8,850 research outputs found
Fixed-Dimensional Energy Games are in Pseudo-Polynomial Time
We generalise the hyperplane separation technique (Chatterjee and Velner,
2013) from multi-dimensional mean-payoff to energy games, and achieve an
algorithm for solving the latter whose running time is exponential only in the
dimension, but not in the number of vertices of the game graph. This answers an
open question whether energy games with arbitrary initial credit can be solved
in pseudo-polynomial time for fixed dimensions 3 or larger (Chaloupka, 2013).
It also improves the complexity of solving multi-dimensional energy games with
given initial credit from non-elementary (Br\'azdil, Jan\v{c}ar, and
Ku\v{c}era, 2010) to 2EXPTIME, thus establishing their 2EXPTIME-completeness.Comment: Corrected proof of Lemma 6.2 (thanks to Dmitry Chistikov for spotting
an error in the previous proof
Hyperplane Separation Technique for Multidimensional Mean-Payoff Games
We consider both finite-state game graphs and recursive game graphs (or
pushdown game graphs), that can model the control flow of sequential programs
with recursion, with multi-dimensional mean-payoff objectives. In pushdown
games two types of strategies are relevant: global strategies, that depend on
the entire global history; and modular strategies, that have only local memory
and thus do not depend on the context of invocation. We present solutions to
several fundamental algorithmic questions and our main contributions are as
follows: (1) We show that finite-state multi-dimensional mean-payoff games can
be solved in polynomial time if the number of dimensions and the maximal
absolute value of the weight is fixed; whereas if the number of dimensions is
arbitrary, then problem is already known to be coNP-complete. (2) We show that
pushdown graphs with multi-dimensional mean-payoff objectives can be solved in
polynomial time. (3) For pushdown games under global strategies both single and
multi-dimensional mean-payoff objectives problems are known to be undecidable,
and we show that under modular strategies the multi-dimensional problem is also
undecidable (whereas under modular strategies the single dimensional problem is
NP-complete). We show that if the number of modules, the number of exits, and
the maximal absolute value of the weight is fixed, then pushdown games under
modular strategies with single dimensional mean-payoff objectives can be solved
in polynomial time, and if either of the number of exits or the number of
modules is not bounded, then the problem is NP-hard. (4) Finally we show that a
fixed parameter tractable algorithm for finite-state multi-dimensional
mean-payoff games or pushdown games under modular strategies with
single-dimensional mean-payoff objectives would imply the solution of the
long-standing open problem of fixed parameter tractability of parity games.Comment: arXiv admin note: text overlap with arXiv:1201.282
Percentile Queries in Multi-Dimensional Markov Decision Processes
Markov decision processes (MDPs) with multi-dimensional weights are useful to
analyze systems with multiple objectives that may be conflicting and require
the analysis of trade-offs. We study the complexity of percentile queries in
such MDPs and give algorithms to synthesize strategies that enforce such
constraints. Given a multi-dimensional weighted MDP and a quantitative payoff
function , thresholds (one per dimension), and probability thresholds
, we show how to compute a single strategy to enforce that for all
dimensions , the probability of outcomes satisfying is at least . We consider classical quantitative payoffs from
the literature (sup, inf, lim sup, lim inf, mean-payoff, truncated sum,
discounted sum). Our work extends to the quantitative case the multi-objective
model checking problem studied by Etessami et al. in unweighted MDPs.Comment: Extended version of CAV 2015 pape
Jamming Games in the MIMO Wiretap Channel With an Active Eavesdropper
This paper investigates reliable and covert transmission strategies in a
multiple-input multiple-output (MIMO) wiretap channel with a transmitter,
receiver and an adversarial wiretapper, each equipped with multiple antennas.
In a departure from existing work, the wiretapper possesses a novel capability
to act either as a passive eavesdropper or as an active jammer, under a
half-duplex constraint. The transmitter therefore faces a choice between
allocating all of its power for data, or broadcasting artificial interference
along with the information signal in an attempt to jam the eavesdropper
(assuming its instantaneous channel state is unknown). To examine the resulting
trade-offs for the legitimate transmitter and the adversary, we model their
interactions as a two-person zero-sum game with the ergodic MIMO secrecy rate
as the payoff function. We first examine conditions for the existence of
pure-strategy Nash equilibria (NE) and the structure of mixed-strategy NE for
the strategic form of the game.We then derive equilibrium strategies for the
extensive form of the game where players move sequentially under scenarios of
perfect and imperfect information. Finally, numerical simulations are presented
to examine the equilibrium outcomes of the various scenarios considered.Comment: 27 pages, 8 figures. To appear, IEEE Transactions on Signal
Processin
Looking at Mean-Payoff through Foggy Windows
Mean-payoff games (MPGs) are infinite duration two-player zero-sum games
played on weighted graphs. Under the hypothesis of perfect information, they
admit memoryless optimal strategies for both players and can be solved in
NP-intersect-coNP. MPGs are suitable quantitative models for open reactive
systems. However, in this context the assumption of perfect information is not
always realistic. For the partial-observation case, the problem that asks if
the first player has an observation-based winning strategy that enforces a
given threshold on the mean-payoff, is undecidable. In this paper, we study the
window mean-payoff objectives that were introduced recently as an alternative
to the classical mean-payoff objectives. We show that, in sharp contrast to the
classical mean-payoff objectives, some of the window mean-payoff objectives are
decidable in games with partial-observation
Unifying Two Views on Multiple Mean-Payoff Objectives in Markov Decision Processes
We consider Markov decision processes (MDPs) with multiple limit-average (or
mean-payoff) objectives. There exist two different views: (i) the expectation
semantics, where the goal is to optimize the expected mean-payoff objective,
and (ii) the satisfaction semantics, where the goal is to maximize the
probability of runs such that the mean-payoff value stays above a given vector.
We consider optimization with respect to both objectives at once, thus unifying
the existing semantics. Precisely, the goal is to optimize the expectation
while ensuring the satisfaction constraint. Our problem captures the notion of
optimization with respect to strategies that are risk-averse (i.e., ensure
certain probabilistic guarantee). Our main results are as follows: First, we
present algorithms for the decision problems which are always polynomial in the
size of the MDP. We also show that an approximation of the Pareto-curve can be
computed in time polynomial in the size of the MDP, and the approximation
factor, but exponential in the number of dimensions. Second, we present a
complete characterization of the strategy complexity (in terms of memory bounds
and randomization) required to solve our problem.Comment: Extended journal version of the LICS'15 pape
Mean-payoff Automaton Expressions
Quantitative languages are an extension of boolean languages that assign to
each word a real number. Mean-payoff automata are finite automata with
numerical weights on transitions that assign to each infinite path the long-run
average of the transition weights. When the mode of branching of the automaton
is deterministic, nondeterministic, or alternating, the corresponding class of
quantitative languages is not robust as it is not closed under the pointwise
operations of max, min, sum, and numerical complement. Nondeterministic and
alternating mean-payoff automata are not decidable either, as the quantitative
generalization of the problems of universality and language inclusion is
undecidable.
We introduce a new class of quantitative languages, defined by mean-payoff
automaton expressions, which is robust and decidable: it is closed under the
four pointwise operations, and we show that all decision problems are decidable
for this class. Mean-payoff automaton expressions subsume deterministic
mean-payoff automata, and we show that they have expressive power incomparable
to nondeterministic and alternating mean-payoff automata. We also present for
the first time an algorithm to compute distance between two quantitative
languages, and in our case the quantitative languages are given as mean-payoff
automaton expressions
Evolutionary games on graphs
Game theory is one of the key paradigms behind many scientific disciplines
from biology to behavioral sciences to economics. In its evolutionary form and
especially when the interacting agents are linked in a specific social network
the underlying solution concepts and methods are very similar to those applied
in non-equilibrium statistical physics. This review gives a tutorial-type
overview of the field for physicists. The first three sections introduce the
necessary background in classical and evolutionary game theory from the basic
definitions to the most important results. The fourth section surveys the
topological complications implied by non-mean-field-type social network
structures in general. The last three sections discuss in detail the dynamic
behavior of three prominent classes of models: the Prisoner's Dilemma, the
Rock-Scissors-Paper game, and Competing Associations. The major theme of the
review is in what sense and how the graph structure of interactions can modify
and enrich the picture of long term behavioral patterns emerging in
evolutionary games.Comment: Review, final version, 133 pages, 65 figure
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