Opinion Dynamics with Preference Matching: How the Desire to Meet Facilitates Opinion Exchange

Steinbacher, Mitja; Steinbacher, Matjaž; Knoppe, Clemens Article — Published Version Opinion Dynamics with Preference Matching: How the Desire to Meet Facilitates Opinion Exchange Computational Economics Provided in Cooperation with: Springer Nature Suggested Citation: Steinbacher, Mitja; Steinbacher, Matjaž; Knoppe, Clemens (2023) : Opinion Dynamics with Preference Matching: How the Desire to Meet Facilitates Opinion Exchange, Computational Economics, ISSN 1572-9974, Springer US, New York, NY, Vol. 64, Iss. 2, pp. 735-768, https://doi.org/10.1007/s10614-023-10455-7 This Version is available at: https://hdl.handle.net/10419/317931 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) Computational Economics (2024) 64:735–768 https://doi.org/10.1007/s10614-023-10455-7 1 3 Opinion Dynamics withPreference Matching: How theDesire toMeet Facilitates Opinion Exchange MitjaSteinbacher1· MatjažSteinbacher2· ClemensKnoppe3 Accepted: 10 August 2023 / Published online: 2 September 2023 © The Author(s) 2023 Abstract The paper reexamines an agent-based model of opinion formation under bounded confidence with heterogeneous agents. The paper is novel in that it extends the standard model of opinion dynamics with the assumption that interacting agents share the desire to exchange opinion. In particular, the interaction between agents in the paper is modeled via a dynamic preferential-matching process wherein agents reveal their preferences to meet according to three features: coherence, opinion difference, and agents’ positive sentiments towards others. Only preferred matches meet and exchange opinion. Through an extensive series of simulation treatments, it follows that the presence of sentiments, on one hand, hardens the matching process between agents, which leads to less communication. But, on the other hand, it increases the diversity in preferred matches between agents and thereby leads to a better-integrated social network structure, which reflects in a reduction of the opinion variance between agents. Moreover, at combinations of (a) high tolerance, (b) low sensitivity of agents to opinion volatility, and (c) low levels of confidence, agents are occasionally drawn away from the consensus, forming small groups that hold extreme opinions. Keywords Opinion dynamics· Sentiments· Bounded confidence· Matching· Simulation treatments· Agent-based model· Social network * Clemens Knoppe [email protected] Mitja Steinbacher mitja.steinbac[email protected] Matjaž Steinbacher matjaz.steinbac[email protected] 1 Faculty ofLaw andBusiness Studies, Catholic Institute, Ljubljana, Slovenia 2 Fund forFinancing theDecommissioning oftheKrško Nuclear Power Plant andDisposal ofRadioactive Waste, Krško, Slovenia 3 Institute ofEconomics, Kiel University, Kiel, Germany 736 M.Steinbacher et al. 1 3 1 Introduction The study of opinion dynamics is a part of a domain of social interaction. Early beginnings of the opinion update theory in social science go back at least to the seminal opinion update model by DeGroot (1974). Since then the study of the opinion dynamics has attracted growing attention across a number of disciplines. In the last decade, the tractability of computer simulation and the availability of sufficient computer power have attributed to the rise of agent-based models of social interaction, in which agents have been transforming from homogeneous factors to more and more comprehensive heterogeneous actors (Macy & Willer, 2002). For instance, see an exemplary agent-based study into the collective behavior of social systems by Epstein and Axtell (1996), where authors grow economic relations from bottom up and show that even simple rules of social interaction among agents can produce stylized macro patterns. Similarly, Axelrod (1997) applied an interactive model of agents with a number of discrete features to study the dissemination of culture. The approach taken in the present paper falls in the domain of models of social interaction, based on opinion formation models with bounded confidence, as originally proposed by Hegselmann and Krause (2002, 2005); Deffuant etal. (2000, 2002). Ultimately, the dynamics in models of social interaction depend on the way agents receive, process, and respond upon information as well as upon their interaction patterns(Macal & North, 2005; Altafini, 2013). Agents’ responses to information and their interaction patterns necessarily evolve within the model. That is, the former are guided by behavioral features of agents that determine their preferences about own state or the state of their neighbors, while the latter depend on the way agents are connected among each other in their social environments and how they choose their counterparts. However, in a current state of the literature, agents update their opinions according to own attitudes towards opinions of their counterparts. These attitudes are usually coupled to the opinion update rule and they might include features, such as stubbornness and confidence (Deffuant et al., 2000; Weisbuch et al., 2002), quorum (Ward et al., 2008), honesty (Dutta & Sen, 2012) or emotions (Schweitzer etal., 2019; Sobkowicz, 2012). This setup prevents, for instance, a study of agents’ desire to meet a particular agent in isolation from the opinion update. In reality, the decision to meet with someone must be mutual for both agents taking place in the opinion exchange and it has to be made prior to their meeting. To study the opinion update without the interference of coupling the opinion adoption to the meeting preferences, this paper sees the opinion update update as subordinated to the preference-based matching process. To achieve this aim, the paper decouples the matching of agents from the opinion update altogether. Here, note that the decoupling does not imply that the desire to meet with someone cannot not be driven by similar influences to those of the opinion update. It only means that the matching and the opinion update are two distinct processes that should stay decoupled. In our case, agents are first classified by their mutual 737 1 3 Opinion Dynamics withPreference Matching: How theDesire… preferences to meet and only those agents that have been mutually paired ultimately take place in the opinion exchange. The classification of pairs proceeds in lines of a matching theory, the approach particularly popular in economics and appropriate for our task at hand. See Gale and Shapley (1962), Roth and Sotomayor (1992) and Roth (1982) for more information on basic fundamentals and the use of the matching theory in economics and social sciences. Only after this matching stage, we adopt a common framework of the opinion update as used in the literature (Deffuant etal., 2000; Hegselmann & Krause, 2002, 2005; Jadbabaie etal., 2003; Blondel etal., 2009). In our case, the ultimate decision to meet an agent or not will be mutual and will be made beforehand by both agents. Each agent will form own lists of preferred matches in her neighborhood. These lists will form a basis for the pairwise matching of agents in preferred pairs. For this end, we will implement Irving (1985) efficient roommate matching algorithm as proposed by Sotomayor (2005), whereby the preference lists for the efficient roommate matching will be implemented as relative radial-basis scores. This turns out to be a computationally simple solution that allows the inclusion of a rich set of behavioral features in the matching process as well as any combination of their eventual co-dependencies. Radial-basis scores as used here in this paper belong to the sub-field of the radial basis kernel (RBK) classification. The RBK is a widely used and highly appreciated technique within a broad scope of the implementation of the artificial intelligence methods (Patle & Chouhan, 2013). This flexibility of the matching theory and its suitability to be dealt with computationally efficient classification techniques seems a promising way of bridging together social psychology, behavioral economics, computational social science, and artificial intelligence not only within the scope of the opinion update, but also within a wider spectrum of models of social interaction and social learning. The preference score depends on the opinion difference, as well as human sentiment, modelled as a random variable, and neighbors’ opinion coherence. In particular, the main focus of this paper is to stress the facilitating role of agents’ positive sentiments of approbation towards each other, first, for the opinion exchange, and, second, for the opinion dynamics as a whole. The sentiments will be added among behavioral features that will shape the matching process between agents during the classification stage. Our motivation is straightforward. Namely, human sentiments of approbation are a long-forgotten concept in economics, despite the fact that they have been rooted in the study of interpersonal relations in economics since the inception of economics as a science (Smith & Wilson, 2019). Moreover, we add a coherence feature, so that agents only listen to those counterparts, whom they consider holding a valid opinion. Validity is defined as opinion persistence, i.e. it controls for the respect of a minimal coherence on part of agents in the opinion model. The coherence is an appealing behavioral feature that has yet to find its way into the main body of the opinion formation theory. Results in this paper are intriguing and yet straightforward. In the presence of positive sentiments of approbation, agents meet less, as it is harder for them to find mutually preferred matches, but the matches that take place are more diverse. That is, agents are more prone to meet with various neighbours rather than always meeting with the same ones. Hence, the positive sentiments facilitate diversity in 738 M.Steinbacher et al. 1 3 the opinion exchange, as agents do not merely reinforce their own beliefs by meeting the same people all of the time. Consequently, the population as a whole exhibits improved global opinion convergence. It is not surprising that this convergence turns out to be more efficient in simulation treatments with a stronger presence of sentiments, where we denoted less opinion exchange and more opinion changes per meeting. And lastly, results show that a greater presence of sentiments facilitates a substantial reduction in the opinion variance while sentiments might even reduce the impact of extremely in-confident agents1 upon others. In addition to that, the coherence feature, if strict enough, prevents extremism by regulating the impact of hyper in-confident. The paper is organized as follows. Section2 discusses related literature. This section intends to explain some of the main theoretical and conceptual backgrounds of the approach taken in this paper and link them to the relevant literature. The discussion in this part tries to stay informative, but it is by no means exhaustive. In particular, the paper is grounded in social sciences and uses a simulation based approach to implement its main methodological contribution, which is a separation of the opinion dynamics in two phases, preference-based matching between agents and pairwise meetings of agents. Due to an interdisciplinary construction of the paper, a bit deeper exposition of theoretical backgrounds is necessary. Section3 presents a standard opinion model within the preference based matching framework. A general formulation is followed by the presentation of the matching between preferred pairs that is developed as an independent preference based ranking. This section keeps a mathematical disposition of the standard opinion model at the minimum, as the model is well known in the literature and does not need much further introduction. However, some additional space in this section is dedicated to the presentation of the matching mechanism, which is at the core of the approach here, particularly to the classification of preference scores by the radial basis kernel. Section4 starts with the presentation of simulation-based treatments and proceeds with a detailed exposition of the main results. The appearance and the role of extremely (hyper) in-confident agents is discussed first and it is followed by the findings about the role of the model’s parameters for the opinion update. The second part of this section is fully devoted to the presence of sentiments in the preference based matching process and how they can facilitate the opinion exchange. Finally, Sect.5 sheds some light on the main conclusions and reveals some suggestions for future research with motivation for a further expansion of models of learning and social interaction. 2 Related Literature 2.1 Opinion Models: Background Early contributions in opinion dynamics modelling, notably Weidlich (1971); Weidlich and Haag (1983) focused on discrete opinions, i.e. opinions s∈[−1, 1] , based on models from physics, such as the Ising spin model. However, the present 1 As defined later in the sequel, extremely in-confident agents adopt changes in their opinions that are larger than overall distances to opinions they receive from their counterpart agents. 739 1 3 Opinion Dynamics withPreference Matching: How theDesire… paper primarily focuses on continuous opinions along the real line. A seminal contribution by DeGroot (1974) provided analytical solutions for the convergence of opinions, i.e. consensus, in the simplest possible form. This model is mathematically equivalent to a heat diffusion process (Weisbuch etal., 2005), albeit discretized in space through the social network structure. Subsequently, this model was expanded upon by Hegselmann and Krause (2002, 2005) (“HK model") and Deffuant et al. (2000), Deffuant et al. (2002) (“DW model"). Both models have in common that they incorporate confidence bounds, i.e. agents only change opinions upon hearing others’ opinions that are not too different, defined by a suitably chosen parameter. In the HK model, agents adopt opinions of all those that are within the confidence bound (or, in networked versions, all neighbors’ opinions that are sufficiently similar). In the DW model, agents interact pairwise and adopt and intermediate opinion, often the unweighted mean of both, as long as opinions are close enough to each other. These models form the basis of the model presented in this paper, as well as a vast body of research exploring additional mechanisms. The basic versions of these models have been comprehensively reviewed and analyzed by Lorenz (2007, 2010). Another closely related research strand concerns social learning (Acemoglu & Ozdaglar, 2011), which employs similar mathematical structures to investigate the convergence of societies toward a ground truth (e.g. Golub & Jackson, 2010). Rather than assuming pure opinions, in these models it is typically assumed that a ground truth exists and convergence towards that truth through social interaction is studied. Convergence features in these contributions are typically studied mathematically, as opposed to the common simulation-based approach in opinion dynamics models. A more comprehensive review of the evolution of opinion formation modelling, including more recent contributions, is presented by Noorazar (2020), who studies milestones of the discrete and continuous opinion models. The paper is focused on extensions of the state of the art opinion models by biased agents, stubborn agents, manipulative agents, the emergence of power, repulsive agents, and various uses of opinion models with different noise-based features. Peralta etal. (2022) provide a condensed and detailed review of the opinion dynamics models classified into models with discrete and continuous opinions. Particularly, empirical validation with data from elections and polls or experimental data is discussed. 2.2 Structures ofSocial Interaction inOpinion Formation Models Opinion formation models are particularly focused on the importance of interactive structures in society that leads from simple behavioral rules to non-trivial macro-level dynamics. The following subsection reviews some contributions that study individual rules of interaction, as well as meso-scale structures, such as diverse network architectures. These models often rely on simulations, as those allow for a wider range of mechanisms that can be studied. 740 M.Steinbacher et al. 1 3 Examples of these lines of research include the study of the spread of extremism in society (Deffuant etal., 2002), the study of crime and social interaction (Glaeser etal., 1996), the spread of cultural traits (Büchel etal., 2014), the study of mass media and public opinions Hu and Zhu (2017), the study of social mobility (Topa & Zenou, 2015). The behavior of agents in these kinds of models is usually driven by features such as uncertainty(Deffuant etal., 2002), prejudice(Friedkin & Johnsen, 1990), opposition(Galam, 2004), influence(Acemoglu etal., 2010; Watts & Dodds, 2007). An important feature in the literature that relates to our approach are heterogeneous confidence bounds. For instance, Lorenz (2010) studied the interaction of closed-minded (small confidence bounds) and open-minded (large confidence bounds) agents and found that, unlike in models of homogeneous bounds, there is a possibility that the consensus can drift off the center of the initial distribution. Kou etal. (2012) motivate heterogeneous confidence bounds with “complex physiological or psychological factors". Finally, Zhang etal. (2017) model time-varying confidence bounds. In our case, confidence bounds are drawn from a normal distribution, allowing for curious examples of extreme in-confidence. Urena etal. (2019) review literature on how trust, reputation and influence propagate on on communication platforms, such as social networks. In line with the idea of trust, Duggins (2014) implements susceptibility to extreme opinions in computational experiments. A high susceptibility score implies that agents exert less influence on others, i.e. they are not trusted as much. The notion of trust and influence is embedded in our model through the coherence feature, whereby agents’ opinions are considered invalid if they fluctuate too much. Rather than reacting to the amplitude of neighbors’ opinions, agents care whether there is much informational content to others’ stances on the subject matter. Since the seminal contribution of DeGroot (1974), opinion formation models are typically implemented on some type of network structure through which agents communicate. A variety of network structures have been applied for this purpose, such as random graphs, Watts-Strogatz small-world networks (Watts & Strogatz, 1998; Steinbacher & Steinbacher, 2019), and scale-free networks (Barabási & Albert, 1999; Das etal., 2014). As a typical prototype that replicates important characteristics of real-world social networks, specifically short distances between agents and high clustering, the underlying network in our model is also a small-world network. For instance, Pan (2012) uses probabilistic approach to study the role of standard network topologies, such as small world networks, star networks,2 and scale-free networks, for the consensus in the opinion update. Alternatively, agents in the opinion update literature have also been placed in a social setting based on the closeness of their opinions or beliefs without the use of network topologies (Glass & Glass, 2021). Particularly interesting is also the emerging literature on dynamic network structures, as in Wu etal. (2022); Kozma and Barrat (2008). In these models, agents can cut connections to neighbors they disagree with and replace those with new links. While we rely on a static network structure, our idea of choosing whom to meet within a given neighborhood, implemented through a preference-based matching 2 As a name suggests, in a star network every agent is connected to a central agent. 741 1 3 Opinion Dynamics withPreference Matching: How theDesire… algorithm (see the following subsection), is closely related to choosing suitable neighborhoods. A potentially interesting extension on the use of network structures in opinion formation models are in multilayer networks. These allow the implementation of different types of connections (family, friends, colleagues), different topics etc. Diakonova etal. (2016) shows that the resulting dynamics from interaction on several layers cannot be reduced to a process on a single layer. This result has been confirmed analytically by Zhang etal. (2017). Battiston etal. (2017) applies a multilayer structure to Axelrod’s model of the cultural dissemination and shows that genuinely novel behavior emerges, which allows for the existence of multiculturality despite the combined pressures of globalization and imitation. A common shortcoming in these models is a lack of empirical validation. A step in this direction has been made by Grimm and Mengel (2020), conducting laboratory experiments to study belief formation and finding that results tend to be inconsistent with the naive learning process in the standard DeGroot (1974) model. 2.3 Preference‑Based Matching Agents in this model do not meet their neighbors at random, as they usually would in most opinion formation models, but through a separate matching process. At the core of this contribution is the idea that agents have preferences over whom to meet and act upon those. The particular matching algorithm that we implement is the roommate matching algorithm by Irving (1985). In the present paper, agents need to mutually agree to the meeting, which resembles the roommate matching problem. The approach has been particularly popularized by Roth and Sotomayor (1992) in the Handbook of Game Theory with Economic Applications. Since then, it has received many implementations in different domains of behavioral game theory (e.g. for more information on behavioral game theory, see the seminal work by Camerer, 2011) and social systems with heterogeneous and interacting agents, such as the ride-sharing matching (Wang etal., 2018), or the mentor-mentee matching on colleges (Haas etal., 2018), for instance. Closely related to the use of the matching of mutually preferred pairs of agents in the present paper, is the stable college admission and marriage problems, initiated long time ago in the works of Gale and Shapley (1962) and McVitie and Wilson (1971). Other related applications is the efficient matching of buyers and sellers in a bipartite network in Kranton and Minehart (2001), as well as job search and match problems (McCall, 1970). The argument to include a preference-based matching process is essentially psychological. In particular, confirmation bias (Wason, 1968; Ross & Anderson, 1982) motivates this behavior of agents to not only be more likely to trust others with similar opinions, implemented by confidence bounds, but also actively seek out information that confirms their beliefs. Confirmation bias has been applied by authors such as Del Vicario etal. (2017), where agents rewire their connections in order to minimize disagreement. It is also the foundation for the emergence of echo chambers due 742 M.Steinbacher et al. 1 3 to the segregation into like-minded groups, as studied by Levy and Razin (2019), as well as Brugnoli etal. (2019), who find empirical evidence for it in Facebook data. On the flip side, one can interpret this mechanism as an avoidance of cognitive dissonance, that could arise through the exposure to opinions that contradict one’s own, as in Li etal. (2020). In the present paper, confirmation bias is implemented through the preference score, which partially consists of the similarity in opinions. As for empirical studies of matching, Arteaga etal. (2022) analyze how online matching platforms shape applicants’ beliefs about schools. The confirmation bias has been assessed in laboratory experiments by (Zou & Xu, 2023), who show that not only similarity in opinions, but also similarity in other aspects matters. In our model, agents do not have other characteristics shaping their identities, but the idea can be subsumed under the notion of sentiments, which form part of the preference score. 2.4 Sentiments inSocial Interaction Our paper tries to enrich the social simulation model of opinion dynamics by adding a notion of a mutual desire to meet in a standard setting of opinion dynamics. In particular, it relates to the concept of mutual self-interest in which cooperative outcomes emerge endogenously. Smith and Wilson (2019) pawed a way for a reconsideration of social interaction between people in economic science. According to their setting, people remain self-interested players, but are bound within social relationships, where they exert mutual feelings of approbation or disapprobation towards each other. Feelings of emotions have been highlighted within agent-based models of social simulation. For instance, Lejmi-Riahi etal. (2019) study emotional experience at workspace, whereby emotional state is assumed to be an important cognitive factor during work activities, such as decision-making, attention, memory, perception and learning. However, the study of emotions within the agent-based models of opinion dynamics is still in its early stage. Authors such as Bagnoli etal. (2007) have coupled the opinion update rule with the the affinity towards other agents. Similarly, Schweitzer etal. (2019) include emotions as drivers for opinion polarization. These models can be seen complimentary to the approach taken in this paper, as we completely decouple sentiments from the opinion update rule. 3 Standard Opinion Model withPreference‑Based Matching 3.1 Standard Opinion Model: AGeneral Formulation In the standard opinion model(Deffuant etal., 2000; Hegselmann & Krause, 2002), agents update opinions at each time by incorporating some fractions of beliefs of their neighbors into their own beliefs. Let agent i have opinion xi(t) lying in some 749 1 3 Opinion Dynamics withPreference Matching: How theDesire… the opinion exchange, the convergent features of the standard opinion update model are at play, driving opinions closer together and reducing variance in opinion thereby. Moreover, given the results of both versions with and without sentiments in the matching process, presented in Fig.2, the minimal variance in the final opinion distribution appears at around 𝜇(𝜆)=0.5 , which represents the well studied point of the strongest convergence in opinions. However, note that the turning point in the opinion variance is not necessarily exactly at 𝜇(𝜆)=0.5 , as the behavioral features of agents in the model are parameterised from symmetric distributions around given expected values. Moreover, in the presence of coherence (i.e. in the second variant), minimal variance appears to happen at slightly lower values of 𝜇(𝜆) . This may be explained by the fact that 𝜆 determines the size of opinion adjustments and the demand for coherence in the opinion of counterpart agents punishes too large movements in their opinions, i.e. this might reduce volatility. In other words, a particular independent set of parameters might be obtained, such that parameters are more favorable to lower variance than parameters in some other set of parameters from these same distributions, thus opinions of agents are not necessarily the least dispersed when 𝜇(𝜆=0.5) . In addition, treatments with sentiments in the matching process (panel on the right in Fig.2) show substantially lower variance than treatments without sentiments in the matching process. This finding corroborates our expected account of the facilitating role of the sentiments for the opinion exchange, a conclusion, to which we will gradually arrive in the sequel. In addition to overall lower variances when sentiments are present, we can observe that the minimum is achieved at a larger range of values of 𝜇(𝜆) . Note an increase in variance at values of 𝜇(𝜆) ≥ 0.9 . Such an increase is likely due to the appearance of some hyper in-confident agents i with 𝜆i>1 . Their appearance is expected at sufficiently high values of 𝜇(𝜆) ; see Table 1 for a general description of parameters that were used in the simulation treatments and Table2 for their values. Hyper in-confident agents adopt changes in their own opinions which are larger in value than the absolute difference between their own opinions and opinions of the agents they were matched with. The presence of hyper in-confident agents in simulation treatments can lead to the occurrence of extreme groups, particularly if these agents get contacted by agents at the tail of the opinion distribution and get dragged away from the center. This increase in variance is facilitated by tolerance of agents, i.e. higher values of 𝜇(𝜃) . While it enables convergence in the presence of “normal" levels of confidence, tolerance also allows for extreme groups on the fringes when hyper in-confident agents are present. As we can see from the bottom panel of Fig.2, referring to the treatments of the second version of the model, which includes coherence in matching and opinion updates, the impact of fringe groups seems to disappear. We do not observe the phase transition in the variance of opinions at a combination of high values of 𝜇(𝜆) and 𝜇(𝜃) , but rather a continuous, well-shaped behavior of the opinion variance throughout the entire range of parameters. Variance is persistently lower the higher the tolerance parameter 𝜃 is. The absence of coherence scoring supports the impact of hyper-in-confident agents in this case. We will elaborate on this finding in a bit more detail in the Sect.4.3. Figure3 shows one such treatment, where extreme opinions are formed: a small group of agents appears in the left corner at opinion values of -4 in the plot on the 750 M.Steinbacher et al. 1 3 right hand side of the Figure. This is the treatment with the highest sentiment bound 𝛼=1 supported by a sufficiently high level of average tolerance 𝜇(𝜃)=0.9 among agents. Namely, agents with higher tolerance are more likely to be affected in the sense that they easily readjust their own opinions. However, as we will see in the latter part of this section, after first studying the role of model parameters for the opinion exchange (Sect. 4.3), the stronger the presence of sentiments in simulation treatments the more agents tolerate diversity in opinion (Sect.4.4). Agents with these preferences show large overall number of opinion changes and promote the convergent forces of the standard opinion model. Despite the presence of sentiments, an extreme group appeared in treatments without coherence in matching and opinion adjustment (Fig.3). This was sufficient to substantially increase the variance of opinions, but the group remained isolated and the convergent features of the model prevailed in leading the society towards a consensus. Moreover, extreme groups only appear in few treatments, and the dispersion of the variance measure among treatments with high values of 𝜇(𝜆) was high. 0.20.40.60.8 λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Sentiment OFF no coherence features Varianceofopinions in t=1000 θ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.20.40.60.8 λ 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Sentiment ON no coherence features Variance of opinions in t=1000 θ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.20.40.60.8 λ 0.16 0.18 0.20 0.22 0.24 Sentiment OFF with coherence features Varianceofopinions in t=1000 averaged over γ,κ θ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.20.40.60.8 λ 0.05 0.10 0.15 0.20 0.25 SentimentON with coherencefeatures Variance of opinions in t=1000 averaged over γ,κ θ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Fig. 2 Variance in opinion as a function of confidence; Left column: simulation treatments without sentiments (Sentiment OFF, 𝛼=0 ); Right column: simulation treatments with sentiments (Sentiment ON, 𝛼=1 ); Top row: simulations without coherence in either matching or opinion formation (averaged over independent simulation runs); Bottom row: simulations with coherence (averaged over values of 𝛾 and 𝜅 ). Note how punishing volatility in opinions avoids the extreme increase in variance for large values of 𝜇(𝜆) 751 1 3 Opinion Dynamics withPreference Matching: How theDesire… In addition, we should also stress that in treatments with coherence in the matching process and opinion validation in the opinion exchange, we might expect some dependence of the opinion dynamics upon the positioning of agents in their neighborhoods. Some agents might get stuck in their social environment, such that their neighbors label them as holding an incoherent opinion early enough in the treatment. Such agents might never get involved in the opinion exchange with others. Anyway, we have not studied the role of social environment and network effects in this paper. 4.3 The Role oftheModel’s Parameters fortheOpinion Exchange In what follows in this part are treatments to study the role of the main parameters in both versions of the standard opinion model with matching and coherence discussed earlier [i.e. see Eqs.(1) and (3]. In this section, we disentangle the effect of the model’s parameters for the presence of convergent forces in the opinion exchange, as captured by the opinion variance. Results are shown in Fig.4. In general, it can be assessed, that as the coherence requirements become stricter (i.e. at lower acceptance boundaries of perceived volatility of others’ opinion 𝜇(𝛾) ), more and more opinions are considered invalid, which limits the convergence of opinions (top left panel). Convergence is driven by opinion changes on the micro-level, which are not tolerated in these set-ups, and hence opinions remain relatively widely dispersed. − 4 − 3 − 2 − 1012 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Density λ=0.9 θ=0.9 SentimentON no coherencefeatures Distribution evolution initialdistribution after 100 steps after 1000 steps Fig. 3 Distribution of opinions: the appearance of extremes (2 agents in this case). Simulation without coherence features in matching or opinion adjustment. It seems that convergence has been reached early on ( t<100 ), as is common in models without matching (e.g. Steinbacher & Steinbacher, 2019). While we have not studied the question in detail, it indicates that asymptotic convergence is achieved 752 M.Steinbacher et al. 1 3 However, we detect a phase transition as the adjustment parameter 𝜇(𝜆) gets large: at low values of 𝜇(𝛾) , i.e. around 0.3, the role of 𝛾 of reducing the opinion variance, which we observed at lower values of 𝜆 , reverses. We now observe a sharp increase in the dispersion of agents’ opinions as 𝜇(𝛾) increases, i.e. as agents care less about the validity of opinions they encounter. The effect can also be observed in the top-right panel in Fig.4: while a regular cyclical pattern occurs throughout the plots at 𝜇(𝛾)<=0.8 , this pattern breaks down at 𝜇(𝛾)=0.9 . At large values of 𝜇(𝜆) , i.e. in the presence of hyper in-confident agents, the variance of opinions sharply increases. The constraint on the size of opinion adjustments is not binding when 𝛾 is large. Hence, at large values of 𝜆 , which is the size of opinion adjustments relative to opinion differences, hyper in-confident agents are not effectively regulated by their social environment and they are able to build groups of extreme opinions away from the general consensus. The bottom panel of plots in Fig.4 reveals the effects of 𝜅 and 𝛾 on the opinion variance and the opinion exchange. We can denote symmetric work of 𝜅 on the variance, which is expected as the parameter measures geometric weights of distant versus recent differences in opinion in the coherence score. The less important are both differences for agents, the lower the variance in opinion, and vice versa, i.e. the more important is either of these differences to agents, the larger the variance in opinion. At 𝜅=0.5 , agents are indifferent between distant and recent changes in opinion. When agents put more emphasis on longer-term persistence, i.e. when 𝜇(𝜅) is large, 𝛾 binds the accumulated opinion adjustments towards the center, rather than individual opinion changes. Hence, agents that start in the tails of the distribution are limited in their ability to converge towards the center. On the other hand, when 𝜇(𝜅) is small, i.e. agents emphasize short-term persistence, any substantial change in opinion adjustment invalidates the opinion of the agent and prevents the matches from adjusting theirs. At intermediate values of 𝜅≈0.5 , agents allow for individual changes, as well as accumulations thereof. Hence, fewer agents’ opinions are invalidated, which supports gradual convergence of opinions, as observed through lower variance. Having said so, the bottom-right plot is more interesting. Remember, the variance is approximately the same at lower and at higher end of the 𝜅 (i.e. in treatments with and without sentiments). The more agents care about long-term changes in opinions, i.e. at larger values of 𝜇(𝜅) , the more relevant the value of 𝛾 becomes. At low values of 𝜅 , agents care mostly about recent changes in opinion, which do not accumulate over time as agents converge towards the centre of the opinion distribution. At larger values however, the accumulated differences begin to matter and create binding boundaries on the adjustment of opinions, reducing the number of opinion changes over the duration of the simulation. Ultimately this boils down to the fact that the existence of memory on the part of agents is a precondition for the concept of coherence. However, we observe that it is not only the existence of memory as such that matters, but also how agents remember, as defined in our case by the Eq.2, through the dependence on the value of 𝜅 . Coherence becomes more binding, and relevant to the willingness of agents 753 1 3 Opinion Dynamics withPreference Matching: How theDesire… to reconsider their opinions, as long-term memory (larger 𝜅 ) gains importance and therefore opinion changes accumulate. And lastly, a striking result that actually comes “for-free" alongside this study, is the variance reducing feature of the coherence parameter 𝛾 . Recall for a moment a sharp increase in the dispersion of opinion in treatments without coherence in the top panel of Fig.2. Namely, what happens here is as follows: once agents are allowed to observe changes in the opinion of others and ignore agents with high propensity to change their opinions, they effectively reduce the dispersion in the opinion across the whole society. This effect can be seen by comparing the upper panel to the lower panel in Fig.2. Agents in the second setting of the opinion model with coherence can neutralize the impact of hyper in-confident agents in the society, effectively preventing extremism. 4.4 Introducing Sentiments: How dothey Facilitate theOpinion Exchange? Finally, let us now turn our attention to the presence of sentiments and study their role as facilitators of the opinion exchange. Note that sentiments in our case are 0.10.20.30.40.50.60.70.80.9 γ 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 SentimentON Varianceofopinions in t=1000 averaged over θ,κ λ 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0.15 0.20 0.25 γ=0.1γ=0.2γ=0.3 0.15 0.20 0.25 γ=0.4γ=0.5γ=0.6 0.25 0.50 0.75 λ 0.15 0.20 0.25 γ=0.7 0.25 0.50 0.75 λ γ=0.8 0.25 0.50 0.75 λ γ=0.9 Variance of opinions in t=1000 Sentiment ON,averagedover θ,κ 0.10.20.30.40.50.60.70.80.9 κ 0.214 0.216 0.218 0.220 0.222 0.224 SentimentOFF Varianceofopinions in t=1000 averaged over λ,θ,γ 0.10.20.30.40.50.60.70. 80 .9 γ 20000 40000 60000 80000 100000 120000 140000 160000 180000 SentimentOFF Number of opinion changes averaged over θ,λ=0.5 κ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Fig. 4 Top panel: variance of opinions as functions of 𝜆 and 𝛾 . Bottom panel (left): opinion variance as a function of 𝜅 ; Bottom panel (right): the total number of opinion changes as a function of 𝛾 and 𝜅 . Sentiment OFF (ON) refers to sentiment bounds 𝛼=0 (1) 754 M.Steinbacher et al. 1 3 independently determined for each agent as a measure of approbation towards other agents. Sentiments enter each agent’s preference score (i.e. see the Eq.4) as a part of the matching that takes place before the meetings between agents are implemented. In what follows, we present the main results of a rich set of treatments obtained at various levels of the upper bound 𝛼 on sentiments. In particular, let us first focus on the number of meetings in treatments with and without coherence features. According to the results in the top panel of Fig.5, the presence of sentiments in the matching process reduces the number of meetings in both treatments with, and without coherence. This happens, because the presence of sentiments broadens the pool of potential preferred matches that can be made. Given that sentiments are independent and non-symmetric between pairs of agents, this complicates the matching between preferred pairs. As a result, they are expected to be more willing to listen to agents that hold more diverse opinions than in the case where the matching of agents depends only upon the opinion-based criteria, such as the absolute opinion difference or coherence. However, this is only one part of the story. Let us now see, i.e. for the same setting, how sentiments interplay with the number of opinion changes. According to the bottom panel of Fig.5, in the presence of sentiments, agents show larger willingness to adopt changes in their opinion than in treatments without sentiments, even though sentiments play no role in the decisions to adopt opinion changes. In addition, note the trajectory of a zero sentiment curve and compare it to the trajectories of nonzero sentiment curves. What we can observe here, is a comparison of two different behaviors. In the zero sentiment case, the number of opinion changes follows the level of tolerance 𝜇(𝜃) in a predictable and well-known way, i.e. the larger the level of tolerance in society, the more agents are willing to change their own opinions. On the other hand, according to our simulation treatments, even a weak presence of sentiments is sufficient to mix preference lists of agents so much that it increases the diversity of mutual pairings between agents, which can dramatically increase the number of opinion changes, particularly at the lowest to medium levels of tolerance 𝜇(𝜃) . This finding permeates through the results of our investigation into sentiment effects: upon adding sentiments into the matching process, we observe a change in the qualitative behavior of the model. This qualitative change appears in Fig.5 as a discontinuous drop in the number of meetings and a difference in the relationship between 𝜃 and the number of opinion changes between sentiment bounds 0 and 0.05. Further increases of the sentiment bound 𝛼 however only create quantitative changes, as we will discuss in detail below. In addition, we can also detect the effect of 𝜃 and its interplay with sentiments. At 𝜇(𝜃)>0.7 (without coherence) and 𝜇(𝜃)>0.8 (model with coherence), the effect of sentiments on bringing diverse people together dissipates as agents at high enough levels of tolerance might start getting more alike faster. The tendency to mix them with diverse agents in such setting can be expected to have negligible effect for an increase in the number of opinion changes. At some point, agents are so much alike that sentiments just play no role anymore. Reinforcing beliefs by meeting similar minded agents in the no-sentiment case starts dominating in terms of the overall 755 1 3 Opinion Dynamics withPreference Matching: How theDesire… number of opinion changes over the tendency of agents to see the diversity in opinion (i.e. at higher sentiment bounds 𝛼 ). Now, let us combine our views of meeting numbers with numbers in opinion changes at Fig.5 and think for a while of the meeting efficiency shown in Fig.6. Again we observe the above-mentioned qualitative difference between the zerosentiment and non-zero sentiment treatments. By adding any level of sentiments into the matching process, the ratio of opinion changes to total meetings increases sharply compared to zero sentiments. Upon increasing sentiments, this ratio continues to rise gradually until 𝛼≈0.3 , after which it decreases. At high values of 𝛼 , agents that are too different in opinions are matched and therefore cannot exchange opinions as frequently as they do at medium levels of 𝛼 . Moreover, the meeting efficiency is lower in the presence of coherence features at all levels of sentiment bound 𝛼 , due to the added constraint on opinion updates. Figure7 shows the relative increase of meeting efficiency between sentiment bounds 0.05 and 0.95 (left panel), and 0 and 1 (right panel) over the tolerance parameter 𝜃 , respectively. Note that comparing of two non-zero levels of 𝛼 yields a qualitatively different result than comparing zero sentiments with a non-zero level. The qualitative 0.00.20.40.60.81.0 sentiment bound α 120000 122500 125000 127500 130000 132500 135000 137500 140000 λ=0.5 no coherence features Number of meetings averaged over θ 0.00.20.40.60. 81 .0 sentimentbound α 122500 125000 127500 130000 132500 135000 137500 λ=0.5 with coherencefeatures Number of meetings averaged over θ,γ,κ 0.10.20.30.40.50.60.70.80.9 θ 50000 100000 150000 200000 250000 λ=0.5 no coherence features Number of opinionchanges α 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1.0 0.10.20.30.40.50.60.70. 80 .9 θ 25000 50000 75000 100000 125000 150000 175000 200000 225000 λ=0.5 with coherencefeatures Number of opinionchanges averaged over γ,κ α 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1.0 Fig. 5 Effects of sentiments on numbers of meetings and opinion changes. Top: Number of meetings; bottom: number of opinion changes; left: model without coherence features in matching and opinion adjustment; right: model with coherence features 756 M.Steinbacher et al. 1 3 difference lies in the difference of comparing levels of sentiments, given they are present, and analyzing the presence of sentiments. However, in both cases the effects of sentiments dissipate as 𝜃 increases. On the one hand, in comparing the treatments when 𝛼=0 with treatments when 𝛼=1 , we can observe a monotonous relationship. The presence of sentiments matters less the more tolerant agents are. Comparing non-zero levels of the sentiment bound 𝛼 (left panel) shows a different interaction between tolerance 𝜃 and sentiments. At very low levels of tolerance 𝜃=0.1 , the increasing sentiments leads to a reduction in meeting efficiency. The agents are too intolerant to accept the added diversity of opinions they are exposed to in their meetings. However, as tolerance increases, we detect an increase in meeting efficiency as a result of increasing the sentiment bound. Hence, low levels of tolerance can be mitigated by sufficient levels of sentiment and a willingness to adopt new opinions upon meeting more diverse agents. The effect approaches 0 as 𝜃≥0.5 . At this level of tolerance, higher levels of sentiment do not add to the meeting efficiency. This stands in contrast to the right plot, which shows the increase in meeting efficiency upon the presence of sentiments: for all levels of 𝜇(𝜃) , the efficiency of meetings is substantially higher. During our discussions so far, we have pointed to the role of sentiments to increase the diversity of meetings. Namely, with the addition of a random variable in the matching process, agents are more willing to meet agents that are not necessarily the most similar in the opinions they hold. On the other hand, without sentiments, they would prefer to meet the most similar agents only, and the preferences would be mutual between pairs of agents. The exchange of opinions would reinforce those preferences as they become more similar in opinion. Upon adding sentiments to the matching process, we can identify an increased diversity of meetings between agents. Figure 8 presents a comparison of the number of unique pairs of agents that have met, i.e. the density of the network of effective interactions, with the total number of edges in the network. The largest increase happens immediately upon adding sentiments to the model, but it increases 0.80 0.81 0.82 0.83 0.84 0.85 0.00.20.40.60.81.0 sentimentbound α 0.62 0.63 0.64 λ=0.5 no coherencefeatures No Sentiment Efficiency of meetings averaged over θ 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.00.20.40.60. 81 .0 sentiment bound α 0.50 0.51 λ=0.5 with coherencefeatures No Sentiment Efficiency of meetings averaged over θ,γ,κ Fig. 6 Meeting efficiency, defined as 1 2 opinion changes meetings , for models with and without coherence features in matching and opinion adjustment 757 1 3 Opinion Dynamics withPreference Matching: How theDesire… further as the sentiment bound 𝛼 increases. Sentiments ensure a well-integrated and connected society, whereas in their absence, the matching process stabilizes and always connects the same pairs. This effect is observed in both versions of the model, with and without coherence features. Last, but not least, after we have identified the facilitating role of the sentiments for the efficiency in the opinion exchange as they allow more diversity in mutual pairings of agents before their meetings. Let us take a look at the dispersion in the opinion among agents at the presence of sentiments in the matching process. The evolution of the variance of opinions is shown in Fig.9. The trajectory of the variance is strikingly clear: a change in variance as a function of varying sentiment bounds 𝛼 follows a continuous and monotonous relationship, except for the jump from zero sentiment case to the first non-zero sentiment case at 𝛼=0.05 . The variance curve shows that the convergent forces increase when agents, due to the impact of positive sentiments, prefer to meet diverse agents. Furthermore, on a side note, the reduction in variance is also driven by 𝜃 , as shown in Fig.10. Without a sufficient level of tolerance among agents, positive sentiments would only contribute to opinion exchange, but not also to the reduction of dispersion in the opinion within the society. Hence, if agents are too intolerant too change their opinions, in almost every meeting, a partial randomization of the matching process (i.e. due to the presence of sentiments), is not able to improve consensus-finding. In particular, at 𝜃∈[0.1, 0.2] , sentiments fail to improve opinion convergence. Only at sufficiently high tolerance levels (i.e. at 𝜃≥0.3 ), the presence of sentiments reduces the dispersion in opinion, with the highest effect being denoted at medium levels of tolerance. In line with these results, the improvement due to the presence of sentiments is less pronounced at high values of 𝜃 , as agents are highly tolerant and opinions might also converge in the absence of sentiments, given some other conditions are met (i.e. such as the absence of hyper in-confident agents). 0.10.20.30.40.50.60.70.80.9 θ −12 −10 −8 −6 −4 −2 0 2 percent increase λ=0.5 no coherence features Relative increase of meetingefficiency between sentimentboundsof0.05and 0.95 0.10.20.30.40.50.60.70. 80 .9 θ 0 20 40 60 80 100 percent increase λ=0.5 no coherencefeatures Relative increase of meetingefficiency between sentimentbounds of 0and 1 Fig. 7 The relative increase between very high and very low levels of sentiment, as regulated by 𝜃 . Left: difference between sentiment bounds 0.05 and 0.95; right: difference 0 and 1 758 M.Steinbacher et al. 1 3 5 Discussion andPerspectives We have disentangled the opinion model and split it into two parts: the preferencebased matching of agents and the opinion exchange. This enabled us to see behind the process of choosing counterpart agents before the opinion exchange takes place. Some of the most standard convergent features from the opinion dynamics literature were also underlined in this paper. For a more detailed discussion about mathematical foundations of convergence in the standard opinion model, see Acemoglu and Ozdaglar (2011) and Acemoglu etal. (2013). In addition, the dynamics in convergence might depend also on the initial distribution of opinion as well as on the inter-connectivity of agents, particularly to their ability to reach out to other agents in the model. It would be interesting to see how the network structure of agents’ interconnections inter-plays with the opinion trajectory. By increasing the sentiment noise in the matching process, we were able to generate a smooth reduction in the opinion variance and an increase in the meeting efficiency. Particularly striking is the reduction in the variance, exhibiting a smooth pattern of a geometric decay. In particular, the presence of sentiments dramatically increases the meeting efficiency, measured as realized opinion changes in an overall number of meetings that took place, while the level of sentiments does not matter as much. In addition to showing the positive effects of sentiments on the opinion convergence, we have shown that the strong presence of positive sentiments facilitates meetings between diverse agents. As the interaction network becomes denser, opinions are propagated through the network more efficiently, strengthening the converging forces. Along these lines, our simulation treatments support the notion that society might reach consensus also at lower tolerance levels, as long as agents are willing to meet due to, for instance, feeling strong positive sentiments towards each other. The finding 0.00.20.40.60.8 sentimentbound α 0.5 0.6 0.7 0.8 0.9λ=0.5 no coherence features Relative densityofthe interactionnetwork 0.00.20.40.60.8 sentimentbound α 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 λ=0.5 with coherencefeatures Relative densityofthe interactionnetwork Fig. 8 The relative density of the interaction networks. Unweighted edges are drawn between agents that have been matched at any point in a simulation run. The total number of links (density of the network) is then compared to to the density of the original network that defines neighborhoods of the agents. Left: model without coherence features in matching and opinion updates; right: model with coherence features 765 1 3 Opinion Dynamics withPreference Matching: How theDesire… Acknowledgements Authors would like to thank anonymous referees during the peer-review of the paper, to seminar participants at the Brown Bag Seminar, Kiel University, Germany, May 19, 2022, and to the discussants at the 25th Annual Workshop on Economic Science with Heterogeneous Interacting Agents (WEHIA 2022), Catania, Italy, June 22–24, 2022, for helpful comments and suggestions. All errors remain the responsibility of authors. Funding Open Access funding enabled and organized by Projekt DEAL. The authors have not disclosed any funding. Declarations Conflict of interest The authors declare no conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. References Acemoglu, D., Como, G., Fagnani, F., & Ozdaglar, A. (2013). Opinion fluctuations and disagreement in social networks. 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