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, Shapley's 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

Complexity of Computing the Shapley Value in Games with Externalities

We study the complexity of computing the Shapley value in games with externalities. We focus on two representations based on marginal contribution nets (embedded MC-nets and weighted MC-nets). Our results show that while weighted MC-nets are more concise than embedded MC-nets, they have slightly worse computational properties when it comes to computing the Shapley value

Axiomatic Analysis of Medial Centrality Measures

We perform the first axiomatic analysis of medial centrality measures. These measures, also called betweenness-like centralities, assess the role of a node in connecting others in the network. We focus on a setting with one target node and several source nodes. We consider three classic medial centrality measures adapted to this setting: Betweenness Centrality, Stress Centrality and Random Walk Betweenness Centrality. While Betweenness and Stress Centralities assume that the information in the network follows shortest paths, Random Walk Betweenness Centrality assumes it moves randomly along the edges. We develop the first axiomatic characterizations of all three measures. Our analysis shows that Random Walk Betweenness, while conceptually different, shares several common properties with classic Betweenness and Stress Centralities.Comment: 31 pages, 5 figure

Wartość Shapleya dla gier z efektami zewnętrznymi i gier na grafach

The Shapley value is one of the most important solution concepts in coalitional game theory. It was originally defined for classical model of a coalitional game, which is relevant to a wide range of economic and social situations. However, while in certain cases the simplicity is the strength of the classical coalitional game model, it often becomes a limitation. To address this problem, a number of extensions have been proposed in the literature. In this thesis, we study two important such extensions – to games with externalities and graph-restricted games. Games with externalities are a richer model of coalitional games in which the value of a coalition depends not only on its members, but also on the arrangement of other players. Unfortunately, four axioms that uniquely determine the Shapley value in classical coalitional games are not enough to imply a unique value in games with externalities. In this thesis, we study a method of strengthening the Null-Player Axiom by using alpha-parameterized definition of the marginal contribution in games with externalities. We prove that this approach yields a unique value for every alpha. Moreover, we show that this method is indeed general, in that all the values that satisfy the direct translation of Shapley’s axioms to games with externalities can be obtained using this approach. Graph-restricted games model naturally-occurring scenarios where coordination between any two players within a coalition is only possible if there is a communication channel between them. Two fundamental solution concepts that were proposed for such a game are the Shapley value and its particular extension – the Myerson value. In this thesis we develop algorithms to compute both values. Since the computation of either value involves visiting all connected induced subgraphs of the graph underlying the game, we start by developing a dedicated algorithm for this purpose and show that it is the fastest known in the literature. Then, we use it as the cornerstone upon which we build algorithms for the Shapley and Myerson values.Wartość Shapleya jest jedną z najważniejszych metod podziału w teorii gier koalicyjnych. Oryginalnie została zdefiniowana w klasycznym modelu gier koalicyjnych, który jest dobrą ilustracją wielu ekonomicznych i społecznych sytuacji. Chociaż prostota jest w wielu przypadkach siłą klasycznego modelu gier koalicyjnych, często staje się jednak też jego ograniczeniem. Aby poradzić sobie z tym problemem, kilka rozszerzeń gier koalicyjnych zostało zaproponowanych w literaturze. W tej rozprawie zajmujemy się dwoma ważnymi rozszerzeniami - do gier z efektami zewnętrznymi oraz gier ograniczonych grafem (ang. graph-restricted games). Gry z efektami zewnętrznymi są bogatszym modelem gier koalicyjnych, w którym wartość koalicji zależy nie tylko od jej członków, ale także od rozmieszczenia innych graczy. Niestety, cztery aksjomaty, które implikują wartość Shapleya w klasycznych grach koalicyjnych, nie są wystarczające, aby implikować unikalną wartość w grach z efektami zewnętrznymi. W tej rozprawie badamy metodę polegającą na wzmocnieniu Aksjomatu Gracza Zerowego (ang. Null-Player Axiom), używając alfa-parametryzowanej definicji wkładu marginalnego. Udowadniamy, że takie podejście daje unikalną wartość dla każdego alfa. Ponadto, pokazujemy że jest ono ogólne: wszystkie wartości, które spełniają bezpośrednie tłumaczenie aksjomatów Shapleya, mogą być uzyskane z użyciem tego podejścia. Gry ograniczone grafem modelują naturalnie pojawiające się sytuacje, w których koordynacja dwóch graczy w ramach koalicji jest możliwa tylko wtedy, gdy istnieje kanał komunikacji między nimi. Dwie podstawowe koncepcje podziału, które zostały zaproponowane dla takich gier to wartość Shapleya oraz jej rozszerzenie - wartość Myersona. W tej rozprawie proponujemy algorytmy do obliczania obu wartości. Ponieważ obliczenie ich opiera się na enumerowaniu wszystkich spójnych indukowanych podgrafów w grafie gry, zaczynamy od opracowania algorytmu dedykowanego do tego celu i pokazujemy, że jest szybszy niż inne algorytmy w literaturze. Potem używamy go jako podstawę algorytmów do obliczania wartości Shapleya i Myersona

Axiomatic Characterization of PageRank

This paper examines the fundamental problem of identifying the most important nodes in a network. We use an axiomatic approach to this problem. Specifically, we propose six simple properties and prove that PageRank is the only centrality measure that satisfies all of them. Our work gives new conceptual and theoretical foundations of PageRank that can be used to determine suitability of this centrality measure in specific applications

How Do Centrality Measures Choose the Root of Trees?

Centrality measures are widely used to assign importance to graph-structured data. Recently, understanding the principles of such measures has attracted a lot of attention. Given that measures are diverse, this research has usually focused on classes of centrality measures. In this work, we provide a different approach by focusing on classes of graphs instead of classes of measures to understand the underlying principles among various measures. More precisely, we study the class of trees. We observe that even in the case of trees, there is no consensus on which node should be selected as the most central. To analyze the behavior of centrality measures on trees, we introduce a property of tree rooting that states a measure selects one or two adjacent nodes as the most important, and the importance decreases from them in all directions. This property is satisfied by closeness centrality but violated by PageRank. We show that, for several centrality measures that root trees, the comparison of adjacent nodes can be inferred by potential functions that assess the quality of trees. We use these functions to give fundamental insights on rooting and derive a characterization explaining why some measure root trees. Moreover, we provide an almost linear time algorithm to compute the root of a graph by using potential functions. Finally, using a family of potential functions, we show that many ways of tree rooting exist with desirable properties

Complexity of Computing the Shapley Value in Games with Externalities

We study the complexity of computing the Shapley value in games with externalities. We focus on two representations based on marginal contribution nets (embedded MC-nets and weighted MC-nets) and five extensions of the Shapley value to games with externalities. Our results show that while weighted MC-nets are more concise than embedded MC-nets, they have slightly worse computational properties when it comes to computing the Shapley value: two out of five extensions can be computed in polynomial time for embedded MC-nets and only one for weighted MC-nets

in Partition Function Games

One of the long-debated issues in coalitional game theory is how to extend the Shapley value to games with externalities (partition-function games). When externalities are present, not only can a player’s marginal contribution—a central notion to the Shapley value—be defined in a variety of ways, but it is also not obvious which axiomatization should be used. Consequently, a number of authors extended the Shapley value using complex and often unintuitive axiomatizations. Furthermore, no algorithm to approximate any extension of the Shapley value to partition-function games has been proposed to date. Given this background, we prove in this paper that, for any well-defined measure of marginal contribution, Shapley’s original four axioms imply a unique value for games with externalities. As an consequence of this general theorem, we show that values proposed by Macho-Stadler et al., McQuillin and Bolger can be derived from Shapley’s axioms. Building upon our analysis of marginal contribution, we develop a general algorithm to approximate extensions of the Shapley value to games with externalities using a Monte Carlo simulation technique.

An Axiomatization of the Eigenvector and Katz Centralities

Feedback centralities are one of the key classes of centrality measures. They assess the importance of a vertex recursively, based on the importance of its neighbours. Feedback centralities includes the Eigenvector Centrality, as well as its variants, such as the Katz Centrality or the PageRank, and are used in various AI applications, such as ranking the importance of websites on the Internet and most influential users in the Twitter social network. In this paper, we study the theoretical underpinning of the feedback centralities. Specifically, we propose a novel axiomatization of the Eigenvector Centrality and the Katz Centrality based on six simple requirements. Our approach highlights the similarities and differences between both centralities which may help in choosing the right centrality for a specific application

Axioms for Distance-Based Centralities

We study the class of distance-based centralities that consists of centrality measures that depend solely on distances to other nodes in the graph. This class encompasses a number of centrality measures, including the classical Degree and Closeness Centralities, as well as their extensions: the Harmonic, Reach and Decay Centralities. We axiomatize the class of distance-based centralities and study what conditions are imposed by the axioms proposed in the literature. Building upon our analysis, we propose the class of additive distance-based centralities and pin-point properties which combined with the axiomatic characterization of the whole class uniquely characterize a number of centralities from the literature