Petri games are a multi-player game model for the synthesis of distributed systems with multiple concurrent processes based on Petri nets. The processes are the players in the game represented by the token of the net. The players are divided into two teams: the controllable system and the uncontrollable environment. An individual controller is synthesized for each process based only on their locally available causality-based information. For one environment player and a bounded number of system players, the problem of solving Petri games can be reduced to that of solving Büchi games. High-level Petri games are a concise representation of ordinary Petri games. Symmetries, derived from a high-level representation, can be exploited to significantly reduce the state space in the corresponding Büchi game. We present a new construction for solving high-level Petri games. It involves the definition of a unique, canonical representation of the reduced Büchi game. This allows us to translate a strategy in the Büchi game directly into a strategy in the Petri game. An implementation applied on six structurally different benchmark families shows in almost all cases a performance increase for larger state spaces.
Petri games have been introduced as a multi-player game model representing causal memory to address the synthesis of distributed systems. For Petri games with one environment player and an arbitrary bounded number of system players, deciding the existence of a safety strategy is EXPTIME-complete. This result forms the basis of the tool ADAM that implements an algorithm for the synthesis of distributed controllers from Petri games. To evaluate the tool, it has been checked on a series of parameterized benchmarks from manufacturing and workflow scenarios. In this paper, we introduce a new possib…
Petri games are a multiplayer game model for the automatic synthesis of distributed systems. We compare two fundamentally different approaches for solving Petri games. The symbolic approach decides the existence of a winning strategy via a reduction to a two-player game over a finite graph, which in turn is solved by a fixed point iteration based on binary decision diagrams (BDDs). The bounded synthesis approach encodes the existence of a winning strategy, up to a given bound on the size of the strategy, as a quantified Boolean formula (QBF). In this paper, we report on initial experience with…
Petri games are a multi-player game model for the synthesis problem in distributed systems, i.e., the automatic generation of local controllers. The model represents causal memory of the players, which are tokens on a Petri net and divided into two teams: the controllable system and the uncontrollable environment. For one environment player and a bounded number of system players, the problem of solving Petri games can be reduced to that of solving Büchi games. High-level Petri games are a concise representation of ordinary Petri games. Symmetries, derived from a high-level representation, can …
In distributed synthesis, we generate a set of process implementations that, together, accomplish an objective against all possible behaviors of the environment. A lot of recent work has focussed on systems with causal memory, i.e., sets of asynchronous processes that exchange their causal histories upon synchronization. Decidability results for this problem have been stated either in terms of control games, which extend Zielonka's asynchronous automata by partitioning the actions into controllable and uncontrollable, or in terms of Petri games, which extend Petri nets by partitioning the toke…
This paper is an exposition of algorithms for finding one or all equilibria of a bimatrix game (a two-player game in strategic form) in the style of a chapter in a graduate textbook. Using labeled "best-response polytopes", we present the Lemke-Howson algorithm that finds one equilibrium. We show that the path followed by this algorithm has a direction, and that the endpoints of the path have opposite index, in a canonical way using determinants. For reference, we prove that a number of notions of nondegeneracy of a bimatrix game are equivalent. The computation of all equilibria of a general b…
The classical Hotelling game is played on a line segment whose points represent uniformly distributed clients. The $n$ players of the game are servers who need to place themselves on the line segment, and once this is done, each client gets served by the player closest to it. The goal of each player is to choose its location so as to maximize the number of clients it attracts. In this paper we study a variant of the Hotelling game where each client $v$ has a tolerance interval, randomly distributed according to some density function $f$, and $v$ gets served by the nearest among the players eli…
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 $ω$-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 exist…
The uniform distribution is an important counterexample in game theory as many of the canonical game dynamics have been shown not to converge to the equilibrium in certain cases. In particular none of the canonical game dynamics converge to the uniform distribution in a form of rock-paper-scissors where the amount an agent can lose is more than the agent can win, despite this fact, it is the unique Nash equilibrium. I will show that certain incentive dynamics are asymptotically stable at the uniform distribution when it is an incentive equilibrium.
We show that, for any sufficiently small fixed $ε> 0$, when both players in a general-sum two-player (bimatrix) game employ optimistic mirror descent (OMD) with smooth regularization, learning rate $η= O(ε^2)$ and $T = Ω(\text{poly}(1/ε))$ repetitions, either the dynamics reach an $ε$-approximate Nash equilibrium (NE), or the average correlated distribution of play is an $Ω(\text{poly}(ε))$-strong coarse correlated equilibrium (CCE): any possible unilateral deviation does not only leave the player worse, but will decrease its utility by $Ω(\text{poly}(ε))$. As an immediate consequence, when th…
The algebraic structures of social organizations: the operad of cooperative games
ENSTA, Institut Polytechnique de Paris（巴黎理工学院 ENSTA）；Conservatoire National des Arts et Métiers（国立工艺院）；Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG（巴黎西岱大学与索邦大学，法国国家科学研究中心，IMJ-PRG）
The behavior of no-regret learning algorithms is well understood in two-player min-max (i.e, zero-sum) games. In this paper, we investigate the behavior of no-regret learning in min-max games with dependent strategy sets, where the strategy of the first player constrains the behavior of the second. Such games are best understood as sequential, i.e., min-max Stackelberg, games. We consider two settings, one in which only the first player chooses their actions using a no-regret algorithm while the second player best responds, and one in which both players use no-regret algorithms. For the former…
While game theory has been transformative for decision-making, the assumptions made can be overly restrictive in certain instances. In this work, we investigate some of the underlying assumptions of rationality, such as mutual consistency and best response, and consider ways to relax these assumptions using concepts from level-$k$ reasoning and quantal response equilibrium (QRE) respectively. Specifically, we propose an information-theoretic two-parameter model called the Quantal Hierarchy model, which can relax both mutual consistency and best response while still approximating level-$k$, QRE…
