So, an opinion profile consists of 2 t M numbers that add up to N , the total number of agents in the population.
We consider the space of all possible opinion profiles, the OPS. The dynamics on this space shows the group-level or aggregate effect of the individual updating by the rule introduced in the previous section. Some opinion profiles act as fixed points: once the population reaches such a state, there is no further dynamics.
By counting states in the OPS and assigning priors probabilities to initial opinion profiles, we can give a probabilistic interpretation to the results.
The analysis in terms of an OPS requires the choice of a particular population size, N ; in the next section, we follow a slightly different approach that does not require this. To simplify the analysis, we leave the number N of agents open and represent all possible opinion profiles for arbitrary N simultaneously, using the opinion density space ODS.
The opinion density coordinates can be viewed as barycentric coordinates, specifying which fraction of the agents adheres to each theory 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.
20th WCP: Defending Longino's Social Epistemology
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. In this case, opinion densities have four digits, which are fractions that sum to 1, so there remain three degrees of freedom. Hence, these opinion densities can be represented using barycentric coordinates in a three-dimensional tetrahedron inside the volume as well as on the surface. On the edges of the tetrahedron, we find populations in which only two of the four theories are represented the other two having density zero.
On the faces of the tetrahedron, we find populations in which one of the four theories is not represented. Inside the volume of the tetrahedron, in each population there is at least one agent for each theory, so none of the density components is zero. Within this triangle, all opinion profiles have zero density at the second position: there are no agents that hold the theory 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. For instance, the redder a point, the larger the fraction of agents that will finally adhere to the inconsistent theory, The results depend on the threshold value D and are presented at the left-hand side of Figure 6.
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For each point, we also indicate after how many steps the final state is reached. We represent this with a gray-scale on the right-hand side of Figure 6. Figure 6.
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On the left-hand side, each position in the ODS is colored depending on its final state see main text for details ; smaller features are indicated with white ellipses and arrows. On the right-hand side, each position in the ODS is given a gray-scale value depending on how many rounds of updating are required for it to reach its final state; smaller features are indicated with ellipses purple for zero, orange for one, and blue for two. On the right-hand side of Figure 6 , we see that all the positions have the color corresponding to the initial opinion profile.
At the left-hand side of Figure 6 , we see that zero steps are required to reach the final state. Both observations confirm that all opinion profiles are fixed points. Because there is no dynamics, it is a situation of indifferent equilibrium. This image is still helpful, because—due to the absence of dynamics—each point in it is colored based on its own coordinates, which can be used as a key to interpret the representation of the results with dynamics.
There are six fixed points. The three consensus positions at the vertices are fixed points, which act as sinks for large portions of the face. The three positions halfway along the edges are fixed points as well. The gray-scale image confirms these findings: the six fixed points do not require any iterations, whereas the others settle after just one update.
Intermediate values for D tend to lead to more complex and interesting behavior. 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, N , by locating a density that is accessible for the N of interest and using the color of that point to determine to which opinion profile it will evolve.
In fact, the results on OPSs in the previous figures do already use the same color convention as that used for the ODS. 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.
Each basin has the same shape with five faces: two equilateral triangles and one rhombus that face the exterior of the ODS and two isosceles right triangles that face the interior of the ODS see also Supplementary Material. Figure 7. Left-hand side: three-dimensional view of ODS with the face shown in previous figures turned toward the right. Right-hand side: exploded view of the same ODS, showing the four basins separately. Each basin has the same shape with five faces: one rhombus, two equilateral triangles, and—facing the interior of the ODS—two isosceles right triangles and a volume that occupies one quarter of the tetrahedral ODS.
For maximal D , the limit probability of arriving at some consensus is 1. Under these conditions, the unstable equilibria on the edges of the basins are isolated points, lines, or areas, which have zero volume and thus zero probability. Whereas the discrete OPS depends on a particular population size, N , the continuous ODS represents the density of theories in populations of arbitrary size.
By considering volumes in the ODS and assigning a prior probability distribution to initial opinion profiles, we can give a probabilistic interpretation to the results, which serve as a good approximation for very large population sizes, but does not apply to small groups. We observe that even if special points such as stable fixed points make up a small portion of the ODS, these points tend to be represented in small populations causing the dynamics to end after few rounds of updating.
Due to social updating, an agent who starts out with a consistent theory about the world may arrive at the inconsistent theory.
Exploring Knowledge as a Social Phenomenon
Even if maintaining consistency at all times is too demanding for non-ideal beings to qualify as a necessary condition for rationality Cherniak, , it is presumably something that rational beings should aim for. This may suggest that social updating is a vice, from the perspective of rationality. However, in our first study Wenmackers et al. 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. Nevertheless, when there is any dynamics at all, many of these opinion profiles evolve to different profiles, some of which have no agents at the inconsistent theory. This is true, in particular, for all the opinion profiles in the blue and green areas, which act as basins for consensus positions on consistent theories.
Once the agents reach consensus on the inconsistent theory, there will be no further dynamics, because all consensus positions are fixed points. Hence, this result may be regarded as a worst case. However, this case study is highly unrealistic for at least three reasons. 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 not expect the agents to adopt an initial theory at random but rather to possess some prior knowledge, such that the distribution of their initial theories is clustered around the true theory which is necessarily a consistent one.
Hence we also expect a preferential position of the opinion profiles in a region around consensus on the true theory. For this reason, investigation of a more complex model, based on a variant of our current update rule EHK , but including evidence-gathering as well as social updating, is high on our to-do list. 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. 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. However, agent B also ought to believe A 's theory, but not vice versa.
We may now suggest an alternative way of determining an agent's peer group: by taking into account also those agents that hold a theory which is within distance D of at least one of the consequences of the first agent's theory. Doing so would help to protect agents against updating to the inconsistent theory. However, it also introduces a preference for less informative theories, so it may hamper the agents' chances of finding the strongest true theory.
Hence, this is a case where different epistemic goals rationality versus finding the truth are in direct conflict with each other and selecting the optimal normative model seems to require meta-norms of rationality. In our previous work Wenmackers et al. This finding is confirmed in the current study. Nevertheless, by studying the dynamical space in general, we have observed certain trends that help to explain the previously obtained results for the probability of consistent-to-inconsistent updating.
In our previous work, we observed that this probability decreases when more independent issues are considered that is, when M increases beyond 2. We are now in a better position to explain the—essentially combinatorial—mechanism behind this finding. This corresponds to the observation in our previous study that the probability of updating to the inconsistent theory is lowered by forming theories over more independent issues higher M.
For a larger number of agents higher N , the dimensions of the belief space remain the same, but the opinion profile has access to more points of this space. As a result, the probability of consensus on the inconsistent theory is lower, too; this is in line with the earlier findings as well. This is confirmed by our study of the ODS: the midpoint of an edge belongs to a line separating two or three basins. Moreover, if the number of agents is a multiple of four, the midpoint of the entire tetrahedron is accessible and acts as a source. So, in the case of an even number of agents, there are more fixed points than in the case of an odd number of agents.
Moreover, for fixed M , there is a limited number of these special points, whereas the total number of accessible points in the belief space rises fast when the number of agents, N , increases. Consequently, the number of these special points as compared to the total number of opinion profiles in the hyper-volume decreases when N increases, which explains the attenuation of the wobble for larger populations. Figure 6 , we see that the majority of opinion densities belong to some basin that is attracted to a sink.