitrary games and that equally divides the payoff in games in which only the grand coalition has a non-zero value. Then, we extend the notion of additivity to graph-restricted games and games with a priori given coalition structure. We prove that the Myerson and Owen values are the only additive solution concepts in these classes, respectively, that satisfy an analogous simple borderline case conditions.

Path Evaluation and Centralities in Weighted Graphs – An Axiomatic Approach

We study the problem of extending the classic centrality measures to weighted graphs. Unfortunately, in the existing extensions, paths in the graph are evaluated solely based on their weights, which is a restrictive and undesirable assumption for a variety of settings. Given this, we define a notion of the path evaluation function that assesses a path between two nodes by looking not only on the sum of edge weights, but also on the number of intermediaries. Using an axiomatic approach, we propose three classes of path evaluation functions. Building upon this analysis, we present the first systematic study how classic centrality measures can be extended to weighted graphs while taking into account an arbitrary path evaluation function. As an application, we use the newly-defined measures to identify the most well-linked districts in a sample public transport network.