Dynamic resource allocation games

In resource allocation games, selfish players share resources that are needed in order to fulfill their objectives. The cost of using a resource depends on the load on it. In the traditional setting, the players make their choices concurrently and in one-shot. That is, a strategy for a player is a subset of the resources. We introduce and study dynamic resource allocation games. In this setting, the game proceeds in phases. In each phase each player chooses one resource. A scheduler dictates the order in which the players proceed in a phase, possibly scheduling several players to proceed concurrently. The game ends when each player has collected a set of resources that fulfills his objective. The cost for each player then depends on this set as well as on the load on the resources in it – we consider both congestion and cost-sharing games. We argue that the dynamic setting is the suitable setting for many applications in practice. We study the stability of dynamic resource allocation games, where the appropriate notion of stability is that of subgame perfect equilibrium, study the inefficiency incurred due to selfish behavior, and also study problems that are particular to the dynamic setting, like constraints on the order in which resources can be chosen or the problem of finding a scheduler that achieves stability

A Hierarchy of Nondeterminism

We study three levels in a hierarchy of nondeterminism: A nondeterministic automaton $\cal A$ is determinizable by pruning (DBP) if we can obtain a deterministic automaton equivalent to $\cal A$ by removing some of its transitions. Then, $\cal A$ is good-for-games (GFG) if its nondeterministic choices can be resolved in a way that only depends on the past. Finally, $\cal A$ is semantically deterministic (SD) if different nondeterministic choices in $\cal A$ lead to equivalent states. Some applications of automata in formal methods require deterministic automata, yet in fact can use automata with some level of nondeterminism. For example, DBP automata are useful in the analysis of online algorithms, and GFG automata are useful in synthesis and control. For automata on finite words, the three levels in the hierarchy coincide. We study the hierarchy for B\"uchi, co-B\"uchi, and weak automata on infinite words. We show that the hierarchy is strict, study the expressive power of the different levels in it, as well as the complexity of deciding the membership of a language in a given level. Finally, we describe a probability-based analysis of the hierarchy, which relates the level of nondeterminism with the probability that a random run on a word in the language is accepting.Comment: 21 pages, 5 figure

A Hierarchy of Nondeterminism

We study three levels in a hierarchy of nondeterminism: A nondeterministic automaton $\cal A$ is determinizable by pruning (DBP) if we can obtain a deterministic automaton equivalent to $\cal A$ by removing some of its transitions. Then, $\cal A$ is good-for-games (GFG) if its nondeterministic choices can be resolved in a way that only depends on the past. Finally, $\cal A$ is semantically deterministic (SD) if different nondeterministic choices in $\cal A$ lead to equivalent states. Some applications of automata in formal methods require deterministic automata, yet in fact can use automata with some level of nondeterminism. For example, DBP automata are useful in the analysis of online algorithms, and GFG automata are useful in synthesis and control. For automata on finite words, the three levels in the hierarchy coincide. We study the hierarchy for B\"uchi, co-B\"uchi, and weak automata on infinite words. We show that the hierarchy is strict, study the expressive power of the different levels in it, as well as the complexity of deciding the membership of a language in a given level. Finally, we describe a probability-based analysis of the hierarchy, which relates the level of nondeterminism with the probability that a random run on a word in the language is accepting.Comment: 21 pages, 5 figure

Research Advances in Chaos Theory

The subject of chaos has invaded practically every area of the natural sciences. Weather patterns are referred to as chaotic. There are chemical reactions and chaotic evolution of insect populations. Atomic and molecular physics have also seen the emergence of the study of chaos in these microscopic domains. This book examines the issue of chaos in nonlinear and dynamical systems, quantum mechanics, biology, and economics