Signed bounded confidence models for opinion dynamics

The aim of this paper is to modify continuous-time bounded confidence opinion dynamics models so that ‘‘changes of opinion’’ (intended as changes of the sign of the initial states) are never induced during the evolution. Such sign invariance can be achieved by letting opinions of different sign localized near the origin interact negatively, or neglect each other, or even repel each other. In all cases, it is possible to obtain sign-preserving bounded confidence models with state-dependent connectivity and with a clustering behavior similar to that of a standard bounded confidence model

Appropriate choice of aggregation operators in fuzzy decision support systems

Fuzzy logic provides a mathematical formalism for a unified treatment of vagueness and imprecision that are ever present in decision support and expert systems in many areas. The choice of aggregation operators is crucial to the behavior of the system that is intended to mimic human decision making. This paper discusses how aggregation operators can be selected and adjusted to fit empirical data&mdash;a series of test cases. Both parametric and nonparametric regression are considered and compared. A practical application of the proposed methods to electronic implementation of clinical guidelines is presented<br /

Discontinuities, Generalized Solutions, and (Dis)agreement in Opinion Dynamics

This paper deals with continuous-time opinion dynamics that feature the interplay of continuous opinions and discrete behaviours. In our model, the opinion of one individual is only influenced by the behaviours of fellow individuals. The key technical difficulty in the study of these dynamics is that the right-hand sides of the equations are discontinuous and thus their solutions must be intended in some generalized sense: in our analysis, we consider both Carath'eodory and Krasovskii solutions. We first prove existence and completeness of Carath'e-o-dory solutions from every initial condition and we highlight a pathological behavior of Carath'eo-do-ry solutions, which can converge to points that are not (Carath'eodory) equilibria. Notably, such points can be arbitrarily far from consensus and indeed simulations show that convergence to non-consensus configurations is very common. In order to cope with these pathological attractors, we then study Krasovskii solutions. We give an estimate of the asymptotic distance of all Krasovskii solutions from consensus and we prove its tightness by an example of equilibrium such that this distance is quadratic in the number of agents. This fact implies that quantization can drastically destroy consensus. However, consensus is guaranteed in some special cases, namely when the communication among the individuals is described by either a complete or a complete bipartite graph

Ultrametricity increases the predictability of cultural dynamics

A quantitative understanding of societies requires useful combinations of empirical data and mathematical models. Models of cultural dynamics aim at explaining the emergence of culturally homogeneous groups through social influence. Traditionally, the initial cultural traits of individuals are chosen uniformly at random, the emphasis being on characterizing the model outcomes that are independent of these (`annealed&#39;) initial conditions. Here, motivated by an increasing interest in forecasting social behavior in the real world, we reverse the point of view and focus on the effect of specific (`quenched&#39;) initial conditions, including those obtained from real data, on the final cultural state. We study the predictability, rigorously defined in an information-theoretic sense, of the \emphsocial content of the final cultural groups (i.e. who ends up in which group) from the knowledge of the initial cultural traits. We find that, as compared to random and shuffled initial conditions, the hierarchical ultrametric-like organization of empirical cultural states significantly increases the predictability of the final social content by largely confining cultural convergence within the lower levels of the hierarchy. Moreover, predictability correlates with the compatibility of short-term social coordination and long-term cultural diversity, a property that has been recently found to be strong and robust in empirical data. We also introduce a null model generating initial conditions that retain the ultrametric representation of real data. Using this ultrametric model, predictability is highly enhanced with respect to the random and shuffled cases, confirming the usefulness of the empirical hierarchical organization of culture for forecasting the outcome of social influence models

Navigating Multidimensional Ideologies with Reddit's Political Compass: Economic Conflict and Social Affinity

The prevalent perspective in quantitative research on opinion dynamics flattens the landscape of the online political discourse into a traditional left--right dichotomy. While this approach helps simplify the analysis and modeling effort, it also neglects the intrinsic multidimensional richness of ideologies. In this study, we analyze social interactions on Reddit, under the lens of a multi-dimensional ideological framework: the political compass. We examine over 8 million comments posted on the subreddits /r/PoliticalCompass and /r/PoliticalCompassMemes during 2020--2022. By leveraging their self-declarations, we disentangle the ideological dimensions of users into economic (left--right) and social (libertarian--authoritarian) axes. In addition, we characterize users by their demographic attributes (age, gender, and affluence). We find significant homophily for interactions along the social axis of the political compass and demographic attributes. Compared to a null model, interactions among individuals of similar ideology surpass expectations by 6%. In contrast, we uncover a significant heterophily along the economic axis: left/right interactions exceed expectations by 10%. Furthermore, heterophilic interactions are characterized by a higher language toxicity than homophilic interactions, which hints at a conflictual discourse between every opposite ideology. Our results help reconcile apparent contradictions in recent literature, which found a superposition of homophilic and heterophilic interactions in online political discussions. By disentangling such interactions into the economic and social axes we pave the way for a deeper understanding of opinion dynamics on social media

HANSON’S AUTOMATED MARKET MAKER

From Hanson’s “market scoring rule,” we derive all the necessary formulae to implement a corresponding automated market maker for a prediction market.  The market maker has many desirable qualities and always stands ready to trade, thus providing liquidity to markets.  The formulae cover all transactions for buying and selling market contracts.  In addition, we address practical concerns like how to correctly treat rounding errors and how to prevent errors that allow traders to cheat the market, and provide a practical numerical example.  We have used Hanson's automated market maker to run many markets at Microsoft