scholarly journals Nash Stable Outcomes in Fractional Hedonic Games: Existence, Efficiency and Computation

2018 ◽  
Vol 62 ◽  
pp. 315-371 ◽  
Author(s):  
Vittorio Bilò ◽  
Angelo Fanelli ◽  
Michele Flammini ◽  
Gianpiero Monaco ◽  
Luca Moscardelli
Keyword(s):  
Social Welfare ◽  
Optimal Solution ◽  
Solution Concept ◽  
Coalition Structure ◽  
Upper And Lower Bounds ◽  
Price Of Stability ◽  
Average Value ◽  
Tree Graphs ◽  
Hedonic Games ◽  
Stable Outcome

We consider fractional hedonic games, a subclass of coalition formation games that can be succinctly modeled by means of a graph in which nodes represent agents and edge weights the degree of preference of the corresponding endpoints. The happiness or utility of an agent for being in a coalition is the average value she ascribes to its members. We adopt Nash stable outcomes as the target solution concept; that is we focus on states in which no agent can improve her utility by unilaterally changing her own group. We provide existence, efficiency and complexity results for games played on both general and specific graph topologies. As to the efficiency results, we mainly study the quality of the best Nash stable outcome and refer to the ratio between the social welfare of an optimal coalition structure and the one of such an equilibrium as to the price of stability. In this respect, we remark that a best Nash stable outcome has a natural meaning of stability, since it is the optimal solution among the ones which can be accepted by selfish agents. We provide upper and lower bounds on the price of stability for different topologies, both in case of weighted and unweighted edges. Beside the results for general graphs, we give refined bounds for various specific cases, such as triangle-free, bipartite graphs and tree graphs. For these families, we also show how to efficiently compute Nash stable outcomes with provable good social welfare.

scholarly journals The Impact of Selfishness in Hypergraph Hedonic Games

2020 ◽  
Vol 34 (02) ◽  
pp. 1766-1773
Author(s):  
Alessandro Aloisio ◽  
Michele Flammini ◽  
Cosimo Vinci
Keyword(s):  
Social Welfare ◽  
Price Of Anarchy ◽  
Maximum Size ◽  
Upper And Lower Bounds ◽  
Worst Case ◽  
The Social ◽  
Worst Case Ratio ◽  
Distinct Node ◽  
Hedonic Games ◽  
The Impact

We consider a class of coalition formation games that can be succinctly represented by means of hypergraphs and properly generalizes symmetric additively separable hedonic games. More precisely, an instance of hypegraph hedonic game consists of a weighted hypergraph, in which each agent is associated to a distinct node and her utility for being in a given coalition is equal to the sum of the weights of all the hyperedges included in the coalition. We study the performance of stable outcomes in such games, investigating the degradation of their social welfare under two different metrics, the k-Nash price of anarchy and k-core price of anarchy, where k is the maximum size of a deviating coalition. Such prices are defined as the worst-case ratio between the optimal social welfare and the social welfare obtained when the agents reach an outcome satisfying the respective stability criteria. We provide asymptotically tight upper and lower bounds on the values of these metrics for several classes of hypergraph hedonic games, parametrized according to the integer k, the hypergraph arity r and the number of agents n. Furthermore, we show that the problem of computing the exact value of such prices for a given instance is computationally hard, even in case of non-negative hyperedge weights.

scholarly journals An Anytime Algorithm for Optimal Coalition Structure Generation

2009 ◽  
Vol 34 ◽  
pp. 521-567 ◽  
Author(s):  
T. Rahwan ◽  
S. D. Ramchurn ◽  
N. R. Jennings ◽  
A. Giovannucci
Keyword(s):  
Optimal Solution ◽  
Coalition Structure ◽  
Upper And Lower Bounds ◽  
Multi Agent Systems ◽  
Worst Case ◽  
Structure Generation ◽  
Anytime Algorithm ◽  
Coalition Structures ◽  
Coalition Structure Generation ◽  
Generation Problem

Coalition formation is a fundamental type of interaction that involves the creation of coherent groupings of distinct, autonomous, agents in order to efficiently achieve their individual or collective goals. Forming effective coalitions is a major research challenge in the field of multi-agent systems. Central to this endeavour is the problem of determining which of the many possible coalitions to form in order to achieve some goal. This usually requires calculating a value for every possible coalition, known as the coalition value, which indicates how beneficial that coalition would be if it was formed. Once these values are calculated, the agents usually need to find a combination of coalitions, in which every agent belongs to exactly one coalition, and by which the overall outcome of the system is maximized. However, this coalition structure generation problem is extremely challenging due to the number of possible solutions that need to be examined, which grows exponentially with the number of agents involved. To date, therefore, many algorithms have been proposed to solve this problem using different techniques ranging from dynamic programming, to integer programming, to stochastic search all of which suffer from major limitations relating to execution time, solution quality, and memory requirements. With this in mind, we develop an anytime algorithm to solve the coalition structure generation problem. Specifically, the algorithm uses a novel representation of the search space, which partitions the space of possible solutions into sub-spaces such that it is possible to compute upper and lower bounds on the values of the best coalition structures in them. These bounds are then used to identify the sub-spaces that have no potential of containing the optimal solution so that they can be pruned. The algorithm, then, searches through the remaining sub-spaces very efficiently using a branch-and-bound technique to avoid examining all the solutions within the searched subspace(s). In this setting, we prove that our algorithm enumerates all coalition structures efficiently by avoiding redundant and invalid solutions automatically. Moreover, in order to effectively test our algorithm we develop a new type of input distribution which allows us to generate more reliable benchmarks compared to the input distributions previously used in the field. Given this new distribution, we show that for 27 agents our algorithm is able to find solutions that are optimal in 0.175% of the time required by the fastest available algorithm in the literature. The algorithm is anytime, and if interrupted before it would have normally terminated, it can still provide a solution that is guaranteed to be within a bound from the optimal one. Moreover, the guarantees we provide on the quality of the solution are significantly better than those provided by the previous state of the art algorithms designed for this purpose. For example, for the worst case distribution given 25 agents, our algorithm is able to find a 90% efficient solution in around 10% of time it takes to find the optimal solution.

scholarly journals On the price of stability of some simple graph-based hedonic games

2020 ◽  
Author(s):  
Christos Kaklamanis ◽  
Panagiotis Kanellopoulos ◽  
Konstantinos Papaioannou ◽  
Dimitris Patouchas
Keyword(s):  
Simple Graph ◽  
Price Of Stability ◽  
Hedonic Games

scholarly journals The combinatorics of discrete time-trees: theory and open problems

2016 ◽  
Author(s):  
Alex Gavryushkin ◽  
Chris Whidden ◽  
Frederick A Matsen
Keyword(s):  
Shortest Paths ◽  
Point Of View ◽  
Upper And Lower Bounds ◽  
Evolutionary Divergence ◽  
Discrete Approximations ◽  
Open Problems ◽  
Graph Theoretic ◽  
Tree Graphs ◽  
Object Of Study ◽  
Rooted Phylogenetic Tree

ABSTRACTA time-tree is a rooted phylogenetic tree such that all internal nodes are equipped with absolute divergence dates and all leaf nodes are equipped with sampling dates. Such time-trees have become a central object of study in phylogenetics but little is known about the parameter space of such objects. Here we introduce and study a hierarchy of discrete approximations of the space of time-trees from the graph-theoretic and algorithmic point of view. One of the basic and widely used phylogenetic graphs, the NNI graph, is the roughest approximation and bottom level of our hierarchy. More refined approximations discretize the relative timing of evolutionary divergence and sampling dates. We study basic graph-theoretic questions for these graphs, including the size of neighborhoods, diameter upper and lower bounds, and the problem of finding shortest paths. We settle many of these questions by extending the concept of graph grammars introduced by Sleator, Tarjan, and Thurston to our graphs. Although time values greatly increase the number of possible trees, we show that 1-neighborhood sizes remain linear, allowing for efficient local exploration and construction of these graphs. We also obtain upper bounds on the r-neighborhood sizes of these graphs, including a smaller bound than was previously known for NNI.Our results open up a number of possible directions for theoretical investigation of graph-theoretic and algorithmic properties of the time-tree graphs. We discuss the directions that are most valuable for phylogenetic applications and give a list of prominent open problems for those applications. In particular, we conjecture that the split theorem applies to shortest paths in time-tree graphs, a property not shared in the general NNI case.

A Study on Cooperative Continuous Static Games without Differentiability under Fuzzy Environment

2022 ◽  
Vol 11 (1) ◽  
pp. 0-0
Keyword(s):  
Fuzzy Numbers ◽  
Optimal Solution ◽  
Solution Concept ◽  
Proper Understanding ◽  
Fuzzy Environment ◽  
The Stability ◽  
The Right ◽  
The Cost ◽  
Stability Sets ◽  
Fuzzy Parameters

In this study, a fuzzy cooperative continuous static game (PQFCCSG) with n players having fuzzy parameters in all of the cost functions and the right- hand-side of constraints is characterized. Their fuzzy parameters are represented by piecewise quadratic fuzzy numbers. The α-pareto optimal solution concept is specified. In addition, the stability sets of the first and second kind without differentiability are conceptualized and established. An illustrated numerical example is discussed for proper understanding and interpretation of the proposed concept.

Runtime analysis of discrete particle swarm optimization algorithms: A survey

2019 ◽  
Vol 61 (4) ◽  
pp. 177-185
Author(s):  
Moritz Mühlenthaler ◽  
Alexander Raß
Keyword(s):  
Particle Swarm Optimization ◽  
Optimization Problems ◽  
Particle Swarm ◽  
Optimal Solution ◽  
Pso Algorithm ◽  
Upper And Lower Bounds ◽  
Swarm Optimization ◽  
Randomized Search Heuristics ◽  
Recent Developments ◽  
Randomized Search

Abstract A discrete particle swarm optimization (PSO) algorithm is a randomized search heuristic for discrete optimization problems. A fundamental question about randomized search heuristics is how long it takes, in expectation, until an optimal solution is found. We give an overview of recent developments related to this question for discrete PSO algorithms. In particular, we give a comparison of known upper and lower bounds of expected runtimes and briefly discuss the techniques used to obtain these bounds.

THE PRICE OF NASH EQUILIBRIA IN MULTICAST TRANSMISSION GAMES

2010 ◽  
Vol 11 (03n04) ◽  
pp. 97-120 ◽  
Author(s):  
VITTORIO BILÒ
Keyword(s):  
Nash Equilibrium ◽  
Cost Sharing ◽  
Price Of Anarchy ◽  
Optimal Solution ◽  
Minimum Cost ◽  
Price Of Stability ◽  
Common Source ◽  
Cost Share ◽  
Worst Case ◽  
The Cost

We consider the problem of sharing the cost of multicast transmissions in non-cooperative undirected networks where a set of receivers R wants to be connected to a common source s. The set of choices available to each receiver r ∈ R is represented by the set of all (s, r)-paths in the network. Given the choices performed by all the receivers, a public known cost sharing method determines the cost share to be charged to each of them. Receivers are selfish agents aiming to obtain the transmission at the minimum cost share and their interactions create a non-cooperative game. Devising cost sharing methods yielding games whose price of anarchy (price of stability), defined as the worst-case (best-case) ratio between the cost of a Nash equilibrium and that of an optimal solution, is not too high is thus of fundamental importance in non-cooperative network design. Moreover, since cost sharing games naturally arise in socio-economical contests, it is convenient for a cost sharing method to meet some constraining properties. In this paper, we first define several such properties and analyze their impact on the prices of anarchy and stability. We also reconsider all the methods known so far by classifying them according to which properties they satisfy and giving the first non-trivial lower bounds on their price of stability. Finally, we propose a new method, namely the free-riders method, which admits a polynomial time algorithm for computing a pure Nash equilibrium whose cost is at most twice the optimal one. Some of the ideas characterizing our approach have been independently proposed in Ref. 10.

scholarly journals Optimasi Proses Klasterisasi di MySQL DBMS dengan Mengintegrasikan Algoritme MIC-Kmeans Menggunakan Bahasa SQL dalam Stored Procedure

2020 ◽  
Vol 7 (2) ◽  
pp. 391
Author(s):  
Issa Arwani
Keyword(s):  
Clustering Algorithms ◽  
Optimal Solution ◽  
Test Results ◽  
Stored Procedure ◽  
Average Value ◽  
Silhouette Coefficient ◽  
Algorithm Mapping ◽  
Kmeans Algorithm ◽  
Time Required ◽  
Mapping Result

<p>Proses klasterisasi data di <em>DBMS</em> akan lebih efisien jika dilakukan langsung di dalam <em>DBMS</em> itu sendiri karena <em>DBMS</em> mendukung untuk pengelolaan data yang baik. <em>SQL-Kmeans</em> merupakan salah satu metode yang sebelumnya telah digunakan untuk mengintegrasikan algoritme klasterisasi <em>K-means</em> ke dalam <em>DBMS</em> menggunakan <em>SQL</em>. Akan tetapi, metode ini juga membawa kelemahan dari algoritme <em>K-means</em> itu sendiri yaitu lamanya iterasi untuk mencapai konvergen dan keakuratan hasil klasterisasi yang belum optimal akibat dari proses inisialisasi <em>centroid</em> awal secara acak. Algoritme <em>Median Initial Centroid (MIC)-Kmeans</em> merupakan pengembangan dari algoritme <em>K-means</em> yang bisa memberikan solusi optimal dalam menentukan awal <em>centroid</em> yang berdampak pada keakuratan dan lamanya iterasi. Dengan keunggulan yang dimiliki algoritme <em>MIC-Kmeans</em>, maka dalam penelitian ini dipilih sebagai alternatif algoritme yang diintegrasikan dalam proses klasterisasi data secara langsung di <em>DBMS</em> menggunakan <em>SQL</em>. Proses integrasinya meliputi 4 tahap yaitu tahap inisialisasi tabel <em>dataset</em>, tahap pemetaan algoritme <em>MIC-Kmeans</em> pada <em>SQL</em> dan tabel <em>dataset</em>, tahap perancangan <em>SQL </em>untuk tiap hasil pemetaan dan tahap implementasi rancangan <em>SQL</em> dalam <em>MySQL</em> <em>stored procedure</em>. Hasil pengujian menunjukkan bahwa metode <em>SQL MIC-Kmeans</em> bisa mengurangi 43% jumlah iterasi dan mengurangi 39% waktu yang dibutuhkan dari metode <em>SQL-Kmeans</em> untuk mencapai konvergen. Selain itu, nilai rata-rata <em>silhouette coefficient </em>metode <em>SQL MIC-Kmeans</em> adalah 0,79 dan masuk dalam kategori <em>strong structure</em> (nilai rentang 0,7 sampai 1). Sedangkan nilai rata-rata <em>silhouette coefficient </em>metode <em>SQL-Kmeans </em>adalah<em> </em>0,68<em> </em>dan masuk dalam kategori <em>medium structure </em>(nilai rentang 0,5 sampai 0,7).</p><p class="Judul2"><strong><em>Abstract</em></strong></p><p class="Judul2"><em>The process of data clustering in the DBMS will be more efficient because the DBMS supports good data management. SQL-Kmeans is a method that has been used to integrate K-means clustering algorithms into DBMS using SQL. However, it carries the weakness of the K-means algorithm itself in the duration of iterations to reach convergence and the accuracy of clustering due to the centroid initialization process randomly. Median Initial Centroid (MIC)-Kmeans algorithm is a development of the K-means algorithm that can provide the optimal solution in determining the initial centroid which has an impact on the accuracy and duration of iterations. With the advantages of the MIC-Kmeans algorithm, the method was chosen as an alternative algorithm to be integrated in the DBMS using SQL  for a clustering. The integration process includes 4 stages, there are dataset initialization, SQL algorithm mapping and dataset table, SQL design for each mapping result, and implementation SQL in the MySQL stored procedure. The test results show that the SQL MIC-Kmeans method can reduce 43% the number of iterations and reduce 39% of the time required from the SQL-Kmeans method to reach convergence. In addition, the average value of the coefficient SQL MIC-Kmeans method is 0.79 and categorized as strong structure (value ranges from 0.7 to 1). While, the average value of the coefficient SQL-Kmeans method is 0.68 and categorized as medium structure (value ranges from 0.5 to 0.7).</em></p>

The Price of Stability of Simple Symmetric Fractional Hedonic Games

2016 ◽  
pp. 220-232 ◽  
Author(s):  
Christos Kaklamanis ◽  
Panagiotis Kanellopoulos ◽  
Konstantinos Papaioannou
Keyword(s):  
Price Of Stability ◽  
Hedonic Games
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