1 Introduction
Cooperative Game Theory provides a mathematical framework for capturing situations where subsets of agents may form a coalition in order to obtain some collective profit or share some collective cost. Formally, a
cooperative game (with transferable utilities) consists of a pair , where is a set of agents called players and is a value function that satisfies . In our context, the value of a coalition represents the profit for if all players in choose to collaborate with (only) each other. The central problem in cooperative game theory is to allocate the total profit of the grand coalition to the individual players in a “fair” way. To this end various solution concepts such as the core, Shapley value or nucleolus have been designed; see [29] for an overview. For example, core solutions try to allocate the total profit such that every coalition gets at least . This is of course not always possible, that is, the core might be empty. This leads to related questions like: “How much do we need to spend in total if we want to give at least to each coalition ?”. In the specific case of simple games (cf. below) where takes only values and, classifying coalitions into “losing” and “winning” coalitions, one may also ask: “How much do we have to give in the worst case to a losing coalition if we want to give at least
to each winning coalition?”As mentioned above, we study simple games. Simple games form a classical class of games, which are well studied; see also the book of Taylor and Zwicker [33]. The notion of being simple means that every coalition either has some equal amount of power or no power at all. Formally, a cooperative game is simple if is a monotone 0–1 function with and , so for all and whenever . In other words, if is simple, then there is a set of winning coalitions that have value and a set of losing coalitions that have value . Note that , and . The monotonicity of implies that subsets of losing coalitions are losing and supersets of winning coalitions are winning. A winning coalition is minimal if every proper subset of is losing, and a losing coalition is maximal if every proper superset of is winning.
A simple game is a weighted voting game
if there exists a payoff vector
such that a coalition is winning if and losing if . Weighted voting games are also known as weighted majority games and form one of the most popular classes of simple games.However, it is easy to construct simple games that are not weighted voting games. We give an example below, but in fact there are many important simple games that are not weighted voting games, and the relationship between weighted voting games and simple games is not yet fully understood. Therefore, Gvozdeva, Hemaspaandra, and Slinko [17] introduced a parameter , called the critical threshold value, to measure the “distance” of a simple game to the class of weighted voting games:
(1) 
A simple game is a weighted voting game if and only if . This follows from observing that each optimal solution of (1) can be scaled to satisfy for all winning coalitions . The scaling enables us to reformulate the critical threshold value as follows:
where
The following concrete example of a simple game that is not a weighted voting game and that has in fact a large value of was given in [13]:
Example 1
Let for some even integer , and let the minimal winning coalitions be the pairs . Then
This means that for every . Then, for every and for at least one of the two loosing coalitions or , we have , showing that . On the other hand, it is easily seen that satisfies for all winning coalitions and for all losing coalitions, showing that . Thus .
This example led the authors of [13] to the following conjecture:
Conjecture 1 ([13])
For every simple game , it holds that .
Our Results.
Section 2 contains our main result. In this section we reformulate and strengthen Conjecture 1 and then we prove the obtained strengthening.
In Section 3 we consider a subclass of simple games based on a natural desirability order [21]. A simple game is complete if the players can be ordered by a complete, transitive ordering , say, , indicating that higher ranked players have more "power" than lower ranked players. More precisely, means that for any coalition . The class of complete simple games properly contains all weighted voting games [15]. For complete simple games, we show an asymptotically upper bound on , namely . This bound matches, up to a factor, the lower bound of in [13], where the bound is conjectured to be tight in [13]. Intuitively, complete simple games are much closer to weighted voting games than arbitrary simple games. So, from this perspective, our result seems to support the hypothesis that is indeed a sensible measure for the distance to weighted voting games.
In Section 4 we discuss some algorithmic and complexity issues. We focus on instances where all minimal winning coalitions have size . We say that such simple games are graphic, as they can conveniently be described by a graph with vertex set and edge set . For graphic simple games we show that computing is NPhard in general, but polynomialtime solvable if the underlying graph is bipartite, or if is known to be small (less than a fixed number ).
Related Work. Due to their practical applications in voting systems, computer operating systems and model resource allocation (see e.g. [3, 7]), structural and computational complexity aspects for solution concepts for weighted voting games have been thoroughly investigated [10, 11, 14, 17, 26].
Another way to measure the distance of a simple game to the class of weighted voting games is to use the dimension of a simple game [32], which is the smallest number of weighted voting games whose intersection equals a given simple game. However, computing the dimension of a simple game is NPhard [8], and the largest dimension of a simple game with players is [24]. Moreover, may be arbitrarily large for simple games with dimension larger than 1. Hence there is no direct relation between the two distance measures. Gvozdeva, Hemaspaandra, and Slinko [17] introduced two other distance parameters as well. One measures the power balance between small and large coalitions. The other one allows multiple thresholds instead of threshold 1 only.
For graphic simple games, it is natural to take the number of players as the input size for answering complexity questions, but in general simple games may have different representations. For instance, one can list all minimal winning coalitions or all maximal losing coalitions. Under these two representations the problem of deciding if , that is, if a given simple game is a weighted voting game, is also polynomialtime solvable. This follows from results of [18, 28], as shown in [14]. The latter paper also showed that the same result holds if the representation is given by listing all winning coalitions or all losing coalitions.
As mentioned, a crucial case in our study is when the simple game is graphic, that is, defined on some graph . In the corresponding matching game a coalition has value equal to the maximum size of a matching in the subgraph of induced by . One of the most prominent solution concepts is the core of a game, defined by . Matching games are not simple games. Yet their core constraints are readily seen to simplify to and for all . Classical solution concepts, such as the core and corerelated ones like least core, nucleolus or nucleon are well studied for matching games, see, for example, [4, 5, 12, 22, 23, 31].
2 The Proof of the Conjecture
To prove Conjecture 1 we reformulate, strengthen and only then verify it. Our approach is inspired by the work of Abdi, Cornuéjols and Lee on identically selfblocking clutters [1]. A coalition is called a cover of if has at least one common player with every coalition in . We call the collection of covers of the blocker of and denote it by ^{1}^{1}1Usually, the notion of a blocker is defined as the collection of minimal covers, but for simplicity of exposition, we define it as the collection of all covers. [9]. We claim that
In order to see this, first suppose that there exists a cover such that . As , this means that . However, as contains no player from , this contradicts our assumption that . Now suppose that there exists a losing coalition such that does not belong to . Then, by definition, there exists a winning coalition with . As , we find that . Then, by the monotonicity property of simple games, must be winning as well, a contradiction.
As , the critical threshold value can be reformulated as follows
Here, stands for the scalar product of two vectors and . To see the last equality, for a cover we can define a corresponding vector by setting if and otherwise.
Conjecture 1 (reformulated) For a simple game with players and the collection of winning coalitions , we have
Next, we prove Theorem 3, which is a strengthening of Conjecture 1. For the proof we need the following straightforward remark, which we leave as an exercise. Here, we write for a vector .
Remark 2
Let be a polyhedron and let be the optimal solution of the program . Then
is an optimal solution of the linear program
.Theorem 3 (Strengthening of Conjecture 1)
For a simple game with players and the collection of winning coalitions , we have
In particular, if is the optimal solution for the program
then
Proof
Consider the unique optimal solution for the program . By Remark 2, is an optimal solution for the program . Thus, is an optimal solution for the program . Thus, we have
finishing the proof.∎
Let us discuss when Conjecture 1 provides a tight upper bound for the critical threshold value. The next theorem shows that if the upper bound in Conjecture 1 is tight, then this fact can be certified in the same way as in Example 1.
Theorem 4
For a simple game with players and the collection of winning coalitions and the collection of losing coalitions , we have
if and only if lies in the convex hull of the characteristic vectors of winning coalitions and lies in the convex hull of the characteristic vectors of losing coalitions.
Proof
Clearly, if lies in the convex hull of the characteristic vectors of winning coalitions and lies in the convex hull of the characteristic vectors of losing coalitions, then for every we have
showing that and hence by Theorem 3.
On the other hand, from the proof of Theorem 3 we know that if then is an optimal solution for with value . Let us show that lies in the convex hull of the characteristic vectors of winning coalitions. To do that consider an optimal dual solution for the program . Using complementary slackness it is straightforward to show that provides coefficients of a convex combination of characteristic vectors of winning coalitions, where the convex combination equals .
In the same way as the proof of Theorem 3, we could show that
where is the optimal solution for the program
Thus, if equals , then and lies in . Hence, if equals , then lies in the convex hull of the characteristic vectors of losing coalitions, finishing the proof. ∎
3 Complete Simple Games
Intuitively, the class of complete simple games is “closer” to weighted voting games than general simple games. The next result quantifies this expectation.
Theorem 3.1
For a complete simple game , it holds that .
Proof
Let be the set of players and assume without loss of generality that . Let be the largest number such that is winning. For , let denote the smallest size of a winning coalition in . Define for and for . Thus, obviously, .
Consider a winning coalition and let be the first player in (with respect to ). If , then and hence . On the other hand, if , then .
For a losing coalition , we conclude that (otherwise would dominate the winning coalition of size in ). So is bounded by . The optimal solution of this maximization problem is and . Hence . Summarizing, we obtain . ∎
In [13] it is conjectured that holds for complete simple games. In the same paper a lower bound of order is given, as well as specific subclasses of complete simple games for which can be proven.
4 Algorithmic Aspects
A fundamental question concerns the complexity of our original problem (1), i.e., the complexity of computing the critical threshold value of a simple game. For general simple games this depends on how the game in question is given, and we refer to Section 1 for a discussion. Here we concentrate on the “graphic” case.
Proposition 1
Computing for bipartite graphs can be done in polynomial time.
Proof
Let be the set of feasible payoffs (satisfying and for ). For , let . Thus . The separation problem for (for any given ) is efficiently solvable. Given , we can check feasibility and whether by solving a corresponding maximum weight independent set problem in the bipartite graph . Thus we can, for any given , apply the ellipsoid method to either compute some or conclude that . Binary search then exhibits the minimum value for which is nonempty; binary search works indeed in polynomial time as the optimal has size polynomially bounded in , which follows from observing that
(2) 
can be computed by solving a linear system of constraints defining an optimal basic solution of the above linear program. ∎
The proof of Proposition 1 also applies to other classes of graphs, such as clawfree graphs (see [6]) in which finding a weighted maximum independent set is polynomialtime solvable. In general, the problem is NPhard.
Proposition 2
Computing for arbitrary graphs is NPhard.
Proof
Let and be two disjoint copies of a graph with independence number . For each and add an edge if and only if or and call the resulting graph . We claim that (thus computing is as difficult as computing ).
First note that the independent sets in are exactly the sets that arise from an independent set in by splitting into two complementary sets and and defining . Hence, on yields where the maximum is taken over all independent sets in . This shows that .
Conversely, let be any feasible payoff in , that is, and for all . Let be a maximum independent set of size in and construct by including for each either or in , whichever has value at least . Then, by construction, is an independent set in with , showing that . ∎
Summarizing, for graphic simple games, computing is as least as hard as computing the size of a maximum independent in . For our last result we assume that is a fixed integer, that is, is not part of the input.
Proposition 3
For every fixed , it is possible to decide if in polynomial time for an arbitrary graph .
Proof
Let for some . By bruteforce, we can check in time if contains vertices that induce disjoint copies of , that is, paths of length for with no edges joining any two of these paths. If so, then the condition implies that one of , say , must receive a payoff , and hence has . As is an independent set, .
Now assume that does not contain disjoint copies of as an induced subgraph, that is, is free. For every , the number of maximal independent sets in a free graphs is due to a result of Balas and Yu [2]. Tsukiyama, Ide, Ariyoshi, and Shirakawa [34] show how to enumerate all maximal independent sets of a graph on vertices and edges using time per independent set. Hence we can find all maximal independent sets of and thus solve, in polynomial time, the linear program (2). Then it remains to check if the solution found satisfies . ∎
5 Conclusions
We have strengthened and proven the conjecture of [13] on simple games (Conjecture 1) and showed a number of computational complexity results for graphic simple games. Moreover, we considered complete simple games and proved a stronger upper bound for this class of games. It remains to tighten the upper bound for complete simple games to if possible. In order to classify simple games, many more subclasses of simple games have been identified in the literature. Besides the two open problems, no optimal bounds for are known for other subclasses of simple games, such as strong, proper, or constantsum games, that is, where , , or for all , respectively.
Acknowledgments. The second and fifth author thank Péter Biró and Hajo Broersma for fruitful discussions on the topic of the paper. The fourth author thanks Ahmad Abdi for valuable and helpful discussions.
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