Strong Nash Equilibria in Games with the Lexicographical Improvement Property

We introduce a class of finite strategic games with the property that every deviation of a coalition of players that is profitable to each of its members strictly decreases the lexicographical order of a certain function defined on the set of strategy profiles. We call this property the Lexicographical Improvement Property (LIP) and show that it implies the existence of a generalized strong ordinal potential function. We use this characterization to derive existence, efficiency and fairness properties of strong Nash equilibria. We then study a class of games that generalizes congestion games with bottleneck objectives that we call bottleneck congestion games. We show that these games possess the LIP and thus the above mentioned properties. For bottleneck congestion games in networks, we identify cases in which the potential function associated with the LIP leads to polynomial time algorithms computing a strong Nash equilibrium. Finally, we investigate the LIP for infinite games. We show that the LIP does not imply the existence of a generalized strong ordinal potential, thus, the existence of SNE does not follow. Assuming that the function associated with the LIP is continuous, however, we prove existence of SNE. As a consequence, we prove that bottleneck congestion games with infinite strategy spaces and continuous cost functions possess a strong Nash equilibrium

Deciding the Borel complexity of regular tree languages

We show that it is decidable whether a given a regular tree language belongs to the class ${\bf \Delta^0_2}$ of the Borel hierarchy, or equivalently whether the Wadge degree of a regular tree language is countable.Comment: 15 pages, 2 figure

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 equation systems 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 $\mu$-calculus model-checking, whose solution can exploit as a key tool Jurdzi\'nski'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 $\mu$-calculi and to the solution of fixpoint equations systems over the reals

Forming Probably Stable Communities with Limited Interactions

A community needs to be partitioned into disjoint groups; each community member has an underlying preference over the groups that they would want to be a member of. We are interested in finding a stable community structure: one where no subset of members $S$ wants to deviate from the current structure. We model this setting as a hedonic game, where players are connected by an underlying interaction network, and can only consider joining groups that are connected subgraphs of the underlying graph. We analyze the relation between network structure, and one's capability to infer statistically stable (also known as PAC stable) player partitions from data. We show that when the interaction network is a forest, one can efficiently infer PAC stable coalition structures. Furthermore, when the underlying interaction graph is not a forest, efficient PAC stabilizability is no longer achievable. Thus, our results completely characterize when one can leverage the underlying graph structure in order to compute PAC stable outcomes for hedonic games. Finally, given an unknown underlying interaction network, we show that it is NP-hard to decide whether there exists a forest consistent with data samples from the network.Comment: 11 pages, full version of accepted AAAI-19 pape