Probability Bounds for Overlapping Coalition Formation

In this work, we provide novel methods which benefit from obtained probability bounds for assessing the ability of teams of agents to accomplish coalitional tasks. To this end, our first method is based on an improvement of the Paley-Zygmund inequality, while the second and the third ones are devised based on manipulations of the two-sided Chebyshev’s inequality and the Hoeffding’s inequality, respectively. Agents have no knowledge of the amount of resources others possess; and hold private Bayesian beliefs regarding the potential resource investment of every other agent. Our methods allow agents to demand that certain confidence levels are reached, regarding the resource contributions of the various coalitions. In order to tackle real-world scenarios, we allow agents to form overlapping coalitions, so that one can simultaneously be part of a number of coalitions. We thus present a protocol for iterated overlapping coalition formation (OCF), through which agents can complete tasks that grant them utility. Agents lie on a social network and their distance affects their likelihood of cooperation towards the completion of a task. We confirm our methods’ effectiveness by testing them on both a random graph of 300 nodes and a real-world social network of 4039 nodes.

[1]  Makoto Yokoo,et al.  Concise Characteristic Function Representations in Coalitional Games Based on Agent Types , 2011, IJCAI.

[2]  Nicholas R. Jennings,et al.  Coalitional stability in structured environments , 2012, AAMAS.

[3]  Robert M Thrall,et al.  Mathematics of Operations Research. , 1978 .

[4]  Roger B. Myerson,et al.  Graphs and Cooperation in Games , 1977, Math. Oper. Res..

[5]  Xiaotie Deng,et al.  On the Complexity of Cooperative Solution Concepts , 1994, Math. Oper. Res..

[6]  Sarit Kraus,et al.  Methods for Task Allocation via Agent Coalition Formation , 1998, Artif. Intell..

[7]  R. Fildes.  Journal of the American Statistical Association : William S. Cleveland, Marylyn E. McGill and Robert McGill, The shape parameter for a two variable graph 83 (1988) 289-300 , 1989 .

[8]  Craig Boutilier,et al.  Coalition formation under uncertainty: bargaining equilibria and the Bayesian core stability concept , 2007, AAMAS '07.

[9]  R. Paley,et al.  A note on analytic functions in the unit circle , 1932, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  Michael Wooldridge,et al.  Computational Aspects of Cooperative Game Theory , 2011, KES-AMSTA.

[11]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[12]  Peter Kulchyski.  and , 2015 .

[13]  Nicholas R. Jennings,et al.  Overlapping Coalition Formation for Efficient Data Fusion in Multi-Sensor Networks , 2006, AAAI.

[14]  T. Komatsu.  On the number of solutions of the Diophantine equation of Frobenius - General case , 2003 .

[15]  Sarit Kraus,et al.  Coalition formation with uncertain heterogeneous information , 2003, AAMAS '03.

[16]  Yoav Shoham,et al.  Bayesian Coalitional Games , 2008, AAAI.

[17]  Jure Leskovec,et al.  {SNAP Datasets}: {Stanford} Large Network Dataset Collection , 2014 .

[18]  Ali Esmaili,et al.  Probability and Random Processes , 2005, Technometrics.

[19]  Nils J. Nilsson,et al.  Artificial Intelligence , 1974, IFIP Congress.

[20]  Simon Gilchrist,et al.  The Macroeconomic Impact of Financial and Uncertainty Shocks , 2016 .

[21]  Nicholas R. Jennings,et al.  Cooperative Games with Overlapping Coalitions , 2010, J. Artif. Intell. Res..

[22]  Craig Boutilier,et al.  Bayesian reinforcement learning for coalition formation under uncertainty , 2004, Proceedings of the Third International Joint Conference on Autonomous Agents and Multiagent Systems, 2004. AAMAS 2004..

[23]  Eli Upfal,et al.  Probability and Computing: Randomized Algorithms and Probabilistic Analysis , 2005 .

[24]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[25]  Edith Elkind,et al.  Overlapping coalition formation games: charting the tractability frontier , 2012, AAMAS 2012.