A Dialogue Game for Coalition Structure Generation with Self-Interested Agents

Coalition structure generation (CSG) for multi-agent systems is a well-studied problem. A vast majority of the previous work and the state-of-the-art approaches to CSG assume a characteristic function form of the coalition values, where a coalition's value is independent of the other coalitions in the coalition structure. Recently, there has been interest in the more realistic partition function form of coalition values, where the value of a coalition is affected by how the other agents are partitioned, via externalities. We argue that in domains with externalities, a distributed/adaptive approach to CSG may be impractical, and that a centralized approach to CSG is more suitable. However, the most recent studies in this direction have focused on cases where all externalities are either always positive or always negative, and results on coalition structure generation in more general settings (in particular, mixed externalities) are lacking. In this paper we propose a framework based on agent-types that incorporates mixed externalities and demonstrate that it includes the previous settings as special cases. We also generalize some previous results in anytime CSG, showing that those results are again special cases. In particular, we extend the existing branch and bound algorithm to this new setting and show empirically that significant pruning can be achieved when searching for the optimal coalition structure. This extends the state-of-the-art in CSG for multi-agent systems.

An important research problem in artificial intelligence is how to organize multiple agents, and coordinate them, so that they can work together to solve problems. Coordinating agents in a multi-agent system can significantly affect the system’s performance—the agents can, in many instances, be organized so that they can solve tasks more efficiently, and consequently benefit collectively and individually. Central to this endeavor is coalition formation—the process by which heterogeneous agents organize and form disjoint groups (coalitions). Coalition formation often involves finding a coalition structure (an exhaustive set of disjoint coalitions) that maximizes the system’s potential performance (e.g., social welfare) through coalition structure generation. However, coalition structure generation typically has no notion of goals. In cooperative settings, where coordination of multiple coalitions is important, this may generate suboptimal teams for achieving and accomplishing the tasks and goals at hand. With this in mind, we consider simultaneously generating coalitions of agents and assigning the coalitions to independent alternatives (e.g., tasks/goals), and present an anytime algorithm for the simultaneous coalition structure generation and assignment problem. This combinatorial optimization problem has many real-world applications, including forming goal-oriented teams. To evaluate the presented algorithm’s performance, we present five methods for synthetic problem set generation, and benchmark the algorithm against the industry-grade solver CPLEX using randomized data sets of varying distribution and complexity. To test its anytime-performance, we compare the quality of its interim solutions against those generated by a greedy algorithm and pure random search. Finally, we also apply the algorithm to solve the problem of assigning agents to regions in a major commercial strategy game, and show that it can be used in game-playing to coordinate smaller sets of agents in real-time.

The coalition formation process, in which a number of independent, autonomous agents come together to act as a collective, is an important form of interaction in multiagent systems. When effective, such coalitions can improve the performance of the individual agents and/or of the system as a whole. However, one of the main problems that hinders the wide spread adoption of coalition formation technologies is the computational complexity of coalition structure generation. That is, once a group of agents has been identified, how can it be partitioned in order tomaximise the social payoff? This problem has been shown to be NP-hard and even finding a sub-optimal solution requires searching an exponential number of solutions. Against this background, this paper reports on a novel anytime algorithm for coalition structure generation that produces solutions that are within a finite bound from the optimal. Our algorithm is benchmarked against Sandholm et al.ýs algorithm [8] (the only other known algorithm for this task that can also establish a worst-case bound from the optimal) and is shown to be up to 10^379 times faster (for systems containing 1000 agents) when small bounds from the optimal are desirable.

Coalition formation is a key topic in multi-agent systems. To date, most work on this problem has concentrated on simple characteristic function games. However, this lacks the notion of tasks which makes it more difficult to apply it in many applications. Dang et showed that this problem was NP-hard and that the minimum number of coalition structures that need to be searched through in order to establish a solution within a bound from the optimal was (2$^{m+{\it n}-1}$-1). Then Dang et presented an algorithm that takes a step further to search those task-based coalition structures whose biggest task-based coalition's cardinality is greater than or equal to ***n (k *** 1)/(k + 1)*** in order to attain the bound K , which is the best result known so far. Against this background, this paper reports on a novel anytime algorithm based on cardinality structure that only have to take a step further to search those task-based coalition structures whose cardinality structure is in the CTCS (n , m , K *). Finally via contrast experiment, the algorithm reported in this paper is obviously better than that of Dang et al. (up to 1019 times faster when n =60,m =40, K =3).

Coalition formation is a fundamental research topic in multi-agent systems. In this context, while it is desirable to generate a coalition structure that maximizes the sum of the values of the coalitions, the space of possible solutions is often too large to allow exhaustive search. Thus, a fundamental open question in this area is the following: Can we search through only a subset of coalition structures, and be guaranteed to find a solution that is within a desirable bound β from optimum? If so, what is the minimum such subset?. To date, the above question has only been partially answered by Sandholm et al. in their seminal work on anytime coalition structure generation [Sandholm et al., 1999]. More specifically, they identified minimum subsets to be searched for two particular bounds: β = n and β = ⌈n/2⌉. Nevertheless, the question remained open for other values of β. In this paper, we provide the complete answer to this question.

How to form effective coalitions is an important issue in multi-agent systems. Coalition Structure Generation ({{mathsf {CSG}}}) is a fundamental problem whose formalization can encompass various applications related to multi-agent cooperation. {{mathsf {CSG}}} involves partitioning a set of agents into coalitions such that the social surplus (i.e., the sum of the values of all coalitions) is maximized. In traditional {mathsf {CSG}}, we are guaranteed that all coalitions will be successfully established, that is, the attendance rate of each agent for joining any coalition is assumed to be 1.0. Having the real world in mind, however, it is natural to consider the uncertainty of agents’ availabilities, e.g., an agent might be available only two or three days a week because of his/her own schedule. Probabilistic Coalition Structure Generation ({{mathsf {PCSG}}}) is an extension of {mathsf {CSG}} where the attendance type of each agent is considered. The aim of this problem is to find the optimal coalition structure which maximizes the sum of the expected values of all coalitions. In {mathsf {PCSG}}, since finding the optimal coalition structure easily becomes intractable, it is important to consider approximation algorithms, i.e., to consider a trade-off between the quality of the returned solution and tractability. In this paper, a formal framework for {mathsf {PCSG}} is introduced. Approximation algorithms for {mathsf {PCSG}} called Bounded Approximation Algorithm based on Attendance Types ({{mathsf {BAAAT}}}) and Involved {mathsf {BAAAT}} ({{mathsf {IBAAAT}}}) are then presented. We prove a priori bounds on the quality of the solution returned by {mathsf {BAAAT}} and {mathsf {IBAAAT}} with respect to the optimum and perform experimental evaluations on a number of benchmarks.

Cooperative Games in Graph Structure

Optimal Coalition Structure Generation (CSG) is a significant research problem in multi-agent systems that remains difficult to solve. This problem has many important applications in transportation, eCommerce, distributed sensor networks and others. The CSG problem is NP-complete and finding the optimal result for n agents needs to check $$O (n^n)$$ possible partitions. The ODP–IP algorithm (Michalak et al. in Artif Intell 230:14–50, 2016) achieves the current lowest worst-case time complexity of $$O (3^n)$$ . In the light of its high computational time complexity, we devise an Imperfect Dynamic Programming (ImDP) algorithm for the CSG problem with runtime $$O (n2^n)$$ given n agents. Imperfect algorithm means that there are some contrived inputs for which the algorithm fails to give the optimal result. We benchmarked ImDP against ODP–IP and proved its efficiency. Experimental results confirmed that ImDP algorithm performance is better for several data distributions, and for some it improves dramatically ODP–IP. For example, given 27 agents, with ImDP for agent-based uniform distribution time gain is 91% (i.e. 49 min).

Finding the optimal coalition structure is an NP-complete problem that is computationally challenging even under quite restrictive assumptions. In this paper, we propose a new algorithm for the Coalition Structure Generation (CSG) problem, which provides good enough quality solutions and that can be run with hundreds of agents. The Fast code-based Algorithm for Coalition Structure generation (FACS) uses a novel representation of the search space of coalition structures and a new code-based search technique. We devise an effective heuristic search method to efficiently explore the space of coalition structures using our code-based technique. Results show that our method outperforms existing state-of-the-art algorithms by multiple orders of magnitude while providing high-quality solutions.

Edge computing (EC) is a distributed computing paradigm that brings computation and data storage closer to the data sources. With the rapid development of EC, offloading scientific computing and data processing tasks from end devices to edge nodes (ENs) can satisfy these tasks’ requirements. However, a single EN with limited computing capacity is usually insufficient to handle these tasks. Hence, using a coalition structure (CS) of many ENs to handle multiple concurrent tasks in an EC environment becomes a feasible scheme. Still, many existing methods cannot be used directly in this scenario because of enormous CS solution space, low search speed, long-running time, poor optimal solution quality, etc. In response, we propose an arbitrary discrete political optimizer (ADPO) algorithm with a discrete recent-past position updating strategy to explore the potential CS space. Unlike other heuristic algorithms, ADPO further improves the better solutions by interacting with each other in the parliamentary affairs phase. After that, we formulate an integer programming (IP) optimization model to alleviate the low search speed or long time consuming. Finally, extensive experiments demonstrate that ADPO is superior to the existing heuristic algorithm. In addition, the IP model excels in exiting algorithms at the running time (milliseconds level) and resource occupied ratio.

One research challenge in multi-agent systems is how to partition a set of agent into coalition structures. The number of different coalitions for a set of agents grows exponential with the number of agents and the number of different partitions grows even faster. The common approach for this problem is to search for the coalition structure that maximizes the system outcome (denoted CS *). Until recently, most algorithms that solve this problem considered that the value of a coalition is given by a characteristic function, in where those values are not influenced by factors that are external to the coalition. More recently, several authors have focused on problems that consider the presence of externalities. In this case, centralized algorithms were developed to search for the CS *, but no algorithm was developed to work in a distributed environment. This paper presents a distributed algorithm for searching coalition structures under presence of externalities.

Randomized coalition structure generation

We consider the issue of representing coalitional games in multi-agent systems with externalities (i.e., in systems where the performance of one coalition may be affected by other co-existing coalitions). In addition to the conventional partition function game representation (PFG), we propose a number of new representations based on a new notion of externalities. In contrast to conventional game theory, our new concept is not related to the process by which the coalitions are formed, but rather to the effect that each coalition may have on the entire system and vice versa. We show that the new representations are fully expressive and, for many classes of games, more concise than the conventional PFG. Building upon these new representations, we propose a number of approaches to solve the coalition structure generation problem in systems with externalities. We show that, if externalities are characterised by various degrees of regularity, the new representations allow us to adapt coalition structure generation algorithms that were originally designed for domains with no externalities, so that they can be used when externalities are present. Finally, building upon Rahwan et al. [16] and Michalak et al. [9], we present a unified method to solve the coalition structure generation problem in any system, with or without externalities, provided sufficient information is available.

This paper presents a new way of formalizing the Coalition Structure Generation problem (CSG), so that we can apply constraint optimization techniques to it. Forming effective coalitions is a major research challenge in AI and multi-agent systems. CSG involves partitioning a set of agents into coalitions so that social surplus is maximized. Traditionally, the input of the CSG problem is a black-box function called a characteristic function, which takes a coalition as an input and returns the value of the coalition. As a result, applying constraint optimization techniques to this problem has been infeasible. However, characteristic functions that appear in practice often can be represented concisely by a set of rules, rather than a single black-box function. Then, we can solve the CSG problem more efficiently by applying constraint optimization techniques to the compact representation directly. We present new formalizations of the CSG problem by utilizing recently developed compact representation schemes for characteristic functions.We first characterize the complexity of the CSG under these representation schemes. In this context, the complexity is driven more by the number of rules rather than by the number of agents. Furthermore, as an initial step towards developing efficient constraint optimization algorithms for solving the CSG problem, we develop mixed integer programming formulations and show that an off-the-shelf optimization package can perform reasonably well, i.e., it can solve instances with a few hundred agents, while the state-of-the-art algorithm (which does not make use of compact representations) can solve instances with up to 27 agents.

Overlapping coalition formation in game theory: A state-of-the-art review