A Faster Algorithm for Solving One-Clock Priced Timed Games

2003

We address the question of existence of equilibrium in general timing games of complete information. Under weak assumptions, any two-player timing game has a subgame perfect -equilibrium, for each > 0. This result is tight. For some classes of games (symmetric games, games with cumulative payoffs), stronger existence results are established. * We thank Elchanan Ben-Porath and Itzhak Gilboa for useful comments.

Lecture Notes in Computer Science, 2007

Three-color parity games capture the disjunction of a Büchi and a co-Büchi condition. The most efficient known algorithm for these games is the progress measures algorithm by Jurdziński. We present an acceleration technique that, while leaving the worst-case complexity unchanged, often leads to considerable speed-ups in games arising in practice. As an application, we consider games played in discrete real time, where players should be prevented from stopping time by always choosing moves with delay zero. The time progress condition can be encoded as a three-color parity game. Using the tool TICC as a platform, we compare the performance of a BDD-based symbolic implementation of the progress measure algorithm with acceleration, and of the symbolic implementation of the classical μ-calculus algorithm of Emerson and Jutla.

Proceedings of the 2018 ACM Conference on Economics and Computation, 2018

Protecting valuable targets from an adversary is an ever-important international concern with far-reaching applications in wildlife protection, border protection, counter-terrorism, protection of ships from piracy, etc. As a successful recent approach, security games cast these issues as two-player games between a defender and an attacker. The defender decides on how to allocate the available resources to protect targets against the attacker who strives to in ict damage on them. The main question of interest here is equilibrium computation. Our focus in this paper is on spatio-temporal security games. However, inspired by the paper of Xu [EC'16], we start with a general model of security games and show that any approximation (of any factor) for the defender's best response (DBR) problem leads to an approximation of the same factor for the actual game. In most applications of security games, the targets are mobile. This leads to a well-studied class of succinct games, namely spatio-temporal security games, that is played in space and time. In such games, the defender has to specify a time-dependent patrolling strategy over a spatial domain to protect a set of moving targets. We give a generalized model of prior spatio-temporal security games that is played on a base graph G. That is, the patrols can be placed on the vertices of G and move along its edges over time. This uni es and generalizes prior spatio-temporal models that only consider speci c spatial domains such as lines or grids. Graphs can further model many other domains of practical interest such as roads, internal maps of buildings, etc. Finding an optimal defender strategy becomes NP-hard on general graphs. To overcome this, we give an LP relaxation of the DBR problem and devise a rounding technique to obtain an almost optimal integral solution. More precisely, we show that one can achieve a (1 − ϵ)-approximation in polynomial time if we allow the defender to use ln(1/ϵ) times more patrols. We later show that this result is in some sense the best possible polynomial time algorithm (unless P=NP). Furthermore, we show that by using a novel dependent rounding technique, the same LP relaxation gives an optimal solution for speci c domains of interest, such as one-dimensional spaces. This result simpli es and improves upon the prior algorithm of Behnezhad et al. [EC'17] on several aspects and can be generalized to other graphs of interest such as cycles. Lastly, we note that most prior algorithms for security games assume that the attacker attacks only once and become intractable for a super-constant number of attacks. Our algorithms are fully polynomial in the input size and work for any given number of attacks. CCS Concepts: • Theory of computation → Algorithmic game theory and mechanism design;

arXiv: Optimization and Control, 2018

Stochastic games are a classical model in game theory in which two opponents interact and the environment changes in response to the players' behavior. The central solution concepts for these games are the discounted values and the value, which represent what playing the game is worth to the players for different levels of impatience. In the present manuscript, we provide algorithms for computing exact expressions for the discounted values and for the value, which are polynomial in the number of pure stationary strategies of the players. This result considerably improves all the existing algorithms, including the most efficient one, due to Hansen, Kouck\'y, Lauritzen, Miltersen and Tsigaridas (STOC 2011).

Proceedings of the 17th international conference on Hybrid systems: computation and control - HSCC '14, 2014

In this paper, we study energy and mean-payoff timed games. The decision problems that consist in determining the existence of winning strategies in those games are undecidable, and we thus provide semi-algorithms for solving these strategy synthesis problems. We then identify a large class of timed games for which our semi-algorithms terminate and are thus complete. We also study in detail the relation between mean-payoff and energy timed games. Finally, we provide a symbolic algorithm to solve energy timed games and demonstrate its use on small examples using HyTech.

2010

In this paper we present an algorithm implemented by MATLAB, and several examples completely realized by this algorithm, based on a method developed by one of the authors to determine the payoff-space of certain normal-form C 1 -games. Specifically, our study is based on a method able to determine the payoff space of normal form C 1games in n dimensions, that is for n-players normal form games whose payoff functions are defined on compact intervals of the real line and of class at least C 1 . In this paper we will determine the payoff space of such normal form C 1 -games in the particular case of two dimensions. The implementation of the algorithm gives the parametric form of the critical zone of a game in the bistrategy space and in the payoff space and their graphical representations. Moreover, we obtain the parametric form of the transformation of the topological boundary of the bistrategy space and of the transformation of the critical zone. The final aim of the program is to plot the entire payoff space of the considered games. One of the main motivations of our paper is that the mixed extension of a bimatrix gamethe most used in the application of Game Theory -is a game of the type considered. For this reason we realized an algorithm that produces the payoff space and the critical zone of a game in normal form supported by a finite family of compact intervals of the real line. Resuming in details, the algorithm returns: the parametric form of the critical zone; the parametric form of the transformation of the topological boundary of the bistrategy space; the parametric form of the transformation of the critical zone. All of them are graphically represented. To prove the efficiency of the algorithm, we show several examples. Our final goal is to provide a valuable tool to study simply but completely normal form C 1 -games in two dimensions.

Games and Economic Behavior, 2008

We consider the following abstraction of competing publications. There are n players in the game. Each player i chooses a point x i in the interval [0, 1], and a player's payoff is the distance from its point x i to the next larger point, or to 1 if x i is the largest. For this game, we give a complete characterization of the Nash equilibrium for the two-player game, and, more important, we give an efficient approximation algorithm to compute numerically the symmetric Nash equilibrium for the n-player game. The approximation is computed via a discrete version of the game. In both cases, we show that the (symmetric) equilibrium is unique. Our algorithmic approach to the n-player game is non-standard in that it does not involve solving a system of differential equations. We believe that our techniques can be useful in the analysis of other timing games.

Information and Computation, 2015

Iterated games are well-known in the game theory literature. We study iterated Boolean games. These are games in which players repeatedly choose truth values for Boolean variables they have control over. Our model of iterated Boolean games assumes that players have goals given by formulae of Linear Temporal Logic (LTL), a formalism for expressing properties of state sequences. In order to represent the strategies of players in such games, we use a finite state machine model. After introducing and formally defining iterated Boolean games, we investigate the computational complexity of their associated game-theoretic decision problems, as well as semantic conditions characterising classes of LTL properties that are preserved by equilibrium points (pure-strategy Nash equilibria) whenever they exist.

2007

In a reachability-time game, players Min and Max choose moves so that the time to reach a final state in a timed automaton is minimised or maximised, respectively. Asarin and Maler showed decidability of reachability-time games on strongly non-Zeno timed automata using a value iteration algorithm. This paper complements their work by providing a strategy improvement algorithm for the problem.

Lecture Notes in Computer Science, 2010

We consider two-player stochastic games over real-time probabilistic processes where the winning objective is specified by a timed automaton. The goal of player is to play in such a way that the play (a timed word) is accepted by the timed automaton with probability one. Player aims at the opposite. We prove that whenever player has a winning strategy, then she also has a strategy that can be specified by a timed automaton. The strategy automaton reads the history of a play, and the decisions taken by the strategy depend only on the region of the resulting configuration. We also give an exponential-time algorithm which computes a winning timed automaton strategy if it exists.

Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11, 2011

Shapley's discounted stochastic games, Everett's recursive games and Gillette's undiscounted stochastic games are classical models of game theory describing two-player zero-sum games of potentially infinite duration. We describe algorithms for exactly solving these games. When the number of positions of the game is constant, our algorithms run in polynomial time.

Electronic Proceedings in Theoretical Computer Science

In this paper, we study algorithms for special cases of energy games, a class of turn-based games on graphs that show up in the quantitative analysis of reactive systems. In an energy game, the vertices of a weighted directed graph belong either to Alice or to Bob. A token is moved to a next vertex by the player controlling its current location, and its energy is changed by the weight of the edge. Given a fixed starting vertex and initial energy, Alice wins the game if the energy of the token remains nonnegative at every moment. If the energy goes below zero at some point, then Bob wins. The problem of determining the winner in an energy game lies in NP ∩ coNP. It is a long standing open problem whether a polynomial time algorithm for this problem exists. We devise new algorithms for three special cases of the problem. The first two results focus on the single-player version, where either Alice or Bob controls the whole game graph. We develop anÕ(n ω W ω) time algorithm for a game graph controlled by Alice, by providing a reduction to the All-Pairs Nonnegative Prefix Paths problem (APNP), where W is the maximum absolute value of any edge weight and ω is the best exponent for matrix multiplication. Thus we study the APNP problem separately, for which we develop anÕ(n ω W ω) time algorithm. For both problems, we improve over the state of the art ofÕ(mn) for small W. For the APNP problem, we also provide a conditional lower bound which states that there is no O(n 3−ε) time algorithm for any ε > 0, unless the APSP Hypothesis fails. For a game graph controlled by Bob, we obtain a near-linear time algorithm. Regarding our third result, we present a variant of the value iteration algorithm, and we prove that it gives an O(mn) time algorithm for game graphs without negative cycles, which improves a previous upper bound. The all-Bob algorithm is randomized, all other algorithms are deterministic.

Information and Computation, 2020

We consider Pareto analysis of reachable states of multi-priced timed automata (MPTA): timed automata equipped with multiple observers that keep track of costs (to be minimised) and rewards (to be maximised) along a computation. Each observer has a constant non-negative derivative which may depend on the location of the MPTA. We study the Pareto Domination Problem, which asks whether it is possible to reach a target location via a run in which the accumulated costs and rewards Pareto dominate a given objective vector. We show that this problem is undecidable in general, but decidable for MPTA with at most three observers. For MPTA whose observers are all costs or all rewards, we show that the Pareto Domination Problem is PSPACE-complete. We also consider an ε-approximate Pareto Domination Problem that is decidable without restricting the number and types of observers. We develop connections between MPTA and Diophantine equations. Undecidability of the Pareto Domination Problem is shown by reduction from Hilbert's 10 th Problem, while decidability for three observers is shown by a translation to a fragment of arithmetic involving quadratic forms.

Electronic Notes in Theoretical Computer Science, 2005

Priced timed (game) automata extend timed (game) automata with costs on both locations and transitions. The problem of synthesizing an optimal winning strategy for a priced timed game under some hypotheses has been shown decidable in . In this paper, we present an algorithm for computing the optimal cost and for synthesizing an optimal strategy in case there exists one. We also describe the implementation of this algorithm with the tool HyTech and present an example.

Lecture Notes in Computer Science, 2005

We consider real-time games where the goal consists, for each player, in maximizing the average reward he or she receives per time unit. We consider zero-sum rewards, so that a reward of +r to one player corresponds to a reward of −r to the other player. The games are played on discrete-time game structures which can be specified using a two-player version of timed automata whose locations are labeled by reward rates. Even though the rewards themselves are zerosum, the games are not, due to the requirement that time must progress along a play of the game. Since we focus on control applications, we define the value of the game to a player to be the maximal average reward per time unit that the player can ensure. We show that, in general, the values to players 1 and 2 do not sum to zero. We provide algorithms for computing the value of the game for either player; the algorithms are based on the relationship between the original, infinite-round game, and a derived game that is played for only finitely many rounds. As memoryless optimal strategies exist for both players in both games, we show that the problem of computing the value of the game is in NP∩coNP.