Abstract
We study a dichotomous decision model, where individuals can make the decision yes or no and can influence the decisions of others. We characterize all decisions that form Nash equilibria. Taking into account the way individuals influence the decisions of others, we construct the decision tilings where the axes reflect the personal preferences of the individuals for making the decision yes or no. These tilings characterize geometrically all the pure and mixed Nash equilibria. We show, in these tilings, that Nash equilibria form degenerated hysteresis with respect to the dynamics, with the property that the pure Nash equilibria are asymptotically stable and the strict mixed equilibria are unstable. These hysteresis can help to explain the sudden appearance of social, political and economic crises. We observe the existence of limit cycles for the dynamics associated to situations where the individuals keep changing their decisions along time, but exhibiting a periodic repetition in their decisions. We introduce the notion of altruist and individualist leaders and study the way that the leader can affect the individuals to make the decision that the leader pretends. © 2014, American Institute of Mathematical Sciences.

Abstract
The Portuguese governmental network comprising all the 776 ministers and junior ministers who were part of the 19 governments between the year 1976 and 2013 is presented and analyzed. The data contain information on connections concerning business and other types of organizations and, to our knowledge, there is no such extensive research in previous literature. Upon the presentation of the data, a social network analysis considering the temporal dimension is performed at three levels of granularity: network-level, subnetwork-level (political groups) and node-level. A discussion based on the results is presented. We conclude that although it fits two of the four preconditions of a small-world model, the Portuguese governmental network is not a small-world network, although presenting an evolution pointing toward becoming one. Also, we use a resilience test to study the evolution of the robustness of the Portuguese governmental network, pinpointing the moment when a set of members became structurally important.

Abstract
We study a finite decision model where the utility function is an additive combination of a personal valuation component and an interaction component. Individuals are characterized according to these two components (their valuation type and externality type), and also according to their crowding type (how they influence others). We study how positive externalities lead to typo symmetries euillbrIn, while negative externalities allow the existence of equillibria that are not type-symmetric. In particular, we show that positive elides lead to euilibria having a unique partition into a minimum number of societies (simi individuals using the same strategy, see 1271); and negat ve externalities I to equilibria with multiple societal partitions, some with the maximum number of societies

Abstract
In two-action generalized polymatrix games, Nash equilibria are support-type-symmetric, i.e., determined by supports for each type of player. We show that such a property does not generalize straightforwardly for games with at least three actions or where interaction weights have different signs (neither all positive nor negative). A non-trivial condition on interaction weights must be satisfied, which may go unnoticed as it is trivially satisfied for: (i) two-action games, (ii) conformity games, and (iii) congestion games. We derive this condition and the corresponding simplified analytic equation for mixed strategies.

Abstract
We show that in finite settings with identical firms and consumers, asymmetric pure price equilibria with positive profits exist. We consider a price competition duopoly for a homogeneous product. Demand stems from a second-stage consumption game at posted prices, with consumers' behavior impacted by negative network effects. We characterize equilibrium prices and demand. In all subgame-perfect pure price equilibria, both firms have positive profits, and in some, firms charge different prices.

Abstract

We implemented Discord as a pedagogical tool in the academic year of 2021/2022 in two mathematics curricular units of the first year of an Informatics Engineering university program. We analyze and discuss the experience, reflecting on usability and influence on learning processes and engagement. We compare the impact of using the platform: 1) when combined with different methodological and pedagogical approaches and 2) to previous years with other (or none) classic virtual forums.