Disagreement behind the veil of ignorance

Abstract

In this paper we argue that there is a kind of moral disagreement that survives the Rawlsian veil of ignorance. While a veil of ignorance eliminates sources of disagreement stemming from self-interest, it does not do anything to eliminate deeper sources of disagreement. These disagreements not only persist, but transform their structure once behind the veil of ignorance. We consider formal frameworks for exploring these differences in structure between interested and disinterested disagreement, and argue that consensus models offer us a solution concept for disagreements behind the veil of ignorance.

Appendix

1.1 Summary of the Lehrer–Wagner model

Let $G = \{1, \ldots N\}$ be a group of agents. The Lehrer–Wagner model is concerned with the problem of estimating an unknown quantity x from the individual estimates x _i of every group member i. This quantity x is thought of as objective and independent of the group members’ cognitive states.

Lehrer and Wagner’s central idea consists in ascribing the agents beliefs about each other’s expertise, or in other words, mutual degrees of respect for the issue at hand. These weights w _ij represent the respect that agent i has for agent j, relative to the subject matter in question, and describe the proportion to which j’s opinion affects i’s revised opinion. The mutual respect assignments are in an N × N matrix W:

$$ W = \left( \begin{array}{llll} w_{11} & w_{12} & \dots & w_{1N} \\ w_{21} & w_{22} & \dots & w_{2N} \\ \dots & \dots & \dots & \dots \\ w_{N1} & w_{N2} & \dots & w_{NN} \end{array}\right). $$

An important mathematical constraint is that the values in each row are nonnegative and normalized so as to sum to 1: ∑ ^N _j=1 w _ij  = 1. Then, W is multiplied with a vector $\vec{x} = (x_{1}, \ldots, x_{N})$ that contains the agents’ individual estimates of x, obtaining a novel updated value for $\vec{x}$:

$$ W \cdot \vec{x} = \left( \begin{array}{l} w_{11} x_{1} + w_{12} x_2+ \ldots + w_{1N} x_N\\ w_{21} x_{1} + w_{22} x_2 + \ldots + w_{2N} x_N \\ \dots \\ w_{N1} x_{1} + w_{N2} x_2 + \ldots + w_{NN} x_N \end{array}\right). $$

In general, however, this procedure will not directly lead to consensus, since the entries of $W \cdot \vec{x}$ differ: $(W\vec{x})_{i} \neq (W\vec{x})_{j}$. However, it can be shown that the iterated application of the pooling procedure represented by W, W ⁿ, converges to the so-called “consensus matrix” $W^{\infty}$. It can also be shown that the individual entries of $W^{\infty} \cdot \vec{x}$ are equal to each other. Thus, repeating the pooling procedure leads to (rational) consensus.