Cooperative games provide a convenient framework to study cooperation among self-interested agents. Formally, a cooperative transferable utility game, or simply a game, is a pair where is a finite set of players and is a characteristic function. We are interested in how players should divide the value of the grand coalition . To capture the idea of a stable or fair payoff division scheme, a number of solution concepts have been developed, such as the core and the Shapley value. Unfortunately, the computational problems associated with many of these solution concepts are often intractable—NP-hard or worse.
There are two ways to circumvent computational intractability in this context. The first approach is to identify interesting classes of games that admit efficient algorithms. For instance, it is well-known that when the characteristic function is convex, an outcome in the core can be computed by a polynomial-time algorithm. However, this algorithm offers no guarantees when the input game is not convex, even if the convexity constraint is only violated at a few points, and hence this approach is of limited value. A more flexible way to cope with hardness is to provide complexity guarantees for all instances so that the guarantee depends on the complexity of the instance, that is, to design algorithms whose running time depends on how well the input instance is structured. In the context of cooperative games, this approach has been pursued by Ieong2005 Ieong2005, who propose a representation formalism, which they call marginal contribution nets (MC-nets), and design an algorithm for computing an allocation in the core whose running time depends on the treewidth of the graph associated with the MC-net representation of the input game.
In this work, we explore the power of the latter approach for two measures of structural complexity of set functions that have been recently developed by Feige2013 Feige2013: the dependency degree and the supermodular degree. Intuitively, the complexity of a set function is measured using the notion of dependency among players at ; such dependencies induce a graph describing the relation between players—the dependency graph. A player’s dependency degree is her degree in this graph; her supermodular degree is her degree in a modified version of this graph, which only takes certain dependencies into account.
Our contribution. We argue that both the dependency degree and the supermodular degree are useful in the context of cooperative games, both analytically and computationally.
We show that several cooperative game theory concepts can be naturally interpreted in terms of a dependency graph. In particular, a players’ dependency degree reflects on her importance in the game: in a simple game, a dummy player is an isolated vertex of the dependency graph and a veto player is a vertex with the maximum degree. We can also relate properties of a game, such as its dimension, to the properties of its dependency graph: for instance, we show that dependency graphs of weighted voting games are clique-trees, namely, chordal graphs.
We then investigate which solution concepts in cooperative games can be computed efficiently if the dependency/supermodular degree is bounded. For simple games, we obtain a number of tractability results with respect to the supermodular degree. Specifically, we prove that simple games admit an efficient algorithm for computing an element of the core or the least core when the supermodular degree is small. Further, while finding an optimal coalition structure is computationally intractable even for weighted voting games, we show that this problem becomes tractable for weighted voting games with small supermodular degree. However, these results do not extend to general games: we prove that the associated separation problem for the least core is NP-hard even for games with constant dependency degree. On the other hand, we show that computing the Shapley value and the Banzhaf value is in FPT with respect to the dependency degree.
We also consider the problem of computing the dependency degree and the supermodular degree given various representations of a game. While intractability turns out to be inevitable in general, we provide polynomial and pseudo-polynomial algorithms for special classes of games.
Related work. Computational aspects of cooperative games have received a considerable amount of attention over the last few decades; we refer the reader to the book of Chalkiadakis2011 Chalkiadakis2011.
Our work is similar in spirit to the complexity study of induced subgraph games or, more broadly, games defined by MC-nets [Deng and Papadimitriou1994, Ieong and Shoham2005, Greco et al.2011, Greco et al.2014, Li and Conitzer2014]. Indeed, each MC-net induces an agent graph, which also aims to capture dependencies among agents. However agent graphs are defined in a purely syntactic manner, by looking at agents’ co-occurrences in the rules of an MC-net, whereas the dependency graph is defined semantically, i.e., in terms of the value of the characteristic function. Moreover, both the dependency degree and the supermodular degree are different from the concepts that are usually used to measure the complexity of an agent graph (such as treewidth).
There are also similarities between our model and Myerson games [Myerson1977, Chalkiadakis, Greco, and Markakis2016, Meir et al.2013, Igarashi2017] where players are located on a graph and coalitions are only allowed to form if they are connected in this graph. However, in Myerson games non-adjacent agents may still depend on each other, and hence an agent’s dependency degree may be high even if her degree in the underlying Myerson graph is small. As a consequence, some problems that are easy for games with small supermodular degree remain hard for games on bounded-degree graphs, even if these graphs are acyclic [Igarashi2017]. Our results for coalition structure generation (Section 6) are similar in spirit to those of Voice2012 Voice2012; we discuss the relationship between their results and ours in Section 6.
The dependency degree and the supermodular degree have been introduced by Feige2013 Feige2013, who showed applications of these measures to the welfare maximization problem. Feldman2014 Feldman2014 generalized these results to function maximization subject to -extendible system constraints (a generalization of the intersection of matroids). These concepts have also been applied in an online setting [Feldman and Izsak2017], and in the context of efficiency of auctions [Feldman et al.2016], optimization of SDN upgrades [Poularakis et al.2017] and committee selection [Izsak2017]. Some related complexity measures are the submodularity ratio [Das and Kempe2011] and MPH [Feige et al.2015].
|General games||Simple games||Weighted voting games|
|Optimal coalition structure||NP-h.||NP-h. for (Th. 11)||FPT wrt p (Cor. 2)|
|Least Core||FPT wrt (Th. 5)||FPT wrt|
|Shapley||FPT wrt (Th. 8)||FPT wrt||FPT wrt|
|Banzhaf||FPT wrt (Th. 9)||FPT wrt||FPT wrt|
We start by defining basic notation and terminology of cooperative games. Recall that a cooperative game is a pair , where is a finite set and is a function from to . Throughout the paper, we assume . For , let
. For a vectorand a subset we use the notation , with the convention that . The subsets of are referred to as coalitions. An imputation for a game is a vector satisfying efficiency : , and individual rationality : for every . For a player and a coalition , we let , that is, is the marginal contribution of to . A player is called a dummy if she does not contribute to any coalition, i.e., for every .
A set function is said to be monotone if for every pair of subsets it holds that implies . A game is said to be simple if is monotone and only takes values in . In a simple game, coalitions of value are said to be winning, and coalitions of value are said to be losing. A winning coalition is said to be minimal if removal of any player from makes it losing, i.e., is losing for every . A player is said to be a veto player if she is present in all winning coalitions, i.e., if a coalition is winning, then . A player is a pivot for a coalition if is losing and is winning.
The core is a classic solution concept in cooperative games. Formally, the core of a game is the set of all imputations such that no coalition has an incentive to defect from , i.e., . As the core can be empty, and, on the other hand, not all outcomes in the core are equally fair, we consider the least core, which can be thought of as the set of most stable outcomes. We first define the excess of a coalition at an imputation as ; intuitively, is the degree of unhappiness of at . The least core of a game is the set of all imputations that minimize the maximum excess, i.e., the set of optimal solutions
of the following linear program:
We also consider solution concepts capturing fairness among players: the Shapley value and the Banzhaf value. The Shapley value of a player in a game is the average of ’s marginal contributions at over all permutations of the players, that is,
The Banzhaf value is the average of ’s marginal contributions at over all coalitions, that is,
Computational setting Throughout the paper, we only consider games such that is computable in time polynomial in . Furthermore, ‘polynomial’ always means polynomial in the number of players . Note that the explicit representation of a game , which lists the values of all coalitions, is not polynomial in ; thus, for our computational results for general games we assume oracle access to the characteristic function . We say that a problem is fixed parameter tractable (FPT) with respect to a parameter if each instance of this problem can be solved in time , where is a computable function that depends on only.
We omit some proofs due to space constraints; the omitted proofs can be found in the full version of the paper [IIE17].
3 Dependency Graphs of Cooperative Games
In this section, we introduce complexity measures representing dependencies among players in a cooperative game [Feige and Izsak2013], and study how well such parameters capture important concepts in cooperative games.
Given a game and two players , we say that player positively depends on player if there exists a coalition such that , i.e., can contribute more to in the presence of . We say that depends on if there exists a coalition such that , i.e., ’s contribution to depends on the presence of . These relations are known to be symmetric [Feige and Izsak2013]; hence, we can capture the dependency relations between players by undirected graphs. Formally, we define the supermodular dependency graph to be an undirected graph where the set of nodes is given by and the set of edges is given by the pairs of players positively depending on each other. The supermodular dependency set of at is
The supermodular degree of is defined as the maximum size of the supermodular dependency set, i.e., . We define the dependency graph to be an undirected graph where the set of nodes is given by and the set of edges is given by the pairs of players depending on each other. The dependency set of at is
The dependency degree of is defined as the maximum size of the dependency set, i.e., .
Consider a simple game with player set and a characteristic function where the set of minimal winning coalitions is . It is easy to see that only such pairs have a positive dependence. Further, player depends on player since can make a positive marginal contribution to the coalition , but her contribution becomes zero in the presence of . A similar argument applies to the pair . The resulting dependency graphs and are depicted in Figure 1.
We will now show that the parameters defined above capture the importance of a player in the underlying game. The theorem below shows that a dummy player corresponds to an isolated node in the dependency graph.
For every cooperative game and every player , the following statements are equivalent:
Player is a dummy player.
iii: Suppose that is a dummy player. Then we have . Now, suppose that there exists a player . Then there is a coalition such that at least one of and is non-zero, contradicting the fact that is a dummy player.
iiiii: This direction holds by the definition.
iiii: Suppose that and . Assume towards a contradiction that for some coalition . Let be a minimal coalition with respect to this property. If , then there is a player such that , which means that can contribute more in the presence of , and hence positively depends on , a contradiction. If , it follows that , a contradiction again. ∎
Positive Dependence in Simple Games
We will now further investigate the structure of dependency graphs in simple games. It turns out that such games can be almost fully characterized by the positive dependence relation. We first observe that in simple games the dependency relation admits a natural interpretation.
Consider a simple game and two players . Player positively depends on player if and only if there exists a coalition such that is winning, but the coalitions and are losing.
Now, in contrast with dummy players, a veto player in a simple game is adjacent to every non-dummy player in the dependency graph.
Let be a veto player in a simple game . Then positively depends on each non-dummy player .
Take any non-dummy player . Then is a pivot for some coalition , i.e., is winning and is losing. Since is a veto player and , we have . Let . Then, the coalition is winning, and coalition is losing. Also, as , coalition is losing. By Lemma 1, we conclude that positively depends on . ∎
We also observe that all minimal winning coalitions correspond to cliques in the supermodular dependency graph.
In a simple game player positively depends on a player if and only if there exists a minimal winning coalition such that .
Suppose that positively depends on . By Lemma 1 there exists a coalition such that is winning, but coalitions and are losing. Choose a minimal coalition with respect to this property. Then, removing any player from makes this coalition losing, and hence is a minimal winning coalition. Conversely, suppose that belong to some minimal winning coalition . Since is a minimal winning coalition, both and are losing, but is winning. By Lemma 1, positively depends on . ∎
Consider a simple game and a minimal winning coalition . Any two distinct players positively depend on each other.
Weighted voting games form a subclass of simple games. Such games can be succinctly represented by the weight of each player, and a quota. Formally, a weighted voting game with a set of players is given by a list of non-negative weights and a quota ; we will write . Its characteristic function is given by
It turns out that the supermodular dependency graph of a weighted voting game has a special structure: we will show that any such graph is a chordal graph. Recall that a graph is said to be chordal if any cycle of four or more nodes has a chord, i.e., an edge that does not belong to the cycle, but connects two of its nodes.
We first state and prove the following lemma.
Consider a weighted voting game where a player positively depends on a player and . Then, player positively depends on player .
Since positively depends on , there is a coalition such that is winning, but and are losing. Now, since , is winning, but and are losing, implying that positively depends on . ∎
For every weighted voting game it holds that its supermodular dependency graph is chordal.
Suppose towards a contradiction that has a chordless cycle of at least four players. Let , where positively depends on for (with the convention that ). Assume without loss of generality that has the maximum weight among the players in and .
First, suppose that there is a player such that . Since positively depends on and , by Lemma 2 this means that positively depends on , a contradiction. Now suppose that for all players we have ; in particular, we have . Since positively depends on , and , by Lemma 2 player positively depends on , a contradiction. ∎
A simple game is the intersection of weighted voting games , , if for every coalition we have if and only if for all . It is known that every simple game can be represented as an intersection of multiple weighted voting games; the minimum number of weighted voting games whose intersection equals to is called the dimension of .
Observe that the game defined in Example 1 has dimension : it can be represented as the intersection of weighted voting games and , where , and all other weights are zero. However, its supermodular dependency graph has a chordless cycle of length four; thus, we cannot guarantee that the supermodular dependency graph of a simple game is chordal beyond dimension .
4 Complexity of Stability-Related Solution Concepts
In this section, we investigate the complexity of computing outcomes in the core and the least core.
For simple games, it is well-known that deciding if the core is not empty or finding an outcome in the core is easy. We now complement this result by showing that computing an element of the least core in simple games is fixed-parameter tractable with respect to the supermodular degree.
Let be a simple game. Given its supermodular dependency graph and oracle access to , we can compute an element of the least core in FPT time with respect to the supermodular degree.
We first check whether the input game admits an imputation, i.e., whether : if not, the least core is empty. Thus, from now on we assume that .
It suffices to show that the separation problem for the linear program is fixed-parameter tractable with respect to the supermodular degree. Fix an and . First, we can clearly check in polynomial time whether is an imputation; thus, in the rest of the proof we assume that this is indeed the case. We need to show that deciding whether the following inequality holds is in FPT with respect to :
Since is an imputation, we have and for all . Now, by non-negativity of and by the fact that for all , the term on the left can be rewritten as
where is the set of all minimal winning coalitions in . It remains to show that the computation of is in FPT with respect to the supermodular degree. Fix and let denote the set of all coalitions in including . By Corollary 1, these coalitions are subsets of , i.e., . By iterating through all and all subsets of , we can compute the value . ∎
When the dependency graph has degree at most , Feige2013 Feige2013 showed that demand queries can be answered in polynomial time; that is, given a characteristic function and a vector , one can efficiently compute a subset maximizing over any subfamily of . This allows us to obtain the following result.
Consider a game whose dependency degree is at most . Given oracle access to the characteristic function , we can decide the non-emptiness of the core or find an element of the least core in time polynomial in .
We will argue that the separation problem for can be solved in time polynomial in . Consider an and a vector . Again, one can decide whether is an imputation in time . Now, if the dependency degree is at most , it follows from the work of Feige2013 Feige2013 that the maximum value over coalitions in can be computed in time polynomial in ; thus, we can compare the maximum with the given and efficiently decide whether satisfies the inequalities in .
It remains to notice that the imputations in the core are exactly the optimal solutions to with replaced with the value , i.e., we can use the same procedure as above to decide whether the core is non-empty. ∎
However, this argument does not extend to general cooperative games: we will now demonstrate that there is a succinctly representable class of games for which the separation problem for is NP-hard, even though the dependency degree of games in this class is bounded by a small constant and their supermodular degree is ,
There exists a family of hypergraph games with dependency degree 7 and supermodular degree 1 for which the separation problem for is NP-hard.
An instance of 3-Regular Independent Set is given by a 3-regular graph and an integer ; it is a ‘yes’-instance if has an independent set of size at least and a ‘no’-instance otherwise. This problem is known to be NP-hard [Garey and Johnson1979]. We will now show how to reduce it to the separation problem for for a family of hypergraph games defined below. For every vertex , we introduce two players , and let . We define a set function by building its hypergraph representation , so that for each the value is computed as the sum of the weights of all hyperedges of the sub-hypergraph induced by . We set for each . For each we connect and by an edge (i.e., a hyperedge of rank 2) of weight 3, and we connect and by an edge of weight . For every edge , we introduce a hyperedge of size 4 and weight containing , , and . Since is 3-regular, and hence . Moreover, the constructed game has dependency degree since is 3-regular, and there is a pair of dependent vertices for every vertex in . Also, the supermodular degree of this game is , as the supermodular dependency set of every vertex only contains the other vertex belonging to the same pair.
Now, we set for each and . Clearly, the vector is an imputation, as we have and the constraint is satisfied for every .
It can be shown that admits an independent set of size if and only if the maximum excess at is at least . Indeed if is an independent set of size in , then is a coalition whose excess at is . Conversely, it can be argued that if there is a coalition whose excess at is then has an independent set of size . ∎
5 Complexity of the Shapley and Banzhaf Values
For the Shapley and Banzhaf values, the following observation is crucial for our analysis: for every player and every coalition it holds that adding players who do not depend on to does not affect ’s marginal contribution to . We formalize this observation in the following lemma. We write .
For every and for every it holds that .
We prove our claim by induction on the size of the set . The claim clearly holds when . Suppose it holds for . Let . Then by the induction hypothesis and by the fact that . Combining these two equations yields
By Lemma 3, one can calculate the Shapley value of each player by iterating over all subsets of her dependency set, computing her marginal contribution, and counting the number of coalitions whose intersection with the dependency set is exactly this subset.
Computing the Shapley value is in FPT with respect to the dependency degree.
The Shapley value can be written as follows:
where . The last equality holds due to Lemma 3. ∎
Similarly, the Banzhaf value can be efficiently computed when the dependency degree is bounded.
Computing the Banzhaf value is in FPT with respect to the dependency degree.
The Banzhaf value can be written as follows:
where the last equality holds due to Lemma 3. ∎
6 Optimal Coalition Structure Generation
So far, we discussed games without coalition structures: we implicitly assume that all players cooperate and are willing to divide the value of the grand coalition. In some settings, however, it may be more efficient to split the players into different teams. The problem of finding the best partition of players, which is referred to as the optimal coalition structure generation problem, has thus been extensively studied (see, e.g., the surveys [Elkind, Rahwan, and Jennings2013, Rahwan et al.2015]). Formally, a coalition structure for is a partition of into disjoint coalitions. A coalition structure for is said to be optimal if the social welfare is maximized.
Not surprisingly, optimal coalition structure generation is NP-hard even for weighted voting games; this can be shown by a straightforward reduction from Partition [Chalkiadakis, Elkind, and Wooldridge2011]. In contrast, we will now argue that if the input game is simple and its supermodular dependency graph has tree-like structure, this problem becomes tractable. To this end, we introduce the notions of tree decomposition and treewidth.
A tree decomposition of a graph is a pair , where is a rooted tree and is a family of subsets of , called bags, where
for every node , the set is nonempty and connected in , and
for every edge , there is a node such that .
For a node of , we let be the union of all bags present in the subtree of rooted at , including . The treewidth of a tree decomposition of is .
Recall that for simple games it holds that minimal winning coalitions form cliques in the supermodular dependency graph. We will now argue that this implies that there is an optimal coalition structure where each winning coalition is contained in some bag of the tree decomposition.
Let be a simple game and let be a tree decomposition of the supermodular dependency graph . For every it holds that every optimal coalition structure for can be transformed into another optimal coalition structure so that for each coalition the following statements hold:
if contains a player with for every child of , then .
can appear in only one subtree, i.e., if is a subset of both and for some children of , then or .
Consider a node . Note that every optimal coalition structure can be transformed into another optimal coalition structure where each winning coalition is minimal. Thus, let be such an optimal coalition structure for . Take an arbitrary coalition . If is losing, then it is clear that can be divided into and for each child of without changing the sum , and hence our claims hold. Now, suppose that is a minimal winning coalition. By Theorem 3, forms a clique.
To show (i), suppose that there is a player such that for every child of . Assume towards a contradiction that and hence there is a player . Observe that no bag , , contains both and , since is not present in for any successor of , and does not appear above . However, since is a clique, is adjacent to in the graph , which means that there is a bag containing both and . This contradicts requirement (ii) of Definition 1.
To show (ii), suppose that is a subset of both and for some children of . Assume towards a contradiction that and ; thus there exist a player and a player . Since the intersection is fully contained in both and , we have , and hence no bag contains both and . However, since is a clique, and are adjacent, contradicting requirement (ii) of Definition 1. ∎
By Lemma 4, one can find an optimal coalition structure for a simple game by trying all possible partitions of each bag and combining them in a bottom-up manner. Before we present the proof of Theorem LABEL:thm:coalitionstructure, we need a few auxiliary definitions.
Consider a cooperative game and a coalition structure for this game. For each subset , we define to be the coalition in containing if such a coalition exists, and otherwise.
A tree decomposition of a graph is nice if for the root of we have , and each node belongs to one of the following types:
Leaf: is a leaf in and .
Introduce: has one child , and for some .
Forget: has one child , and for some .
Join: has two children such that .
We are now ready to prove Theorem 10.
Consider a simple game . There exists an algorithm that, given oracle access to and a tree decomposition of the supermodular dependency graph with treewidth , computes an optimal coalition structure in time .
First, recall that in time we can transform a given tree decomposition of treewidth with nodes into a nice one with the same treewidth and nodes [Cygan et al.2015]. In what follows, let denote such a decomposition.
Now, we give a dynamic program over the tree decomposition as follows. For each node , each coalition structure of , and each subset of , we define to be the maximum value such that is a coalition structure of where all the coalitions in are extended in without changing the coalitions in , i.e., for all , , and for all , . Starting from the leaves and going up to the root, we will fill out the dynamic programming table. The case where is a leaf corresponds to the base case of the recurrence; we then compute values for a non-leaf node based on the values for the children of . We will finally obtain , which is the value we want to compute.
Leaf: If is a leaf node, then we have only one value .
Introduce: Suppose is an introduce node with child such that . Let be a coalition in containing , and let . We claim that is given by the value of an optimal coalition structure of the subtree rooted at the child and the marginal contribution of to the coalition , namely,
To see this, let be a partition of that attains the maximum in the definition of and satisfies condition (i) of Lemma 4. Since , we have and hence . Then, it follows that is a partition considered in the definition of , and we have . Hence,
Conversely, let be a partition of that attains the maximum in the definition of . Then, since , but , the coalition remains the same in , i.e., ; thus, partition is considered in the definition of , and
Forget: Suppose is a forget node with child such that . Then, it can be verified that is given by
where and for .
Join: Finally, suppose that is a join node with children such that . Then, is the maximum value of
over all the pairs where is a disjoint union of and ; intuitively, each specifies how a coalition in will be extended to subtrees and .
To show the correctness of our algorithm, let be a partition of that attains the maximum in the definition of and satisfies condition (ii) of Lemma 4.
Let be the set of coalitions in that remain the same, i.e., . Observe that by condition (ii) of Lemma 4, each coalition either remains the same in or is extended to a subtree rooted at or , that is, or for some . We now divide into two families and depending on how has been extended in ; specifically, we let and be subsets of such that can be represented as a disjoint union of and , and
includes all the coalitions in such that (S) is contained in only, i.e., and ; and
includes all the coalitions in such that (S) is contained in only, i.e., and .
Let , . For , set