Local Dynamics in Bargaining Networks via Random-Turn Games

We present a new technique for analyzing the rate of convergence of local dynamics in bargaining networks. The technique reduces balancing in a bargaining network to optimal play in a randomturn game. We analyze this game using techniques from martingale and Markov chain theory. We obtain a tight polynomial bound on the rate of convergence for a nontrivial class of unweighted graphs (the previous known bound was exponential). Additionally, we show this technique extends naturally to many other graphs and dynamics.

[1]  J. Nash.  THE BARGAINING PROBLEM , 1950, Classics in Game Theory.

[2]  L. Shapley,et al.  The assignment game I: The core , 1971 .

[3]  E. Berlekamp,et al.  Winning Ways for Your Mathematical Plays , 1983 .

[4]  K. Cook,et al.  The Distribution of Power in Exchange Networks: Theory and Experimental Results , 1983, American Journal of Sociology.

[5]  Sharon C. Rochford,et al.  Symmetrically pairwise-bargained allocations in an assignment market , 1984 .

[6]  David Williams,et al.  Probability with Martingales , 1991, Cambridge mathematical textbooks.

[7]  Karen S. Cook,et al.  Power in exchange networks: a power-dependence formulation , 1992 .

[8]  M. Maschler.  The bargaining set, kernel, and nucleolus , 1992 .

[9]  T. Raghavan,et al.  An algorithm for finding the nucleolus of assignment games , 1994 .

[10]  J. Propp,et al.  Richman games , 1995, math/9502222.

[11]  U. Faigle,et al.  An efficient algorithm for nucleolus and prekernel computation in some classes of TU-games , 1998 .

[12]  David Willer.  Network Exchange Theory , 1999 .

[13]  J. Propp,et al.  Combinatorial Games under Auction Play , 1999 .

[14]  Daniël Paulusma,et al.  Complexity aspects of cooperative games , 2001 .

[15]  Margarida Corominas-Bosch,et al.  Bargaining in a network of buyers and sellers , 2004, J. Econ. Theory.

[16]  Kamal Jain,et al.  A polynomial time algorithm for computing an Arrow-Debreu market equilibrium for linear utilities , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[17]  Y. Peres,et al.  Tug-of-war and the infinity Laplacian , 2006, math/0605002.

[18]  Xiaotie Deng,et al.  Settling the Complexity of Two-Player Nash Equilibrium , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[19]  Fernando Charro,et al.  A mixed problem for the infinity Laplacian via Tug-of-War games , 2007, 0706.4267.

[20]  Fan Chung Graham,et al.  Internet and Network Economics, Third International Workshop, WINE 2007, San Diego, CA, USA, December 12-14, 2007, Proceedings , 2007, WINE.

[21]  Dmitry M. Malioutov,et al.  Linear programming analysis of loopy belief propagation for weighted matching , 2007, NIPS.

[22]  Tanmoy Chakraborty,et al.  Bargaining Solutions in a Social Network , 2008, WINE.

[23]  Éva Tardos,et al.  Balanced outcomes in social exchange networks , 2008, STOC.

[24]  Vahab S. Mirrokni,et al.  Uncoordinated two-sided matching markets , 2009, SECO.

[25]  Yifeng Yu.  Maximal and minimal solutions of an Aronsson equation: L∞ variational problems versus the game theory , 2009, 0906.0625.

[26]  Paul W. Goldberg,et al.  The Complexity of Computing a Nash Equilibrium , 2009, SIAM J. Comput..

[27]  Sanjeev Khanna,et al.  Network bargaining: algorithms and structural results , 2009, EC '09.

[28]  Yuval Peres,et al.  Convergence of Local Dynamics to Balanced Outcomes in Exchange Networks , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[29]  Tanmoy Chakraborty,et al.  A behavioral study of bargaining in social networks , 2010, EC '10.

[30]  Moez Draief,et al.  Bargaining dynamics in exchange networks , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[31]  Yashodhan Kanoria,et al.  An FPTAS for Bargaining Networks with Unequal Bargaining Powers , 2010, WINE.

[32]  Nikhil R. Devanur,et al.  Monotonicity in bargaining networks , 2010, SODA '10.

[33]  Nicole Immorlica,et al.  The Cooperative Game Theory Foundations of Network Bargaining Games , 2010, ICALP.

[34]  Yashodhan Kanoria,et al.  Fast convergence of natural bargaining dynamics in exchange networks , 2010, SODA '11.

[35]  Christian Borgs,et al.  Belief Propagation for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions , 2007, SIAM J. Discret. Math..