68,048 research outputs found
The Complexity of All-switches Strategy Improvement
Strategy improvement is a widely-used and well-studied class of algorithms
for solving graph-based infinite games. These algorithms are parameterized by a
switching rule, and one of the most natural rules is "all switches" which
switches as many edges as possible in each iteration. Continuing a recent line
of work, we study all-switches strategy improvement from the perspective of
computational complexity. We consider two natural decision problems, both of
which have as input a game , a starting strategy , and an edge . The
problems are: 1.) The edge switch problem, namely, is the edge ever
switched by all-switches strategy improvement when it is started from on
game ? 2.) The optimal strategy problem, namely, is the edge used in the
final strategy that is found by strategy improvement when it is started from
on game ? We show -completeness of the edge switch
problem and optimal strategy problem for the following settings: Parity games
with the discrete strategy improvement algorithm of V\"oge and Jurdzi\'nski;
mean-payoff games with the gain-bias algorithm [14,37]; and discounted-payoff
games and simple stochastic games with their standard strategy improvement
algorithms. We also show -completeness of an analogous problem
to edge switch for the bottom-antipodal algorithm for finding the sink of an
Acyclic Unique Sink Orientation on a cube
Synthesising Strategy Improvement and Recursive Algorithms for Solving 2.5 Player Parity Games
2.5 player parity games combine the challenges posed by 2.5 player
reachability games and the qualitative analysis of parity games. These two
types of problems are best approached with different types of algorithms:
strategy improvement algorithms for 2.5 player reachability games and recursive
algorithms for the qualitative analysis of parity games. We present a method
that - in contrast to existing techniques - tackles both aspects with the best
suited approach and works exclusively on the 2.5 player game itself. The
resulting technique is powerful enough to handle games with several million
states
Non-oblivious Strategy Improvement
We study strategy improvement algorithms for mean-payoff and parity games. We
describe a structural property of these games, and we show that these
structures can affect the behaviour of strategy improvement. We show how
awareness of these structures can be used to accelerate strategy improvement
algorithms. We call our algorithms non-oblivious because they remember
properties of the game that they have discovered in previous iterations. We
show that non-oblivious strategy improvement algorithms perform well on
examples that are known to be hard for oblivious strategy improvement. Hence,
we argue that previous strategy improvement algorithms fail because they ignore
the structural properties of the game that they are solving
Choosing Products in Social Networks
We study the consequences of adopting products by agents who form a social
network. To this end we use the threshold model introduced in Apt and Markakis,
arXiv:1105.2434, in which the nodes influenced by their neighbours can adopt
one out of several alternatives, and associate with such each social network a
strategic game between the agents. The possibility of not choosing any product
results in two special types of (pure) Nash equilibria.
We show that such games may have no Nash equilibrium and that determining the
existence of a Nash equilibrium, also of a special type, is NP-complete. The
situation changes when the underlying graph of the social network is a DAG, a
simple cycle, or has no source nodes. For these three classes we determine the
complexity of establishing whether a (special type of) Nash equilibrium exists.
We also clarify for these categories of games the status and the complexity
of the finite improvement property (FIP). Further, we introduce a new property
of the uniform FIP which is satisfied when the underlying graph is a simple
cycle, but determining it is co-NP-hard in the general case and also when the
underlying graph has no source nodes. The latter complexity results also hold
for verifying the property of being a weakly acyclic game.Comment: 15 pages. Appeared in Proc. of the 8th International Workshop on
Internet and Network Economics (WINE 2012), Lecture Notes in Computer Science
7695, Springer, pp. 100-11
Social Network Games with Obligatory Product Selection
Recently, Apt and Markakis introduced a model for product adoption in social
networks with multiple products, where the agents, influenced by their
neighbours, can adopt one out of several alternatives (products). To analyze
these networks we introduce social network games in which product adoption is
obligatory.
We show that when the underlying graph is a simple cycle, there is a
polynomial time algorithm allowing us to determine whether the game has a Nash
equilibrium. In contrast, in the arbitrary case this problem is NP-complete. We
also show that the problem of determining whether the game is weakly acyclic is
co-NP hard.
Using these games we analyze various types of paradoxes that can arise in the
considered networks. One of them corresponds to the well-known Braess paradox
in congestion games. In particular, we show that social networks exist with the
property that by adding an additional product to a specific node, the choices
of the nodes will unavoidably evolve in such a way that everybody is strictly
worse off.Comment: In Proceedings GandALF 2013, arXiv:1307.416
An Exponential Lower Bound for the Latest Deterministic Strategy Iteration Algorithms
This paper presents a new exponential lower bound for the two most popular
deterministic variants of the strategy improvement algorithms for solving
parity, mean payoff, discounted payoff and simple stochastic games. The first
variant improves every node in each step maximizing the current valuation
locally, whereas the second variant computes the globally optimal improvement
in each step. We outline families of games on which both variants require
exponentially many strategy iterations
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