Disagreement, resilience, and the limits of principle

An interactive simulation

The present model simulates possible responses to peer disagreement. It examines two different kinds of response: first, agents can update by adjusting their beliefs in the direction of their peers. Specifically, agents control which weight w they assign to their peer's belief. Second, agents can respond by lowering their confidence that they are right. Following a proposal from Steglich-Petersen, such higher-order doubt ought to be reflected in their belief’s resilience r, which tracks how responsive it is to incoming evidence. In this model, possible responses of disagreements can be represented as points in the (w, r) space. But which point, or region, produces the best results? The model shows that there is no uniform answer. Optimal responses depend on the type of disagreement in question. But we need not abandon the search for principles, as long as the principles are appropriately fine-tuned. The goal of this model is to surface such principles.

Some background: the Hegselmann–Krause framework

The formal study of how epistemic communities form and revise beliefs has developed largely through agent-based models—computer simulations in which individual agents update their beliefs according to simple rules, and the collective dynamics are observed over many rounds (Šešelja 2022). Models of this kind have been used to study network effects on scientific inquiry (Zollman 2007, 2010), the epistemic costs of conformity (Weatherall & O’Connor 2021), and the impact of biased or deceptive agents on group belief formation (Holman & Bruner 2015; Weatherall et al. 2018).

A particularly influential framework for modeling belief dynamics is the Hegselmann–Krause (HK) bounded confidence model (Hegselmann & Krause 2002, 2009). In its basic form, the HK model represents agents as holding real-valued opinions on some parameter—say, a probability or a numerical estimate—and updating by averaging over the opinions of those agents whose views fall within a confidence interval ε of their own. Agents that are too far apart simply do not influence each other. Extensions of this model have incorporated truth-seeking, cognitive division of labor, and noisy evidence (Hegselmann & Krause 2009; Douven & Kelp 2011), and it has become a standard reference point for formal work in social epistemology.

One important feature of the HK approach is worth drawing out explicitly. Much of the philosophical literature on disagreement frames the problem in terms of credences: two agents assign different degrees of belief to the same proposition. But as Douven (2010) notes, most real disagreements are not like this. They are disagreements between agents who hold beliefs with different content—not contradictory beliefs (which cannot both be true and cannot both be false), but contrary ones, which cannot both be true but can both be false. Two economists disagree not by assigning different probabilities to a single claim, but by asserting different values for the same quantity. The HK model captures this naturally: agents hold positions on a continuous scale, and disagreement is a difference in those positions, not a difference in the confidence with which a shared proposition is held.

This model

The present model adopts this framework and builds on it in several directions, keeping the exposition close to the basic HK setup.

As a toy example, the two economists might not just disagree in the sense that they assign different probabilities to the proposition ‘A recession is coming.’ They might also have different opinions about the magnitude of this recession. There is a true value τ. Each agent’s estimate is a noisy signal of that true value—their opinion is τ plus some random error, drawn from a normal distribution with variance σ². We say the agents are epistemic peers not because they are exactly equally reliable (i.e., their opinions have the same variances), but in the weaker sense that neither has any reason to think the other is better or worse. This is modeled by sampling their variances from the same prior distribution—an inverse-gamma distribution, which is a standard Bayesian prior on variance. The agents are symmetric in expectation, without being numerically identical.

Given their observations and this prior on variance, there is a well-defined posterior distribution over τ. Importantly, the agents are not assumed to know this distribution, and they are not assumed to update by computing the posterior expectation. Instead, they update according to a weighted average of their two opinions, governed by a steadfastness parameter w:

After this peer update, a second round of updating occurs, representing the agents’ sensitivity to further evidence. Agents receive a noisy signal of τ—drawn from a normal distribution centered on the true value—and update by blending their current opinion with the signal, governed by a signal weight parameter a:

The parameter a corresponds, roughly, to the inverse of resilience: an agent with low a is highly resilient, barely moving in response to new evidence; an agent with high a is highly responsive. This is the mechanism Steglich-Petersen (2019) associates with the rational response to higher-order defeat, and which I formalize using weighted conditionalization in my own work: after doubting yourself, you should update more strongly on new evidence.

There is one further element. An inquiry closes—in the sense of zetetic inertia (Friedman 2020)—when the remaining disagreement between the two agents falls below a threshold ε. Once inquiry is closed, agents no longer update on the incoming signal, treating the question as settled.

What the simulation shows

The heatmap displays the average accuracy (measured as mean absolute error or mean squared error) of the agents’ final opinions across the full space of (w, a) pairs, evaluated over many sampled worlds. Brighter regions correspond to lower error—better collective accuracy.

The central observation is that the optimal region of the (w, a) space is not fixed. It shifts depending on:

Readers are invited to convince themselves that no single (w, a) pair dominates across all parameter settings. Highly conciliatory responses work well when the signal is reliable and the prior variance is low; more steadfast responses do better when the signal is noisy and agents have good reason to trust their own estimates. The appropriate degree of resilience depends on the same factors. This is consistent with Kelly’s (2010) contention that the disagreement debate asks the wrong question when it seeks a universal answer, and with the simulation-based argument in Douven and Kelp (2011) that neither conciliation nor steadfastness is unconditionally best.

The model is an idealization in the sense standard to this literature (Šešelja 2022). It abstracts away from the specific content of agents’ reasons, from the structure of communication networks, and from multi-agent dynamics. Its value lies not in direct empirical application, but in making the normative landscape visible: showing how the costs and benefits of different updating dispositions depend on structural features of the epistemic situation.