Parity Games Research Articles

Strategic decompositions of normal form games: Zero-sum games and potential games

We introduce new classes of games, called zero-sum equivalent games and zero-sum equivalent potential games, and prove decomposition theorems involving these classes of games. Two games are “strategically equivalent” if, for every player, the payoff differences between two strategies (holding other players' strategies fixed) are identical. A zero-sum equivalent game is a game that is strategically equivalent to a zero-sum game; a zero-sum equivalent potential game is a potential game that is strategically equivalent to a zero-sum game. We also call a game “normalized” if the sum of one player's payoffs, given the other players' strategies, is zero. One of our main decomposition results shows that any normal form game, whether the strategy set is finite or continuous, can be uniquely decomposed into a zero-sum normalized game, a zero-sum equivalent potential game, and an identical interest normalized game, each with distinctive equilibrium properties.

Solution of an EPQ model for imperfect production process under game and neutrosophic fuzzy approach

This article deals with an Economic Production Quantity (EPQ) deteriorating inventory model for non-random uncertain environment. It includes rework process, screening of imperfect items and partial backlogging. The items are partially serviceable, because at the time of production some items are found to be defective which cannot be recoverable or serviceable. At first, we develop a cost minimization problem under several assumptions related to imperfect items and rework process under certain linear constraints. We solve the crisp model (primal nonlinear problem) first, and then we convert this model into equivalent game problem taking the help of the theories related to strong and weak duality theorem. However, this game problem consists of the Lagrangian function that correspond a nonlinear objective function subject to some linear constraints. The main objective of the study is to develop a solution procedure of the problem associated to an imperfect process where all unit cost components might increase or decrease neutrosophically. Thus, according to the experiences gained by the decision maker (DM) we fuzzify all cost components as sub-neutrosophic offset. To defuzzify the model we have utilized the sine cuts of neutrosophic fuzzy numbers followed by a solution procedure developed in solving the matrix game exclusively. To validate the model, a numerical example is studied then we have compared the optimal results among the original problem, the equivalent game problem and the game problem under neutrosophic environment explicitly. Our findings reveal that under negative α-cuts the value of the objective function assumes lower and higher values. Finally, sensitivity analysis, graphical illustrations, conclusions and scope of future works have been discussed.

Simple Characterizations of Potential Games and Zero-sum Equivalent Games

We provide several tests to determine whether a game is a potential game or whether it is a zero-sum equivalent game-a game which is strategically equivalent to a zero-sum game in the same way that a potential game is strategically equivalent to a common interest game. We present a unified framework applicable for both potential and zero-sum equivalent games by deriving a simple but useful characterization of these games. This allows us to re-derive known criteria for potential games, as well as obtain several new criteria. In particular, we prove (1) new integral tests for potential games and for zero-sum equivalent games, (2) a new derivative test for zero-sum equivalent games, and (3) a new representation characterization for zero-sum equivalent games.

Deciding Parity Games in Quasi-polynomial Time

It is shown that the parity game can be solved in quasi-polynomial time. The parameterized parity game---with $n$ nodes and $m$ distinct values (a.k.a. colors or priorities)---is proven to be in th...

Robust worst cases for parity games algorithms

The McNaughton-Zielonka divide et impera algorithm is the simplest and most flexible approach available in the literature for determining the winner in a parity game. Despite its theoretical exponential worst-case complexity and the negative reputation as a poorly effective algorithm in practice, it has been shown to rank among the best techniques for solving such games. Also, it proved to be resistant to a lower bound attack, even more than the strategy improvements approaches, but finally Friedmann provided a family of games on which the algorithm requires exponential time. An easy analysis of this family shows that a simple memoization technique can help the algorithm solve the family in polynomial time. The same result can also be achieved by exploiting an approach based on the dominion-decomposition techniques proposed in the literature. These observations raise the question whether a suitable combination of dynamic programming and game-decomposition techniques can improve on the exponential worst case of the original algorithm. In this paper we answer this question negatively, by providing a robustly exponential worst case, showing that no possible intertwining of the above mentioned techniques can help mitigating the exponential nature of the divide et impera approaches. The resulting worst case is even more robust than that, since it serves as a lower bound for progress measures based algorithms as well, such as Small Progress Measure and its quasi-polynomial variant recently proposed by Jurdziński and Lazic.

Practical synthesis of reactive systems from LTL specifications via parity games

The synthesis of reactive systems from linear temporal logic (LTL) specifications is an important aspect in the design of reliable software and hardware. We present our adaption of the classic automata-theoretic approach to LTL synthesis, implemented in the tool Strix which has won the two last synthesis competitions (Syntcomp2018/2019). The presented approach is (1) structured, meaning that the states used in the construction have a semantic structure that is exploited in several ways, it performs a (2) forward exploration such that it often constructs only a small subset of the reachable states, and it is (3) incremental in the sense that it reuses results from previous inconclusive solution attempts. Further, we present and study different guiding heuristics that determine where to expand the on-demand constructed arena. Moreover, we show several techniques for extracting an implementation (Mealy machine or circuit) from the witness of the tree-automaton emptiness check. Lastly, the chosen constructions use a symbolic representation of the transition functions to reduce runtime and memory consumption. We evaluate the proposed techniques on the Syntcomp2019 benchmark set and show in more detail how the proposed techniques compare to the techniques implemented in other leading LTL synthesis tools.

A Parity Game Tale of Two Counters

Parity games are simple infinite games played on finite graphs with a winning condition that is expressive enough to capture nested least and greatest fixpoints. Through their tight relationship to the modal mu-calculus, they are used in practice for the model-checking and synthesis problems of the mu-calculus and related temporal logics like LTL and CTL. Solving parity games is a compelling complexity theoretic problem, as the problem lies in the intersection of UP and co-UP and is believed to admit a polynomial-time solution, motivating researchers to either find such a solution or to find superpolynomial lower bounds for existing algorithms to improve the understanding of parity games. We present a parameterized parity game called the Two Counters game, which provides an exponential lower bound for a wide range of attractor-based parity game solving algorithms. We are the first to provide an exponential lower bound to priority promotion with the delayed promotion policy, and the first to provide such a lower bound to tangle learning.

Simple Fixpoint Iteration To Solve Parity Games

A naive way to solve the model-checking problem of the mu-calculus uses fixpoint iteration. Traditionally however mu-calculus model-checking is solved by a reduction in linear time to a parity game, which is then solved using one of the many algorithms for parity games. We now consider a method of solving parity games by means of a naive fixpoint iteration. Several fixpoint algorithms for parity games have been proposed in the literature. In this work, we introduce an algorithm that relies on the notion of a distraction. The idea is that this offers a novel perspective for understanding parity games. We then show that this algorithm is in fact identical to two earlier published fixpoint algorithms for parity games and thus that these earlier algorithms are the same. Furthermore, we modify our algorithm to only partially recompute deeper fixpoints after updating a higher set and show that this modification enables a simple method to obtain winning strategies. We show that the resulting algorithm is simple to implement and offers good performance on practical parity games. We empirically demonstrate this using games derived from model-checking, equivalence checking and reactive synthesis and show that our fixpoint algorithm is the fastest solution for model-checking games.

Performance analysis of one‐step prediction‐based cognitive jamming in jammer‐radar countermeasure model

Strong adaptive radar, such as cognitive radar (CNR), can perform various missions while ensuring its own security in electronic warfare, via detecting environments and changing the radar parameters in real time. Unfortunately, most of the current military countermeasures, such as jamming-based electronic countermeasures, have rarely been related to jamming for CNR. Since the behaviours of radar in the traditional design of the jammer-radar scenario are always static, it is easy to create a subjective or local optimal jamming effect. In order to dynamically analyse the execution process of a complete jamming radar mission, this work establishes an equivalent attack-defence game in which the radar is regarded as a defence decision agent, and the jammer is an attack decision agent. The attributes of the game's players, the rules of the game, and the conditions for the end of the game are set clearly by setting reasonable parameters. After searching for antagonism strategies by exhaustive method, it can be found that the survivability of the predictive cognitive jamming is much stronger than that of the normal jamming based on real-time sampling data of radars. This conclusion is demonstrated through a 1 ms simulation of the game process.

Resource-Aware Automata and Games for Optimal Synthesis

We consider quantitative notions of parity automaton and parity game aimed at modelling resource-aware behaviour, and study (memory-full) strategies for exhibiting accepting runs that require a minimum amount of initial resources, respectively for winning a game with minimum initial resources. We also show how such strategies can be simplified to consist of only two types of moves: the former aimed at increasing resources, the latter aimed at satisfying the acceptance condition.

Infinite-duration Bidding Games

<?tight?>Two-player games on graphs are widely studied in formal methods, as they model the interaction between a system and its environment. The game is played by moving a token throughout a graph to produce an infinite path. There are several common modes to determine how the players move the token through the graph; e.g., in turn-based games the players alternate turns in moving the token. We study the bidding mode of moving the token, which, to the best of our knowledge, has never been studied in infinite-duration games. The following bidding rule was previously defined and called Richman bidding. Both players have separate budgets , which sum up to 1. In each turn, a bidding takes place: Both players submit bids simultaneously, where a bid is legal if it does not exceed the available budget, and the higher bidder pays his bid to the other player and moves the token. The central question studied in bidding games is a necessary and sufficient initial budget for winning the game: a threshold budget in a vertex is a value t ∈ [0, 1] such that if Player 1’s budget exceeds t , he can win the game; and if Player 2’s budget exceeds 1 − t , he can win the game. Threshold budgets were previously shown to exist in every vertex of a reachability game, which have an interesting connection with random-turn games—a sub-class of simple stochastic games in which the player who moves is chosen randomly. We show the existence of threshold budgets for a qualitative class of infinite-duration games, namely parity games, and a quantitative class, namely mean-payoff games. The key component of the proof is a quantitative solution to strongly connected mean-payoff bidding games in which we extend the connection with random-turn games to these games, and construct explicit optimal strategies for both players.

An upper bound for the restricted online Ramsey number

The restricted (m,n;N)-online Ramsey game is a game played between two players, Builder and Painter. The game starts with N isolated vertices. Each turn Builder picks an edge to build and Painter chooses whether that edge is red or blue, and Builder aims to create a red Km or blue Kn in as few turns as possible. The restricted online Ramsey number r̃(m,n;N) is the minimum number of turns that Builder needs to guarantee her win in the restricted (m,n;N)-online Ramsey game. We show that if N=r(n,n), r̃(n,n;N)≤N2−Ω(NlogN),motivated by a question posed by Conlon, Fox, Grinshpun and He. The equivalent game played on infinitely many vertices is called the online Ramsey game. As almost all known Builder strategies in the online Ramsey game end up reducing to the restricted setting, we expect further progress on the restricted online Ramsey game to have applications in the general case.

Learning to Entangle Radio Resources in Vehicular Communications: An Oblivious Game-Theoretic Perspective

This paper studies the problem of non-cooperative radio resource scheduling in a vehicle-to-vehicle communication network. The technical challenges lie in high vehicle mobility and data traffic variations. Over the discrete scheduling slots, each vehicle user equipment (VUE)-pair competes with other VUE-pairs in the coverage of a road side unit (RSU) for the limited frequency to transmit queued packets. The frequency allocation at the beginning of each slot by the RSU is regulated following a sealed second-price auction. Each VUE-pair aims to optimize the expected long-term performance. Such interactions among VUE-pairs are modelled as a stochastic game with a semi-continuous global network state space. By defining a partitioned control policy, we transform the stochastic game into an equivalent game with a global queue state space of finite size. We adopt an oblivious equilibrium (OE) to approximate the Markov perfect equilibrium (MPE), which characterizes the optimal solution to the equivalent game. The OE solution is theoretically proven to be with an asymptotic Markov equilibrium property. Due to the lack of a priori knowledge of network dynamics, we derive an online algorithm to learn the OE policies. Numerical simulations validate the theoretical analysis and show the effectiveness of the proposed online learning algorithm.

An ordered approach to solving parity games in quasi-polynomial time and quasi-linear space

Parity games play an important role in model checking and synthesis. In their paper, Calude et al. have recently shown that these games can be solved in quasi-polynomial time. We show that their algorithm can be implemented efficiently: we use their data structure as a progress measure, allowing for a backward implementation instead of a complete unravelling of the game. To achieve this, a number of changes have to be made to their techniques, where the main one is to add power to the antagonistic player that allows for determining her rational move without changing the outcome of the game. We provide a first implementation for a quasi-polynomial algorithm, test it on small examples, and provide a number of side results, including minor algorithmic improvements, a quasi-bi-linear complexity in the number of states and edges for a fixed number of colours, matching lower bounds for the algorithm of Calude et al., and a complexity index associated to our approach, which we compare to the recently proposed register index.

Communication under language barriers

We study the welfare effect of language barriers in communication. Specifically, we compare the equilibrium welfare in a game with language barriers to that in the equivalent game without language barriers. We show how and why language barriers may (weakly) improve welfare by providing two positive results. First, in a game with any language barriers, we prove that if we allow for N-dimensional communication, any equilibrium outcome of the equivalent game without language barriers can be replicated. Second, for any payoff primitive, we provide a welfare ranking for several noisy-communication devices, including language barriers, that generalizes the results in Goltsman et al. (2009). In particular, our results imply that there always exist some language barriers whose maximal equilibrium welfare (always weakly and sometimes strictly) dominates any noisy-talk equilibrium (and hence also any cheap-talk equilibrium) under no language barriers.

Fixpoint games on continuous lattices

Many analysis and verifications tasks, such as static program analyses and model-checking for temporal logics, reduce to the solution of systems of equations over suitable lattices. Inspired by recent work on lattice-theoretic progress measures, we develop a game-theoretical approach to the solution of systems of monotone equations over lattices, where for each single equation either the least or greatest solution is taken. A simple parity game, referred to as fixpoint game, is defined that provides a correct and complete characterisation of the solution of systems of equations over continuous lattices, a quite general class of lattices widely used in semantics. For powerset lattices the fixpoint game is intimately connected with classical parity games for µ-calculus model-checking, whose solution can exploit as a key tool Jurdziński’s small progress measures. We show how the notion of progress measure can be naturally generalised to fixpoint games over continuous lattices and we prove the existence of small progress measures. Our results lead to a constructive formulation of progress measures as (least) fixpoints. We refine this characterisation by introducing the notion of selection that allows one to constrain the plays in the parity game, enabling an effective (and possibly efficient) solution of the game, and thus of the associated verification problem. We also propose a logic for specifying the moves of the existential player that can be used to systematically derive simplified equations for efficiently computing progress measures. We discuss potential applications to the model-checking of latticed µ-calculi.

Memoryless determinacy of infinite parity games: Another simple proof

In 1998 Zielonka simplified the proofs of memoryless determinacy of infinite parity games. In 2018 Haddad simplified some proofs of memoryless determinacy of finite parity games. This article adapts Haddad's technique for infinite parity games. Two proofs are given, a shorter one and a more constructive one. None of them uses Zielonka's traps and attractors.

A delayed promotion policy for parity games

Parity games are two-player infinite-duration games on graphs that play a crucial role in various fields of theoretical computer science. Finding efficient algorithms to solve these games in practice is widely acknowledged as a core problem in formal verification, as it leads to efficient solutions of the model-checking and satisfiability problems of expressive temporal logics, e.g., the modal μCalculus. Their solution can be reduced to the problem of identifying sets of positions of the game, called dominions, in each of which a player can force a win by remaining in the set forever. Recently, a novel technique to compute dominions, called priority promotion, has been proposed, which is based on the notions of quasi dominion, a relaxed form of dominion, and dominion space. The underlying framework is general enough to accommodate different instantiations of the solution procedure, whose correctness is ensured by the nature of the space itself. In this paper we propose a new such instantiation, called delayed promotion, that tries to reduce the possible exponential behaviors exhibited by the original method in the worst case. The resulting procedure often outperforms both the state-of-the-art solvers and the original priority promotion approach.

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.

Adaptive dynamic surface-based differential games for a class of pure-feedback nonlinear systems with output constraints

ABSTRACTThis paper investigates the zero-sum differential game problem for a class of uncertain nonlinear pure-feedback systems with output constraints and unknown external disturbances. A barrier Lyapunov function is introduced to tackle the output constraints. By constructing an affine variable at each dynamic surface control design step rather than utilising the mean-value theorem, the tracking control problem for pure-feedback systems can be transformed into an equivalent zero-sum differential game problem for affine systems. Then, the solution of associated Hamilton–Jacobi–Isaacs equation can be obtained online by using the adaptive dynamic programming technique. Finally, the whole control scheme that is composed of a feedforward dynamic surface controller and a feedback differential game control strategy guarantees the stability of the closed-loop system, and the tracking error is remained in a bounded compact set. The simulation results demonstrate the effectiveness of the proposed control scheme.