21,956 research outputs found
Equilibria, Fixed Points, and Complexity Classes
Many models from a variety of areas involve the computation of an equilibrium
or fixed point of some kind. Examples include Nash equilibria in games; market
equilibria; computing optimal strategies and the values of competitive games
(stochastic and other games); stable configurations of neural networks;
analysing basic stochastic models for evolution like branching processes and
for language like stochastic context-free grammars; and models that incorporate
the basic primitives of probability and recursion like recursive Markov chains.
It is not known whether these problems can be solved in polynomial time. There
are certain common computational principles underlying different types of
equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP.
Representative complete problems for these classes are respectively, pure Nash
equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria
in 2-player normal form games, and (mixed) Nash equilibria in normal form games
with 3 (or more) players. This paper reviews the underlying computational
principles and the corresponding classes
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
Constant Rank Bimatrix Games are PPAD-hard
The rank of a bimatrix game (A,B) is defined as rank(A+B). Computing a Nash
equilibrium (NE) of a rank-, i.e., zero-sum game is equivalent to linear
programming (von Neumann'28, Dantzig'51). In 2005, Kannan and Theobald gave an
FPTAS for constant rank games, and asked if there exists a polynomial time
algorithm to compute an exact NE. Adsul et al. (2011) answered this question
affirmatively for rank- games, leaving rank-2 and beyond unresolved.
In this paper we show that NE computation in games with rank , is
PPAD-hard, settling a decade long open problem. Interestingly, this is the
first instance that a problem with an FPTAS turns out to be PPAD-hard. Our
reduction bypasses graphical games and game gadgets, and provides a simpler
proof of PPAD-hardness for NE computation in bimatrix games. In addition, we
get:
* An equivalence between 2D-Linear-FIXP and PPAD, improving a result by
Etessami and Yannakakis (2007) on equivalence between Linear-FIXP and PPAD.
* NE computation in a bimatrix game with convex set of Nash equilibria is as
hard as solving a simple stochastic game.
* Computing a symmetric NE of a symmetric bimatrix game with rank is
PPAD-hard.
* Computing a (1/poly(n))-approximate fixed-point of a (Linear-FIXP)
piecewise-linear function is PPAD-hard.
The status of rank- games remains unresolved
On the Complexity of Nash Equilibria of Action-Graph Games
We consider the problem of computing Nash Equilibria of action-graph games
(AGGs). AGGs, introduced by Bhat and Leyton-Brown, is a succinct representation
of games that encapsulates both "local" dependencies as in graphical games, and
partial indifference to other agents' identities as in anonymous games, which
occur in many natural settings. This is achieved by specifying a graph on the
set of actions, so that the payoff of an agent for selecting a strategy depends
only on the number of agents playing each of the neighboring strategies in the
action graph. We present a Polynomial Time Approximation Scheme for computing
mixed Nash equilibria of AGGs with constant treewidth and a constant number of
agent types (and an arbitrary number of strategies), together with hardness
results for the cases when either the treewidth or the number of agent types is
unconstrained. In particular, we show that even if the action graph is a tree,
but the number of agent-types is unconstrained, it is NP-complete to decide the
existence of a pure-strategy Nash equilibrium and PPAD-complete to compute a
mixed Nash equilibrium (even an approximate one); similarly for symmetric AGGs
(all agents belong to a single type), if we allow arbitrary treewidth. These
hardness results suggest that, in some sense, our PTAS is as strong of a
positive result as one can expect
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
Dynamical selection of Nash equilibria using Experience Weighted Attraction Learning: emergence of heterogeneous mixed equilibria
We study the distribution of strategies in a large game that models how
agents choose among different double auction markets. We classify the possible
mean field Nash equilibria, which include potentially segregated states where
an agent population can split into subpopulations adopting different
strategies. As the game is aggregative, the actual equilibrium strategy
distributions remain undetermined, however. We therefore compare with the
results of Experience-Weighted Attraction (EWA) learning, which at long times
leads to Nash equilibria in the appropriate limits of large intensity of
choice, low noise (long agent memory) and perfect imputation of missing scores
(fictitious play). The learning dynamics breaks the indeterminacy of the Nash
equilibria. Non-trivially, depending on how the relevant limits are taken, more
than one type of equilibrium can be selected. These include the standard
homogeneous mixed and heterogeneous pure states, but also \emph{heterogeneous
mixed} states where different agents play different strategies that are not all
pure. The analysis of the EWA learning involves Fokker-Planck modeling combined
with large deviation methods. The theoretical results are confirmed by
multi-agent simulations.Comment: 35 pages, 16 figure
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