Enrico Malizia

On the complexity of core, kernel, and bargaining set

Coalitional games model scenarios where players can collaborate by forming coalitions in order to obtain higher worths than by acting in isolation. A fundamental issue of coalitional games is to single out the most desirable outcomes in terms of worth distributions, usually called solution concepts. Since decisions taken by realistic players cannot involve unbounded resources, recent computer science literature advocated the importance of assessing the complexity of computing with solution concepts. In this context, the paper provides a complete picture of the complexity issues arising with three prominent solution concepts for coalitional games with transferable utility, namely, the core, the kernel, and the bargaining set, whenever the game worth-function is represented in some reasonably compact form. The starting points of the investigation are the settings of graph games and of marginal contribution nets, where the worth of any coalition can be computed in polynomial time in the size of the game encoding and for which various open questions were stated in the literature. The paper answers these questions and, in addition, provides new insights on succinctly specified games, by characterizing the computational complexity of the core, the kernel, and the bargaining set in relevant generalizations and specializations of the two settings. Concerning the generalizations, the paper shows that dealing with arbitrary polynomial-time computable worth functions-no matter of the specific game encoding being considered-does not provide any additional source of complexity compared to graph games and marginal contribution nets. Instead, only for the core, a slight increase in complexity is exhibited for classes of games whose worth functions encode NP-hard optimization problems, as in the case of certain combinatorial games. As for specializations, the paper illustrates various tractability results on classes of bounded treewidth graph games and marginal contribution networks.

Subsidies, stability, and restricted cooperation in coalitional games

Cooperation among automated agents is becoming increasingly important in various artificial intelligence applications. Coalitional (i.e., cooperative) game theory supplies conceptual and mathematical tools useful in the analysis of such interactions, and in particular in the achievement of stable outcomes among self-interested agents. Here, we study the minimal external subsidy required to stabilize the core of a coalitional game. Following the Cost of Stability (CoS) model introduced by Bachrach et al. [2009a], we give tight bounds on the required subsidy under various restrictions on the social structure of the game. We then compare the extended core induced by subsidies with the least core of the game, proving tight bounds on the ratio between the minimal subsidy and the minimal demand relaxation that each lead to stability.

The Complexity of the Nucleolus in Compact Games

The nucleolus is a well-known solution concept for coalitional games to fairly distribute the total available worth among the players. The nucleolus is known to be NP-hard to compute over compact coalitional games, that is, over games whose functions specifying the worth associated with each coalition are encoded in terms of polynomially computable functions over combinatorial structures. In particular, hardness results have been exhibited over minimum spanning tree games, threshold games, and flow games. However, due to its intricate definition involving reasoning over exponentially many coalitions, a nontrivial upper bound on its complexity was missing in the literature and looked for. This article faces this question and precisely characterizes the complexity of the nucleolus, by exhibiting an upper bound that holds on any class of compact games, and by showing that this bound is tight even on the (structurally simple) class of graph games. The upper bound is established by proposing a variant of the standard linear-programming based algorithm for nucleolus computation and by studying a framework for reasoning about succinctly specified linear programs, which are contributions of interest in their own. The hardness result is based on an elaborate combinatorial reduction, which is conceptually relevant for it provides a “measure” of the computational cost to be paid for guaranteeing voluntary participation to the distribution process. In fact, the pre-nucleolus is known to be efficiently computable over graph games, with this solution concept being defined as the nucleolus but without guaranteeing that each player is granted with it at least the worth she can get alone, that is, without collaborating with the other players. Finally, this article identifies relevant tractable classes of coalitional games, based on the notion of type of a player. Indeed, in most applications where many players are involved, it is often the case that such players do belong in fact to a limited number of classes, which is known in advance and may be exploited for computing the nucleolus in a fast way.

Non-transferable utility coalitional games via mixed-integer linear constraints

Coalitional games serve the purpose of modeling payoff distribution problems in scenarios where agents can collaborate by forming coalitions in order to obtain higher worths than by acting in isolation. In the classical Transferable Utility (TU) setting, coalition worths can be freely distributed amongst agents. However, in several application scenarios, this is not the case and the Non-Transferable Utility setting (NTU) must be considered, where additional application-oriented constraints are imposed on the possible worth distributions. In this paper, an approach to define NTU games is proposed which is based on describing allowed distributions via a set of mixed-integer linear constraints applied to an underlying TU game. It is shown that such games allow non-transferable conditions on worth distributions to be specified in a natural and succinct way. The properties and the relationships among the most prominent solution concepts for NTU games that hold when they are applied on (mixed-integer) constrained games are investigated. Finally, a thorough analysis is carried out to assess the impact of issuing constraints on the computational complexity of some of these solution concepts.

Achieving new upper bounds for the hypergraph duality problem through logic

The hypergraph duality problem Dual is defined as follows: given two simple hypergraphs G and H, decide whether H consists precisely of all minimal transversals of G (in which case we say that G is the dual of H). This problem is equivalent to decide whether two given non-redundant monotone DNFs are dual. It is known that DUAL, the complementary problem to Dual, is in GC (log2 n, PTIME), where GC(f(n), C) denotes the complexity class of all problems that after a nondeterministic guess of O(f(n)) bits can be decided (checked) within complexity class C. It was conjectured that DUAL is in GC(log2 n, LOGSPACE). In this paper we prove this conjecture and actually place the DUAL problem into the complexity class GC(log2 n, TC0) which is a subclass of GC(log2 n, LOGSPACE). We here refer to the logtime-uniform version of TC0, which corresponds to FO(COUNT), i.e., first order logic augmented by counting quantifiers. We achieve the latter bound in two steps. First, based on existing problem decomposition methods, we develop a new nondeterministic algorithm for DUAL that requires to guess O(log2 n) bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in FO(COUNT).

Hard and easy k -typed compact coalitional games: the knowledge of player types marks the boundary

Coalitional games model scenarios where rational agents can form coalitions so as to obtain higher worths than by acting in isolation. Once a coalition forms and obtains its worth, the problem of how this worth can be fairly distributed has to be faced. Desirable worth distributions are usually referred to as solution concepts. Recent research pointed out that, while reasoning problems involving such solution concepts are hard in general for games specified in compact form (e.g., graph games), some of them, in particular the core, become tractable when agents come partitioned into a fixed number k of types, i.e., of classes of strategically equivalent players. The paper continues along this line of research, by firstly showing that two other relevant solution concepts, the kernel and the nucleolus, are tractable in this setting and independently of the specific game encoding, provided worth functions are given as a polynomial-time computable oracles. Then, it analyzes a different setting where games are still k-typed but the actual player partitioning is not a-priori known. Within this latter setting, the paper addresses the question about how efficiently strategic equivalence between pairs of players can be recognized, and reconsiders the computational complexity of the core, the kernel, and the nucleolus. All such problems and notions emerged to be intractable, thereby evidencing that the knowledge of player types marks the boundary of tractability for reasoning about k-typed coalitional games.