8,185 research outputs found
Efficient cooperation by exchanging favors
We study chip-strategy equilibria in two-player repeated games. Intuitively, in these equilibria players exchange favors by taking individually suboptimal actions if these actions create a "gain" for the opponent larger than the player's "loss" from taking them. In exchange, the player who provides a favor implicitly obtains from the opponent a chip that entitles the player to receiving a favor at some future date. Players are initially endowed with a number of chips, and a player who runs out of chips is no longer entitled to receive any favors until she provides a favor to the opponent, in which case she receives one chip back.We show that such simple chip strategies approximate efficient outcomes in a class of repeated symmetric games with incomplete information, in which each player has two possible types, when discounting vanishes. This class includes many important applications, studied in numerous previous papers, such as favor exchange model of Mobius (2001), repeated auctions, andthe repeated version of Spulber duopolies of Athey and Bagwell (2001), among others. We also show the limitation of chip strategies. For example, if players have more than two types, then such simple chip strategies may not approximate efficient outcomes even in symmetric games
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
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
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
- …