Eisenberg-Gale markets: algorithms and structural properties

We define a new class of markets, the Eisenberg-Gale markets. This class contains Fisher's linear market, markets from the resource allocation framework of Kelly kelly, as well as numerous interesting new markets.We obtain combinatorial, strongly polynomial algorithms for severalmarkets in this class. Our algorithms have a simple description as ascending price auctions. Our algorithms lead to insights into the efficiency, fairness, rationality of solutions, and competition monotonicity of these markets. A classification of Eisenberg-Gale markets w.r.t. these properties reveals a surprisingly rich set of possibilities.

[1]  H. Scarf,et al.  How to Compute Equilibrium Prices in 1891 , 2005 .

[2]  Nimrod Megiddo,et al.  Optimal flows in networks with multiple sources and sinks , 1974, Math. Program..

[3]  S. Fujishige,et al.  A Strongly Polynomial-Time Algorithm for Minimizing Submodular Functions (Algorithm Engineering as a New Paradigm) , 1999 .

[4]  Vijay V. Vazirani,et al.  Approximation algorithms for metric facility location and k-Median problems using the primal-dual schema and Lagrangian relaxation , 2001, JACM.

[5]  Frank Kelly,et al.  Charging and rate control for elastic traffic , 1997, Eur. Trans. Telecommun..

[6]  Christos H. Papadimitriou,et al.  Worst-case equilibria , 1999 .

[7]  Lun Li,et al.  Cross-layer optimization in TCP/IP networks , 2005, IEEE/ACM Transactions on Networking.

[8]  Steven H. Low,et al.  Optimization flow control—I: basic algorithm and convergence , 1999, TNET.

[9]  Vijay V. Vazirani,et al.  Market equilibria for homothetic, quasi-concave utilities and economies of scale in production , 2005, SODA '05.

[10]  Lloyd S. Shapley,et al.  On balanced sets and cores , 1967 .

[11]  Vijay V. Vazirani,et al.  Equitable cost allocations via primal-dual-type algorithms , 2002, STOC '02.

[12]  Frank Kelly,et al.  Rate control for communication networks: shadow prices, proportional fairness and stability , 1998, J. Oper. Res. Soc..

[13]  Kiyohito Nagano,et al.  On Convex Minimization over Base Polytopes , 2007, IPCO.

[14]  Y. L. Chen,et al.  Multicommodity Network Flows with Safety Considerations , 1992, Oper. Res..

[15]  Alexander Schrijver,et al.  A Combinatorial Algorithm Minimizing Submodular Functions in Strongly Polynomial Time , 2000, J. Comb. Theory B.

[16]  Nikhil R. Devanur,et al.  Market equilibrium via a primal-dual-type algorithm , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[17]  W. T. Tutte.  On the Problem of Decomposing a Graph into n Connected Factors , 1961 .

[18]  Nikhil R. Devanur,et al.  On Competitiveness in Uniform Utility Allocation Markets , 2007, WINE.

[19]  E. Eisenberg.  Aggregation of Utility Functions , 1961 .

[20]  H. Groenevelt.  Two algorithms for maximizing a separable concave function over a polymatroid feasible region , 1991 .

[21]  Ashish Goel,et al.  Approximate majorization and fair online load balancing , 2001, TALG.

[22]  V. Jacobson,et al.  Congestion avoidance and control , 1988, CCRV.

[23]  QUTdN QeO,et al.  Random early detection gateways for congestion avoidance , 1993, TNET.

[24]  Satoru Iwata,et al.  A combinatorial strongly polynomial algorithm for minimizing submodular functions , 2001, JACM.

[25]  E. Eisenberg,et al.  CONSENSUS OF SUBJECTIVE PROBABILITIES: THE PARI-MUTUEL METHOD, , 1959 .

[26]  Vijay V. Vazirani,et al.  A primal-dual algorithm for computing Fisher equilibrium in the absence of gross substitutability property , 2005, Theor. Comput. Sci..

[27]  Bruno Codenotti,et al.  Efficient Computation of Equilibrium Prices for Markets with Leontief Utilities , 2004, ICALP.

[28]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[29]  C. Nash-Williams.  Edge-disjoint spanning trees of finite graphs , 1961 .

[30]  Nikhil R. Devanur,et al.  New Results on Rationality and Strongly Polynomial Time Solvability in Eisenberg-Gale Markets , 2006, WINE.