Coordination Games on Weighted Directed Graphs

We study strategic games on weighted directed graphs, where each player’s payoff is defined as the sum of the weights on the edges from players who chose the same strategy, augmented by a fixed nonnegative integer bonus for picking a given strategy. These games capture the idea of coordination in the absence of globally common strategies. We identify natural classes of graphs for which finite improvement or coalition-improvement paths of polynomial length always exist, and consequently a (pure) Nash equilibrium or a strong equilibrium can be found in polynomial time. The considered classes of graphs are typical in network topologies: simple cycles correspond to the token ring local area networks, whereas open chains of simple cycles correspond to multiple independent rings topology from the recommendation G.8032v2 on Ethernet ring protection switching. For simple cycles, these results are optimal in the sense that without the imposed conditions on the weights and bonuses, a Nash equilibrium may not even exist. Finally, we prove that determining the existence of a Nash equilibrium or of a strong equilibrium is NP-complete already for unweighted graphs, with no bonuses assumed. This implies that the same problems for polymatrix games are strongly NP-hard.

On the Logic of Belief and Propositional Quantification

We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is something that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss.

Sets, rules and natural classes: [ ] vs. { }

We discuss a set-theoretic treatment of segments as sets of valued features and of natural classes as intensionally defined sets of sets of valued features. In this system, the empty set { } corresponds to a completely underspecified segment, and the natural class [ ] corresponds to the set of all segments, making a feature ± Segment unnecessary. We use unification, a partial operation on sets, to implement feature-filling processes, and we combine unification with set subtraction to implement feature-changing processes. We show how unification creates the illusion of targeting only underspecified segments, and we explore the possibility that only unification rules whose structural changes involve a single feature are UG-compatible. We show that no such Singleton Set Restriction can work with rules based on set subtraction. The system is illustrated using toy vowel harmony systems and a treatment of compensatory lengthening as total assimilation.

Algorithmic properties of first-order modal logics of finite Kripke frames in restricted languages

We study the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate modal logics of finite Kripke frames on the logics’ algorithmic properties. A finite frame is a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that most predicate modal logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $\varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter. This applies to the logics of finite frames of the predicate extensions of the sublogics of propositional modal logics $\textbf{GL}$, $\textbf{Grz}$ and $\textbf{KTB}$—among them, $\textbf{K}$, $\textbf{T}$, $\textbf{D}$, $\textbf{KB}$, $\textbf{K4}$ and $\textbf{S4}$.

Ontological Categories and the Transversality Requirement

Which categories of entities qualify as ontological categories? Which combinations of categories qualify as adequate systems of ontological categories? These are the two questions the author focuses on in this article. Contrary to the usual praxis in contemporary ontological literature, he addresses both questions conjointly. First, the author presents some problems of characterizing ontological categories in purely extensional terms, i.e. as widely inclusive natural classes. Second, he introduces the transversality requirement: ontological categories should be individually and naturally domain-transversal, i.e. ontological categories must be neutral concerning different scientific disciplines like physics, biology and mathematics. As a result, ontological categories must have instances in any domain of reality. Finally, the author checks the adequacy of some systems of ontological categories according to this criterion and meets some possible objections.

Almost Group Envy-free Allocation of Indivisible Goods and Chores

We consider a multi-agent resource allocation setting in which an agent's utility may decrease or increase when an item is allocated. We take the group envy-freeness concept that is well-established in the literature and present stronger and relaxed versions that are especially suitable for the allocation of indivisible items. Of particular interest is a concept called group envy-freeness up to one item (GEF1). We then present a clear taxonomy of the fairness concepts. We study which fairness concepts guarantee the existence of a fair allocation under which preference domain. For two natural classes of additive utilities, we design polynomial-time algorithms to compute a GEF1 allocation. We also prove that checking whether a given allocation satisfies GEF1 is coNP-complete when there are either only goods, only chores or both.

Uncountable structures are not classifiable up to bi-embeddability

Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic 164(12) (2013) 1454–1492], we show that whenever [Formula: see text] is a cardinal satisfying [Formula: see text], then the embeddability relation between [Formula: see text]-sized structures is strongly invariantly universal, and hence complete for ([Formula: see text]-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or groups. This fully generalizes to the uncountable case the main results of [A. Louveau and C. Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc. 357(12) (2005) 4839–4866; S.-D. Friedman and L. Motto Ros, Analytic equivalence relations and bi-embeddability, J. Symbolic Logic 76(1) (2011) 243–266; J. Williams, Universal countable Borel quasi-orders, J. Symbolic Logic 79(3) (2014) 928–954; F. Calderoni and L. Motto Ros, Universality of group embeddability, Proc. Amer. Math. Soc. 146 (2018) 1765–1780].

Evolution of social norms and correlated equilibria

Social norms regulate and coordinate most aspects of human social life, yet they emerge and change as a result of individual behaviors, beliefs, and expectations. A satisfactory account for the evolutionary dynamics of social norms, therefore, has to link individual beliefs and expectations to population-level dynamics, where individual norms change according to their consequences for individuals. Here, we present a model of evolutionary dynamics of social norms that encompasses this objective and addresses the emergence of social norms. In this model, a norm is a set of behavioral prescriptions and a set of environmental descriptions that describe the expected behaviors of those with whom the norm holder will interact. These prescriptions and descriptions are functions of exogenous environmental events. These events have no intrinsic meaning or effect on the payoffs to individuals, yet beliefs/superstitions regarding them can effectuate coordination. Although a norm’s prescriptions and descriptions are dependent on one another, we show how they emerge from random accumulations of beliefs. We categorize the space of social norms into several natural classes and study the evolutionary competition between these classes of norms. We apply our model to the Game of Chicken and the Nash Bargaining Game. Furthermore, we show how the space of norms and evolutionary stability are dependent on the correlation structure of the environment and under which such correlation structures social dilemmas can be ameliorated or exacerbated.