On the computational complexity of qualitative coalitional games

Abstract We study coalitional games in which agents are each assumed to have a goal to be achieved, and where the characteristic property of a coalition is a set of choices , with each choice denoting a set of goals that would be achieved if the choice was made. Such qualitative coalitional games ( qcg s) are a natural tool for modelling goal-oriented multiagent systems. After introducing and formally defining qcg s, we systematically formulate fourteen natural decision problems associated with them, and determine the computational complexity of these problems. For example, we formulate a notion of coalitional stability inspired by that of the core from conventional coalitional games, and prove that the problem of showing that the core of a qcg is non-empty is D p 1 -complete. (As an aside, we present what we believe is the first “natural” problem that is proven to be complete for D p 2 .) We conclude by discussing the relationship of our work to other research on coalitional reasoning in multiagent systems, and present some avenues for future research.

[1]  Randal E. Bryant,et al.  Symbolic Boolean manipulation with ordered binary-decision diagrams , 1992, CSUR.

[2]  Michael Luck,et al.  Cooperation Structures , 1997, IJCAI.

[3]  David S. Johnson,et al.  A Catalog of Complexity Classes , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

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

[5]  Sarit Kraus,et al.  Feasible Formation of Coalitions Among Autonomous Agents in Nonsuperadditive Environments , 1999, Comput. Intell..

[6]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[7]  Jim Kadin.  The Polynomial Time Hierarchy Collapses if the Boolean Hierarchy Collapses , 1988, SIAM J. Comput..

[8]  Barbara Messing,et al.  An Introduction to MultiAgent Systems , 2002, Künstliche Intell..

[9]  Simon Parsons,et al.  Qualitative methods for reasoning under uncertainty , 2001 .

[10]  Celia Wrathall,et al.  Complete Sets and the Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[11]  Michael Wooldridge,et al.  Introduction to multiagent systems , 2001 .

[12]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[13]  Vaughan R. Pratt,et al.  The effect of basis on size of Boolean expressions , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[14]  Randal E. Bryant,et al.  Graph-Based Algorithms for Boolean Function Manipulation , 1986, IEEE Transactions on Computers.

[15]  Alan H. Bond,et al.  Readings in Distributed Artificial Intelligence , 1988 .

[16]  Dov M. Gabbay,et al.  More on Non-Cooperation in Dialogue Logic , 2001, Log. J. IGPL.

[17]  Mihalis Yannakakis,et al.  The complexity of facets (and some facets of complexity) , 1982, STOC '82.

[18]  Vincent Conitzer,et al.  Complexity of determining nonemptiness of the core , 2003, EC '03.

[19]  Moshe Tennenholtz,et al.  On Cooperation in a Multi-Entity Model , 1989, IJCAI.

[20]  Jesús Mario Bilbao Arrese,et al.  Complexity in cooperative game theory , 2000 .

[21]  V. M. Khrapchenko.  Method of determining lower bounds for the complexity of P-schemes , 1971 .

[22]  Paul E. Dunne.  Prevarication in dispute protocols , 2003, ICAIL.

[23]  Gerhard Weiss,et al.  Multiagent Systems , 1999 .

[24]  Claus-Peter Schnorr.  The network complexity and the Turing machine complexity of finite functions , 2004, Acta Informatica.

[25]  A. Razborov.  Lower bounds on monotone complexity of the logical permanent , 1985 .

[26]  Michael J. Fischer,et al.  Relations Among Complexity Measures , 1979, JACM.

[27]  Moshe Tennenholtz,et al.  Multi-entity Models , 1996, Machine Intelligence 14.

[28]  Tuomas Sandholm,et al.  Distributed rational decision making , 1999 .

[29]  H. Keiding,et al.  Effectivity functions in social choice , 1991 .

[30]  Albert A. Mullin,et al.  Review: O. B. Lupanov, On a Method of Circuit Synthesis , 1970 .

[31]  Richard Chang,et al.  The Boolean Hierarchy and the Polynomial Hierarchy: A Closer Connection , 1996, SIAM J. Comput..

[32]  Jan van Leeuwen,et al.  Handbook of Theoretical Computer Science, Vol. A: Algorithms and Complexity , 1994 .

[33]  Sarit Kraus,et al.  Task Allocation Via Coalition Formation Among Autonomous Agents , 1995, IJCAI.

[34]  Patrick Brézillon,et al.  Lecture Notes in Artificial Intelligence , 1999 .

[35]  Onn Shehory,et al.  Coalition structure generation with worst case guarantees , 2022 .

[36]  Sarit Kraus,et al.  Coalition Formation Among Autonomous Agents: Strategies and Complexity , 1993, MAAMAW.

[37]  Peter Haddawy,et al.  Utility Models for Goal‐Directed, Decision‐Theoretic Planners , 1998, Comput. Intell..

[38]  Ariel Rubinstein,et al.  A Course in Game Theory , 1995 .

[39]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[40]  Paul E. Dunne,et al.  The Complexity of Boolean Networks , 1988 .

[41]  Marc Pauly,et al.  A Modal Logic for Coalitional Power in Games , 2002, J. Log. Comput..

[42]  F. Zagare.  Game Theory: Concepts and Applications , 1984 .