263 research outputs found
Complexity Theory, Game Theory, and Economics: The Barbados Lectures
This document collects the lecture notes from my mini-course "Complexity
Theory, Game Theory, and Economics," taught at the Bellairs Research Institute
of McGill University, Holetown, Barbados, February 19--23, 2017, as the 29th
McGill Invitational Workshop on Computational Complexity.
The goal of this mini-course is twofold: (i) to explain how complexity theory
has helped illuminate several barriers in economics and game theory; and (ii)
to illustrate how game-theoretic questions have led to new and interesting
complexity theory, including recent several breakthroughs. It consists of two
five-lecture sequences: the Solar Lectures, focusing on the communication and
computational complexity of computing equilibria; and the Lunar Lectures,
focusing on applications of complexity theory in game theory and economics. No
background in game theory is assumed.Comment: Revised v2 from December 2019 corrects some errors in and adds some
recent citations to v1 Revised v3 corrects a few typos in v
Nash equilibria, gale strings, and perfect matchings
This thesis concerns the problem 2-NASH of finding a Nash equilibrium of a bimatrix
game, for the special class of so-called “hard-to-solve” bimatrix games. The term “hardto-solve” relates to the exponential running time of the famous and often used Lemke–
Howson algorithm for this class of games. The games are constructed with the help of
dual cyclic polytopes, where the algorithm can be expressed combinatorially via labeled
bitstrings defined by the “Gale evenness condition” that characterise the vertices of these
polytopes.
We define the combinatorial problem “Another completely labeled Gale string” whose
solutions define the Nash equilibria of any game defined by cyclic polytopes, including
the games where the Lemke–Howson algorithm takes exponential time. We show that
“Another completely labeled Gale string” is solvable in polynomial time by a reduction to
the “Perfect matching” problem in Euler graphs. We adapt the Lemke–Howson algorithm
to pivot from one perfect matching to another and show that again for a certain class
of graphs this leads to exponential behaviour. Furthermore, we prove that completely
labeled Gale strings and perfect matchings in Euler graphs come in pairs and that the
Lemke–Howson algorithm connects two strings or matchings of opposite signs.
The equivalence between Nash Equilibria of bimatrix games derived from cyclic polytopes, completely labeled Gale strings, and perfect matchings in Euler Graphs implies that
counting Nash equilibria is #P-complete. Although one Nash equilibrium can be computed in polynomial time, we have not succeeded in finding an algorithm that computes
a Nash equilibrium of opposite sign. However, we solve this problem for certain special cases, for example planar graphs. We illustrate the difficulties concerning a general
polynomial-time algorithm for this problem by means of negative results that demonstrate
why a number of approaches towards such an algorithm are unlikely to be successful
Finding Nash equilibria of bimatrix games
This thesis concerns the computational problem of finding one Nash equilibrium of a bimatrix game, a two-player game in strategic form. Bimatrix games are among the most basic models in non-cooperative game theory, and finding a Nash equilibrium is important for their analysis.
The Lemke—Howson algorithm is the classical method for finding one Nash equilib-rium of a bimatrix game. In this thesis, we present a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, the games are constructed using pairs of dual cyclic polytopes with 2d suitably labelled facets in d-space. The construc-tion is extended to two classes of non-square games where, in addition to exponentially long Lemke—Howson computations, finding an equilibrium by support enumeration takes exponential time on average.
The Lemke—Howson algorithm, which is a complementary pivoting algorithm, finds at least one solution to the linear complementarity problem (LCP) derived from a bimatrix game. A closely related complementary pivoting algorithm by Lemke solves more general LCPs. A unified view of these two algorithms is presented, for the first time, as far as we know. Furthermore, we present an extension of the standard version of Lemke's algorithm that allows one more freedom than before when starting the algorithm
Computing the Equilibria of Bimatrix Games using Dominance Heuristics
We propose a formulation of a general-sum bimatrix game as a bipartite
directed graph with the objective of establishing a correspondence between the
set of the relevant structures of the graph (in particular elementary cycles)
and the set of the Nash equilibria of the game. We show that finding the set of
elementary cycles of the graph permits the computation of the set of
equilibria. For games whose graphs have a sparse adjacency matrix, this serves
as a good heuristic for computing the set of equilibria. The heuristic also
allows the discarding of sections of the support space that do not yield any
equilibrium, thus serving as a useful pre-processing step for algorithms that
compute the equilibria through support enumeration
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