## 1 Introduction

## 2 Aggregation rules and deliberative norms

### 2.1 Ranking opinion aggregation rules

### 2.2 Evaluating social deliberation designs

## 3 A binary classification problem

### 3.1 The model

### 3.2 The non-deliberation design

^{1}

^{2}

### 3.3 The deliberation designs

^{3}

^{4}The horizontal line represents individual or dictatorship competence, which is equal to majority competence with \(n=1\). For design (1), we see that the majority rule underperforms the dictatorship rule for all \(n>1\), but for design (2), the majority rule outperforms the dictatorship rule for all positive \({\mathbb {N}}\backslash \{1,2,4,6\}\).

### 3.4 Discussion

^{5}

## 4 A numerical estimation problem

### 4.1 The model

^{6}It is important to distinguish between the decision problem from the point of view of the agents, who do not know the target value they are estimating, and the decision problem from the point of view of the modeler, who does know it. We will be concerned only with the latter and therefore we can calculate quantities based on the target value of any arbitrary d in order to evaluate how the agents are expected to fare.

^{7}Let the partition \(Z=\left\{ Z_{j}\right\} _{j\in n}\) be a coarsening of \({\dot{Z}}\), such that each member \(Z_{j}\in Z\) denotes a different deliberation design. An unknown probability distribution \(P\left( Z_{j}\mid D\right) \) is defined over the partition Z.

^{8}An individual deliberator working on a specific decision problem realization d under deliberation design \(Z_{j}\) is modeled by random variable \(X_{i}\), with some probability distribution \(P\left( X_{i}\mid d,Z_{j}\right) \).

^{9}Alternatively, I use \(X_{i,d,Z_{j}}\) for the random variable \(X_{i}\) distributed according to \(P\left( X_{i}\mid d,Z_{j}\right) \), where the latter probability distribution is defined over the sub-sample space that corresponds with the realization of the specific decision problem instance d under the specific deliberation design \(Z_{j}\). Further, I model each individual deliberator \(X_{i,D,Z}\) or \(X_{i,d,Z_{j}}\) as an unbiased Gaussian random variable that scatters around the target value t with some variance. Being unbiased, the deliberators are just as likely to underestimate the truth as they are to overestimate it. In contrast to the model from the previous section, deliberators need not be interchangeable and can differ with respect to their variance around the target. Let us now consider how introducing social deliberation might change the accuracy of a single collective judgment.

### 4.2 An instance of tragic accuracy improvement

Deliberator | a | b | c | d | e | f | g | h | i | j | avg | t |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Round 1 | 85 | 82 | 80 | 78 | 70 | 55 | 50 | 45 | 40 | 39 | 62.4 | 60 |

Round 2 | 84 | 75 | 79 | 77 | 69 | 59 | 59 | 60 | 59 | 55 | 67.6 | 60 |

Improvement | +1 | +7 | +1 | +1 | +1 | +4 | +9 | +15 | +19 | +16 | \(-\)5.2 |

### 4.3 An improvement of collective competence under dictatorship rule

^{10}

^{11}For any decision problem instance d:

### 4.4 Deteriorating collective competence under the averaging rule

^{12}

### 4.5 Discussion

^{13}Accordingly, even if two deliberators are perfectly correlated, a reduction in \({\overline{Var}}\left( X_{1:n,d,Z_{j}}\mid d\right) \) decreases the upper bound on the error component \({\overline{Cov}}\left( X_{1:n,d,Z_{j}}\mid d\right) \). However, for any positive value for the individual variance and positive value for average pair-wise covariance, there will be a threshold number of deliberators n above which \(\frac{1}{n}{\overline{Var}}\left( X_{1:n,d,Z_{j}}\mid d\right) <\left( 1-\frac{1}{n}\right) {\overline{Cov}}\left( X_{1:n,d,Z_{j}}\mid d\right) \). If that occurs, the deliberation design reduces the collective competence under the simple averaging rule.

^{14}However, this does not mean that simple averaging could not be outperformed by a different aggregation rule.