4,655 research outputs found
Approachability in Stackelberg Stochastic Games with Vector Costs
The notion of approachability was introduced by Blackwell [1] in the context
of vector-valued repeated games. The famous Blackwell's approachability theorem
prescribes a strategy for approachability, i.e., for `steering' the average
cost of a given agent towards a given target set, irrespective of the
strategies of the other agents. In this paper, motivated by the multi-objective
optimization/decision making problems in dynamically changing environments, we
address the approachability problem in Stackelberg stochastic games with vector
valued cost functions. We make two main contributions. Firstly, we give a
simple and computationally tractable strategy for approachability for
Stackelberg stochastic games along the lines of Blackwell's. Secondly, we give
a reinforcement learning algorithm for learning the approachable strategy when
the transition kernel is unknown. We also recover as a by-product Blackwell's
necessary and sufficient condition for approachability for convex sets in this
set up and thus a complete characterization. We also give sufficient conditions
for non-convex sets.Comment: 18 Pages, Submitted to Dynamic Games and Application
Approachability in Stackelberg Stochastic Games with Vector Costs
The notion of approachability was introduced by Blackwell [1] in the context
of vector-valued repeated games. The famous Blackwell's approachability theorem
prescribes a strategy for approachability, i.e., for `steering' the average
cost of a given agent towards a given target set, irrespective of the
strategies of the other agents. In this paper, motivated by the multi-objective
optimization/decision making problems in dynamically changing environments, we
address the approachability problem in Stackelberg stochastic games with vector
valued cost functions. We make two main contributions. Firstly, we give a
simple and computationally tractable strategy for approachability for
Stackelberg stochastic games along the lines of Blackwell's. Secondly, we give
a reinforcement learning algorithm for learning the approachable strategy when
the transition kernel is unknown. We also recover as a by-product Blackwell's
necessary and sufficient condition for approachability for convex sets in this
set up and thus a complete characterization. We also give sufficient conditions
for non-convex sets.Comment: 18 Pages, Submitted to Dynamic Games and Application
On an unified framework for approachability in games with or without signals
We unify standard frameworks for approachability both in full or partial
monitoring by defining a new abstract game, called the "purely informative
game", where the outcome at each stage is the maximal information players can
obtain, represented as some probability measure. Objectives of players can be
rewritten as the convergence (to some given set) of sequences of averages of
these probability measures. We obtain new results extending the approachability
theory developed by Blackwell moreover this new abstract framework enables us
to characterize approachable sets with, as usual, a remarkably simple and clear
reformulation for convex sets. Translated into the original games, those
results become the first necessary and sufficient condition under which an
arbitrary set is approachable and they cover and extend previous known results
for convex sets. We also investigate a specific class of games where, thanks to
some unusual definition of averages and convexity, we again obtain a complete
characterization of approachable sets along with rates of convergence
Robust approachability and regret minimization in games with partial monitoring
Approachability has become a standard tool in analyzing earning algorithms in
the adversarial online learning setup. We develop a variant of approachability
for games where there is ambiguity in the obtained reward that belongs to a
set, rather than being a single vector. Using this variant we tackle the
problem of approachability in games with partial monitoring and develop simple
and efficient algorithms (i.e., with constant per-step complexity) for this
setup. We finally consider external regret and internal regret in repeated
games with partial monitoring and derive regret-minimizing strategies based on
approachability theory
Approachability of Convex Sets in Games with Partial Monitoring
We provide a necessary and sufficient condition under which a convex set is
approachable in a game with partial monitoring, i.e.\ where players do not
observe their opponents' moves but receive random signals. This condition is an
extension of Blackwell's Criterion in the full monitoring framework, where
players observe at least their payoffs. When our condition is fulfilled, we
construct explicitly an approachability strategy, derived from a strategy
satisfying some internal consistency property in an auxiliary game. We also
provide an example of a convex set, that is neither (weakly)-approachable nor
(weakly)-excludable, a situation that cannot occur in the full monitoring case.
We finally apply our result to describe an -optimal strategy of the
uninformed player in a zero-sum repeated game with incomplete information on
one side
Attainability in Repeated Games with Vector Payoffs
We introduce the concept of attainable sets of payoffs in two-player repeated
games with vector payoffs. A set of payoff vectors is called {\em attainable}
if player 1 can ensure that there is a finite horizon such that after time
the distance between the set and the cumulative payoff is arbitrarily
small, regardless of what strategy player 2 is using. This paper focuses on the
case where the attainable set consists of one payoff vector. In this case the
vector is called an attainable vector. We study properties of the set of
attainable vectors, and characterize when a specific vector is attainable and
when every vector is attainable.Comment: 28 pages, 2 figures, conference version at NetGCoop 201
Strong and safe Nash equilibrium in some repeated 3-player games
We consider a 3-player game in the normal form, in which each player has two
actions. We assume that the game is symmetric and repeated infinitely many
times. At each stage players make their choices knowing only the average
payoffs from previous stages of all the players. A strategy of a player in the
repeated game is a function defined on the convex hull of the set of payoffs.
Our aim is to construct a strong Nash equilibrium in the repeated game, i.e. a
strategy profile being resistant to deviations by coalitions. Constructed
equilibrium strategies are safe, i.e. the non-deviating player payoff is not
smaller than the equilibrium payoff in the stage game, and deviating players'
payoffs do not exceed the non-deviating player payoff more than a positive
constant which can be arbitrary small and chosen by the non-deviating player.
Our construction is inspired by Smale's good strategies described in
\cite{smale}, where the repeated Prisoner's Dilemma was considered. In proofs
we use arguments based on approachability and strong approachability type
results.Comment: 19 page
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