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*Seamus Bradley*

## Aggregation

### 1. Impossibility theorems

#### Syllabus

1) Impossibility theorems |
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1.1) Voting |

1.2) Propositional aggregation |

1.3) Probabilities |

1.4) The general impossibility idea |

2) Aggregating imprecise probabilities |
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2.1) Walley’s desiderata |

2.2) Some methods for aggregating IP models |

2.3) Belief merging for propositions |

In the first part we introduce the concept of aggregation and quickly sketch some classic impossibility theorems for aggregation. Aggregation is the process of collecting a number of agents’ opinions and using them to create a group opinion. There are a number of purposes you might have for aggregating the opinions of a number of agents:

- A rough summary of the individuals views
- A compromise opinion in order to make a decision
- A consensus reached after discussion
- The opinion of an external decision maker taking account of the individuals views
- The opinion of a group member taking account of the other individuals views

#### 1.1 Voting

One of the first impossibility theorems for aggregation is Arrow’s Impossibility Theorem for voting systems. This limits aggregation of preference rankings. Arrow’s theorem gave rise to a large field of social choice theory, and many other impossibility results (e.g. Gibbard Satterthwaite theorem) have followed. See for a recent overview of the field.

#### 1.2 Propositional aggregation

Instead of aggregating preference rankings, we can consider aggregating agents’ opinions, their propositional knowledge. Here, too, we have some important limitations.

#### 1.3 Probabilities

Finally, we have some results that limit what is possible when it comes to aggregating probability judgements. On the one hand, we have results that generalise impossibilities from the propositional case, and on the other we have characterisations of various probabilistic aggregation rules

#### 1.4 The general impossiblity data

Finally we discuss the proof strategy common to many impossibility proofs, and some other general impossibility proofs.

### Aggregating imprecise probabilities

#### 2.1 Walley’s desiderata

Peter Walley suggested a number of desiderata that we would like aggregation of IP models to satisfy. Many of these are similar to ideas we’ve seen in earlier sections.

#### 2.2 Some methods for aggregating IP models

We discuss some of the proposals for aggregating IP models, including those by Moral and Sagrado, Stewart and Quintana and de Cooman and Troffaes.

#### 2.3 Belief merging for propositions

One standard approach to aggregating propositional information is by using “merging operators”. (The belief merging framework can also be used to generate an impossibility result). This approach can be generalised to the theory of “Belief Models” which includes many IP models as special cases, thus we can generate methods for aggregating IP models through exploiting well known merging results.