The interaction between a system and the components modeling its environment is traditionally modeled by a multi-player game played on a finite graph. In zero-sum games, the players have conflicting objectives, and it is clear that increasing the memory of the environment players can only make it harder for the system to win. In non-zero-sum games, the objectives of the players may overlap. There, typical questions concern the stability of the game and the equilibria the players may reach. In particular, in rational synthesis (RS), the goal is to find an equilibrium that satisfies the objective of the system. We study how the memory of the environment players may affect the existence of an RS solution. As we show, the picture is diverse, even when the objectives of all players are memoryless. On the one hand, when stability amounts to a Nash equilibrium (NE), then increasing the memory of the environment may only help the system to suggest an RS solution. On the other hand, when the notion of stability involves deviations by coalitions of environment players, for example in a strong Nash equilibrium (SNE), then increasing their memory may sometimes enable and sometimes prevent the existence of an RS solution. We study memory bounds for the players, showing that the memory required may be polynomial in an NE-RS solution and exponential in an SNE-RS solution. We also solve the SNE-RS problem, show that it is PSPACE-complete, and relate the differences between NE and SNE with the differences between cooperative and non-cooperative RS.

Abstract We introduce and study non-zero-sum multi-player games with weighted multiple objectives . In these games, the objective of each player consists of a set $$\alpha $$ α of underlying objectives and a weight function $$w: 2^\alpha \rightarrow \mathbb {Z}$$ w : 2 α → Z that maps each subset X of $$\alpha $$ α to the utility of the player when exactly all the objectives in X are satisfied. We study the existence and synthesis of stable outcomes with desired utilities for the players. The problem generalizes rational synthesis and enables the synthesis of outcomes that satisfy wellness, fairness, and priority requirements. We study the extension of the game by payments , with which players can incentivize each other to follow strategies that are beneficial for the paying player. We show how such payments can be used in order to repair systems. We study the complexity of the setting for various classes of weight functions. In particular, general weight functions are related to Muller objectives, and the synthesis problem for them is PSPACE-complete. We study non-decreasing, additive, positive, and other classes of weight functions, and the way they affect the memory required for the players and the complexity of the synthesis problem.

Each language ⁎ L ⊆ Σ ⁎ induces an infinite sequence { P r ( L , n ) } n = 1 ∞ , where for all n ≥ 1 , the value P r ( L , n ) ∈ [ 0 , 1 ] is the probability of a word of length n to be in L , assuming a uniform distribution on the letters in Σ. Previous studies of { P r ( L , n ) } n = 1 ∞ for a regular language L , concerned zero-one laws, density, and accumulation points. We study monotonicity of { P r ( L , n ) } n = 1 ∞ , possibly in the limit. We show that monotonicity may depend on the distribution of letters, study how operations on languages affect monotonicity, and characterize classes of languages for which the sequence is monotonic. We extend the study to languages L of infinite words, where we study the probability of lasso-shaped words to be in L and consider two definitions for P r ( L , n ) . The first refers to the probability of prefixes of length n to be extended to words in L , and the second to the probability of word w of length n to be such that w ω is in L . Thus, in the second definition, monotonicity depends not only on the length of w , but also on the words being periodic. We also study the complexity of calculating P r ( L , n ) for the various definitions.