16 research outputs found
The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions
We show that the widely used homotopy method for solving fixpoint problems,
as well as the Harsanyi-Selten equilibrium selection process for games, are
PSPACE-complete to implement. Extending our result for the Harsanyi-Selten
process, we show that several other homotopy-based algorithms for finding
equilibria of games are also PSPACE-complete to implement. A further
application of our techniques yields the result that it is PSPACE-complete to
compute any of the equilibria that could be found via the classical
Lemke-Howson algorithm, a complexity-theoretic strengthening of the result in
[Savani and von Stengel]. These results show that our techniques can be widely
applied and suggest that the PSPACE-completeness of implementing homotopy
methods is a general principle.Comment: 23 pages, 1 figure; to appear in FOCS 2011 conferenc
An Empirical Study of Finding Approximate Equilibria in Bimatrix Games
While there have been a number of studies about the efficacy of methods to
find exact Nash equilibria in bimatrix games, there has been little empirical
work on finding approximate Nash equilibria. Here we provide such a study that
compares a number of approximation methods and exact methods. In particular, we
explore the trade-off between the quality of approximate equilibrium and the
required running time to find one. We found that the existing library GAMUT,
which has been the de facto standard that has been used to test exact methods,
is insufficient as a test bed for approximation methods since many of its games
have pure equilibria or other easy-to-find good approximate equilibria. We
extend the breadth and depth of our study by including new interesting families
of bimatrix games, and studying bimatrix games upto size .
Finally, we provide new close-to-worst-case examples for the best-performing
algorithms for finding approximate Nash equilibria
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
The Simplex Algorithm is NP-mighty
We propose to classify the power of algorithms by the complexity of the
problems that they can be used to solve. Instead of restricting to the problem
a particular algorithm was designed to solve explicitly, however, we include
problems that, with polynomial overhead, can be solved 'implicitly' during the
algorithm's execution. For example, we allow to solve a decision problem by
suitably transforming the input, executing the algorithm, and observing whether
a specific bit in its internal configuration ever switches during the
execution. We show that the Simplex Method, the Network Simplex Method (both
with Dantzig's original pivot rule), and the Successive Shortest Path Algorithm
are NP-mighty, that is, each of these algorithms can be used to solve any
problem in NP. This result casts a more favorable light on these algorithms'
exponential worst-case running times. Furthermore, as a consequence of our
approach, we obtain several novel hardness results. For example, for a given
input to the Simplex Algorithm, deciding whether a given variable ever enters
the basis during the algorithm's execution and determining the number of
iterations needed are both NP-hard problems. Finally, we close a long-standing
open problem in the area of network flows over time by showing that earliest
arrival flows are NP-hard to obtain
A Generalized Training Approach for Multiagent Learning
This paper investigates a population-based training regime based on
game-theoretic principles called Policy-Spaced Response Oracles (PSRO). PSRO is
general in the sense that it (1) encompasses well-known algorithms such as
fictitious play and double oracle as special cases, and (2) in principle
applies to general-sum, many-player games. Despite this, prior studies of PSRO
have been focused on two-player zero-sum games, a regime wherein Nash
equilibria are tractably computable. In moving from two-player zero-sum games
to more general settings, computation of Nash equilibria quickly becomes
infeasible. Here, we extend the theoretical underpinnings of PSRO by
considering an alternative solution concept, -Rank, which is unique
(thus faces no equilibrium selection issues, unlike Nash) and applies readily
to general-sum, many-player settings. We establish convergence guarantees in
several games classes, and identify links between Nash equilibria and
-Rank. We demonstrate the competitive performance of
-Rank-based PSRO against an exact Nash solver-based PSRO in 2-player
Kuhn and Leduc Poker. We then go beyond the reach of prior PSRO applications by
considering 3- to 5-player poker games, yielding instances where -Rank
achieves faster convergence than approximate Nash solvers, thus establishing it
as a favorable general games solver. We also carry out an initial empirical
validation in MuJoCo soccer, illustrating the feasibility of the proposed
approach in another complex domain
Unit vector games
McLennan and Tourky (2010) showed that “imitation games” provide a new view of the computation of Nash equilibria of bimatrix games with the Lemke–Howson algorithm. In an imitation game, the payoff matrix of one of the players is the identity matrix. We study the more general “unit vector games”, which are already known, where the payoff matrix of one player is composed of unit vectors. Our main application is a simplification of the construction by Savani and von Stengel (2006) of bimatrix games where two basic equilibrium-finding algorithms take exponentially many steps: the Lemke–Howson algorithm, and support enumeration
The Complexity of All-switches Strategy Improvement
Strategy improvement is a widely-used and well-studied class of algorithms
for solving graph-based infinite games. These algorithms are parameterized by a
switching rule, and one of the most natural rules is "all switches" which
switches as many edges as possible in each iteration. Continuing a recent line
of work, we study all-switches strategy improvement from the perspective of
computational complexity. We consider two natural decision problems, both of
which have as input a game , a starting strategy , and an edge . The
problems are: 1.) The edge switch problem, namely, is the edge ever
switched by all-switches strategy improvement when it is started from on
game ? 2.) The optimal strategy problem, namely, is the edge used in the
final strategy that is found by strategy improvement when it is started from
on game ? We show -completeness of the edge switch
problem and optimal strategy problem for the following settings: Parity games
with the discrete strategy improvement algorithm of V\"oge and Jurdzi\'nski;
mean-payoff games with the gain-bias algorithm [14,37]; and discounted-payoff
games and simple stochastic games with their standard strategy improvement
algorithms. We also show -completeness of an analogous problem
to edge switch for the bottom-antipodal algorithm for finding the sink of an
Acyclic Unique Sink Orientation on a cube