Determinization and Memoryless Winning Strategies

We show that Safra's determinization ofω-automata with Streett (strong fairness) acceptance condition also gives memoryless winning strategies in infinite games, for the player whose acceptance condition is the complement of the Streett condition. Both determinization and memorylessness are essential parts of known proofs of Rabin's tree automata complementation lemma. Also, from Safra's determinization construction, along with its memoryless winning strategy extension, a single exponential complementation of Streett tree automata follows. A different single exponential construction and proof first appeared in [N. Klarlund (1992), Progress measures, immediate determinacy, and a subset construction for tree automata,in“Proceedings, 7th IEEE Symposium on Logics in Computer Science”].     

Uniform Tauberian theorem in differential games

This paper establishes the uniform Tauberian theorem for differential zero-sum games. Under rather mild conditions imposed on the dynamics and running cost, two parameterized families of games are considered, i.e., the ones with the payoff functions defined as the Cesaro mean and Abel mean of the running cost. The asymptotic behavior of value in these games is investigated as the game horizon tends to infinity and the discounting parameter tends to zero, respectively. It is demonstrated that the uniform convergence of value on an invariant subset of the phase space in one family implies the uniform convergence of value in the other family and that the limit values in the both families coincide. The dynamic programming principle acts as the cornerstone of proof.     

On the determinacy of concurrent games on event structures with infinite winning sets

We consider nondeterministic concurrent games played on event structures and study their determinacy problem—the existence of winning strategies. It is known that when the winning conditions of the games are characterised by a collection of finite winning sets/plays, a restriction (called race-freedom) on the boards where the games are played guarantees determinacy. However the games may no longer be determined when the winning sets are infinite. This paper provides a study of concurrent games and nondeterministic winning strategies by analysing conditions that ensure determinacy when infinitely many events are played, that is, when the winning sets are infinite. The main result is a determinacy theorem for a class of games with a bounded concurrency property and infinite winning sets shown to be finitely decidable.     

Uniform Solution of Parity Games on Prefix-Recognizable Graphs

Walukiewicz gave in 1996 a solution for parity games on pushdown graphs: he proved the existence of pushdown strategies and determined the winner with an EXPTIME procedure. We give a new presentation and a new algorithmic proof of these results, obtain a uniform solution for parity games (independent of their initial configuration), and extend the results to prefix-recognizable graphs. The winning regions of the players are proved to be effectively regular, and winning strategies are computed.     

Game-theoretic analysis of an ancient Chinese horse race problem

This paper analyzes a legendary Chinese horse race problem involving the King of Qi and General Tianji which took place more than 2000 years ago. In this problem each player owns three horses of different speed classes and must choose the sequence of horses to compete against each other. Depending on the payoffs received by the players as a result of the horse races, we analyze two groups of constant-sum games. In each group, we consider three separate cases where the outcomes of the races are (i) deterministic, (ii) probabilistic within the same class, and (iii) probabilistic across classes. In the first group, the player who wins the majority of races receives a one-unit payoff. For this group we show analytically that the three different games with non-singular payoff matrices have the same solution where each player has a unique optimal mixed strategy with equal probabilities. For the second group of games where the payoff to a player is the total number of races his horses have won, we use linear programming with non-numeric data to show that the solution of the three games are mixed strategies given as a convex combination of two extreme points. We invoke results from information theory to prove that to maximize the opponent's “entropy” the players should use the equal probability mixed strategy that was found for the one-unit games.     

Influence of initial conditions in 2\times 2 symmetric games

It is known that quantum game is characterized by the payoff matrix as well as initial states of the quantum objects used as carriers of information in a game. Further, the initial conditions of the quantum states influence the strategies adopted by the quantum players. In this paper, we identify the necessary condition on the initial states of quantum objects for converting symmetric games into potential games, in which the players acquire the same payoff matrix. The necessary condition to preserve the symmetric type and potential type of the game is found to be the same. The present work emphasizes the influence of the initial states in the quantization of games.     

Structural relations of symmetry among players in strategic games

The notions of symmetry and anonymity in strategic games have been formalized in different ways in the literature. We propose a combinatorial framework to analyze these notions, using group actions. Then, the same framework is used to define partial symmetries in payoff matrices. With this purpose, we introduce the notion of the role a player plays with respect to another one, and combinatorial relations between roles are studied. Building on them, we define relations directly between players, which provide yet another characterization of structural symmetries in the payoff matrices of strategic games.     

Equilibrium Trajectories in Dynamical Bimatrix Games with Average Integral Payoff Functionals

This paper considers models of evolutionary non-zero-sum games on the infinite time interval. Methods of differential game theory are used for the analysis of game interactions between two groups of participants. We assume that participants in these groups are controlled by signals for the behavior change. The payoffs of coalitions are defined as average integral functionals on the infinite horizon. We pose the design problem of a dynamical Nash equilibrium for the evolutionary game under consideration. The ideas and approaches of non-zero-sum differential games are employed for the determination of the Nash equilibrium solutions. The results derived in this paper involve the dynamic constructions and methods of evolutionary games. Much attention is focused on the formation of the dynamical Nash equilibrium with players strategies that maximize the corresponding payoff functions and have the guaranteed properties according to the minimax approach. An application of the minimax approach for constructing optimal control strategies generates dynamical Nash equilibrium trajectories yielding better results in comparison to static solutions and evolutionary models with the replicator dynamics. Finally, we make a comparison of the dynamical Nash equilibrium trajectories for evolutionary games with the average integral payoff functionals and the trajectories for evolutionary games with the global terminal payoff functionals on the infinite horizon.     

Strategy synthesis for multi-dimensional quantitative objectives

Multi-dimensional mean-payoff and energy games provide the mathematical foundation for the quantitative study of reactive systems, and play a central role in the emerging quantitative theory of verification and synthesis. In this work, we study the strategy synthesis problem for games with such multi-dimensional objectives along with a parity condition, a canonical way to express $\omega $ -regular conditions. While in general, the winning strategies in such games may require infinite memory, for synthesis the most relevant problem is the construction of a finite-memory winning strategy (if one exists). Our main contributions are as follows. First, we show a tight exponential bound (matching upper and lower bounds) on the memory required for finite-memory winning strategies in both multi-dimensional mean-payoff and energy games along with parity objectives. This significantly improves the triple exponential upper bound for multi energy games (without parity) that could be derived from results in literature for games on vector addition systems with states. Second, we present an optimal symbolic and incremental algorithm to compute a finite-memory winning strategy (if one exists) in such games. Finally, we give a complete characterization of when finite memory of strategies can be traded off for randomness. In particular, we show that for one-dimension mean-payoff parity games, randomized memoryless strategies are as powerful as their pure finite-memory counterparts.     

A game theoretical perspective on the quantum probabilities associated with a GHZ state

In the standard approach to quantum games, players’ strategic moves are local unitary transformations on an entangled state that is subsequently measured. Players’ payoffs are then obtained as expected values of the entries in the payoff matrix of the classical game on a set of quantum probabilities obtained from the quantum measurement. In this paper, we approach quantum games from a diametrically opposite perspective. We consider a classical three-player symmetric game along with a known expression for a set of quantum probabilities relevant to a tripartite Einstein–Podolsky–Rosen (EPR) experiment that depends on three players’ directional choices in the experiment. We define the players’ strategic moves as their directional choices in an EPR setting and then express their payoff relations in the resulting quantum game in terms of their directional choices, the entries of the payoff matrix, and the quantum probability distribution relevant to the tripartite EPR experiment.     

猜数游戏与编码理论

本文揭示了一类猜数游戏与编码理论中的奇偶校验码以及海明码之间的联系,并利用奇偶校验码和海明码,提出了这类游戏的解决方案。     

Discounting Infinite Games But How and Why?

In a recent paper de Alfaro, Henzinger and Majumdar [Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Discounting the future in systems theory. In ICALP 2003, volume 2719 of LNCS, pages 1022–1037. Springer, 2003] observed that discounting successive payments, the procedure that is employed in the classical stochastic game theory since the seminal paper of Shapley [L.S. Shapley. Stochastic games. Proceedings Nat. Acad. of Science USA, 39:1095–1100, 1953], is also pertinent in the context of much more recent theory of stochastic parity games [L. de Alfaro and R. Majumdar. Quantitative solution to omega-regular games. In STOC'01, pages 675–683. ACM Press, 2001. final version to appear in Journal of Computer and System Sciences, L. de Alfaro, T.A. Henzinger, and O. Kupferman. Concurrent reachability games. In FOCS'98, pages 564–575. IEEE Computer Society Press, 1998, L. de Alfaro and T.A. Henzinger. Concurrent ω-regular games. In LICS'00, pages 142–154. IEEE Computer Society Press, 2000] which were proposed as a tool for verification of probabilistic systems. We show that, surprisingly perhaps, the particular discounting used in [Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Discounting the future in systems theory. In ICALP 2003, volume 2719 of LNCS, pages 1022–1037. Springer, 2003] is in fact very close to the original ideas of Shapley. This observation allows to realize that the specific discounting of [Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Discounting the future in systems theory. In ICALP 2003, volume 2719 of LNCS, pages 1022–1037. Springer, 2003] suffers in fact from some needless restrictions. We advocate that dropping the constraints imposed in [Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Discounting the future in systems theory. In ICALP 2003, volume 2719 of LNCS, pages 1022–1037. Springer, 2003] leads to a more general and elegant theory that includes parity and mean payoff games as particular limit cases.     

Decentralized network topologies in multirobot systems

This paper studies the problem of forming and updating the network topology in a multirobot system that is simultaneously engaged in a given task. The contribution of this paper is to propose a decentralized model of how the network may evolve based on network-related payoff functions and pairwise games. As such, pairwise games provide a practical and general scheme for contacting other robots and revising the network topology. A network is deemed acceptable by all the robots using pairwise stability and pairwise Nash equilibrium. As an application, we consider networks that are generated with mutual link-based payoff functions and show that – under some assumptions regarding changes in the configuration states – each game is ensured of converging to a pairwise stable network. This approach is then integrated with a common robotic task where the network is critical to successful task completion. The resulting performance is evaluated with respect to a variety of measures including task completion, network density, and the average payoff along with comparative results with all-to-all communication.     

Strategy construction for parity games with imperfect information

We consider two-player parity games with imperfect information in which strategies rely on observations that provide imperfect information about the history of a play. To solve such games, i.e., to determine the winning regions of players and corresponding winning strategies, one can use the subset construction to build an equivalent perfect-information game. Recently, an algorithm that avoids the inefficient subset construction has been proposed. The algorithm performs a fixed-point computation in a lattice of antichains, thus maintaining a succinct representation of state sets. However, this representation does not allow to recover winning strategies.In this paper, we build on the antichain approach to develop an algorithm for constructing the winning strategies in parity games of imperfect information. One major obstacle in adapting the classical procedure is that the complementation of attractor sets would break the invariant of downward-closedness on which the antichain representation relies. We overcome this difficulty by decomposing problem instances recursively into games with a combination of reachability, safety, and simpler parity conditions. We also report on an experimental implementation of our algorithm; to our knowledge, this is the first implementation of a procedure for solving imperfect-information parity games on graphs.     

Reaching pareto-optimality in prisoner’s dilemma using conditional joint action learning

We consider the learning problem faced by two self-interested agents repeatedly playing a general-sum stage game. We assume that the players can observe each other’s actions but not the payoffs received by the other player. The concept of Nash Equilibrium in repeated games provides an individually rational solution for playing such games and can be achieved by playing the Nash Equilibrium strategy for the single-shot game in every iteration. Such a strategy, however can sometimes lead to a Pareto-Dominated outcome for games like Prisoner’s Dilemma. So we prefer learning strategies that converge to a Pareto-Optimal outcome that also produces a Nash Equilibrium payoff for repeated two-player, n-action general-sum games. The Folk Theorem enable us to identify such outcomes. In this paper, we introduce the Conditional Joint Action Learner (CJAL) which learns the conditional probability of an action taken by the opponent given its own actions and uses it to decide its next course of action. We empirically show that under self-play and if the payoff structure of the Prisoner’s Dilemma game satisfies certain conditions, a CJAL learner, using a random exploration strategy followed by a completely greedy exploitation technique, will learn to converge to a Pareto-Optimal solution. We also show that such learning will generate Pareto-Optimal payoffs in a large majority of other two-player general sum games. We compare the performance of CJAL with that of existing algorithms such as WOLF-PHC and JAL on all structurally distinct two-player conflict games with ordinal payoffs.     

基于动态模糊联盟合作博弈的区间模糊Shapley值

利用模糊数学相关理论,针对n人合作博弈中支付函数是模糊三角函数的情形,对经典Shapley值提出的三条公理进行了拓展,并构造了区间模糊Shapley值。考虑到盟友在合作结束后需要对具体的联盟收益进行分配,利用构造的区间模糊Shapley值隶属函数给出了确定的收益分配方案。最后利用实例对该方法的有效性和可行性进行了说明。     

Certifying DFA Bounds for Recognition and Separation

Innovations in Systems and Software Engineering - The automation of decision procedures makes certification essential. We suggest to use determinacy of turn-based two-player games with regular...     

A proof of the determinacy property of the data flow schema

Previous proofs of data flow determinacy proceeded indirectly and relied on results from abstract semantics. In this paper, a direct proof of the determinacy of data flow schema is given. This proof does not use high powered metamathematical theorems and consequently is accessible to all researchers in data flow, not just those intimately familiar with deep semantic results. Our proof proceeds in three stages. First, the data flow schema which contains instances of Link, Sink and Operator actors is proved to be determinate by reduction to the results of Karp and Miller (1966). Second, data flow schema containingSwitch actors, in addition to the above actors, are proved to be determinate. Finally, data flow schema which additionally contain Apply actors are proven to be determinate.     

Evaluation of memoryless simplification

We investigate the effectiveness of the memoryless simplification approach described by Lindstrom and Turk (1998). Like many polygon simplification methods, this approach reduces the number of triangles in a model by performing a sequence of edge collapses. It differs from most recent methods, however, in that it does not retain a history of the geometry of the original model during simplification. We present numerical comparisons showing that the memoryless method results in smaller mean distance measures than many published techniques that retain geometric history. We compare a number of different vertex placement schemes for an edge collapse in order to identify the aspects of the memoryless simplification that are responsible for its high level of fidelity. We also evaluate simplification of models with boundaries, and we show how the memoryless method may be tuned to trade between manifold and boundary fidelity. We found that the memoryless approach yields consistently low mean errors when measured by the Metro mesh comparison tool. In addition to using complex models for the evaluations, we also perform comparisons using a sphere and portions of a sphere. These simple surfaces turn out to match the simplification behaviors for the more complex models that we used