2.1. Zollman’s two-armed Bandit model

In examining how credibility excess may influence epistemic outcomes, I will rely on modifications to the two-armed bandit model first introduced into the philosophy of science by Zollman (Reference Zollman2007).Footnote ³ As motivation for his model, Zollman considers a community of medical researchers who, at regular intervals, must choose between a well-understood method and a newly introduced method for treating their patients. Since the researchers feel obliged to maximize their patients’ chances for a favorable outcome, they always treat their patients using the method they currently believe is best. This assessment, however, may change over time as researchers update their credence as to the relative superiority of the new method of treatment based on reported treatment outcomes.

In modeling this scenario, Zollman uses a network of $N$ agents who, at regular intervals, decide to perform one of two possible actions. By stipulation, action A has a success rate of $0.5$ and action B (the “B”etter of the two actions) has a success rate of $0.5 + \varepsilon $ . While agents know action A’s success rate with complete certainty, they are only certain that action B’s success rate is either $\varepsilon $ better or worse than that of action A.Footnote ⁴

When deciding which action to perform, agents opt for the action they currently believe is more likely to be successful – this assessment is based on their credence that the success rate of action B is superior to that of action A. Epistemically, however, agents are only better off if action B is performed, as it is only then that evidence can be gathered as to its relative rate of success.

In simulating how such a community epistemically evolves over time, each agent is initialized with a randomly selected credence as to the superiority of action B in the interval $\left( {0,1} \right)$ . Deciding which action to perform based on this credence, a simulation round commences with each agent performing their preferred action $n$ times. Agents then report the observed successes and failures of their actions, sharing these results with their immediate network neighbors. Using the standard Bayesian method in conjunction with the reported outcomes they have gathered, agents update their credence concerning the superiority of action B. The round ends with agents once again selecting their preferred action based on their updated credence.Footnote ⁵

Given enough simulation rounds, only two stable outcomes are possible for fully connected networks of agents. If all agents choose to perform action A in a given round, fully stable incorrect consensus results. Since, in such a scenario, new evidence regarding action B will never be produced going forward, credence values for all agents will remain fixed below $0.5$ . Correct consensus, the only other possible stable outcome, is taken to have been achieved when all agents have credence values above some threshold – with this threshold set at $0.99$ for the simulations discussed in this paper.Footnote ⁶ Correct consensus is only approximately stable in that an arbitrarily long sequence of unlucky results when performing action B could, at any point, result in every agent switching to action A. This possibility, however, becomes increasingly unlikely the higher each agent’s credence becomes. Polarized outcomes, where agents stably differ in their preferred choice of action, are precluded in this setup.

Running simulations with agents placed in several standard network configurations (e.g. wheel, cycle, complete), Zollman (Reference Zollman2007) uses his model to investigate how changes in how widely reported action results are shared impacts the success of an epistemic community. He finds that, for at least some range of parameters, the more connected the agents, the less frequently, but faster, (approximately) stable correct consensus is reached.Footnote ⁷

2.2. Incorporating reliability and credibility

Zollman’s motivating example involves medical researchers whose choice between treatment options is influenced by past reported outcomes. Determining a treatment’s success or failure, however, is often a less than straightforward task. A patient’s condition can improve for reasons unrelated to treatment, medical tests can produce misleading results, researchers can be more or less skilled at interpreting symptoms, and so on. Reported outcomes are then often less than fully accurate, with levels of accuracy varying depending on disease, treatment, and researcher-specific factors.

Incorporating this real-world feature into Zollman’s modeling framework, we will take the reliability, $r,$ of an agent to be the probability that an action’s success or failure will be reported by the agent as such (with $1 - r$ providing the probability that an agent will misreport an action’s outcome as its opposite). Recall that the objective rate of success of action B is $0.5 + \varepsilon $ . When action B is performed $n$ times, successes will then follow a binomial distribution with parameters $n$ and $0.5 + \varepsilon $ . The successes reported by an agent with reliability $r$ will follow a binomial distribution with parameters $n$ and $0.5 + \left( {2r - 1} \right)\varepsilon $ instead.Footnote ⁸

Figure 1 provides a concrete illustration of the distributions involved when $n = 150$ , $\varepsilon = 0.3$ , and $r = 0.6$ . The dark gray graph shows the “Actual” distribution of successes when action B is performed $150$ times, while the light gray graph shows the distribution of successes “Reported” by an agent with reliability $0.6$ .

Figure 1.

Distributions of actual successes, reported successes, and successes assumed by an updating agent, when $n = 150$ , $\varepsilon = 0.3$ , $r = 0.6$ , and $c = 0.8$ .

Given potential inaccuracies in reported outcomes, it makes sense for an agent to factor in a reporting agent’s reliability when updating on the information they provide. Take credibility, $c$ , to be this assumed reliability. When the credibility assigned by an updating agent to a reporting agent matches the reporting agent’s actual reliability (i.e. $c = r$ ), credibility can be said to have been “correctly” ascribed; when $c \lt r$ , credibility has been under-ascribed; and when $c \gt r$ , credibility has been over-ascribed.Footnote ⁹

Under the assumption that action B is superior, an updating agent who assigns credibility $c$ to an agent performing action B will take reported successes to follow a binomial distribution with parameters $n$ and $0.5 + \left( {2c - 1} \right)\varepsilon $ . The dark red graph in Figure 1 shows this “Assumed” distribution when $c = 0.8$ . Since, in this case, credibility is over-ascribed (i.e. $c \gt r$ ), but not maximally so (i.e. $c \lt 1$ ), the bulk of the “Assumed” distribution falls between the bulk of the dark gray “Actual” and the light gray “Reported” distributions. Under the assumption that action B is inferior (i.e. the objective success rate of action B is $0.5 - \varepsilon $ ), the updating agent will take the light red graph to give the distribution of reported successes instead. This second distribution, also needed when performing Bayesian updating, is simply the “Assumed” distribution horizontally reflected about its center (i.e. about $x = {n \over 2} = 75$ ).

2.3. Over-ascribing credibility within a dominant group

With the Zollman model suitably modified to allow for less than maximally high levels of both reliability and credibility, an examination of the systematic over-ascription of credibility by a group of agents now becomes possible.

In modeling such a scenario, agents will be divided into two mutually exclusive groups, the “dominant” group and the “marginalized” group, with the parameter $f$ used to specify the fraction of agents in the marginalized group. For simplicity in interpreting results, agents will be arranged in a complete network (i.e. all outcome reports will be shared with the entire community), and every agent will be assigned the same reliability, $r$ . When both the updating and reporting agents are members of the dominant group, the credibility ascribed by the updating agent to the reporting agent will be set to ${c_{dominant}}$ . In all other cases, credibility will be set to the reporting agent’s actual reliability (i.e. $r$ ). The over-ascription of credibility within the dominant group can then be simulated by specifying a ${c_{dominant}}$ value in the interval $\left( {r,1} \right]$ .Footnote ¹⁰

A concern at this point may be whether it is realistic to assume that it is only members of the dominant group that over-ascribe credibility. Should members of the marginalized group be taken to over-ascribe credibility as well? Using data from the well-known Race Attitude Implicit Association Test, Morehouse and Banaji (Reference Morehouse and Banaji2024) report that while White Americans on average exhibit a clear pro-White bias, Black Americans on average display very little implicit racial bias.Footnote ¹¹ More generally, Morehouse and Banaji (Reference Morehouse and Banaji2024, 31) conclude that “unlike members of socially advantaged groups, who consistently display implicit in-group preferences, members of socially disadvantaged groups typically do not.” Taking within-group over-ascription of credibility to be a reflection of in-group preferences, the modeling assumption that it is only members of the dominant group who over-ascribe credibility seems like a plausible one.

When an updating agent over-ascribes, rather than correctly ascribes, credibility to a reporting agent, credence updates are amplified. Provided $r$ is greater than $0.5$ (i.e. the reporting agent is more likely to accurately report an action result than misreport it), the over-ascribing agent will update in the same direction as a similarly epistemically positioned correctly ascribing agent, but with a magnitude that is greater.Footnote ¹² In this sense, members of the dominant group can be said to be “overly aggressive” or “over-react” when updating on outcomes reported by dominant group members.

3.1. Accuracy and speed in reaching correct consensus

We begin by examining how the over-ascription of credibility within the dominant group impacts both the frequency of correct consensus and the speed with which correct consensus is achieved. In order to eliminate noisy comparisons involving success rates near the upper limit of $1$ , only $N$ , $f$ , $\varepsilon $ , and $n$ values where the frequency of correct consensus in baseline simulations was below $0.99$ will be considered in this subsection.

Figure 2 shows typical results for the fraction of simulations that reach correct consensus as ${C_{dominant}}$ is varied, with colored lines corresponding to different fractions of agents in the marginalized group. As can be seen, regardless of the marginalized group’s relative size, the fraction of simulations ending in correct consensus decreases as ${c_{dominant}}$ increases from the baseline value of $0.85$ to the maximally over-ascribed value of $1$ . This effect is more pronounced the smaller the relative size of the marginalized group (i.e. the lower the value of $f$ ). Both of these trends hold for all parameter values considered.

Figure 2.

Fraction of simulations reaching correct consensus over a range of ${c_{dominant}}$ and $f$ values, when $N = 12$ , $n = 10$ , and $\varepsilon = 0.01$ .

This decrease in accuracy is accompanied by an increase in the speed with which epistemic success is achieved. For all parameter values considered, the entire community reaches correct consensus in less rounds on average when the dominant group over-ascribes credibility than in the baseline scenario. This effect is more pronounced as the dominant group gets larger, with average rounds to correct consensus decreasing as $f$ decreases for all combinations of parameters.

In explaining these results, it is useful to recall Zollman’s (Reference Zollman2007) observation that, when connectivity in an epistemic community increases, accuracy is sacrificed for speed. A similar tradeoff is at play here. Credence changes that result from updating on reported outcomes are amplified when credibility is over-ascribed. In non-baseline scenarios, members of the dominant group then over-react to reported outcomes provided by dominant group members. When these reports accurately reflect the superiority of action B, dominant group members achieve higher credence values earlier. This increases the speed with which correct consensus is typically achieved. When agents are unlucky and these reported outcomes suggest that action A is superior instead, dominant group members are quicker to lower their credence and abandon the better action. Correct consensus occurs less frequently as a result. Since there are more agents over-reacting to more data as the dominant group gets larger, these effects become more pronounced as $f$ gets smaller.

3.2. Average rounds to threshold credence

As discussed in the previous subsection, when credibility is over-ascribed within the dominant group, correct consensus is reached less frequently, but is typically achieved in fewer simulation rounds. The cost associated with this tradeoff of accuracy for speed is one borne by the entire community – with all agents reaching, or failing to reach, high credence in the relevant true proposition, regardless of group membership. In this subsection, we will examine the distribution of the related speed benefit among members of the two groups.

Simulations are taken to end in correct consensus when all agents have credence values concerning the superiority of action B above $0.99$ . Measuring how quickly agents in the dominant group and marginalized group reach this threshold provides a sensible way of comparing their relative speed in achieving epistemic success. In order to avoid noisy comparisons involving values very close to the lower limit of $1$ , only $N$ , $f$ , $\varepsilon $ , and $n$ values where the baseline average rounds to correct consensus was above $1.1$ will be considered in this subsection.

Figure 3 shows typical simulation results for the average rounds to threshold credence as ${c_{dominant}}$ is varied from the baseline value of $0.85$ to the maximally over-ascribed value of $1$ . The major takeaway, seen for all parameter values considered, is that agents in the dominant group reach threshold credence faster than agents in the marginalized group, with this difference increasing as ${c_{dominant}}$ increases. While average rounds to threshold credence trends down in all cases for the dominant group as the over-ascription of credibility increases, there is no clear trend across parameter values in the speed with which agents in the marginalized group reach this same threshold.

Figure 3.

Average rounds to $0.99$ credence for agents in the dominant and marginalized groups over a range of ${c_{dominant}}$ values, when $N = 24$ , $f = {1 \over 2}$ , $n = 1$ , and $\varepsilon = 0.02$ .

A straightforward explanation can be given as to why this occurs. Recall that, in non-baseline simulations, members of the dominant group update more aggressively on reported outcomes provided by dominant group members. Members of the dominant group will then tend to front-run members of the marginalized group when it comes to adjusting their credence based on general trends in evidence – these trends typically reflect the superiority of action B in this model.Footnote ¹³ When correct consensus is eventually reached, members of the dominant group will then tend to reach high credence values sooner. This is despite the fact that the exact same information is being reported to all members of the community.

Consistent with this explanation, the relative speed advantage enjoyed by dominant group members gets larger as the relative size of the dominant group gets larger (i.e. as $f$ gets smaller). When there is a higher fraction of agents in the dominant group, there is typically more information available per round to aggressively update on, increasing the observed effect.

3.3. Average credence

While average rounds to threshold credence measure how quickly agents in both the dominant and marginalized groups come to have high confidence that the proposition in question is true, it is also worth examining how credence values differ over the entire learning period. To this end, the time-weighted average credence for simulations ending in correct consensus was calculated for members of both groups.Footnote ¹⁴ As in the previous subsection, only $N$ , $f$ , $\varepsilon $ , and $n$ values where the baseline average rounds to correct consensus was above $1.1$ will be considered.

Figure 4 shows typical average credence values for agents in both the dominant group and marginalized group as ${c_{dominant}}$ is varied from the baseline value of $0.85$ to the maximally over-ascribed value of $1$ . As can be seen, dominant group agents typically have higher average credence values than marginalized group agents, with this difference increasing as ${c_{dominant}}$ increases. While this difference is not large in terms of magnitude, it is consistent for all parameter values considered.

Figure 4.

Average credence for agents in the dominant and marginalized groups over a range of ${c_{dominant}}$ values, when $N = 48$ , $f = {1 \over {12}}$ , $n = 1$ , and $\varepsilon = 0.05$ .

The explanation for this effect is once again that members of the dominant group update more aggressively on reported results provided by dominant group members than updating agents in the marginalized group. Since these reported action results typically push credence values in the correct epistemic direction, members of the dominant group will tend to have higher average credence values when the entire successful learning period is considered. This provides a second way in which members of the dominant group are relatively advantaged by the over-ascription of credibility.