Core Membership Computation for Succinct Representations of Coalitional Games

Core Membership Computation for Succinct Representations of Coalitional Games Xi Alice Gao May 11, 2009 Abstract In this paper, I compare and contrast two formal results on the computational complexity of core membership determination in two compact representations of coalitional games. Conitzer and Sandholm [1] proposed the multi-issue representations of coalitional games. This representation attempts to decompose a coalitional games into a set of sub-games. Later, Ieong and Shoham [3] proposed the marginal contribution nets (MC-nets) representation of coalitional games. The MC-nets representation attempts to use boolean logic to reduce the size of the coalitional game representation. In this paper, I compare the core membership results for these two coalitional game representations, discuss their implications, and suggest possible future research directions. In particular, these two papers seem to suggest two seemingly contradictory research directions for the core-membership problem. Conitzer and Sandholm [1] argued that stability concepts like the core for coalitional games should take into account of the computational complexity of finding a beneficial deviation for a particular agent. However, Ieong and Shoham [3] argued that it is important to search for more succinct representations of coalitional games which could overcome the computational hardness of the general core membership problem. 1 Introduction In coalitional game theory, the basic modeling unit is a group of agents, known as a coalition, rather than the individual agents. In coalitional games, agents can benefit by cooperating with each other and receiving higher payoff by working in a group rather than working individually. Coalitional games assign a payoff to each group of agents in the game. A naive representation of a coalitional game is to enumerate the payoff to every possible set of agents. This requires space exponential in the number of agents in the game, and is clearly not practical for a game with a large number of agents. Notice that the problem regarding the representation of games is analogous to the problem raised by the normal form representation of games in non-cooperative game theory. Even though the normal form representation is completely expressive, it fails to provide compact representations for many games with practical purposes. We can not perform efficient computations for a game which cannot be represented concisely in the first place. This problem leads to the development of many succinct representations of non-cooperative games. Therefore, developing succinct representations of coalitional games is critical for being able to reason about these games efficiently. For coalitional games, several solution concepts have been proposed. The two most important ones are the Shapley value and the core. In this paper, I focus on results concerning the core. The core is a solution concept for stability. If a payoff vector for the agents is in the core of a coalitional game, then no group of 1

players has an incentive to break away from the grand coalition and form their own coalition. Two related questions have been raised regarding the concept of the core. The first one asks whether the core for a given coalition game is empty. A nonempty core signifies that there exist some outcome of the game that is stable against possible deviations by a group of agents. Another related question regarding the core asks whether a given payoff vector is inside the core of a coalitional game. In the following sections, I first introduce the two compact representations proposed. Next, I describe the results in both papers regarding the computational complexity of determining core membership for these succinct representations. Finally, I discuss the implications of these results and suggest future work in this area. 2 Succinct representations of coalitional games 2.1 Technical background A coalitional game can be represented by the pair (N, v) where N is a set of agents, and v : 2 N R is a function that maps each group of agents S N to a real-valued payoff This is known as the characteristic form. This definition makes two important assumptions. First, utilities are transferable among agents in a coalition. Second, the payoff for for a coalition is not affected by agents outside of the coalition. Ieong and Shoham [3] proposed four criteria for evaluating the quality of coalitional game representations. First, expressivity is concerned with the breadth of the class of coalitional games covered by the representation. Next, conciseness asks for the space requirement of the representation. Moreover, efficiency refers to whether there exist efficient algorithms for these representations, and simplicity requires the representations to be easy to understand for the users. Based on these criteria, the ideal representation should be able to express any coalitional games, use very little space, admit efficient computations, and be easy to use. I use these criteria as guidelines for evaluating the two succinct representations of coalitional games. 2.2 The multi-issue representation The multi-issue representation decomposes the coalitional game into a number of distinct issues. More formally, the multi-issue representation defines a set of characteristic functions (v 1, v 2,..., v T ) where each v i : 2 N R is a decomposition of the characteristic function v over T issues, and all the v i s satisfy the property that for any S N, v(s) = T i=1 v i(s). A useful way to think about multi-issue representation is to use the following example [1]. Consider a scenario in which certain tasks must be performed by a set of agents. Each agent are capable of performing a number of tasks. Alternatively, we could understand this setting as that each task could only be completed by any member of a group of agents in the game. Accomplishing a certain task generates some value. Therefore, each different coalition of agents could have a different collective skill set and are therefore capable of performing a subset of tasks available. 2

Also, to explain why this representation is more concise over the naive representation, we need to recognize the following property of this representation: The characteristic function v i for issue i only concerns a subset of agents C i A if v i (S 1 ) = v i (S 2 ) whenever C i S 1 and C i S 2. Intuitively, under this representation, the characteristic function v i for each issue i need only be defined for a subset of agents who are concerned with this particular issue. Therefore, the multi-issue representation only needs to define T i=1 2 Ci values instead of the 2 N values using the naive representation. This property makes the multi-issue representation exponentially more concise than the naive representation assuming that the C i are small. Also, notice that the multi-issue representation is fully expressive, meaning that it can represent any arbitrary coalitional game. Intuitively, if there is no easy to decompose a given coalitional game into multiple issues, we could always treat the entire coalition game as a single big issue. 2.3 The marginal contribution nets representation The MC-nets representation scheme attempts to use the power of boolean logic to represent features of coalitional games. The basic idea is to use set of rules. These rules define mapping from pattern to value as shown below: Rule: Pattern value The pattern is simply a boolean expression which specifies a logical statement over a subset of agents. For instance, a simple pattern could be a conjunction of agents which specifies that all of agents in the conjunction must be present for the rule to apply. A rule applies to a group of agents S if S meets the satisfies the pattern specified. Then the payoff of a group of agents is defined to be sum over the values of all rules that apply to the group. The flexibility of MC-nets is inherently due to the flexibility of boolean logical expressions. Therefore, MCnets could be easily extended by allowing more and more complicated boolean expressions in the pattern. Ieong and Shoham [3] only concerned themselves with conjunctions of boolean expressions specifying the presence and absence of certain agents. The expressions specifying presence of agents are called positive literals and the ones for absence of agents are called negative literals. A typical pattern is in the following form: {p 1 p 2... p m n 1 n 2... n n } A rule containing such a pattern will only apply to a group if the group contains all the agents p i s and does not contain all the agents n i s. Given the definition of MC-nets, Ieong and Shoham [3] discussed several propositions regarding the representation power of MC-nets. First, MC-nets was proven to be fully expressive. For any arbitrary coalitional game, Ieong and Shoham [3] showed that the set of rules for MC-nets could be constructed by using a similar idea as the inclusion-exclusion principle. Moreover, Ieong and Shoham [3] discussed two propositions regarding a comparison of representation power of the multi-issue and the MC-nets representations. First, they showed that marginal contribution networks use at most a linear factor (in the number of agents) more space than multi-issue representation for any game. 3

This proposition implies that, on average, for arbitrary coalitional games, the space requirement for using MC-nets is comparable to the space requirement of the multi-issue representation. Second, they showed that, for certain coalitional games, MC-nets can be exponentially more concise than multi-issue representation. The proof for this proposition used the unit game in which the value of any nonempty coalition is one. The unit game can be represented in O( N ) space using MC-nets with negative literals. However, the multi-issue representation will require space O(2 n ) to represent the unit game since there is no decomposition of this game into distinct issues. Thus, MC-nets has a relative advantage over the multi-issue scheme in terms of representation power alone. 3 The core membership results In this paper, I focus on results in the two papers regarding the problem of checking whether a payoff vector is in the core. Formally, given a coalitional game and a payoff vector x, the core-membership question asks whether x is in the core. By definition of the core, a payoff vector is in the core if no subcoalition has an incentive to break away from the grand coalition. 3.1 Core membership in multi-issue domains In their paper, Conitzer and Sandholm [1] proved that given an arbitrary coalitional game, it is NP-complete to determine whether there exists a subcoalition having an incentive to break away from the grand coalition. Such a subcoalition is referred to as a blocking coalition. This problem can be formally defined as follows: Given a characteristic function with a decomposition v = T i=1 v i. Each v i is only defined over members of a subset of agents C i N, and each v i over the 2 Ci is defined naively by enumerating the payoff value for each possible S i C i. Additionally, we are given a payoff vector x : N R specifying the payoff for each agent in the grand coalition. The question that we asks is whether there is some blocking coalition S such that v(s) > n N x(n). Conitzer and Sandholm [1] showed that this problem is NP-complete by a reduction from the VERTEX- COVER problem. For their proof, they only worked on a special case of the general CORE-MEMBERSHIP problem. This special case specifies that C i = 3, i and v i for each issue i only takes binary values, and all the v i s are increasing and superadditive. The definitions for increasing and superadditive are omitted here since they are not critical for understanding the proof. Notice that proving that this special case is NP-hard is sufficiently to guarantee that the general problem is also NP-hard. A sketch of the proof is as follows: Proof Sketch First, this problem is in NP since for any given coalition S, we could compute v(s) and n N x(n) in polynomial time and check if the former is larger. To show that this problem is NP-hard, a reduction from the VERTEX-COVER problem is used. The VERTEX-COVER problem is defined as follows: Given a graph G = (V, E), and an integer r > 0, determine whether there exists a vertex cover for G of size at most r. A vertex cover is a set of vertices such that each edge of the graph is incident to at least one vertex of the set. (1) Given an instance of the vertex cover problem, an instance of the core-membership problem can be 4

constructed as follows: For each vertex v V, defines an agent n v. Also define one additional agent n 0. Every edge e E represents an issue i e and T = E. For each issue i e, the function v ie is only defined for members of the set C ie = {n 0 } {n v : v is endpoint of some edge e}. The function v ie is defined to be 1 for any subset of C ie containing the agent n 0 and at least one other agent n v. Otherwise, v ie is defined to be 0. Finally, the payoff vector x is defined as: x(n 0 ) = T 1 2, and for any v V, x(n v) = 1 2(r+1/2). (2) Given a solution to the VERTEX-COVER instance, we have that there exists a set W V of vertices such that W is a valid vertex cover and W r. Then consider the set S = {n 0 } {n v : v W }. By definition of the representation, we could easily check that v ie (S) = 1 for all issues since all edges in the graph has at least one endpoint in W. Thus we have v(s) = T. On the other hand, n S x(n) = T 1/2 + W 1 2(r+1/2) T 1/2 + r 1 2(r+1/2) < T 1/2 + r 1 2r = T = v(s) Thus, S is a blocking coalition and we have found a solution to the CORE-MEMBERSHIP problem instance. (3) Given a solution to the CORE-MEMBERSHIP problem instance, we have a subset S N such that v(s) > n S x(n). By definition, S must contain a 0. Otherwise, we will have v(s) = 0 and S could not possibly be a blocking coalition. Now consider the set W = {v : a v S}. We can derive as follows: 1 x(n) = T 1/2 + W 2(r + 1/2) n S T v(s) > n S x(n) and W is an integer W r Additionally, since v(s) x(n 0 ) = T 1/2, v ir = 1 for every issue i e. Thus, W covers all edges and is a valid solution the corresponding VERTEX-COVER problem instance. End of Proof Sketch 3.2 Core membership in MC-nets In [3], Ieong and Shoham presented two results regarding core membership in MC-nets. For the first result, they proved that the general core-membership problem for MC-nets is conp-complete since it follows from the hardness results of a previously proposed graphical form of coalitional games. The MC-nets representation can be seen as a generalization of the graphical form proposed in [2]. Nevertheless, Ieong and Shoham [3] developed an algorithm in an effort to overcome the computational hardness of the core-membership problem. In summary, their core-membership algorithm utilized tree decomposition techniques and runs in time exponential only in the treewidth of the agent graph. Therefore, for graphs of small treewidth, this algorithm presents a tractable solution to determine whether a payoff vector is in the core. 5

To explain the core-membership algorithm in detail, we first review some necessary concepts in tree decomposition and treewidth. A tree decomposition converts an arbitrary graph G to a tree T. Each node in T represents a subset of vertices in G. Also, the tree T needs to satisfy several properties to be a valid tree decomposition: All the vertices in G must be present in T For any edge in G, there exists a node in tree T containing both endpoints of this edge. (Running intersection property) If a node X j is on the path from node X i to node X k in T, then X i X k X j (X j must contain at least the intersection of X i and X k. The treewidth of a tree decomposition is defined to be the maximum cardinality over all sets in the nodes of tree T minus one. Then the treewidth of a graph is defined as the minimum treewidth over all tree decompositions of the graph. A tree decomposition T can be converted into a nice tree decomposition of the same treewidth and of size linear in that of T. A nice tree decomposition specifies a rooted tree and has several properties: The leaf nodes i have one element in its set X i. The introduce nodes have one child j and has one less element in their set than their child node. The forget nodes have one child j and has one more element in their set than their child node. The join nodes have two children j and k and have exactly same set of elements as each of its children. To introduce the core membership algorithm, Ieong and Shoham [3] further defined two key concepts as follows: The excess of a coalition S is defined as x(s) v(s), where x is a given payoff vector and v is the characteristic form. Intuitively, the excess measures how close the group S is to violating the core condition. The naive approach for checking whether a payoff vector belongs to the core is to check that the excesses of all groups are non-negative. This algorithm, however, takes advantage of the tree decomposition to make such inferences in a structured manner. The reserve of a coalition S relative to a coalition U is the minimum excess over all coalitions T sch that S T U. This reserve is denoted by r(s, U). The group T with minimum excess is denoted as arg r(s, U). U is called the limiting set of the reserve and S is the base set of the reserve. Given a nice tree decomposition of the graph, the algorithm keeps track of the reserves of all non-empty subsets at each node in the tree. For each node, these reserve values are referred to as the r-values of a node. Ieong and Shoham [3] proved that the payoff vector x is in the core if and only if the r-values for all nodes in the tree are non-negative. Proof Sketch If the reserve at some node i for some group S is negative, then there exists a coalition T with negative excess. hence the payoff vector is not in the core. This is the easy direction of the proof. For the more involved direction, assume that some payoff vector x is not in the core, then there exists a coalition R with negative excess. We need to find a node in the tree composition with negative reserve value. First of all, if R and the root node are not disjoint sets, then the reserve value for some group in the root is negative. Otherwise, if R and the root node X root are disjoint, we remove all agents in X root from all the nodes in the tree. By the running intersection property, the sets of nodes in the remaining tree 6

are disjoint. We can consider that these remaining nodes constitute a forest of trees. To find a node with negative reserve value, we just need to compute the reserve values for each individual tree in this forest. For each tree in the forest, we use the same technique we just described to further decompose the tree. Therefore, by induction, if the excess for R is negative, then the reserve at some node must be negative. End of Proof Sketch The authors gave an algorithm for computing the reserve value for the four types of nodes in the a nice tree decomposition. They also proved that the reserve value computation for each type of node is correct. To summarize, each step in the reserve value computation at each node takes time at most exponential in the number of agents in the node. Therefore, the core-membership algorithm runs in time exponential only int he treewidth of the graph. 4 Discussion and future work These two results offer us some insights into the computational complexity of the core membership problem for two different succinct representations of coalitional games. First of all, we have to admit that the general core-membership problem is hard (NP-complete or conpcomplete) even with these succinct representations. One reason for this fact that is both representations are fully expressive. So in a sense, we are essentially considering the general core-membership problem for general coalitional games with a different way to write down the game rules. This idea raises the issue of the trade-off between expressiveness and computational efficiency. Even though these compact representations make it possible to represent a subset of coalitional games using a reasonable amount of space, their expressive power might have caused them to be not helpful in alleviating the computational hardness of the core-membership problem. Thus, a natural future direction is to consider coalitional game representations that are not necessarily fully expressive. In practice, we often encounter particular types of games that can be represented as coalitional games with special structure. Therefore, some compact representations and corresponding algorithms that are tailored to these special games might be helpful in solving these practical scenarios of coalitional games. From a practical standpoint, solving games that are useful in practice using a not fully expressive representation is perhaps more important than developing a fully expressive representation that does not make the computational problem easier to solve. An interesting argument that I thought about while reading these results are related to the goal for considering the core membership problem. Conitzer and Sandholm [1] argued in their paper that the computational complexity of the core membership problem serves to increase the stability of the grand coalition. This argument is in sharp contrast to the effort by Ieong and Shoham [3] in developing an algorithm in order to overcome the computational hardness of the core membership problem. Perhaps before we dive into the investigation of the core-membership problem, we should carefully consider our goal in considering this computational problem in the first place. For instance, from a mechanism design perspective, it is potentially beneficial to ensure that the core membership determination is a hard computational problem. This computational difficulty not only illustrates that the core may be an unnecessarily strong solution concept, it also suggests that agents will face a difficult problem if they want to determine whether they should break away from the grand coalition. Thus, the computational hardness here serves to ensure the stability of the payoff vector for a practical scenario. This type of stability is particularly important if we take into account of possible manipulations by the agents. Obviously, we need to recognize that NP-completeness is a worst-case measure of hardness and this hardness result might not be a significant barrier if the problem instance is small enough. The paper by Conitzer and Sandholm [1] suggested future research directions alone these lines. They posed the question of whether 7

it is possible to design new payoff distribution schemes that are hard to manipulate. Also, they raised the question of whether it is possible to construct stability concepts that take into account of the complexity of finding a beneficial deviation. On the other hand, from an algorithmic analysis perspective, it is natural to keep searching for efficient algorithms for solving the core membership problem. Given a particular coalitional game, it would be nice to be able to find the payoff vectors in the core efficiently. This is useful information to determine the stability property of the given game. Hence, searching for tractable algorithms for the core membership problem in coalitional games is still a meaningful and important research direction to pursue. In their paper, Ieong and Shoham [3] suggested a few ideas for extending the MC-nets representation to make it more concise. These ideas are relevant for this particular paper. However, I would propose a different research direction concerning the big picture of solving the core membership problem in coalitional games. I don t think making the representation more and more concise is meaningful after a certain point. Rather, we should try to reason about the underlying reasons for the computational hardness of the core membership problem. From these results, I think that it might be more meaningful to search for representations which are not fully expressive, but nonetheless allow efficient algorithms for the core membership problems. It would be ideal if these representations are specially tailored to certain practical examples of coalitional games. References [1] Conitzer, V., and Sandholm, T. Computing shapley values, manipulating value division schemes, and checking core membership in multi-issue domains. In AAAI (2004), pp. 219 225. [2] Deng, X., and Papadimitriou, C. H. On the complexity of cooperative solution concepts. Math. Oper. Res. 19, 2 (1994), 257 266. [3] Ieong, S., and Shoham, Y. Marginal contribution nets: a compact representation scheme for coalitional games. In EC 05: Proceedings of the 6th ACM conference on Electronic commerce (New York, NY, USA, 2005), ACM, pp. 193 202. 8