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Networks, Communication and Hierarchy: Applications to Cooperative Games

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Frontiers of Dynamic Games

Part of the book series: Trends in Mathematics ((TM))

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Abstract

Agents participating in different kind of organizations, usually take different positions in some network structure. Two well-known network structures are hierarchies and communication networks. We give an overview of the most common models of communication and hierarchy restrictions in cooperative games, compare different network structures with each other and discuss network structures that combine communication as well as hierarchical features. Throughout the survey, we illustrate these network structures by applying them to cooperative games with restricted cooperation.

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Notes

  1. 1.

    For union stable systems where all singletons are feasible, Algaba et al. [4] unified two important lines of restricted cooperation: the one introduced by Myerson [54] and the one initiated by Faigle [39]. Moreover, the relationship among union stable systems and hypergraphs is established by Algaba et al. [7].

  2. 2.

    As an alternative to restricting the coalitions (i.e. sets of players) that are feasible, Faigle and Kern [40] introduce games under precedence constraints where (1) the game is defined on a restricted domain that is determined by the hierarchy, and (2) the possible orders in which coalitions can be formed is restricted. In this setting Algaba et al. [12] define a class of new values in which the removal of certain ‘irrelevant’ players does not effect the payoffs of the remaining players. A comparison between the two models is given in Algaba and van den Brink [1].

  3. 3.

    Another common expression of the Shapley value is using the so-called Harsanyi dividends (see Harsanyi [47]) but we will not use that in this survey.

  4. 4.

    For TU-games, van den Brink [68] axiomatized the Shapley value by efficiency, the null player property and a fairness axiom that requires that the payoffs of two players change by the same amount if to a game v we add another game w such that the two players are symmetric in game w. Deleting an edge from a communication graph, the two players on the deleted edge are symmetric in the difference game between the two communication restricted games.

  5. 5.

    Another type of motivation for a solution is by strategic implementation. A strategic implementation of the Myerson value can be found in Slikker [65], who modified the bidding mechanism of Pérez-Castrillo and Wettstein [61] for the Shapley value to communication graph games.

  6. 6.

    Another application of union stable systems can be found in Algaba et al. [19].

  7. 7.

    Algaba et al. [2] inductively apply the union stability operator to any network structure \(\mathscr {G} \subseteq 2^N\) which always ends up in a union stable system.

  8. 8.

    From a computational point of view, Algaba et al. [8] study the complexity of the Myerson value for games on a union stable system by means of the Harsanyi dividends. Polynomial time algorithms for computing the Myerson value in weighted voting games restricted by a tree are given in Fernández et al. [41].

  9. 9.

    The Myerson value for games on a union stable system is a particular case of the class of Harsanyi power solutions for games on a union stable system, as introduced by Algaba et al. [11], which generalizes the class of Harsanyi power solutions for communication graph games presented by van den Brink et al. [77]. Moreover, the class of Harsanyi power solutions has been studied, for a particular case of union stable systems derived from the family of winning coalitions associated with a voting game, in Algaba et al. [19]. In fact, this setting allows for studying situations in which there exists a feedback between the economic influence of each coalition of agents and its political power.

  10. 10.

    Other characterizations of the Myerson value for games on a union stable system can be found in Algaba et al. [10].

  11. 11.

    In van den Brink et al. [76] union closed systems are considered where the only requirement of a set of feasible coalitions is that it is union closed. They exploit the property that in such network structures every feasible coalition has a unique largest feasible subset.

  12. 12.

    Core properties are considered in Derks and Gilles [34].

  13. 13.

    Axiomatizations of the two permission values using conjunctive, respectively disjunctive, fairness together with efficiency, additivity, the inessential player property, the necessary player property and (weak) structural monotonicity can be found in van den Brink [66], van den Brink [67]. These axioms have a natural interpretation in several applications such as, for example, polluted river problems of Ni and Wang [56] and Dong et al. [37], see van den Brink et al. [80].

  14. 14.

    The disjunctive fairness axiom is stronger and also requires an equal change in the payoff of every ‘complete superior’ of the predecessor on the arc, being those superiors that are on every path from a top player to this predecessor.

  15. 15.

    Similar as mentioned for disjunctive fairness in the previous footnote, the full axiom also requires equal changes on the payoffs of every ‘complete superior’ of these other predecessors.

  16. 16.

    Games on intersection closed set systems are studied in Beal et al. [23]. Intersection closedness is one of the characterizing properties of the important network structures called convex geometries. Convex geometries are a combinatorial abstraction of convex sets introduced by Edelman and Jamison [38]. A network structure \(\mathscr {G} \subseteq 2^N\) is a convex geometry if it satisfies the following properties: (1) (feasible empty set) \(\emptyset \in \mathscr {G}\), (2) (intersection closed) \(S,T\in \mathscr {G}\) implies that \(S\cap T\in \mathscr {G}\), and (3) (augmentation’) \(S\in \mathscr {G}\) with S ≠ N, implies that there exists i ∈ N ∖ S such that \(S\cup \{i\}\in \mathscr {G}\). (In Sect. 5, we consider another augmentation property.)

  17. 17.

    Although van den Brink et al. [79] consider permission tree games, they distinguish solutions that are based on a communication or hierarchy approach. For cycle-free communication graph games, Demange [32] introduces the so-called hierarchical outcomes, that are extreme points of the core of the restricted game if the game is superadditive and the graph is cycle-free. Nonemptiness of the core of a superadditive, cycle-free graph game was shown, independently, in Demange [31] and Le Breton et al. [51]. Herings et al. [48] consider the average of the hierarchical outcomes.

  18. 18.

    The interior operator is characterized by: (1) \(int_{\mathscr {A}}(\emptyset )=\emptyset \), (2) \(int_{\mathscr {A}}(S)\subseteq S\), (3) if S ⊆ T then \(int_{\mathscr {A}}(S)\subseteq int_{\mathscr {A}}(T)\), (4) \(int_{\mathscr {A}}(int_{\mathscr {A}}(S))=int_{\mathscr {A}}(S)\), and (5) if \(i,j\in int_{\mathscr {A}}(S)\) and \(j\in int_{\mathscr {A}}(S\setminus \{i\})\) then \(i\notin int_{\mathscr {A}}(S\setminus \{j\})\).

  19. 19.

    In antimatroid \(\mathscr {A}\), the path S is covered by path T if S ⊂ T with |T| = |S| + 1, and the unique extreme player of T is the player in T ∖ S. For a coalition S to be deleted leaving behind an antimatroid, the deleted coalition should be a path (otherwise, union closedness will be violated) that is not covered by a path (otherwise accessibility will be violated, since if path S is covered by path T ⊃ S with |T| = |S| + 1, after deleting S from the antimatroid, T has no any extreme player).

  20. 20.

    One can see that applying this fairness to the sets \(\varPhi ^c_D\), respectively \(\varPhi ^d_D\), of a game with permission structure gives the corresponding conjunctive, respectively disjunctive, fairness as follows. Deleting an arc (i, j), with |P D(j)|≥ 2, in a permission structure leads to more (respectively less) feasible coalitions in \(\varPhi ^c_D\) (respectively \(\varPhi ^d_D\)) and every coalition that is ‘gained’ (respectively ‘lost’) contains player j and all other predecessors h ∈ P D(j) ∖{i} (respectively player i).

  21. 21.

    They characterize the Shapley value for games on an antimatroid by this fairness axiom together with axioms generalizing efficiency, additivity, the inessential player property and the necessary player property for games with a permission structure as mentioned in Footnote 11.

  22. 22.

    For results of games on antimatroids we refer to Algaba et al. [5, 6]. For antimatroids that are not normal, similar results can be stated restricted to the class of players that belong to at least one feasible coalition.

  23. 23.

    A graph is cycle-complete if, whenever there is a cycle, the subgraph on that cycle is complete.

  24. 24.

    The context makes clear if we consider paths in a communication graph or paths in an antimatroid.

  25. 25.

    Notice the difference with the augmentation property that is used as a defining property of a convex geometry, see Footnote 14.

  26. 26.

    Applications of weighted graphs, considering weights on links as well as on nodes can be found in, for example, Lindelauf et al. [52], who measure the importance of terrorists based on their centrality in a terrorist network.

  27. 27.

    The other characterizing properties are that ρ(S) ⊆ S for all S ⊆ N, and ρ(S) ⊆ ρ(T) for all S ⊆ T ⊆ N.

  28. 28.

    For example, in the line permission tree D = {(1,  2),  (2,  3)} on N = {1,  2,  3} the feasible part of coalition {2,  3} is singleton {3} since its direct predecessor is in the coalition, while the predecessor of 2 is not in the coalition. In this case ρ({1,  2,  3}) = {1,  2,  3}, ρ({1,  2} = {1,  2}, ρ({1,  3}) = ρ({1}) = {1}, ρ({2,  3}) = {3} and ρ({2}) = ρ({3}) = ∅. Thus ρ(ρ({2,  3})) = ρ({3}) = ∅ ≠ {3} = ρ({2,  3}).

  29. 29.

    Although they are not games with a permission structure, the airport games of Littlechild and Owen [53], respectively, the joint liability problems of Dehez and Ferey [30] are the duals of auction games, respectively polluted river games. This also makes the graph game approach useful to these applications, in particular for self-dual solutions such as the Shapley value. A comparison between these four applications, based on anti-duality relations, can be found in Oishi et al. [58].

  30. 30.

    The introduction of specific networks as, for instance, coloured networks in the setting of TU-games in Algaba et al. [16] allows us to analyze profit sharing problems in an intermodal transport system. More general networks are the so-called labelled networks as presented by Algaba et al. [17] which curiously coincide with the set of museum pass games Ginsburgh and Zang [45] as shown in Algaba et al. [18].

  31. 31.

    Specifically, they satisfy (1) feasible empty set, (2) augmentation’, and (3) a weaker intersection property requiring that \(S,T\in \mathscr {F}\) with S ∪ T ≠ N, implies that \(S\cap T\in \mathscr {F}\).

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Acknowledgements

Authors are grateful to the editors for the invitation to participate in this volume and two anonymous referees for their revision. Financial support from the Ministerio de Ciencia, Innovación y Universidades (MCIU), the Agencia Estatal de Investigación (AEI) and the Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-097965-B-I00 is gratefully acknowledged.

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Authors and Affiliations

  1. Matemática Aplicada II and Instituto de Matemáticas de la Universidad de Sevilla (IMUS), Escuela Superior de Ingenieros, Sevilla, Spain

    Encarnación Algaba

  2. Department of Economics and Tinbergen Institute, VU University Amsterdam, Amsterdam, The Netherlands

    René van den Brink

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  1. Encarnación Algaba
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  2. René van den Brink
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Correspondence to René van den Brink .

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  1. St. Petersburg State University, St. Petersburg, Russia

    Leon A. Petrosyan

  2. Institute of Applied Mathematical Research, Russian Academy of Sciences, Petrozavodsk, Russia

    Vladimir V. Mazalov

  3. Graduate School of Management, St. Petersburg State University, St. Petersburg, Russia

    Nikolay A. Zenkevich

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Algaba, E., van den Brink, R. (2021). Networks, Communication and Hierarchy: Applications to Cooperative Games. In: Petrosyan, L.A., Mazalov, V.V., Zenkevich, N.A. (eds) Frontiers of Dynamic Games. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-93616-7_1

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