NDLI: Root counts of semi-mixed systems, and an application to counting nash equilibria

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Root counts of semi-mixed systems, and an application to counting nash equilibria

Content Provider	ACM Digital Library
Author	Vidunas, Raimundas Emiris, Ioannis Z.
Abstract	Semi-mixed algebraic systems are those where the equations can be partitioned into subsets with common Newton polytopes. We are interested in counting roots of semi-mixed multihomogeneous systems, where both variables and equations can be partitioned into blocks, and each block of equations has a given degree in each block of variables. The motivating example is counting the number of totally mixed Nash equilibria in games of several players. Firstly, this paper relates and unifies the BKK and multivariate Bézout bounds for semi-mixed systems, through mixed volumes and matrix permanents. Permanent expressions for BKK bounds hold for all multihomogeneous systems, without any requirement of semi-mixed structure, as well as systems whose Newton polytopes are products of polytopes in complementary subspaces. Secondly, by means of a novel asymptotic analysis, the complexity of a combinatorial geometric algorithm for semi-mixed volumes (i.e., mixed volumes of semi-mixed systems) is explored and juxtaposed to the complexities of computing permanents, or using generating functions (via MacMahon's Master theorem), or orthogonal polynomials.
Starting Page	154
Ending Page	161
Page Count	8
File Format	PDF
ISBN	9781450325011
DOI	10.1145/2608628.2608679
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2014-07-23
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Generating function Multihomogeneous bézout bound Nash equilibrium Matrix permanent Mixed volume
Content Type	Text
Resource Type	Article

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