12,783 research outputs found
Function Approximation for Solving Stackelberg Equilibrium in Large Perfect Information Games
Function approximation (FA) has been a critical component in solving large
zero-sum games. Yet, little attention has been given towards FA in solving
\textit{general-sum} extensive-form games, despite them being widely regarded
as being computationally more challenging than their fully competitive or
cooperative counterparts. A key challenge is that for many equilibria in
general-sum games, no simple analogue to the state value function used in
Markov Decision Processes and zero-sum games exists. In this paper, we propose
learning the \textit{Enforceable Payoff Frontier} (EPF) -- a generalization of
the state value function for general-sum games. We approximate the optimal
\textit{Stackelberg extensive-form correlated equilibrium} by representing EPFs
with neural networks and training them by using appropriate backup operations
and loss functions. This is the first method that applies FA to the Stackelberg
setting, allowing us to scale to much larger games while still enjoying
performance guarantees based on FA error. Additionally, our proposed method
guarantees incentive compatibility and is easy to evaluate without having to
depend on self-play or approximate best-response oracles.Comment: To appear in AAAI 202
Playing Anonymous Games using Simple Strategies
We investigate the complexity of computing approximate Nash equilibria in
anonymous games. Our main algorithmic result is the following: For any
-player anonymous game with a bounded number of strategies and any constant
, an -approximate Nash equilibrium can be computed
in polynomial time. Complementing this positive result, we show that if there
exists any constant such that an -approximate
equilibrium can be computed in polynomial time, then there is a fully
polynomial-time approximation scheme for this problem.
We also present a faster algorithm that, for any -player -strategy
anonymous game, runs in time and computes an -approximate equilibrium. This algorithm follows from the
existence of simple approximate equilibria of anonymous games, where each
player plays one strategy with probability , for some small ,
and plays uniformly at random with probability .
Our approach exploits the connection between Nash equilibria in anonymous
games and Poisson multinomial distributions (PMDs). Specifically, we prove a
new probabilistic lemma establishing the following: Two PMDs, with large
variance in each direction, whose first few moments are approximately matching
are close in total variation distance. Our structural result strengthens
previous work by providing a smooth tradeoff between the variance bound and the
number of matching moments
Computing Approximate Nash Equilibria in Polymatrix Games
In an -Nash equilibrium, a player can gain at most by
unilaterally changing his behaviour. For two-player (bimatrix) games with
payoffs in , the best-known achievable in polynomial time is
0.3393. In general, for -player games an -Nash equilibrium can be
computed in polynomial time for an that is an increasing function of
but does not depend on the number of strategies of the players. For
three-player and four-player games the corresponding values of are
0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general
-player games where a player's payoff is the sum of payoffs from a number of
bimatrix games. There exists a very small but constant such that
computing an -Nash equilibrium of a polymatrix game is \PPAD-hard.
Our main result is that a -Nash equilibrium of an -player
polymatrix game can be computed in time polynomial in the input size and
. Inspired by the algorithm of Tsaknakis and Spirakis, our
algorithm uses gradient descent on the maximum regret of the players. We also
show that this algorithm can be applied to efficiently find a
-Nash equilibrium in a two-player Bayesian game
Algorithms for generalized potential games with mixed-integer variables
We consider generalized potential games, that constitute a fundamental subclass of generalized Nash equilibrium problems. We propose different methods to compute solutions of generalized potential games with mixed-integer variables, i.e., games in which some variables are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how different potential functions can be used to select equilibria. We propose a new Gauss–Southwell algorithm to compute approximate equilibria of any generalized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches
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
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