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Path-Disruption Games: Bribery and a Probabilistic Model

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Abstract

Path-disruption games, recently introduced by Bachrach and Porat, are coalitional games played on graphs where one or multiple adversaries each seeks to reach a given target vertex from a given source vertex, while a coalition of agents seeks to prevent that from happening by blocking every path from the source to the target for each adversary. These coalitional games model, for instance, security issues in computer networks. Inspired by bribery in voting, we introduce the notion of bribery for path-disruption games. We analyze the question of how hard it is to decide whether the adversaries can bribe some of the agents such that no coalition will form that blocks all paths for them. We show that this problem is NP-complete for a single adversary and complete for \({\Sigma }^{\mathrm {P}}_{2} = \text {NP}^{\text {NP}}\), the second level of the polynomial hierarchy, for the case of multiple adversaries. We also expand the model by allowing uncertainty about the targets: In probabilistic path-disruption games, we assign to each vertex the probability that an adversary wants to reach it, and we study the complexity of problems related to common solution concepts (such as the core and the ε-core) and other properties of such games.

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Notes

  1. Two players i and j in a coalitional game G = (N, v) are said to be symmetric if for all coalitions \(C \subseteq N \smallsetminus \{i,j\}\), we have v(C ∪ {i}) = v(C ∪ {j}).

  2. MCVC, a decision problem mentioned by Bachrach and Porat [5], is defined as follows: Given a graph G = (U, E), m vertex pairs (s j , t j ), 1 ≤ j ≤ m, a weight function \(w:U\to \mathbb {Q}_{\geq 0}\), and a bound \(K\in \mathbb {Q}_{\geq 0}\), does there exist a subset U ′⊆U such that \({\sum }_{u\in U^{\prime }} w(u)\leq K\) and the induced subgraph \(G|_{U\smallsetminus U^{\prime }}\) contains no path linking a pair (s j , t j ), 1 ≤ j ≤ m? It is known that MCVC belongs to P for problem instances with m < 3, yet is NP-complete for problem instances with m ≥ 3. The related optimization problem for m < 3 can be solved in polynomial time using the same algorithm as the decision problem with a corresponding output (see also Dahlhaus et al. [18]).

  3. By “possible target” we mean each vertex that has a positive probability of being a target.

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Acknowledgments

We are grateful to the journal reviewers for their helpful comments, and we also thank the reviewers for the conferences ADT’11, ECAI’12, STAIRS’12, and AAMAS’14 again for their helpful comments on previous versions of parts of this paper. This work was supported in part by Deutsche Forschungsgemeinschaft grants RO 1202/12-1, RO-1202/14-1, and RO-1202/14-2, the European Science Foundation’s EUROCORES program LogICCC, and by COST Action IC1205 on Computational Social Choice. This work was done in part while the second author was visiting Stanford University and University of Rochester, and he thanks the hosts, Yoav Shoham and Lane A. Hemaspaandra, for their warm hospitality.

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  1. Institut für Informatik, Heinrich-Heine-Universität Düsseldorf, 40225, Düsseldorf, Germany

    Anja Rey & Jörg Rothe

  2. Computer Science Department, Stanford University, Stanford, CA, 94305, USA

    Adrian Marple

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  1. Anja Rey
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Preliminary versions of parts of this paper appear in the proceedings of the 2nd International Conference on Algorithmic Decision Theory (ADT’11) [47], the 20th European Conference on Artificial Intelligence (ECAI’12) and the 6th European Starting AI Researcher Symposium (STAIRS’12) [48], and the 13th International Conference on Autonomous Agents and Multiagent Systems (AAMAS’14) [36].

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Rey, A., Rothe, J. & Marple, A. Path-Disruption Games: Bribery and a Probabilistic Model. Theory Comput Syst 60, 222–252 (2017). https://doi.org/10.1007/s00224-016-9669-1

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