Parrondo games as lattice gas automata

Parrondo games are coin flipping games with the surprising property that alternating plays of two losing games can produce a winning game. We show that this phenomenon can be modelled by probabilistic lattice gas automata. Furthermore, motivated by the recent introduction of quantum coin flipping games, we show that quantum lattice gas automata provide an interesting definition for quantum Parrondo games.Comment: 12 pages, plain TeX, 10 PostScript figures included with epsf.tex (ignore the under/overfull \vbox error messages); for related work see http://math.ucsd.edu/~dmeyer/research.htm

Obligation Blackwell Games and p-Automata

We recently introduced p-automata, automata that read discrete-time Markov chains. We used turn-based stochastic parity games to define acceptance of Markov chains by a subclass of p-automata. Definition of acceptance required a cumbersome and complicated reduction to a series of turn-based stochastic parity games. The reduction could not support acceptance by general p-automata, which was left undefined as there was no notion of games that supported it. Here we generalize two-player games by adding a structural acceptance condition called obligations. Obligations are orthogonal to the linear winning conditions that define winning. Obligations are a declaration that player 0 can achieve a certain value from a configuration. If the obligation is met, the value of that configuration for player 0 is 1. One cannot define value in obligation games by the standard mechanism of considering the measure of winning paths on a Markov chain and taking the supremum of the infimum of all strategies. Mainly because obligations need definition even for Markov chains and the nature of obligations has the flavor of an infinite nesting of supremum and infimum operators. We define value via a reduction to turn-based games similar to Martin's proof of determinacy of Blackwell games with Borel objectives. Based on this definition, we show that games are determined. We show that for Markov chains with Borel objectives and obligations, and finite turn-based stochastic parity games with obligations there exists an alternative and simpler characterization of the value function. Based on this simpler definition we give an exponential time algorithm to analyze finite turn-based stochastic parity games with obligations. Finally, we show that obligation games provide the necessary framework for reasoning about p-automata and that they generalize the previous definition

On the Succinctness of Good-for-MDPs Automata

Good-for-MDPs and good-for-games automata are two recent classes of nondeterministic automata that reside between general nondeterministic and deterministic automata. Deterministic automata are good-for-games, and good-for-games automata are good-for-MDPs, but not vice versa. One of the question this raises is how these classes relate in terms of succinctness. Good-for-games automata are known to be exponentially more succinct than deterministic automata, but the gap between good-for-MDPs and good-for-games automata as well as the gap between ordinary nondeterministic automata and those that are good-for-MDPs have been open. We establish that these gaps are exponential, and sharpen this result by showing that the latter gap remains exponential when restricting the nondeterministic automata to separating safety or unambiguous reachability automata.Comment: 18 page

Revisiting Robustness in Priced Timed Games

Priced timed games are optimal-cost reachability games played between two players---the controller and the environment---by moving a token along the edges of infinite graphs of configurations of priced timed automata. The goal of the controller is to reach a given set of target locations as cheaply as possible, while the goal of the environment is the opposite. Priced timed games are known to be undecidable for timed automata with $3$ or more clocks, while they are known to be decidable for automata with $1$ clock. In an attempt to recover decidability for priced timed games Bouyer, Markey, and Sankur studied robust priced timed games where the environment has the power to slightly perturb delays proposed by the controller. Unfortunately, however, they showed that the natural problem of deciding the existence of optimal limit-strategy---optimal strategy of the controller where the perturbations tend to vanish in the limit---is undecidable with $10$ or more clocks. In this paper we revisit this problem and improve our understanding of the decidability of these games. We show that the limit-strategy problem is already undecidable for a subclass of robust priced timed games with $5$ or more clocks. On a positive side, we show the decidability of the existence of almost optimal strategies for the same subclass of one-clock robust priced timed games by adapting a classical construction by Bouyer at al. for one-clock priced timed games

Best Response Games on Regular Graphs

With the growth of the internet it is becoming increasingly important to understand how the behaviour of players is affected by the topology of the network interconnecting them. Many models which involve networks of interacting players have been proposed and best response games are amongst the simplest. In best response games each vertex simultaneously updates to employ the best response to their current surroundings. We concentrate upon trying to understand the dynamics of best response games on regular graphs with many strategies. When more than two strategies are present highly complex dynamics can ensue. We focus upon trying to understand exactly how best response games on regular graphs sample from the space of possible cellular automata. To understand this issue we investigate convex divisions in high dimensional space and we prove that almost every division of $k-1$ dimensional space into $k$ convex regions includes a single point where all regions meet. We then find connections between the convex geometry of best response games and the theory of alternating circuits on graphs. Exploiting these unexpected connections allows us to gain an interesting answer to our question of when cellular automata are best response games

A Comparison of BDD-Based Parity Game Solvers

Parity games are two player games with omega-winning conditions, played on finite graphs. Such games play an important role in verification, satisfiability and synthesis. It is therefore important to identify algorithms that can efficiently deal with large games that arise from such applications. In this paper, we describe our experiments with BDD-based implementations of four parity game solving algorithms, viz. Zielonka's recursive algorithm, the more recent Priority Promotion algorithm, the Fixpoint-Iteration algorithm and the automata based APT algorithm. We compare their performance on several types of random games and on a number of cases taken from the Keiren benchmark set.Comment: In Proceedings GandALF 2018, arXiv:1809.0241