319 research outputs found
Stochastic uncoupled dynamics and Nash equilibrium
In this paper we consider dynamic processes, in repeated games, that are subject to the natural informational restriction of uncoupledness. We study the almost sure convergence to Nash equilibria, and present a number of possibility and impossibility results. Basically, we show that if in addition to random moves some recall is introduced, then successful search procedures that are uncoupled can be devised. In particular, to get almost sure convergence to pure Nash equilibria when these exist, it su±ces to recall the last two periods of play.Uncoupled, Nash equilibrium, stochastic dynamics, bounded recall
Self-stabilizing uncoupled dynamics
Dynamics in a distributed system are self-stabilizing if they are guaranteed
to reach a stable state regardless of how the system is initialized. Game
dynamics are uncoupled if each player's behavior is independent of the other
players' preferences. Recognizing an equilibrium in this setting is a
distributed computational task. Self-stabilizing uncoupled dynamics, then, have
both resilience to arbitrary initial states and distribution of knowledge. We
study these dynamics by analyzing their behavior in a bounded-recall
synchronous environment. We determine, for every "size" of game, the minimum
number of periods of play that stochastic (randomized) players must recall in
order for uncoupled dynamics to be self-stabilizing. We also do this for the
special case when the game is guaranteed to have unique best replies. For
deterministic players, we demonstrate two self-stabilizing uncoupled protocols.
One applies to all games and uses three steps of recall. The other uses two
steps of recall and applies to games where each player has at least four
available actions. For uncoupled deterministic players, we prove that a single
step of recall is insufficient to achieve self-stabilization, regardless of the
number of available actions
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
Directional learning and the provisioning of public goods
We consider an environment where players are involved in a public goods game
and must decide repeatedly whether to make an individual contribution or not.
However, players lack strategically relevant information about the game and
about the other players in the population. The resulting behavior of players is
completely uncoupled from such information, and the individual strategy
adjustment dynamics are driven only by reinforcement feedbacks from each
player's own past. We show that the resulting "directional learning" is
sufficient to explain cooperative deviations away from the Nash equilibrium. We
introduce the concept of k-strong equilibria, which nest both the Nash
equilibrium and the Aumann-strong equilibrium as two special cases, and we show
that, together with the parameters of the learning model, the maximal
k-strength of equilibrium determines the stationary distribution. The
provisioning of public goods can be secured even under adverse conditions, as
long as players are sufficiently responsive to the changes in their own payoffs
and adjust their actions accordingly. Substantial levels of public cooperation
can thus be explained without arguments involving selflessness or social
preferences, solely on the basis of uncoordinated directional (mis)learning.Comment: 7 two-column pages, 3 figures; accepted for publication in Scientific
Report
The Query Complexity of Correlated Equilibria
We consider the complexity of finding a correlated equilibrium of an
-player game in a model that allows the algorithm to make queries on
players' payoffs at pure strategy profiles. Randomized regret-based dynamics
are known to yield an approximate correlated equilibrium efficiently, namely,
in time that is polynomial in the number of players . Here we show that both
randomization and approximation are necessary: no efficient deterministic
algorithm can reach even an approximate correlated equilibrium, and no
efficient randomized algorithm can reach an exact correlated equilibrium. The
results are obtained by bounding from below the number of payoff queries that
are needed
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