176 research outputs found
Efficient Decomposition of Bimatrix Games (Extended Abstract)
Exploiting the algebraic structure of the set of bimatrix games, a
divide-and-conquer algorithm for finding Nash equilibria is proposed. The
algorithm is fixed-parameter tractable with the size of the largest irreducible
component of a game as parameter. An implementation of the algorithm is shown
to yield a significant performance increase on inputs with small parameters.Comment: In Proceedings SR 2014, arXiv:1404.041
Separable and Low-Rank Continuous Games
In this paper, we study nonzero-sum separable games, which are continuous
games whose payoffs take a sum-of-products form. Included in this subclass are
all finite games and polynomial games. We investigate the structure of
equilibria in separable games. We show that these games admit finitely
supported Nash equilibria. Motivated by the bounds on the supports of mixed
equilibria in two-player finite games in terms of the ranks of the payoff
matrices, we define the notion of the rank of an n-player continuous game and
use this to provide bounds on the cardinality of the support of equilibrium
strategies. We present a general characterization theorem that states that a
continuous game has finite rank if and only if it is separable. Using our rank
results, we present an efficient algorithm for computing approximate equilibria
of two-player separable games with fixed strategy spaces in time polynomial in
the rank of the game
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Reformulations of mathematical programming problems as linear complementarity problems
A family of complementarity problems are defined as extensions of the well known Linear Complementarity Problem (LCP). These are
(i.) Second Linear Complementarity Problem (SLCP) which is an LCP extended by introducing further equality restrictions and unrestricted variables,
(ii.) Minimum Linear Complementarity Problem (MLCP) which is an
LCP with additional variables not required to be complementary and with a linear objective function which is to be minimized,
(iii.) Second Minimum Linear Complementarity Problem (SMLCP) which is an MLCP but the nonnegative restriction on one of each pair of complementary variables is relaxed so that it is allowed to be unrestricted in value.
A number of well known mathematical programming problems, namely quadratic programming (convex, nonconvex, pseudoconvex nonconvex), bilinear programming, game theory, zero-one integer programming, the fixed charge problem, absolute value programming, variable separable programming are reformulated as members of this family of four complementarity problems
On Computability of Equilibria in Markets with Production
Although production is an integral part of the Arrow-Debreu market model,
most of the work in theoretical computer science has so far concentrated on
markets without production, i.e., the exchange economy. This paper takes a
significant step towards understanding computational aspects of markets with
production.
We first define the notion of separable, piecewise-linear concave (SPLC)
production by analogy with SPLC utility functions. We then obtain a linear
complementarity problem (LCP) formulation that captures exactly the set of
equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production,
and we give a complementary pivot algorithm for finding an equilibrium. This
settles a question asked by Eaves in 1975 of extending his complementary pivot
algorithm to markets with production.
Since this is a path-following algorithm, we obtain a proof of membership of
this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of
existence of equilibrium (i.e., without using a fixed point theorem),
rationality, and oddness of the number of equilibria. We further give a proof
of PPAD-hardness for this problem and also for its restriction to markets with
linear utilities and SPLC production. Experiments show that our algorithm runs
fast on randomly chosen examples, and unlike previous approaches, it does not
suffer from issues of numerical instability. Additionally, it is strongly
polynomial when the number of goods or the number of agents and firms is
constant. This extends the result of Devanur and Kannan (2008) to markets with
production.
Finally, we show that an LCP-based approach cannot be extended to PLC
(non-separable) production, by constructing an example which has only
irrational equilibria.Comment: An extended abstract will appear in SODA 201
On Sparse Discretization for Graphical Games
This short paper concerns discretization schemes for representing and
computing approximate Nash equilibria, with emphasis on graphical games, but
briefly touching on normal-form and poly-matrix games. The main technical
contribution is a representation theorem that informally states that to account
for every exact Nash equilibrium using a nearby approximate Nash equilibrium on
a grid over mixed strategies, a uniform discretization size linear on the
inverse of the approximation quality and natural game-representation parameters
suffices. For graphical games, under natural conditions, the discretization is
logarithmic in the game-representation size, a substantial improvement over the
linear dependency previously required. The paper has five other objectives: (1)
given the venue, to highlight the important, but often ignored, role that work
on constraint networks in AI has in simplifying the derivation and analysis of
algorithms for computing approximate Nash equilibria; (2) to summarize the
state-of-the-art on computing approximate Nash equilibria, with emphasis on
relevance to graphical games; (3) to help clarify the distinction between
sparse-discretization and sparse-support techniques; (4) to illustrate and
advocate for the deliberate mathematical simplicity of the formal proof of the
representation theorem; and (5) to list and discuss important open problems,
emphasizing graphical-game generalizations, which the AI community is most
suitable to solve.Comment: 30 pages. Original research note drafted in Dec. 2002 and posted
online Spring'03 (http://www.cis.upenn.
edu/~mkearns/teaching/cgt/revised_approx_bnd.pdf) as part of a course on
computational game theory taught by Prof. Michael Kearns at the University of
Pennsylvania; First major revision sent to WINE'10; Current version sent to
JAIR on April 25, 201
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