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Edited by: Shira Elqayam, De Montfort University, UK

Reviewed by: David E. Over, Durham University, UK; Rainer Hegselmann, Bayreuther Forschungszentrums Modellierung und Simulation sozioökonomischer Phänomene, Germany

*Correspondence: Igor Douven, Faculty of Philosophy, University of Groningen, Oude Boteringestraat 52, 9712 GL Groningen, Netherlands e-mail:

This article was submitted to Cognitive Science, a section of the journal Frontiers in Psychology.

† Address from October 2014: Institute of Philosophy, University of Leuven, Leuven, Belgium

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Both in philosophy and in psychology, human rationality has traditionally been studied from an “individualistic” perspective. Recently, social epistemologists have drawn attention to the fact that epistemic interactions among agents also give rise to important questions concerning rationality. In previous work, we have used a formal model to assess the risk that a particular type of social-epistemic interactions lead agents with initially consistent belief states into inconsistent belief states. Here, we continue this work by investigating the dynamics to which these interactions may give rise in the population as a whole.

This paper aims to show the importance of a social perspective in the study of human rationality. While, as will be seen, the work we present relies on computer simulations, we believe it may inspire further empirical research by social scientists. Computer simulations, such as those to be presented, form a bridge between normative models and descriptive results. The simulations depend on a theoretical model with various parameters. Some combinations of the parameters may be optimal for the attainment of one or more norms, whereas other combinations of parameters may give a good approximation to an epistemic group of real people. If the model parameters can be linked to variables in the real world, this may enable us to give practical advice for increasing rationality in social settings.

In previous work, we studied a formal model of a type of epistemic interactions in which agents whose belief states are in some sense close together compromise by settling on a kind of “averaging” belief state. We showed that compromising in this way carries the risk of leading agents with initially consistent belief states to become inconsistent. Although it was shown in the same paper how this risk could be minimized, it might nonetheless be considered as a reason for banishing the designated kind of interactions. Here, we continue the previous work by investigating the dynamics of a population as a whole to which epistemic compromising may give rise. We pay special attention to the conditions under which such compromising may lead to a consensus among the members of a population. This is intended to shed new light on the question of whether it is at all rational to interact epistemically in the said kind of way.

In their study of human rationality, philosophers as well as psychologists of reasoning have tended to focus on individual thinkers in isolation from their social environment. Which beliefs an individual ought to hold and how an individual ought to change his beliefs have traditionally been regarded as questions that are independent of which beliefs other individuals hold or how other individuals change their beliefs. This is at least somewhat surprising, given that we are so obviously members of a community of individuals who pursue by and large the same epistemic goals, who frequently engage in common activities to gather new evidence, who constantly exchange information, who often (have to) rely on the words of others, who regularly seek each other's advice in epistemic matters, and who sometimes put great effort into trying to influence one another's opinions. In fact, we see these kinds of behavior not just in everyday life, but also, and even especially, in the practice of science, which many regard as producing the—in some sense—best and most valuable knowledge. Doubtlessly, there are more and less rational ways of engaging in these various activities, and it would seem part of the business of philosophy, as well as of that of psychology, to sort out which are which.

At least in philosophy, there is a growing awareness that the general neglect of the group level in studying human rationality has created a serious gap in our understanding indeed, and philosophers have begun to correct this lacuna^{1}

The popularity of social epistemology being on the rise, it is easily missed that we still do not know whether, on balance, the possible benefits of social-epistemic interactions outweigh their possible costs. Indeed, various philosophers and also sociologists have enthusiastically reported about the “wisdom of the crowds,” in the context of which it has been asserted that the aggregated opinions of a group of laypeople is often closer to the truth than the opinions of individual experts (Surowiecki, ^{2}

In trying to give cost–benefit analyses of diverse types of social epistemic interactions, and also for related purposes, a number of social epistemologists have recently started using computer simulations for studying communities of epistemically interacting artificial agents, where the agents typically adapt their beliefs (fully or partially) on the basis of information about the beliefs of other agents in the community. It has been argued that, insofar as these methods capture central aspects of the epistemic interactions between

By far the most research on rationality is concerned either with developing an experimentally informed descriptive model of actual human thinking or with developing a theoretically-oriented normative model of idealized human thinking. We present a study that nominally falls in the first category, in that we study opinion dynamics with the help of computer experiments concerning epistemically interacting agents. But it would be more accurate to say that our study falls somewhere on the continuum between descriptive and normative work. The agents that we model are inspired by particular aspects of human thinking (such as the observation that humans have opinions on multiple topics, some of which are logically independent, and some of which are logically connected) and human epistemic interaction (most notably, that in practice we allow others' beliefs to influence our own as well as try to make our beliefs influence those of others), but—as will emerge—they clearly lack other characteristics of human thinking. So, it is not a purely descriptive study. The agents in the simulations do follow a prescribed way of revising their opinions and never fail to adhere to it, but they are still non-ideal thinkers; for example, they may come to believe an inconsistency without realizing this. Hence, it is not a purely normative model either.

As mentioned in the introduction, we here continue work that we have started elsewhere (Douven, ^{3}

The most widely known formal model for studying the effects of epistemic interactions on the belief states of individual agents is probably the model developed in Hegselmann and Krause (

The basic version of the HK model assumes communities of agents that are trying to determine the value τ of some unspecified parameter by repeatedly and simultaneously averaging over the opinions of those agents that are within their so-called Bounded Confidence Interval (BCI)^{4}_{i}_{i}_{i}

It is a limitation of the HK model that it considers only agents whose belief states consist, at any given point in time, of just one value. In Riegler and Douven (

Given _{M} = 2^{M} possible worlds that we can distinguish between. In turn, this means that there are _{M} = 2^{wM}^{5}

In this model, agents revise their theory of the world by taking into account the theories held by certain other agents in the community, comparable to how the agents in the HK model update. However, now the BCI is defined in a slightly more complicated way. To quantify the distance between two theories, the so-called Hamming distance δ between the corresponding bit strings is used: this distance is given by the number of locations in which these strings differ. The BCI is then defined by placing a threshold value

^{2} possible worlds: the world in which

The update rule for theories in this model—so, basically the analog of (HK)—is a bitwise operation in two steps: (1) averaging and (2) rounding. In step (1), for each bit of the theory, a straight average is taken of the corresponding bit of those agents that are within the agent's BCI (note that this includes the agent himself). In general, the result is a value in the interval [0, 1] rather than just a 0 or 1. Hence the need for step (2): in case the average is greater than 1/2, the corresponding bit is updated to 1; in case the average is less than 1/2, the corresponding bit is updated to 0; and in case the average is exactly equal to 1/2, the corresponding bit keeps its initial value.

More formally, the _{i}

1. 1100 | 4. 1000 | 7. 0000 |

2. 1101 | 5. 1101 | 8. 1101 |

3. 0001 | 6. 0001 | 9. 0001 |

Assume that _{1}(0) = {1, 2, 4, 5, 8}. Agent 1 will update his theory to _{1}(1) = 1101, given that all agents in _{1}(0) deem the first world possible, and hence _{1}(1)[1] = 1; all but one of the peers deem the second world possible, so _{1}(1)[2] = 1; all peers deem the third world impossible, so _{1}(1)[3] = 0; and although _{1} initially deems the fourth world _{1}(0) deem that world _{1}(1)[4] = 1. ◊

In Wenmackers et al. (^{M}) were also shown to decrease the chance of updating to the inconsistent theory.

The mere possibility of arriving at an inconsistent theory—even though it has a low probability—might be thought to discredit EHK. But this would be to overlook that the update rule can have compensating advantages. The extension of the HK model that was studied in Riegler and Douven (_{7}(0) = {3, 4, 6, 7, 9} and that averaging-and-rounding over the corresponding belief states results in a consistent belief state, to wit, _{7}(1) = 0001.) However, to give a more systematic answer to the question of which advantages updating via (EHK) may have, more must be known about the properties of this update rule.

To take further steps toward determining which properties (EHK) has, beyond the ones presented in Wenmackers et al. (

Our investigations in the following focus on the case in which there is only one binary property that the agents consider to form their theory about the world (i.e.,

We model a group of _{M}^{M} possible worlds (or combinations of the properties being true or false in the world). Each agent holds a theory about the world, which can be represented as a string of _{M}^{tM} such theories. Agents consider as epistemic peers those agents who currently hold a “sufficiently similar” theory, which means that the number of bits that are different between the agent's own theory and that of a potential peer is less than a certain threshold, called the bound of confidence

Our goal is to investigate how the opinions in the population as a whole change over time due to the iterated application of (EHK) by the individual agents. To achieve this, we first need to identify the relevant belief space, by which we mean the phase space in which we can represent the opinion dynamics of the entire group of agents. An _{M} components, which sum to _{00}, _{01}, _{10}, _{11}〉, which can be represented in a three-dimensional tetrahedron. For a representation of the tetrahedral OPS with two (

To elaborate, if there are two agents (

If there are three agents (

For any fixed number of _{M} − 1 dimensions, or the multiset coefficient of choosing _{M} options). If there are four or more agents, then the points in the OPS also occupy the interior volume of the tetrahedron. (For four agents, this concerns only the central point 〈1, 1, 1, 1〉).

In principle, the OPS for any particular

We view the OPS equipped with the two-step social update rule (EHK) (with

The lower the threshold _{M}

We will represent the dynamics on the OPS by arrows that point from an initial opinion profile toward the corresponding final state. For the sake of illustration, we consider populations in which none of the agents hold theory 01, so that we can limit ourselves to one face of the OPS. First, suppose that there are just two agents. In this case, there is no dynamics, irrespective of the value for

If there are three agents, then for

Figure

So far, we have considered the opinion dynamics for a fixed number of agents in the population. If we continue the above analysis for ever larger population sizes, predictable patterns appear, such as (for

The vertices act as sinks, but the number of points attracted to them depends on the BCI.

If the number of agents is even, the midpoint of the edges is accessible and acts as a source (for other points on the edge).

If the number of agents is a multiple of three, the midpoint of each of the faces is accessible and acts as a source (for

If the number of agents is a multiple of four, the midpoint of the entire tetrahedron is accessible and acts as a source.

This suggests a different way of studying the opinion dynamics: instead of considering populations with a particular population size, one can consider populations in general and ask, for each possible opinion, which

We can give a probabilistic interpretation of the previous results. For instance, we may be interested in the probability that the agents in the population reach a consensus on a particular theory. If we assume that each initial opinion profile is equally likely (uniform prior probability), then the probability of reaching consensus on a particular theory is equal to the number of opinion profiles in the basin of this consensus position divided by the total number of points in the OPS. (For a non-uniform prior probability, we may compute a similar fraction based on weighted sums).

For

In our previous paper (Wenmackers et al., _{M} − 2 dimensions, or the multiset coefficient of choosing _{M} − 1 options). For

An (anonymous) opinion profile specifies the number of agents that holds each of the theories. So, an opinion profile consists of 2^{tM} numbers that add up to

To simplify the analysis, we leave the number _{M} components. We represented the components of a particular opinion profile ^{6}

Another way of looking at the transition from OPS to ODS is as follows: we can track the dynamics for a large set of different population sizes and represent the accumulated data in a single tetrahedral grid. In the limit where we combine the OPSs for all (infinitely many) finite population sizes, this accumulative OPS becomes continuous instead of a discrete grid. Hence, the ODS is a continuous space in _{M} − 1 dimensions. (There are _{M} components of the opinion density vector, which are fractions that sum to 1, so there remain _{M} − 1 degrees of freedom).

To visualize the ODS, we have written an additional program in Object Pascal. Although the ODS represents a continuous space, numerical methods require it to be discretized, such that the program only encounters density vectors which have four rational indices. By multiplying the four rational indices of an opinion profile by their least common denominator, we compute an opinion profile that is representative of that density. The evolution of this profile is computed as before. The numerical result is indicated by means of colors (as explained below).

We consider the ODS equipped with (EHK) as update rule (for particular values of

As before, we focus on the case with

Also similar as before, we only represent a single face of the tetrahedral ODS: the triangle with vertices at (0, 0, 0, 1), (0, 0, 1, 0), and (1, 0, 0, 0), with the “double” edge at the bottom. Within this triangle, all opinion profiles have zero density at the second position: there are no agents that hold the theory 01.

For each position in the chosen triangle, we compute the (normalized) opinion profile that it will ultimately evolve to. We represent this by a color. Specifically, the color (

The results for

The results for

Intermediate values for

Recall that for a fixed number of agents, not all points of the continuous ODS are accessible. Once you have computed the opinion dynamics for the ODS, you can use the results to construct the dynamics on an OPS for a fixed number of agents,

Similarly to the discussion of the OPS results, we also give a probabilistic interpretation of the results concerning the ODS. If we assume that each initial opinion profile is equally likely (uniform prior probability), then the probability of reaching consensus on a particular theory is equal to the volume of the basin associated with this consensus position divided by the total volume of the OPS. At least, this fraction expresses the limit probability associated with an infinite population size, in which the relative importance of special points (unstable equilibria) is vanishingly small.

In Figure

Since the four basins have the same shape and size and together fill the entire volume of the ODS, they each correspond to a relative volume of 1/4. Under the assumption of a uniform prior, the limit of the probability of arriving at a particular consensus for exceedingly large populations is 1/4 (

In particular, in the infinite population limit there is a probability of 1/4 of arriving at the inconsistent theory. However, if we only consider opinion densities where the inconsistent theory initially has zero density (which are all represented at the a single face of the ODS), the probability of evolving to an opinion profile with a non-zero density at the inconsistent theory (let alone unit density at this position) is zero (at least for

Whereas the discrete OPS depends on a particular population size,

Due to social updating, an agent who starts out with a consistent theory about the world may arrive at the inconsistent theory. Even if maintaining consistency at all times is too demanding for non-ideal beings to qualify as a necessary condition for rationality (Cherniak,

Our current study of the opinion dynamics on the belief space reveals another virtue of the social updating process: even if an agent starts out at the inconsistent theory, the agent's opinion may change—to one of the consistent theories—due to the social update rule. This could already be seen on the basis of Example 3.2, but the results depicted in Figure

We have given a probabilistic interpretation of the results on the belief space (OPS and ODS). We have seen that in the limit for an infinite population size and for large BCIs (_{M}

First, the assumption of a uniform prior on the opinion profiles does not apply to real cases. Observe that if the agents were to pick out their initial theory at random, the distribution of initial anonymous opinion profiles would be higher around the center of the belief space. (For larger populations, there are more combinations of individual theories that lead to an anonymous opinion profile, in which all theories are represented almost evenly.) More importantly, however, we do

Second, in many practical situations relevant population sizes tend to be small (just think of the last meeting you attended), such that the infinite population limit does not apply well to them. In smaller populations, the relative importance of unstable equilibria (which do not lead to consensus) is more pronounced.

Third, modeling belief states as theories of the world only has practical relevance when _{M}

For all these reasons, we estimate the probability of arriving at a consensus on the inconsistent theory to be very small in a realistic setting—in any case well below 1/4.

The mechanism for social updating may also be criticized in the following way. If agents' belief states are theories, their beliefs are closed under the consequence relation. So, illustrating with theories for the case of

In our previous work (Wenmackers et al.,

For _{M} − 1) and the basin that is attracted by the sink corresponding to consensus on the inconsistency becomes a smaller fraction of its total (hyper-)volume (equal to 1/_{M}_{M}

For belief spaces with a fixed number of agents (with

Moreover, for fixed

Additionally, as the number

While the model studied in this paper is idealized in several respects, it is not completely unrealistic. Even if real agents do not generally compromise with their peers exactly in the way our artificial agents do, real agents do tend to influence each other's belief states, whether consciously or not. Idealized models can give information about such processes, much in the way in which the Ideal Gas Law gives information about the behavior of real gases. Also, there are several ways to make the model more realistic, for instance, as indicated earlier, by providing the agents with direct evidence about the truth, which in our model could be added as a driving force, directed toward a particular theory, or—equivalently—as an external potential directed toward one of the vertices of the ODS, corresponding to consensus on a theory with exactly one non-zero bit.

But even in its present, idealized form, the model we have studied demonstrates that there may be issues of rationality specifically arising from the way or ways we interact epistemically with fellow inquirers. We will be content if this sways some traditional (“individualistic”) epistemologists as well as some psychologists to take the social level into consideration in their studies of rationality. For the latter group, we note that already the current model suggests a number of seemingly worthwhile empirical studies, focusing on how real people influence one another's belief states, on which factors determine whether people regard someone as their peer (in the technical sense used here), and on whether whatever epistemic interactions take place in reality tend to aid the achievement of people's epistemic goals.

Sylvia Wenmackers's work was financially supported by a Veni-grant from the Netherlands Research Organization (NWO project 639.031.244 “Inexactness in the exact sciences”).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We are greatly indebted to Christopher von Bülow for very helpful comments on a previous version of this paper. We are also thankful for helpful comments by the guest editor Shira Elqayam and the two referees.

The Supplementary Material for this article can be found online at:

^{1}Psychologists have recently begun to explore connections between rationality and argumentation, which can also be regarded as an appreciation of the fact that the social bears on questions of rationality. For a particularly noteworthy contribution in this vein, see Mercier and Sperber (

^{2}See Andler (

^{3}In our previous work, we concentrated on the possibility of ending up at the inconsistent theory, because believing a contradiction is generally considered as irrational. As a referee remarked, if the entire epistemic community ends up at the tautological theory, which represents a complete lack of knowledge about the world, this may also be considered as a vicious result—although not irrational

^{4}To forestall misunderstanding, it is worth mentioning that the word “Bounded” in “Bounded Confidence Interval” refers to the fact that the confidence intervals in the HK model have a lower and upper bound; in particular, the word is not meant to suggest any connection with Simon's notion of bounded rationality (see, e.g., Simon,

^{5}To avoid later disappointment, we note already at this juncture that, while we are introducing a general framework for representing theories, our own later investigations of this framework will focus on the

^{6}The notion of barycentric coordinates, which comes from geometry, may need some introduction. We first introduce the concept of a simplex. A two-dimensional simplex is a triangle: a figure with three vertices. In three dimensions, a simplex is a tetrahedron: a figure with four vertices. In general, in ^{tM} = 4 vertices. Hence, for the simplest case with ^{tM} − 1 = 3 dimensions. To indicate a particular point inside a