Theory of Computing Systems

, Volume 60, Issue 2, pp 222–252

Path-Disruption Games: Bribery and a Probabilistic Model

Article

DOI: 10.1007/s00224-016-9669-1

Cite this article as:
Rey, A., Rothe, J. & Marple, A. Theory Comput Syst (2017) 60: 222. doi:10.1007/s00224-016-9669-1
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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.

Keywords

Game theory Path-disruption games Bribery Complexity 

1 Introduction

Mice love cheese. Suppose a mouse is sitting in the basement of a big house and the cheese is in the pantry, so the mouse wants to get there through a labyrinthine warren of mouse burrows, passages, and holes in the walls. However, whenever it comes out of a hole and enters a room, the mouse is in danger, as a large population of cats lives in this house as well. Cats love to play with mice, and their goal is to block every single path from the basement to the pantry.

Bachrach and Porat [5] introduce these cat-and-mouse games as path-disruption games, coalitional games in which agents (cats) are placed on the vertices of a graph (corresponding to the rooms in the house; the burrows and passages in the walls correspond to the edges of the graph) and form coalitions (subsets of the cat population). A coalition wins the game if it can block every path from a given source vertex (the basement) to a given target vertex (the pantry). They also consider variants of this game, by allowing vertex costs and multiple adversaries, each represented by a pair of a source and a target vertex. Real-world applications include security issues in computer networks where a coalition of agents seeks to prevent, by blocking all paths, a malicious intruder reaching a target computer from a source computer (see also the closely related zero-sum security games on graphs studied by Jain et al. [32], and the network security games studied by, e.g., Washburn and Wood [62]; Smith and Lim [59]). In computer science, such situations may also occur in the field of multiagent systems (see, e.g., Shoham and Leyton-Brown [58]).

The computational analysis of social-choice-theoretic scenarios (a vibrant field known as computational social choice, see, e.g., Brandt et al. [15, 16] and Rothe [53]) and game-theoretic scenarios (a field known as algorithmic game theory, see, e.g., Nisan et al. [40]; Chalkiadakis et al. [17]; Faliszewski et al. [30]; Elkind and Rothe [21]) have become areas of increasing interest in recent years. In particular, coalitional games such as weighted voting games [4, 22, 45, 49], network flow games [6, 7, 46], path coalitional games [3], hedonic games [2, 19, 50, 51, 61, 64, 65], etc. have been analyzed from a computational complexity point of view. Uncertainty has been studied in strategic games with regard to noise on both the intruder and the defender side [67]. Hespanha et al. [31] study probabilistic pursuit-evasion games. Here, probability is related to finding an evader at a certain point of time.

In cooperative game theory, a key question is to analyze the stability of games, that is, to determine which coalitions will form and how to divide the payoff within a coalition (see, e.g., Bachrach et al. [9], for the cost of stability in coalitional games). Now suppose that the mice from our previous example found a box of cat food and place some of it in some of the rooms. Sure enough, if a cat agent is no longer hungry after being bribed by enough cat food, it will let the mouse pass the room it is guarding. However, different amounts of cat food may be required for different cats to be sated and the mice have only a limited budget available to bribe them, so they need to think hard about where to place the cat food in order to safely move to their targets. Inspired by bribery in the context of voting (see, e.g., Faliszewski et al. [27, 28]), we introduce the notion of bribery in path-disruption games. While in computational social choice bribery has been studied intensely (see, e.g., Elkind et al. [23]; Schlotter et al. [56]; Faliszewski et al. [29]; Baumeister and Rothe [11]; Faliszewski and Rothe [26]), and has been adapted to other fields such as judgement aggregation as well (see, e.g., Endriss et al. [24, 25] and Baumeister et al. [13, 14]), we are not aware of a bribery model in the context of cooperative games.

In voting, bribery is distinguished from manipulation and control scenarios: While in manipulation some of the voters themselves change their ballots strategically in order to obtain a desired outcome and while in control the election chair modifies the structure of an election (e.g., by adding or deleting candidates or voters) in order to reach a certain goal, in bribery an external agent tries to change the outcome to her advantage by paying a certain amount of money, without exceeding a given budget, to some of the voters for them to change their ballots. Adapting bribery to our setting of path-disruption games, note that the agents (the cats, in our example) collaborate while, at the same time, they want to win against their adversary (the mouse) who can actively interfere with the situation by seeking to bribe some of the agents. In this sense, this game combines aspects of both cooperative and noncooperative game theory. Relatedly, Bachrach et al. [8, 10] study the reliability of players in cooperative games. In the case of no costs for blocking a vertex, this problem is related to well-studied graph problems such as generalized connectivity problems (see, e.g., Alon et al. [1]; Li and Mao [35]).

We will analyze the complexity of the problem of whether the adversaries in a path-disruption game with costs can bribe some of the agents such that no coalition will be formed that prevents the adversaries from reaching their targets. We show that this problem is NP-complete for a single adversary. For the case of multiple adversaries, we show that this problem is complete for \({{\Sigma }_{2}^{p}} = \text {NP}^{\text {NP}}\), the second level of the polynomial hierarchy [38, 60]. These hardness results can be seen as complexity shields against bribery: Although bribery is possible in general for path-disruption games, it may be intractable for the adversary to find out whether it is successful in a given game.

Furthermore, we expand the model of path-disruption game by allowing uncertainty about the target vertices. Suppose some cheese can be found not only in the pantry, but also in other rooms of the house. Then the cats don’t know for sure where the mouse is heading to and which paths to block. Rather, every room in the house is a potential target that the mouse seeks to reach with a certain given probability.

Organization

In Section 2, we provide some preliminaries and formally define path-disruption games. Section 3 introduces a more general class of games, the probabilistic path-disruption games, and discusses some of their properties. In Section 4, we introduce the notion of bribery in path-disruption games and present our complexity results for the corresponding problems. In Section 5, we study stability concepts and other game-theoretic properties for probabilistic path-disruption games. Finally, conclusions and future work are given in Section 6 (see Table 1 in this section for a summary of results).

2 Preliminaries

In this section, we present the needed notions from cooperative game theory, graph theory, and complexity theory in Section 2.1 and the formal model of path-disruption game in Section 2.2.

2.1 Basic Notions of Cooperative Game Theory and Complexity Theory

A coalitional game with transferable utilities\(\mathcal {G}=(N,v)\) consists of a (finite) set of players, N, and a coalitional function, \(v:2^{N}\to \mathbb {R}\). Every subset CN is called a coalition. When considering a multiagent application (see, e.g., Shoham and Leyton-Brown [58]), players in a coalitional game are often referred to as agents. Here, the terms agent and player are used synonymously.

A coalitional game \(\mathcal {G}=(N,v)\) is called increasing or monotonic if v(A) ≤ v(B) for ABN. \(\mathcal {G}\) is said to be simple if it is monotonic and each coalition CN either wins (v(C) = 1) or loses (v(C) = 0) the game, i.e., v(C) ∈ {0,1} for all CN. For more background on these fundamental notions of cooperative game theory, we refer the reader to the textbooks by Chalkiadakis et al. [17], Osborne and Rubinstein [41], Peleg and Sudhölter [43], and Shoham and Leyton-Brown [58] and to the book chapter by Elkind and Rothe [21]. Common solution concepts and related problems in the context of probabilistic path-disruption games will be defined in Section 5.

Let \(\mathbb {R}\), \(\mathbb {R}_{\geq 0}\), and \(\mathbb {Q}_{\geq 0}\) denote the set of real numbers, nonnegative real numbers, and nonnegative rational numbers, respectively. Let \(\mathbb {N}\smallsetminus \{0\} = \{1, 2, \ldots \}\) denote the set of positive integers.

A graph G = (U, E) can be either directed or undirected. We analyze path-disruption games on undirected graphs, as this is the more demanding case regarding the computational hardness results (even though, of course, this is not the case for upper bounds): Given an undirected graph, we can simply reduce the problem to the more general case of a directed graph by substituting each undirected edge {x, y} by the two directed edges (x, y) and (y, x). Given a graph G = (U, E), we denote the subgraph induced by a subset of edgesEE by \( G|_{E^{\prime }} = (U,E^{\prime }) \) and the subgraph induced by a subset of verticesUU by \(G|_{U^{\prime }} = (U^{\prime },\{\{x,y\}\in E\,\mid \: x\in U^{\prime }\text { and } y\in U^{\prime }\})\).

We assume the reader is familiar with the standard notions and notation of complexity theory, such as the complexity classes P and NP, the notion of (polynomial-time many-one) reducibility, denoted by \({\leq _{\mathrm {m}}^{\mathrm {p}}}\), and hardness and completeness with respect to \({\leq _{\mathrm {m}}^{\mathrm {p}}}\). \({{\Sigma }_{2}^{p}}={\text {NP}}^{{\text {NP}}}\) and \({{\Pi }_{2}^{p}}={\text {coNP}}^{{\text {NP}}}\) are the second level of the polynomial hierarchy (see Meyer and Stockmeyer [38]; Stockmeyer [60]). We make use of the well-known quantifier characterization of \({{\Sigma }^{\mathrm {P}}_{2}}\) [38, 66]: A problem A is in Σ2P if and only if there exists a set B ∈ P and a polynomial p such that for each input x ∈ Σ,
$$x\in A\iff (\exists y\in{\Sigma}^{\ast}:|y|\leq p(|x|))\, (\forall z\in{\Sigma}^{\ast}:|z|\leq p(|x|))\, [(x,y,z)\in B], $$
where Σ denotes the set of all strings over some alphabet Σ and |s| denotes the length of a string s ∈ Σ. For further reading, we refer to the textbooks by Papadimitriou [42] and Rothe [52].

2.2 Path-Disruption Games

Following Bachrach and Porat [5], we define several types of path-disruption games on graphs. Given a graph G = (U, E) with n = ∥U∥ vertices, each agent iN = {1,…, n} represents vertex ui. Moreover, there are m adversaries who each want to travel from a source vertex sj in U (renaming some of the members of U) to a target vertex tj in U, 1 ≤ jm. We say a coalition CNblocks a path fromsj to tj if there is no path from sj to tj in the induced subgraph \(G|_{U\smallsetminus \{u_{i}\,\mid \: i\in C\}}\) or if sj or tj are not even in \(U\smallsetminus \{u_{i}\,\mid \: i\in C\}\). Four types of path-disruption games are distinguished: those with a single adversary and with multiple adversaries, and for both with and without costs. We denote path-disruption games with costs by PDGC, and path-disruption games without costs by PDG. The most general game is the model with several adversarial players and costs for each vertex to be blocked. Letting \(c:U\to \mathbb {R}_{\geq 0}\) be a function defining the cost for each vertex to be blocked, \(c(C)={\sum }_{i\in C}c(u_{i})\) denotes coalition C’s cost in total for each CN.

Definition 1

Given an undirected graph G = (U, E) with n = ∥U∥ vertices and with an adversary associated with each of the pairs (s1, t1),…,(sm, tm), a cost function \(c:U\to \mathbb {R}_{\geq 0}\), a reward \(R\in \mathbb {R}_{\geq 0}\), a PDGC-Multiple is defined by players N = {1,…, n}, where i represents ui, 1 ≤ in, and the coalitional function
$$\begin{array}{@{}rcl@{}} v(C)&=&\left\{\begin{array}{ll} R-\mu(C) &\text{if } \mu(C)<\infty \\ 0 & \text{otherwise} \end{array}\right. \end{array} $$
with
$$\begin{array}{@{}rcl@{}} \mu(C)&=&\left\{\begin{array}{ll} \min\{c(B) \,\mid \: B\subseteq C\text{ and } \tilde{v}(B)=1\}& \text{if } \tilde{v}(C)=1 \\ \infty & \text{otherwise}, \end{array}\right. \end{array} $$
where
$$\begin{array}{@{}rcl@{}} \tilde{v}(C) &=&\left\{\begin{array}{ll} 1 & \text{if }C\text{ blocks each path from }s_{j}\text{ to }t_{j}, \text{for each } j,\,1\leq j\leq m \\ 0 & \text{otherwise.} \end{array}\right. \end{array} $$

Letting m = 1, we have a restriction to a single adversary, namely PDGC-Single. Letting c(ui) = 0 for all i, 1 ≤ in, R = 1, and, thus, \(v(C)=\tilde {v}(C)\), the simple games without costs, PDG-Multiple and PDG-Single, are defined. We say a coalition CNwins the game if \(\tilde {v}(C)=1\), and loses it otherwise.

In Definition 1, weights and bounds are real numbers; however, to make the problems for these games suitable for computer processing (and to define their complexity in a reasonable way), we will henceforth assume that all weights and bounds are rational numbers.

Example 1

As an example, consider the following game with five players in N = {1,2,3,4,5}, one adversary, and without costs, played on the graph given in Fig. 1.
Fig. 1

A single-adversary path-disruption game without costs

It can be seen that, for instance, the coalition consisting of player 5 alone is not successful, but coalitions {2,3} and {1} are.

Adding costs to this setting, e.g., c(u1) = 4, c(u2) = 1, c(u3) = 2, c(u4) = 3, c(u5) = 2, and a reward R = 4, {1} on its own does not have a positive value any more, but any coalition including {2, 3} does.

3 Probabilistic Path-Disruption Games

We now expand the setting of path-disruption games by introducing uncertainty to the targets. First, we will define probabilistic path-disruption games formally and then we will study the complexity of a number of their game-theoretic properties.

In path-disruption games as defined in Section 2.2, it is assumed that an adversary has one certain target. More generally, we consider the case where the actual target is unknown, though we have probabilities that indicate where the intruder might go. Let us define the notion of probabilistic path-disruption game in its most general variant first, with costs and multiple adversaries.

Definition 2

We are given an undirected graph G = (U, E) with n vertices and m adversaries, each sitting on a given vertex sjU, 1 ≤ jm. Let U = {u1,…, un} (some vertices are additionally labeled by sj, 1 ≤ jm), and consider every ui as a potential target vertex. Let pj, i be the probability that adversary j (placed on sj) wants to reach ui, where \({\sum }_{i=1}^{n} p_{j,i} = 1\) for each j, 1 ≤ jm. Further, we are given a cost function \(c: U \to \mathbb {R}_{\geq 0}\), and a reward R. Using this domain, we define a probabilistic path-disruption game with costs (PPDGC) and multiple adversaries as follows. Let N = {1,…, n} be the set of agents, where i represents ui. To define the coalitional function v, let CN be a coalition of agents (i.e., each iC is placed on ui) and let
$$\begin{array}{@{}rcl@{}} v(C) = \tilde{v}(C)\cdot(R-\mu(C)) \end{array} $$
with
$$\begin{array}{@{}rcl@{}} \mu(C) = \left\{\begin{array}{ll} \min\{c(B) \,\mid \: B \subseteq C \text{ and } \tilde{v}(B)=\tilde{v}(C)\} &\text{if }\tilde{v}(C) > 0,\\ - 1 & \text{otherwise}, \end{array}\right. \end{array} $$
where
$$\begin{array}{@{}rcl@{}} \tilde{v}(C) =\prod\limits_{j=1}^{m}\sum\limits_{i=1}^{n} p_{j,i} \cdot w(C,j,i) \end{array} $$
and
$$\begin{array}{@{}rcl@{}} w(C,j,i) = \left\{\begin{array}{ll} 1 & \text{if }C\text{ blocks each path from }s_{j}\text{ to }t_{i},\\ 0 & \text{otherwise}. \end{array}\right. \end{array} $$
Note that if \(\tilde {v}(C)=0\) then the minimal costs do not influence v(C), so they can be any number. If for each j, 1 ≤ im, there exists exactly one i, 1 ≤ in, such that pj, i = 1 (and we thus have pj, k = 0 for all ki), we obtain the multiple-adversary PDGCs by Bachrach and Porat [5] as defined in Definition 1. The probabilistic analogues of their other three variants of path-disruption games are defined as follows. A PPDG with multiple adversaries and without costs is defined as above, except that neither a cost function nor a reward is given and the coalitional function itself is defined by
$$v(C) = \prod\limits_{j=1}^{m}\sum\limits_{i=1}^{n} p_{j,i} \cdot w(C,j,i). $$
The models with a single adversary with or without costs are obtained from the above two variants by setting m = 1.

Example 2

Consider the same graph as in Fig. 1, except that now the adversary’s target is uncertain. We only know that remaining at the source or moving to the destination u2 is impossible (i.e., has zero probability), moving to u4 is most likely (with a probability of 0.5), and the other two vertices are equally likely targets (with a probability of 0.25 each). This example is visualized in Fig. 2.
Fig. 2

A single-adversary probabilistic path-disruption game without costs

Now, considering the same costs and the same reward as in Example 1 (i.e., c(u1) = 4, c(u2) = 1, c(u3) = 2, c(u4) = 3, c(u5) = 2, and R = 4), player 5, for instance, has a more central role with \(v(\{5\})=\frac {1}{4}\cdot (4-2)=\frac {1}{2} \). The values of {1} and {2,3} remain zero and one, respectively. Coalition {3,4} even has a negative value.

Having defined a new type of game, it is natural to ask about its basic game-theoretic properties. First, note that probabilistic path-disruption games (even without costs) are not simple, as soon as one of the given probabilities pj, i is strictly between 0 and 1. Second, Bachrach and Porat [5] show that path-disruption games without costs are monotonic, and we observe that their proof easily extends to the probabilistic case. In addition, we show below that not even the original path-disruption games with costs are monotonic.

Proposition 1

PPDGs without costs are monotonic, whereas PDGCs and PPDGCs in general are not.

Proof

In a given PPDG without costs, for all i, 1 ≤ in, for all j, 1 ≤ jm, and for all A and B, ABN, it holds that w(A, j, i) ≤ w(B, j, i), since a coalition can never block fewer paths than any of its subcoalitions. Thus
$$v(A)=\prod\limits_{j=1}^{m}\sum\limits_{i=1}^{n} p_{j,i} \cdot w(A,j,i) \leq \prod\limits_{j=1}^{m}\sum\limits_{i=1}^{n} p_{j,i} \cdot w(B,j,i) =v(B). $$
Nonmonotonicity of the cost case can be shown by the following single-adversary example. Let G = ({u1,…, u4},{{u1, u2},{u1, u4},{u2, u3},{u3, u4}}), s = u1, p1,4 = 1, c(ui) = 1 for all i, 1 ≤ i ≤ 4, and R = 1. Although {2}⊆{2,3}, it holds that v({2})=0>−1 = v({2,3}). □

Next, we turn to three other basic properties of coalitional games.

Definition 3

A coalitional game with transferable utility \({{\mathcal {G}}=(N,v)}\) is said to be
  1. 1.

    a constant-sum game if it satisfies that \(v(C)+v(N\smallsetminus C) = v(N)\) for each coalition CN;

     
  2. 2.

    superadditive if it satisfies that v(CD) ≥ v(C) + v(D) for each two disjoint coalitions C, DN;

     
  3. 3.

    convex if its coalitional function v is supermodular, i.e., if it satisfies that v(CD) + v(CD) ≥ v(C) + v(D) for each two coalitions C, DN.

     

In particular, the grand coalition can be expected to form in a superadditive game because any two coalitions can merge without loss. However, superadditive games can still have an empty core. The core is a central stability concept for cooperative games (to be considered in Section 5.2 in more detail). Intuitively, if the core of a game is empty then the game is unstable: There are players who have an incentive to deviate from the grand coalition. On the other hand, every convex game is a superadditive game whose core is guaranteed to be nonempty [57].

Proposition 2

Neither path-disruption games nor probabilistic path-disruption games, not even in the simplest case (without costs and for a single adversary), are constant-sum, convex, or superadditive.

Proof

It is enough to prove these three claims for path-disruption games, even without costs and with only a single adversary, and the corresponding claims for the other cases follow immediately.

By the following counterexample such a game is not a constant-sum game. Consider the graph given in Fig. 3a, and the game on that graph with six players, without costs, with a reward of 1, and with a single adversary wishing to travel from s to t. Observe that v(C) = 1, but also v(N) = 1 and \(v(N\smallsetminus C)=1\) for the coalition C with two players as shown in the figure, so \(v(C)+v(N\smallsetminus C) = 1+1 \neq 1 = v(N)\).

By the same counterexample, it holds that v(CD) = 1<1+1 = v(C) + v(D) for the disjoint coalitions C and D shown in Fig. 3a. Hence, path-disruption games (even without cost and with a single adversary) are not superadditive, and thus they are not convex either. □

Fig. 3

Counterexamples for the proofs of Proposition 2 and Theorem 1

The previous two propositions show that path-disruption games and probabilistic path-disruption games share all properties considered there, which raises the question of whether the probabilistic case really provides novel features. To be specific, given a probabilistic path-disruption game, we are interested in whether we can always find a (nonprobabilistic) path-disruption game that is strategically equivalent in the following sense: Two coalitional games \({{\mathcal {G}}=(N,v)}\) and \({\mathcal {G}}^{\prime }=(N,v^{\prime })\) are strategically equivalent if there exist α>0 and \(\beta :N\to \mathbb {R}\) such that \(v^{\prime }(C)=\alpha v(C)+{\sum }_{i\in C}\beta (i)\) holds for each CN. The following proposition shows that path-disruption games and probabilistic path-disruption games are not strategically equivalent in general, and thus the latter are a reasonable new model.

Theorem 1

There exists a probabilistic path-disruption game without costs and with a single adversary that is not strategically equivalent to any nonprobabilistic path-disruption game (with the same number of players, without costs, and with a single adversary).

Proof

Consider the game \({{\mathcal {G}}=(N,v)}\) with three players, played on the graph in Fig. 3b without costs and a reward of 1. The coalitional function of this game is
with 2 and 3 being symmetric players.1 Assume that there exists a nonprobabilistic path-disruption game \({\mathcal {G}}^{\prime }=(N,v^{\prime })\) without costs and with a single adversary such that \({\mathcal {G}}^{\prime }\) and \({\mathcal {G}}\) are strategically equivalent. By definition, there exist α>0 and \(\beta :N\to \mathbb {R}\) such that \(v(C)=\alpha v^{\prime }(C)+{\sum }_{i\in C}\beta (i)\) holds for each coalition CN. Consider the following two cases.
Case 1:
Let s = u1 be the same starting point of the adversary. The coalitional function then is
with a, b, c ∈ {0,1} and ca and cb by monotonicity. The equations 1 = α⋅1 + β(1), 1 = α⋅1 + β(1) + β(2), and 1 = α⋅1 + β(1) + β(3) imply that β(2) = β(3)=0. Therefore, we obtain a = b = c = 1 and \(\alpha =\frac {1}{2}\). This, however, contradicts 1 = α⋅1 + β(2) + β(3).
Case 2:
Let the adversary start at a different vertex, say at s = u2. In this case, the coalitional function is
with a, b, c ∈ {0,1} and ca and cb by monotonicity. From \(\frac {1}{2}=\alpha \cdot 1+\beta (2)\) and 1 = α⋅1 + β(1) + β(2) we obtain \(\beta (1)=\frac {1}{2}\). Therefore, we have a contradiction from 1 = α⋅1 + β(2) + β(3) and \(1=\alpha \cdot 1+\frac {1}{2}+\beta (2)+\beta (3)\).

This completes the proof. □

Bachrach and Porat [5] define various problems for path-disruption games, each related to some game-theoretic notion (see, e.g., Chalkiadakis et al. [17]; Elkind and Rothe [21]; Peleg and Sudhölter [43]; Shoham and Leyton-Brown [58]), and study their complexity. We obtain more general analogous problems for probabilistic path-disruption games, so any lower bound for the more special variant of a problem immediately is inherited by its generalized variant. On the other hand, upper bounds known for problems on nonprobabilistic path-disruption games may be invalid for their more general analogues, or if they are valid, they might be harder to prove. We define these problems for probabilistic path-disruption games and the notions they are based on, and we discuss their complexity in Section 5. We start here by analyzing how hard it is to compute the value of a coalition in a given (compactly represented) probabilistic path-disruption game. Although the model is more general, we can reduce this problem to that in the original (nonprobabilistic) setting in polynomial time.

Theorem 2

Given a single-adversary PPDGC and a coalition, its value can be computed in polynomial time.

Proof

We show this rather unexpected proposition by the following construction: Given a PPDGC \({\mathcal {G}}\) with a single adversary, consisting of G = (U, E), sU, \(c:U\to \mathbb {Q}_{\geq 0}\), \(R\in \mathbb {Q}_{\geq 0}\), and p1, i, 1 ≤ in, and given a coalition CN, computing \(\tilde {v}(C)\) involves at most n computations of w(C,1, i) which, in turn, can be determined in polynomial time using a graph accessibility algorithm. Either \(\tilde {v}(C)=0\), then we can return 0 as the value of C; or, \(\tilde {v}(C)>0\). Then we consider the graph G = (U, E) with U = U ∪ {un+1} and
$$E^{\prime} = E\cup\{\{u_{i},u_{n+1}\}\,\mid \: 1\leq i\leq n\text{ with } p_{1,i}>0\text{ and }w(C,1,i)=1\}. $$
Let t = un+1. Define a new cost function \(c^{\prime }:U\to \mathbb {Q}_{\geq 0}\), by setting c(ui) = c(ui) if iC, and \(c^{\prime }(u_{i})=1+{\sum }_{i\in C}c(u_{i})\) otherwise. The latter is a sufficiently high rational value that, intuitively, can be read as “infinite costs.” Now, we determine the minimal costs κ (regarding the cost function c) needed to disrupt all paths from s to t in G. This can be done in polynomial time using the known algorithm for the problem MultipairCut with Vertex Costs (MCVC, for short) for m = 1, which runs in polynomial time.2
In order to calculate \(v(C)=\tilde {v}(C)\cdot (R-\kappa )\), we now show that κ = μ(C). By construction of G, C blocks all paths from s to t. Since all other vertices have greater costs, the vertices with minimal costs κ correspond to the players in C. It holds that
$$\begin{array}{@{}rcl@{}} \mu(C)=\min\{c(B)\,\mid \: B\subseteq C \text{ and } \tilde{v}(B)=\tilde{v}(C)\}, \end{array} $$
which is equal to the minimum costs of a coalition BC that blocks the same possible targets as C.3 Since t is only connected to the possible targets blocked by C, this is equal to
$$\min\{c(B)\,\mid \: B\subseteq C\text{ and }B \text{ blocks all paths from }s \text{ to } t\text{ in }G^{\prime}\}, $$
which in turn is equal to κ by definition. □

Using Theorem 2, the fact that \(\tilde {v}(C)\) can be computed in polynomial time for a coalition C even for multiple adversaries, and a corresponding result for PDGC-Multiple by Bachrach and Porat [5], we obtain the following.

Corollary 1

In a PPDG without costs, a coalition’s value can be determined in polynomial time, but it is NP-hard to decide whether the value of a coalition is greater than a given value, for the case of multiple adversaries with costs.

4 Bribery in Path-Disruption Games with Costs

In this section, we introduce the notion of bribery in path-disruption games and study its complexity both for a single adversary and for multiple adversaries for the cost case. For alternative approaches on how bribery might be defined for cooperative games, see Section 6.

4.1 Definitions

Given a PDG-Single or PDGC-Single, can the adversary who wants to travel from source s to target t, bribe a coalition BN of agents such that no coalition CN will be formed that blocks each path from s to t? What if there are multiple adversaries? There are several possibilities to define the related decision problems. Considering the simplest form of path-disruption game (with a single adversary, without costs, and with constant prices for each agent and an infinite budget for the adversary), the answer is yes if and only if (G, s, t) ∈ GAP, where GAP is the graph accessibilty problem (see Savitch [55]; Jones [33]): Given a graph G and two distinct vertices, a source vertex s and a target vertex t, can t be reached via a path from s? This problem can be solved in nondeterministic logarithmic space (and thus in deterministic polynomial time). The equivalence holds, since bribery of all agents on a path from s to t (if there exists one) will guarantee the adversary a safe travel.

If, on the other hand, the number of agents the adversary can bribe is limited by a number k, bribery is possible if and only if there is a path from s to t with length at most k, which is decidable in polynomial time.

Therefore, we consider bribery only for path-disruption games with costs in the following. Here, in contrast to the no-cost case, even if a limited budget may not allow bribing the players on each vertex in any path from the source to the target, successful bribery might still be possible. The formal definition of this problem for a single adversary is:

PDGC-Single-Bribery

Given:

A PDGC-Single \({\mathcal {G}=(N,v)}\), a price function

 

\(\pi :U\to \mathbb {Q}_{\geq 0}\), and a budget \(K\in \mathbb {Q}_{\geq 0}\).

Question:

Is there a coalition BN such that

 

\({\sum }_{i\in B} \pi (u_{i})\leq K\) and no coalition \(C\!\subseteq \! N\!\smallsetminus \!B\)

 

has a value v(C) > 0?

Bribery in the multiple-adversary case can be defined analogously:

PDGC-Multiple-Bribery

Given:

A PDGC-Multiple \({\mathcal {G}=(N,v)}\), a price function

 

\(\pi :U\to \mathbb {Q}_{\geq 0}\), and a budget \(K\in \mathbb {Q}_{\geq 0}\).

Question:

Is there a coalition BN such that

 

\({\sum }_{i\in B} \pi (u_{i})\leq K\) and no coalition \(C\!\subseteq \! N\!\smallsetminus \!B\)

 

has a value v(C)>0?

Example 3

Consider the same graph, adversary, and costs as in Example 1 and Fig. 1. Moreover, let π(u1) = 3 and π(u2) = π(u3) = π(u4) = π(u5) = 1 be the vertices’ prices and K = 1 the briber’s budget. In this example, bribery is not successful, since either v({4})=1 or v({2,3})=1. If K = 2 and the prices are the same, however, bribery is successful, e.g., for the coalition {2,4}.

Let PPDGC-Single-Bribery and PPDGC-Multiple-Bribery denote the corresponding problems where the input game is a PPDGC with single and multiple adversaries, respectively.

In the following two sections, we give our complexity results for the bribery problems in path-disruption games with a single adversary and with multiple adversaries.

4.2 Complexity of Bribery in Single-Adversary Path-Disruption Games

Theorem 3 classifies PDGC-Single-Bribery in terms of its complexity.

Theorem 3

PDGC-Single-Briberyis NP-complete.

Proof

First we show that the problem is in NP. Given a PDGC consisting of a graph G = (U, E), a cost function \(c:U\to \mathbb {Q}_{\geq 0}\), a reward \(R\in \mathbb {Q}_{\geq 0}\), a source and a target vertex, s, tU, a price function \(\pi :U\to \mathbb {Q}_{\geq 0}\), and a bound \(K\in \mathbb {Q}_{\geq 0}\), we can nondeterministically guess a coalition BN, N = {1,…, n}, n = ∥U∥. Obviously, it can be tested in polynomial time whether \({\sum }_{i\in B}\pi (u_{i})\leq K\). If this inequality fails to hold, bribery of B is not possible. Otherwise, we need to test whether it holds for all coalitions \(C\subseteq N\smallsetminus {B}\) that v(C) ≤ 0. That is the case if and only if either \(\tilde {v}(C)=0\) or Rμ(C) < . We can test this property by the following algorithm. Let \(c^{\prime }:U\to \mathbb {Q}_{\geq 0}\) be a new cost function with
$$\begin{array}{@{}rcl@{}} c^{\prime}(u_{i})=\left\{\begin{array}{ll} c(u_{i})&\text{if }i\notin B\\ R & \text{if }i\in B. \end{array}\right. \end{array} $$
Note that c can be constructed in polynomial time. Determine the minimal cost κ needed to separate s from t regarding c. This can be done by means of the polynomial-time algorithm solving the problem MCVC (recall Footnote 2 in the proof of Theorem 2).
If κR, we have that for all \(C\subseteq N\smallsetminus {B}\),
$$\begin{array}{@{}rcl@{}} v(C)=\left\{\begin{array}{ll} R-\kappa\leq 0&\text{if }\tilde{v}(C)=1\\ 0&\text{if }\tilde{v}(C)=0. \end{array}\right. \end{array} $$

Thus, for all \(C\subseteq N\smallsetminus {B}\), the coalitional function is at most 0 and bribery is possible. If, on the other hand, κ < R, then there exists a minimal winning coalition CN with μ(C) = κ, v(C) = Rκ>0. Since we defined c(ui) = R for all iB, C is a subset of \(N\smallsetminus B\). Therefore, bribery of B is not possible.

Next we show that PDGC-Single-Bribery is NP-hard. We prove this by means of a reduction from the NP-complete problem Partition that is based on the reduction Partition\({\leq _{\mathrm {m}}^{\mathrm {p}}}\)MaxCut by Karp [34]. We start with this first step in order to obtain a restricted variant of MaxCut, from which onwards we can perform the second step of the reduction.

Partition is the problem of deciding, given a nonempty sequence of positive integers A = (a1,…, an) such that \({\sum }_{i=1}^{n} a_{i}\) is even, whether there is a subset A⊆{1,…, n} such that \({\sum }_{i\in A^{\prime }} a_{i} = {\sum }_{i\in \{1,\dots ,n\}\setminus A^{\prime }} a_{i}\). MaxCut is also a partitioning problem, but now the question is whether the vertex set U of a given graph G = (U, E) with edge weights \(w:E\to \mathbb {N}\smallsetminus \{0\}\) can be partitioned into two disjoint vertex subsets U1, U2U such that the total weight of the edges crossing this cut \({\sum }_{\{x,y\}\in E,x\in U_{1},y\in U_{2}}w(\{x,y\})\) is at least as large as a given value \(K\in \mathbb {N}\smallsetminus \{0\}\).

The reduction works as follows. Given an instance A = (a1, a2,…, am) of Partition, create the following MaxCut instance: G = (U, E), where U = {u1, u2,…, um} and E = {{ui, uj} ∣ ui, ujU, ij}, \(w:E^{\prime }\to \mathbb {N}\smallsetminus \{0\}\) with w({ui, uj}) = aiaj, and \(K=\frac {S^{2}}{4}\) with \({S={\sum }_{i=1}^{m} a_{i}}\). Obviously, the MaxCut property is satisfied if and only if A belongs to Partition.

Next, given A and G, we create the following instance X of PDGC-Single-Bribery. The path-disruption game consists of graph G = (U, E), where
$$\begin{array}{@{}rcl@{}} U&=&U^{\prime}\cup\{u_{m+1},u_{m+2}\} \cup\{u_{m+2+i},u_{2m+2+i}\,\mid \: 1\leq i\leq m\}\\ &&\cup\left\{u_{3m+2+j}\,\middle| \: e_{j}\in E^{\prime}, 1\leq j\leq \frac{m(m-1)}{2}\right\},\\ E&=&\{\{u,u_{3m+2+j}\},\{u_{3m+2+j},v\}\,\mid \: \{u,v\}=e_{j}\in E^{\prime}\}\\ &&\cup\{\{u_{m+1},u_{m+2+i}\},\{u_{m+2+i},u_{i}\}\,\mid \: 1\leq i\leq m\}\\ &&\cup\{\{u_{i},u_{2m+2+i}\},\{u_{2m+2+i},u_{m+2}\}\,\mid \: 1\leq i\leq m\}, \end{array} $$
and furthermore of source vertex s = um+1, target vertex t = um+2, reward \(R=\left (\frac {S^{2}}{2}\right )+S\), and cost function \(c:U\to \mathbb {Q}_{\geq 0}\), defined by
$$\begin{array}{@{}rcl@{}} c(u_{i})=\left\{\begin{array}{ll} R&\text{if }1\leq i\leq m+2\\ a_{j}&\text{if }m+3\leq i\leq 2m+2,\,i=m+2+j\\ a_{j}\cdot\left(\frac{S}{2}+1\right) &\text{if }2m+3\leq i\leq 3m+2,\,i=2m+2+j\\ w(e_{j})&\text{if } 3m+3\leq i\leq n,\,i=3m+2+j \end{array}\right. \end{array} $$
with \(n=3m+2+\frac {m(m-1)}{2}\). Moreover, let \(K=\frac {S}{2}\) and let the price function \(\pi :U\to \mathbb {Q}_{\geq 0}\) be defined by
$$\begin{array}{@{}rcl@{}} \pi(u_{i})=\left\{\begin{array}{ll} K+1&\text{if }1\leq i\leq m+2\\ a_{j}&\text{if }m+3\leq i\leq 2m+2,\,i=m+2+j\\ K+1&\text{if }2m+3\leq i\leq n. \end{array}\right. \end{array} $$
Figure 4 illustrates this construction, where square vertices denote the source and the target and diamond-shaped vertices have prices below the budget.
Fig. 4

Construction of the PDGC-Single-Bribery instance X

We claim that
$$ A\in{\textsc{Partition}}\iff X\in{\textsc{PDGC-Single-Bribery}}. $$
(1)
Only if: Suppose APartition. Then there is a subset A⊆{1,…, m} with
$$\sum\limits_{i\in A^{\prime}}a_{i} = \sum\limits_{i\in \{1,\dots,m\}\smallsetminus A^{\prime}}a_{i}=\frac{S}{2}. $$
We show that bribery is possible for coalition
$$B=\{m+2+i\,\mid \: i\in A^{\prime}\}\subseteq N. $$
First, note that
$$\sum\limits_{m+2+i\in B}\pi(u_{m+2+i})=\sum\limits_{i\in A^{\prime}}\pi(u_{m+2+i})=\sum\limits_{i\in A^{\prime}}a_{i}=\frac{S}{2}= K. $$
Second, we need to prove that for each coalition \(C\subseteq N\smallsetminus {B}\), v(C) ≤ 0. Let C be an arbitrary coalition of \(N\smallsetminus B\). If \(\tilde {v}(C)=0\), then v(C) = 0 by definition. Otherwise, C contains a minimal winning subcoalition CC with \(\tilde {v}(C^{\prime })=1\) and \(\mu (C)={\sum }_{i\in C^{\prime }}c(u_{i})\).
If C contains an agent situated on a vertex in {u1,…, um+2}, then μ(C) ≥ R, so v(C) ≤ 0. Thus we may assume that C∩{1,…, m+2} = . C must contain {2m+2 + iiA}; otherwise, a path from s = um+1 over um+2 + i, ui, and u2m+2 + i to t = um+2 for an iA, is not blocked. For all i, \(i\in \{1,\dots ,m\}\smallsetminus A^{\prime }\), we have that m+2 + i or 2m+2 + i has to be in C. Define \(\tilde {A}_{1} = \{i\mid i\in A\smallsetminus A^{\prime },\, 2m+2+i\in C^{\prime }\}\), \(x = {\sum }_{i\in \tilde {A}_{1}}a_{i}\leq \frac {S}{2}\), and let \(\tilde {A}_{2}\) be the set containing the remaining \(i\notin A^{\prime }\cup \tilde {A}_{1}\). Consequently, \(\{m+2+i\,\mid \: i\in \tilde {A}_{2}\}\subseteq C^{\prime }\). If \(\tilde {A}_{2}=\emptyset \), then C = {2m+2 + i ∣ 1 ≤ im} with \({\sum }_{i\in C^{\prime }}c(u_{i})=S\cdot (\frac {S}{2}+1)=R\). So assume that \(\tilde {A}_{2}\neq \emptyset \). If \(\tilde {A}_{1}=\emptyset \), then \(\{m+2+i\,\mid \: i\in \{1,\dots ,m\}\smallsetminus A^{\prime }\}\subseteq C^{\prime }\). C is a minimal winning coalition if and only if additionally \(\{3m+2+j\,\mid \: e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime },\, j_{1}\in A^{\prime }, j_{2}\notin A^{\prime }\}\) are in C. Thus
$$\begin{array}{@{}rcl@{}} \mu(C)&=&\sum\limits_{i\in A^{\prime}}c(u_{2m+2+i}) +\sum\limits_{i\notin A^{\prime}}c(u_{m+2+i}) +\sum\limits_{j_{1}\in A^{\prime}} {\sum}_{\overset{j_{2}\notin A^{\prime}}{e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime}}}c(u_{3m+2+j})\\ &=&\sum\limits_{i\in A^{\prime}}a_{i}\cdot\left(\frac{S}{2}+1\right) +\sum\limits_{i\notin A^{\prime}}a_{i} +\sum\limits_{j_{1}\in A^{\prime}} \sum\limits_{\overset{j_{2}\notin A^{\prime}}{e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime}}} w(e_{j})\\ &=&\frac{S}{2}\cdot\left(\frac{S}{2}+1 \right) +\frac{S}{2} +\sum\limits_{j_{1}\in A^{\prime}}\sum\limits_{j_{2}\notin A^{\prime}} a_{j_{1}}\cdot a_{j_{2}}\\ &=&\frac{S^{2}}{4}+S+\frac{S}{2}\cdot\frac{S}{2} =\frac{S^{2}}{2}+S=R. \end{array} $$
Assume that \(\tilde {A}_{1}\neq \emptyset \). In order to block all paths, it must be the case that
$$\{3m+2+j\,\mid \: e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime},\, j_{1}\in A^{\prime}, j_{2}\in\tilde{A}_{2}\}\subseteq C^{\prime} $$
and
$$\{3m+2+j\,\mid \: e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime},\, j_{1}\in\tilde{A}_{1}, j_{2}\in\tilde{A}_{2}\}\subseteq C^{\prime}. $$
C is not minimal if it contains both m+2 + i and 2m+2 + i for some i, 1 ≤ im. If this were the case for some \(i\in \tilde {A}_{1}\), then either the same subset of {3m+2 + j, ejE} would be in C, which would make m+2 + i redundant; or we would have
$$\begin{array}{@{}rcl@{}} \{3m+2+j\,\mid \: e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime},\, j_{1}\in A^{\prime}, j_{2}\in\tilde{A}_{2}\} & \subseteq & C^{\prime},\\ \{3m+2+j\,\mid \: e_{j}=\{u_{j_{1}},u_{i}\}\in E^{\prime},\, j_{1}\in A^{\prime}\} & \subseteq & C^{\prime},\\ \{3m+2+j\,\mid \: e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime},\, j_{1}\in\tilde{A}_{2}, j_{2}\in\tilde{A}_{1},\,j_{2}\neq i\} & \subseteq & C^{\prime}, \text{ and}\\ \{3m+2+j\,\mid \: e_{j}=\{u_{i},u_{j_{2}}\}\in E^{\prime},\, j_{2}\in\tilde{A}_{1}\} & \subseteq & C^{\prime}, \end{array} $$
which makes blocking of 2m+3 + i unnecessary and is in fact the same case as \(\tilde {A}_{1}=\tilde {A}_{1}\smallsetminus \{i\}\). Thus we have
$$\begin{array}{@{}rcl@{}} \mu(C)-R&=& \sum\limits_{i\in A^{\prime}}c(u_{2m+2+i}) +\sum\limits_{i\in\tilde{A}_{1}}c(u_{2m+2+i}) +\sum\limits_{i\in\tilde{A}_{2}}c(u_{m+2+i})\\ &&+\sum\limits_{j_{1}\in A^{\prime}} \sum\limits_{\overset{{j_{2}}\in\tilde{A}_{2}}{e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime}}} c(u_{3m+2+j})\\ &&+\sum\limits_{{j_{1}}\in\tilde{A}_{1}} \sum\limits_{\overset{{j_{2}}\in\tilde{A}_{2}}{e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime}}} c(u_{3m+2+j}) -\frac{S^{2}}{2}-S \end{array} $$
$$\begin{array}{@{}rcl@{}} &=& \sum\limits_{i\in A^{\prime}}a_{i}\cdot\left(\frac{S}{2}+1\right) +\sum\limits_{i\in\tilde{A}_{1}}a_{i}\cdot\left(\frac{S}{2}+1\right) +\sum\limits_{i\in\tilde{A}_{2}}a_{i}\\ &&+\sum\limits_{{j_{1}}\in A^{\prime}} \sum\limits_{\overset{{j_{2}}\in\tilde{A}_{2}}{e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime}}} w(e_{j}) +\sum\limits_{{j_{1}}\in\tilde{A}_{1}} \sum\limits_{\overset{{j_{2}}\in\tilde{A}_{2}}{e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime}}} w(e_{j})\\ &&-\frac{S^{2}}{2}-S\\ &=&\frac{S}{2}\cdot\left(\frac{S}{2}+1 \right) +x\cdot\left(\frac{S}{2}+1 \right) +\left(\frac{S}{2}-x\right) +\sum\limits_{{j_{1}}\in A^{\prime}}\sum\limits_{{j_{2}}\in\tilde{A}_{2}} a_{j_{1}}a_{j_{2}}\\ &&+\sum\limits_{{j_{1}}\in\tilde{A}_{1}}\sum\limits_{{j_{2}}\in\tilde{A}_{2}} a_{j_{1}}a_{j_{2}} -\frac{S^{2}}{2}-S\\ &=&\frac{S^{2}}{4}+S+x\cdot\frac{S}{2} +\frac{S}{2}\cdot\left(\frac{S}{2}-x\right) +x\cdot\left(\frac{S}{2}-x\right) -\frac{S^{2}}{2}-S\\ &=&-x^{2}+\frac{S}{2}x =-x\left(x-\frac{S}{2}\right), \end{array} $$
so μ(C)−R is a function of x. For each x with \(0\leq x\leq \frac {S}{2}\), it holds that μ(C)−R ≥ 0. Therefore, bribery is possible.
If: Suppose that X belongs to PDGC-Single-Bribery. Then there exists a coalition BN with
$$ \sum\limits_{i\in B}\pi(u_{i})\leq K $$
(2)
and for all coalitions \(C\subseteq N\smallsetminus {B}\), we have that
$$ \text{either }\tilde{v}(C)=0\text{ or }\mu(C)\geq R. $$
(3)
Since all other vertices have a price greater than K, B is a subset of {m+3,…,2m+2}. Assume that B = . Then \(C=\{m+3,\dots ,2m+2\}\subseteq N\smallsetminus B\) is a minimal winning coalition with \(\tilde {v}(C)=1\) and
$$\mu(C)=\sum\limits_{i=1}^{m} c(u_{m+2+i})=\sum\limits_{i=1}^{m} a_{i}= S < \frac{S^{2}}{2} +S = R. $$
That is a contradiction to Condition (3). On the other hand, Condition (2) implies that
$$\sum\limits_{i\in B}\pi(u_{i})=\sum\limits_{m+2+i\in B}a_{i}\leq K=\frac{S}{2}, $$
and, in particular, B ≠ {m+3,…,2m+2}. This leads to the following two cases.
Case 1:
\({\sum }_{i\in B}\pi (u_{i})<K=\frac {S}{2}\). Let x denote
$$\sum\limits_{i\in B}\pi(u_{i})=\sum\limits_{m+2+i\in B}a_{i}=\sum\limits_{i\in B}c(u_{i}), $$
so \(0<x<\frac {S}{2}\). Then
$$\begin{array}{@{}rcl@{}} C & = & \{2m+2+i\,\mid \: m+2+i\in B\}\cup\\ && \{m+2+i\,\mid \: 1\leq i\leq m, m+2+i\notin B\}\cup \\ && \{3m+2+j\,\mid \: e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime},\, m+2+j_{1}\in B, m+2+j_{2}\notin B\} \end{array} $$
is a winning coalition in \(N\smallsetminus B\). Note that no subcoalition of C is successful, which implies that \(\mu (C)={\sum }_{i\in C}c(u_{i})\). Therefore, C satisfies
$$\begin{array}{@{}rcl@{}} \mu(C)-R&=&\sum\limits_{i\in C}c(u_{i})-\frac{S^{2}}{2}-S\\ &=&\sum\limits_{m+2+i\in B}c(u_{2m+2+i}) +\sum\limits_{\overset{i=1}{m+2+i\notin B}}^{m} c(u_{m+2+i})\\ &&+\sum\limits_{m+2+j_{1}\in B}\sum\limits_{\overset{m+2+j_{2}\notin B}{e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime}}}c(u_{3m+2+j}) -\frac{S^{2}}{2}-S\\ &=&x\cdot\left(\frac{S}{2}+1\right)+(S-x)-\frac{S^{2}}{2}-S\\ &&+\sum\limits_{m+2+j_{1}\in B}\sum\limits_{\overset{m+2+j_{2}\notin B}{e_{j}=\{u_{j_{1}},u_{j_{2}}\}\in E^{\prime}}}w(e_{j})\\ &=&x\cdot\frac{S}{2}-\frac{S^{2}}{2}+x\cdot(S-x)\\ &=&-x^{2}+\frac{3S}{2}x-\frac{S^{2}}{2}\\ &=&-(x-S)\left(x-\frac{S}{2}\right) \end{array} $$
where the penultimate equation holds by the definition of c. For x with \(0<x<\frac {S}{2}\), it holds that μ(C)−R < 0, which again is a contradiction to Condition (3).
Case 2:
\({\sum }_{i\in B}\pi (u_{i})=K\). That is,
$$\sum\limits_{m+2+i\in B}a_{i}=\frac{S}{2}. $$
Thus there exists a partition into A = {aim+2 + iB} and \(\{1,\dots ,m\}\smallsetminus A^{\prime }\) with
$$\sum\limits_{a_{i}\in A^{\prime}}a_{i}=\sum\limits_{a_{i}\in \{1,\dots,m\}\smallsetminus A^{\prime}}a_{i}=\frac{S}{2}. $$

This concludes the proof of (1). The observation that the construction described above can be done in polynomial time completes the proof of the theorem. □

4.3 Complexity of Bribery in Multiple-Adversary Path-Disruption Games

The complexity of PDGC-Multiple-Bribery is classified in Theorem 4.

Theorem 4

PDGC-Multiple-Briberyis\({{\Sigma }_{2}^{p}}\)-complete.

Proof

PDGC-Multiple-Bribery belongs to \({{\Sigma }_{2}^{p}}={\text {NP}}^{{\text {NP}}}\), since given an instance X consisting of graph G, adversaries (sj, tj), 1 ≤ jm, cost function c, reward R, price function π, and budget K, X is a positive instance if and only if
$$(\exists B\subseteq N)(\forall C\subseteq N\smallsetminus{B})(\forall D\subseteq C) \left[\sum\limits_{i\in B}\pi(u_{i})\leq K \text{ and } v(D)\leq0\right], $$
which is equivalent to
$$(\exists B\subseteq N)(\forall D\subseteq N\smallsetminus B) \left[\sum\limits_{i\in B}\pi(u_{i})\leq K \text{ and } \left(\tilde{v}(D)=0 \text{ or } \sum\limits_{i\in D}c(u_{i})\geq R\right)\right]. $$
The property in brackets can obviously be tested in polynomial time. Thus the problem satisfies the quantifier characterization of \({{\Sigma }_{2}^{p}}\).

In order to show Σ2P-hardness, we reduce from the well-known \({{\Sigma }_{2}^{p}}\)-complete problem QBF2 (see Meyer and Stockmeyer [38]; Stockmeyer [60]), which asks whether a given quantified boolean formula F = (∃X) (∀Y) f(X, Y) is valid, where X is a set of p boolean variables, Y is a set of q boolean variables, and f(X, Y) is a disjunction of k implicants, \(f(X,Y) = \bigvee _{i=1}^{k} (u_{i} \wedge v_{i} \wedge w_{i})\), and ui, vi, and wi, 1 ≤ ik, are literals over XY. That is, we ask whether there exists an assignment to the variables of X such that for all variable assignments to Y, f evaluates to true. Note that we assume, without loss of generality, that every implicant has exactly three literals.

The graph G for the game that is part of the PDGC-Multiple-Bribery instance to be constructed from F, is built from the three graphs, G1, G2, and G3, shown in Fig. 5. Again, let square vertices denote the source and target vertices, and let diamond-shaped vertices denote those with prices below the budget. In particular, G is constructed from G1, G2, and G3 by identifying, for each occurrence of a literal ui, vi, or wi in f, the vertex in G3 representing this literal with the vertex representing the corresponding variable (xX or yY) or its negation (¬x or ¬y) in G1 or G2. For instance, if the ith implicant occurring in f is (x1y2∧¬x3), then ui = x1, vi = y2. and wi = ¬x3, that is, x1 is connected to ai and \(a_{i}^{\prime }\), y2 is connected to bi and \(b_{i}^{\prime }\), and ¬x3 is connected to ci and \(c_{i}^{\prime }\).
Fig. 5

Three graphs for the reduction proving Theorem 4

The players on vertices in G1 labeled with X variables or their negations (diamond shape) are bribable for a price of 1 but have 0 cost (they are free to participate in a coalition if not bribed), sources sj and targets tj (rectangle shape) have a cost of 6k + q+1 and a price of p+1, and all other vertices (circle shape) have cost 1 and a price of p+1. Let K = p be the briber’s budget and R = 6k + q+1 be the reward. Thus the adversaries can bribe up to p players having price 1 (note that all other players—those not on vertices in G1 labeled with X variables or their negations—are too expensive to bribe).

We show that bribery is possible if and only if the original quantified formula F is valid.

A coalition C can only have a positive value if it contains a successful subcoalition with at most 6k + q players other than those on vertices labeled xX or ¬x that are not bribed, and with no player on a source or target vertex, as their costs are too high to allow μ(C) < R.

Intuitively, the purpose of graph G1 in Fig. 5a is to enforce consistency on the part of the briber, the purpose of G2 in Fig. 5b is for consistency of the coalition, and the purpose of G3 in Fig. 5c is to enforce the implicants.

Consider the case where the briber does not play consistently, i.e., either plays (a) neither x nor ¬x or (b) both x and ¬x, for an xX. In case (b), since the number of bribable players is limited by p, there is some other xX such that case (a) holds. In case (a), a coalition C consisting of the players corresponding to x and ¬x, to any consistent assignment to the variables in Y, and to vertices ai, \(a^{\prime }_{i}\), bi, \(b^{\prime }_{i}\), ci, and \(c^{\prime }_{i}\) for all i, 1 ≤ ik, can form to block all paths from sj to tj for 1 ≤ j ≤ 7. Since this sums up to μ(C) = 6k + q < R (i.e., v(C)>0), inconsistent bribery must fail.

Now assume that the briber plays consistently. In this case, the coalition must include all players on vertices di, 1 ≤ i ≤ 2k; otherwise, there would be a path from s1 to t1. Note that, by construction, for all i, 1 ≤ ik, at least two players on vertices in {ai, bi, ci} must be part of the blocking coalition, since otherwise there is either a path from s4 to t4 or from s5 to t5.

Likewise, at least two players on vertices in \(\{a^{\prime }_{i}, b^{\prime }_{i}, c^{\prime }_{i}\}\) must participate in a blocking coalition due to paths from s6 to t6 or from s7 to t7. Also notice that if the player on ai is not in the blocking coalition, the ones on \(b^{\prime }_{i}\) and \(c^{\prime }_{i}\) must be because of paths from s3 to t3 (and again symmetric statements can be made for \(a^{\prime }_{i}\), bi, etc.).

Furthermore, for each yY, either the player on y or ¬y must be part of the blocking coalition; otherwise, there is a path from s2 to t2.

Altogether, this forces the coalition to include 6k + q vertices with cost 1 each (circle shape), leaving for each i, 1 ≤ ik, two players on one of \(\{a_{i}, a^{\prime }_{i}\}\), \(\{b_{i}, b^{\prime }_{i}\}\), or \(\{c_{i},c^{\prime }_{i}\}\) out of the blocking coalition, and for each yY, a player on one of y or ¬y out as well. Therefore, the blocking coalition represents a consistent assignment to the variables in Y and, for each implicant, cancels the effect of two out of three of its literals.

In this case, the briber can only succeed if for some implicant all three literals are satisfied, no matter what assignment the coalition chooses. This, of course, happens if and only if the original quantified boolean formula F is valid. □

For probabilistic path-disruption games, the lower bounds are implied by Theorems 3 and 4.

Similarly to the nonprobabilistic case, bribery is possible in a given instance of PPDGC-Multiple-Bribery if and only if
$$(\exists B\subseteq N) (\forall D\subseteq N\smallsetminus B) \left[\sum\limits_{i\in B}\pi(u_{i})\leq K\text{ and } \left(\tilde v(D)=0\text{ or } c(D)\geq R \right) \right]. $$
Thus the upper bounds remain valid with similar arguments as above.

Corollary 2

For bribery in probabilistic path-disruption games the following holds:
  1. 1.

    PPDGC-Single-Briberyis NP-complete and

     
  2. 2.

    PPDGC-Multiple-Briberyis Σ2P-complete.

     

5 Stability

In this section, we analyze verification and existence problems of stability concepts in probabilistic path-disruption games. See Table 1 for a summary of results as well as a comparison with known results for the original model.

5.1 Veto Players

A player of high significance in a game is a veto player: No coalition has a positive value without it. Questions of interest include:
  1. 1.

    testing whether a given player has the veto property,

     
  2. 2.

    testing whether there exist any veto players,

     
  3. 3.

    the counting problem asking how many veto players there are, and

     
  4. 4.

    the corresponding search problem where the task is to find the veto players.

     

We can show that in a PPDG without costs, a player i is a veto player if and only if it is placed on a vertex with pj, i = 1. If at least two vertices have a positive probability of being a target, a player on any of these vertices can be part of a coalition that has a positive value without the other players necessarily being contained. Thus we can decide in polynomial time whether a given player in a given PPDG without costs is a veto player; hence, all veto players can be found in polynomial time. The role of the players placed on the adversaries source vertices is similar to that of a veto player: Every coalition CN that contains all players sitting on source vertices has value v(C) = 1. Thus the general model does not yield a higher complexity than the original model. For PPDGCs, these problems are most likely less efficient to solve, since monotonicity cannot be utilized here. Deciding whether a given player is a veto player is in coNP in this case.

5.2 Core

The perhaps most popular solution concept is the core of a game, the set of all payoff vectors (i.e., a distribution of a game’s total payoff among the players) that stabilize the game, i.e., the payoff of each coalition is no less than its value. In particular, the grand coalition is always successful for path-disruption games without costs. Therefore, it makes sense to seek to stabilize the coalition structure consisting of the grand coalition. The following central problems related to the core are commonly studied:
  1. 1.

    Given a game and a payoff vector, is it in the core of the game?

     
  2. 2.

    Is the core of a given game empty?

     
  3. 3.

    Given a game, compute its core.

     

Proposition 3

For a given single-adversary PPDG without costs, these three questions can be solved in polynomial time.

Proof

Observe that the core of a PPDG with a single adversary and without costs is nonempty if and only if an agent placed on a vertex is a veto player. Moreover, in this case, the core consists of only one element. If there is a small probability for at least one other target than the certain target vertex itself, the core is empty. Hence, the core can be computed in polynomial time, and it thus can be decided in polynomial time whether the core is empty, and also whether a given payoff vector belongs to it. □

In this respect, the model of PPDG behaves like a simple game, even though in general it does not. In the multiple-adversary and no-cost case, for a fixed number m of adversaries, deciding whether a payoff vector is in the core of a given PPDG can also be done in polynomial time. On the other hand, if m is not fixed, this cannot be shown straightforwardly. In contrast to the original (nonprobabilistic) model of PDG, we suspect this problem (in the no-cost and multiple-adversary case) to be coNP-complete. Even with costs the upper bound holds.

Proposition 4

The problem of deciding whether a given payoff vector is in the core of a PPDGC is in coNP.

Proof

Let q be a given payoff vector, and let a PPDGC be given as described in Definition 2. Note that for each coalition CN, there exists a coalition CC with μ(C) = μ(C) = c(C) ≤ c(C). Therefore,
$$\begin{array}{@{}rcl@{}} R-c(C) & \leq & R-\mu(C) = R-c(C^{\prime}) = v(C^{\prime}) \text{ and }\\ \mathbf{q}(C^{\prime}) & \leq & \mathbf{q}(C). \end{array} $$
Consequently, q being in the core of the game implies that
$$R-c(C)\leq R-\mu(C)=v(C)\leq \mathbf{q}(C), $$
for each CN. If q is not in the core of the game, there exists a coalition CN with v(C)>q(C). For the corresponding CN, it holds that
$$R-c(C^{\prime})=R-\mu(C)=v(C)>\mathbf{q}(C)\geq \mathbf{q}(C^{\prime}). $$
Thus we only need to test whether Rc(C) ≤ q(C) for all coalitions CN, which can be done in coNP. □

5.3 ε-Core

A weaker form of the core is the ε-core of a game, where a certain deficit not exceeding a bound ε is allowed. Maschler et al. [37] introduce the least core of a game as its minimal nonempty ε-core. Let d(C) = v(C)−q denote the deficit of a coalition C, for a payoff vector q. Note that the least core of a coalitional game is never empty. Problems of interest here are:
  1. 1.

    Given a game \(\mathcal {G}\), a payoff vector q and a rational bound ε, is the maximal deficit at most ε, or, equivalently, is q in the ε-core of \(\mathcal {G}\)?

     
  2. 2.

    Compute the least core of a given game.

     
An imputation is a payoff vector q = (q1,…, qn) satisfying efficiency (i.e., \({\sum }_{i=1}^{n} q_{i} = v(N)\)) and individual rationality (i.e., qiv({i}) for each iN). If only imputations are allowed in the ε-core (as, e.g., Bachrach and Porat [5] require in their definition of the least core), then the least core of a PPDG with a single adversary and without costs is equal to its core, and thus computable in polynomial time. In general, this does not hold.

Theorem 5

For multiple adversaries and with or without costs, it is coNP-complete to decide whether a given payoff vector is in the ε-core of a given probabilistic path-disruption game for a given ε.

Proof

Testing whether maxCNd(C) ≤ ε is equivalent to testing if for every coalition CN it holds that q(C) ≥ v(C)−ε. Thus, in order to solve the complement of our problem in NP, we can guess a coalition CN nondeterministically and test in polynomial time (see Theorem 2 and Corollary 1) whether q(C) < v(C)−ε. This shows membership in coNP.

We prove coNP-hardness by means of a reduction from \(\overline {\textsc {MCVC}}\) (again, see Footnote 2 in the proof of Theorem 2). Without loss of generality, we can assume that the bound K and the vertex weights w(u), uU, in an MCVC instance are natural numbers, since in the reduction from MaxCut to MultiterminalCut by Dahlhaus et al. [18, Theorem 3] weights and bounds are also natural numbers. Given such an MCVC instance X consisting of a graph G = (U, E) and m vertex pairs (sj, tj), sj, tjU, weight function w, and a bound K, we construct an instance with the same graph G, adversaries sitting on sj, 1 ≤ jm, and probabilities pj, i = 1 if ui = tj for 1 ≤ jm and uiU, and pj, i = 0 otherwise. Moreover, we have
$$\mathbf{q}=\left(\frac{w(u_{1})}{{\sum}_{i=1}^{n} w(u_{i})},\dots, \frac{w(u_{n})}{{\sum}_{i=1}^{n} w(u_{i})}\right) \quad \text{and}\quad \varepsilon=1-\frac{K+\frac{1}{2}}{{\sum}_{i=1}^{n} w(u_{i})}. $$
Obviously, this construction can be done in polynomial time. Note that q is a pre-imputation (i.e., a payoff vector satisfying efficiency—\({\sum }_{i=1}^{n} q_{i}=v(N)=1\)—though possibly not individual rationality). We now verify that the given instance X is not in MCVC if and only if q belongs to the ε-core of the constructed game.
Only if: Suppose that the given instance does not belong to MCVC, that is, for all subsets UU blocking all paths from sj to tj, 1 ≤ jm, it holds that \({\sum }_{u_{\ell }\in U^{\prime }}w(u_{\ell })>K\). By construction and the condition that all weights and K are natural numbers, it follows that \({\sum }_{u_{\ell }\in U^{\prime }}w(u_{\ell }) > K+\frac {1}{2}\). Thus, for all coalitions CN with a positive value (that is, for each adversary j, 1 ≤ jm, C blocks each path from sj to the only possible target), it holds that
$$\sum\limits_{\ell\in C} q_{\ell} =\frac{1}{{\sum}_{i=1}^{n} w(u_{i})}\sum\limits_{\ell\in C}w(u_{\ell})\ >\ \frac{K+\frac{1}{2}}{{\sum}_{i=1}^{n} w(u_{i})}\ =\ 1-\varepsilon. $$
This means that
$$d(C)\ =\ v(C)-\mathbf{q}(C)\ =\ 1-\sum\limits_{\ell\in C}q_{\ell}\ <\ 1-(1-\varepsilon)=\varepsilon. $$
Since the grand coalition N with d(N) = 0 has a positive value and each coalition that cannot disrupt all adversaries has a deficit at most 0, the maximal deficit is less than ε, so in the constructed instance q belongs to the ε-core, as desired.
If: Let the maximal deficit of a coalition in the game be at most ε, that is, for all coalitions CN it holds that d(C) ≤ ε. In particular, for each coalition with a positive value, we have v(C)−q(C) = 1−q(C) ≤ ε. Thus, for all subsets of vertices UU blocking each path from sj to tj, 1 ≤ jm, we have \({\sum }_{u_{\ell }\in U^{\prime }}q_{\ell } \geq 1-\varepsilon \), which implies
$$\sum\limits_{u_{\ell}\in U^{\prime}}\frac{w(u_{\ell})}{{\sum}_{i=1}^{n} w(u_{i})}\ \geq\ 1-\left(1-\frac{K+\frac{1}{2}}{{\sum}_{i=1}^{n} w(u_{i})}\right) \ =\ \frac{K+\frac{1}{2}}{{\sum}_{i=1}^{n} w(u_{i})}, $$
which, in turn, implies \({\sum }_{u_{\ell }\in U^{\prime }}w(u_{\ell })\geq K+\frac {1}{2} > K\). Thus there is no subset of vertices satisfying the conditions of MCVC.

For the case with costs, note that the coNP-hardness lower bound is trivially inherited from the case without costs. On the other hand, the coNP upper bound can be shown similarly as in the proof of Proposition 4. □

This problem becomes solvable in polynomial time if there are no costs and the game’s domain is restricted to be a complete graph.

5.4 Dummy Players

A player of little significance in a game is a dummy player. There are different interpretations of what that means and, thus, different definitions. One notion is: A player is said to be a dummy if adding her does not change the value of any coalition at all (see, e.g., Dubey and Shapley [20]). Such a player is sometimes also referred to as a null player [43]. A second notion in the literature is: A player is a dummy if adding her changes the value of each coalition only to her own value. Formally, the first notion says that a player iN in a game \(\mathcal {G}=(N,v)\) is a dummy player if v(C ∪ {i})−v(C) = 0 for each CN, and the second notion says that iN is a dummy player if v(C ∪ {i})−v(C) = v({i}) for each CN (see, e.g., Shoham and Leyton-Brown [58]).

Given a game and a player i, is i a dummy player? The coNP-hardness proof for this problem in a PDG, due to Bachrach and Porat [5], can be adapted to apply to PPDGs, too. The corresponding coNP upper bound holds straightforwardly.

Proposition 5

For both notions of dummy player, the problem of whether a given player in a given PPDG is a dummy player is coNP-complete.

On trees and complete graphs, however, this problem becomes solvable in polynomial time. The best known upper bound for this problem for PPDGCs is \({{\Pi }_{2}^{p}}\). (The technique that was useful for the core cannot be adapted straightforwardly to apply here.)

6 Conclusions and Future Work

The conceptual contributions of this paper are two-fold. First, we have introduced bribery in path-disruption games, a class of cooperative games defined by Bachrach and Porat [5]; second, we have extended their model to a probabilistic setting with uncertainty about the targets the adversaries wish to reach (Table 1).
Table 1

Complexity results for stability and bribery problems in (probabilistic) path-disruption games

Bachrach and Porat [5]

a direct consequence of Bachrach and Porat [5]

Let us first discuss the complexity of bribery. While bribery is easy in path-disruption games without costs, we have shown that in path-disruption games with costs bribery is NP-complete in the single-adversary case and is \({{\Sigma }_{2}^{p}}\)-complete in the multiple-adversary case. On the one hand, these results complete the picture of the complexity of bribery problems in path-disruption games with costs; on the other hand, they provide a \({{\Sigma }_{2}^{p}}\)-completeness result for a natural problem in game theory, which are far rarer than NP-completeness results in this area. Two recent other such results are due to Woeginger [65] and Peters [44], who show that recognizing (strict) core stability in additive hedonic games is \({{\Sigma }_{2}^{p}}\)-complete, too.

Our computational hardness results may be interpreted as providing protection against bribery attacks on PDGCs. Therefore, in light of the motivation of these games in terms of network security issues, these are positive results. One may naturally wonder to what extent the result for the multiple-adversary case (stated as Theorem 4) provides deeper insight into the computional complexity of PDGC-Multiple-Bribery, given that its NP-hardness already follows from the single-adversary case with costs (stated as Theorem 3). As pointed out by Woeginger [63], however, \({{\Sigma }_{2}^{p}}\)-hardness indeed provides a much better protection than merely NP-hardness, since most of the common methods used to circumvent NP-hardness—such as approximation, fixed-parameter tractability, or typical-case analyses (see, e.g., the survey by Rothe and Schend [54], for a discussion of applying such methods to NP-hard voting problems)—are far less applicable to circumvent \({{\Sigma }_{2}^{p}}\)-hardness.

Regarding our expansion of path-disruption games by allowing uncertainty about the adversaries’ targets, we have discussed the complexity of problems related to various solution concepts and other properties of these more general games.

For future work, it might be interesting to determine the complexity of problems not settled here or by Bachrach and Porat [5], and to consider other problems related to solution or stability concepts for path-disruption games and probabilistic path-disruption games (see, e.g., Chalkiadakis et al. [17]), including the cost of stability due to Bachrach et al. [9] or the stable sets due to Neumann and Morgenstern [39]. Furthermore, the restriction to special classes of graphs (not only trees and complete graphs, but also planar graphs or graphs with properties that can often be found in real life, like “small worlds”) might be interesting to investigate. For example, Bachrach and Porat [5] analyze path-disruption games on trees with the result that very often problems that are hard in general become solvable in polynomial time for trees. We suspect that PDGC-Multiple-Bribery is NP-complete when restricted to planar graphs, in contrast to the general problem. Still, this would mean the problem is computationally intractable.

Next to studying the computational complexity of the general problem, in future work one might explore finding subsets of the stability problems in which they are tractable by analyzing the underlying graph. Moreover, it would be interesting to vary the model of bribery and to study the resulting problems in terms of their complexity.

Additionally, in future work, one can study other possible models of bribery for cooperative games. For instance, one might consider a different cost model, such as a cost model for completely removing a player, or for changing the value of all coalitions containing a certain agent by a certain amount additively or by a certain factor multiplicatively. So far we have focused on letting the graph structure and the coalitional function remain the same while changing the bribed agents’ intention to participate in a coalition. In the context of voting, known variations of bribery include, e.g., microbribery [27, 28], swap bribery [23], extension bribery [12], and various versions of “campaign management” (see, e.g., Schlotter et al. [56]; Faliszewski et al. [29]; Faliszewski and Rothe [26]). In the context of path-disruption games, one variation might be to consider multiple, independently concurring bribes; another one to define the costs of blocking a vertex in a graph and the prices for bribing the corresponding agents in relation to each other. This might be analyzed in connection with the stability of the game (in terms of various stability concepts) and might lead to a new perspective on the topic.

Footnotes
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 (sj, tj), 1 ≤ jm, a weight function \(w:U\to \mathbb {Q}_{\geq 0}\), and a bound \(K\in \mathbb {Q}_{\geq 0}\), does there exist a subset UU 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 (sj, tj), 1 ≤ jm? 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.

 

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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Conflict of interests

The authors declare that they have no conflict of interest.

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institut für InformatikHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany
  2. 2.Computer Science DepartmentStanford UniversityStanfordUSA

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