Optimal Coalition Structures Research Articles

Optimal coalition structures for probabilistically monotone partition function games

For cooperative games with externalities, the problem of optimally partitioning a set of players into disjoint exhaustive coalitions is called coalition structure generation, and is a fundamental computational problem in multi-agent systems. Coalition structure generation is, in general, computationally hard and a large body of work has therefore investigated the development of efficient solutions for this problem. However, the existing methods are mostly limited to deterministic environments. In this paper, we focus attention on uncertain environments. Specifically, we define probabilistically monotone partition function games, a subclass of the well-known partition function games in which we introduce uncertainty. We provide a constructive proof that an exact optimum can be found using a greedy approach, present an algorithm for finding an optimum, and analyze its time complexity.

Optimisation for Coalitions Formation Considering the Fairness in Flexibility Market Participation

This paper proposes a coalitional game-theoretical model for consumers’ flexibility coalition formation, supported by an optimization model based on differential evolution. Traditionally, the participation in conventional electricity markets used to be limited to large producers and consumers. The final end-users contract their energy supply with retailers, since due to the smaller quantity available for trading, they cannot participate in electricity market transactions. Nowadays, the growing concept of local electricity market brings many advantages to the end-users. The flexibility negotiation considering local areas is an important procedure for network operators and it is incorporating a local electricity market opportunity. A coalition formation model to facilitate small players participation in the flexibility market proposed by the network operator is addressed in this work. The inclusion of Shapley value in the proposed model enables finding the best coalition structures considering the fairness of the coalitions in addition to the potential income achieved by the consumers when selling their flexibility. An optimization model based on differential evolution is also proposed as the way to find the optimal coalition structures based on the multi-criteria specifications.

Computing optimal coalition structures in polynomial time

The optimal coalition structure determination problem is in general computationally hard. In this article, we identify some problem instances for which the space of possible coalition structures has a certain form and constructively prove that the problem is polynomial time solvable. Specifically, we consider games with an ordering over the players and introduce a distance metric for measuring the distance between any two structures. In terms of this metric, we define the property of monotonicity, meaning that coalition structures closer to the optimal, as measured by the metric, have higher value than those further away. Similarly, quasi-monotonicity means that part of the space of coalition structures is monotonic, while part of it is non-monotonic. (Quasi)-monotonicity is a property that can be satisfied by coalition games in characteristic function form and also those in partition function form. For a setting with a monotonic value function and a known player ordering, we prove that the optimal coalition structure determination problem is polynomial time solvable and devise such an algorithm using a greedy approach. We extend this algorithm to quasi-monotonic value functions and demonstrate how its time complexity improves from exponential to polynomial as the degree of monotonicity of the value function increases. We go further and consider a setting in which the value function is monotonic and an ordering over the players is known to exist but ordering itself is unknown. For this setting too, we prove that the coalition structure determination problem is polynomial time solvable and devise such an algorithm.

A fast approximation algorithm for solving the complete set packing problem

We study the complete set packing problem (CSPP) where the family of feasible subsets may include all possible combinations of objects. This setting arises in applications such as combinatorial auctions (for selecting optimal bids) and cooperative game theory (for finding optimal coalition structures). Although the set packing problem has been well-studied in the literature, where exact and approximation algorithms can solve very large instances with up to hundreds of objects and thousands of feasible subsets, these methods are not extendable to the CSPP since the number of feasible subsets is exponentially large. Formulating the CSPP as an MILP and solving it directly, using CPLEX for example, is impossible for problems with more than 20 objects. We propose a new mathematical formulation for the CSPP that directly leads to an efficient algorithm for finding feasible set packings (upper bounds). We also propose a new formulation for finding tighter lower bounds compared to LP relaxation and develop an efficient method for solving the corresponding large-scale MILP. We test the algorithm with the winner determination problem in spectrum auctions, the coalition structure generation problem in coalitional skill games, and a number of other simulated problems that appear in the literature.

Randomized coalition structure generation

Randomization can be employed to achieve constant factor approximations to the coalition structure generation problem in less time than all previous approximation algorithms. In particular, this manuscript presents a new randomized algorithm that can generate a 2 3 approximate solution in O ( n 2.587 n ) time, improving upon the previous algorithm that required O ( n 2.83 n ) time to guarantee the same performance. Also, the presented new techniques allow a 1 4 approximate solution to be generated in the optimal time of O ( 2 n ) and improves on the previous best approximation ratio obtainable in O ( 2 n ) time of 1 8 . The presented algorithms are based upon a careful analysis of the sizes and numbers of coalitions in the smallest optimal coalition structures. An empirical analysis of the new randomized algorithms compared to their deterministic counterparts is provided. We find that the presented randomized algorithms generate solutions with utility comparable to what is returned by their deterministic counterparts (in some cases producing better results on average). Moreover, a significant speedup was found for most approximation ratios for the randomized algorithms over the deterministic algorithms. In particular, the randomized 1 2 approximate algorithm runs in approximately 22.4 % of the time required for the deterministic 1 2 approximation algorithm for problems with between 20 and 27 agents.

Computing optimal coalition structures in non-linear logistics domains

We study computing optimal coalition structures in non-linear logistics domains, where coalition values are not known a priori and computing them is NP-hard. In our setting, the common goal of the agents is to minimise the system|s cost. Agents perform two steps: 1) deliberate appropriate coalitions; 2) exchange computed coalitions and generate coalition structures. We apply the concept of best coalition introduced in Sombattheera and Ghose (2008), to work in the non-linear logistics domains. We provide an algorithm and explain via examples to show how it works. Lastly, we show the empirical results of our algorithm in terms of elapsed time and number of coalition structures generated.