Speaker: Shaun White (PhD student, Department of Mathematics)
Topic: Applications of the Gibbard-Satterthwaite Theorem to voting systems
When: 2:30-3:30, Tuesday 17 September
Where: Room 5115, OGGB
Abstract:

The Gibbard-Satterthwaite Theorem is one of social choice theory’s most notable results. Social choice theorists usually present the theorem as a statement about voting systems. Consequently, political scientists have shown considerable interest in the theorem and its applications.

The theorem applies to many voting systems, but it doesn’t apply to all voting systems. If we ask “which systems does the theorem apply to?”, the social choice theorist and the political scientist will give what appear to be different answers. This is partly because social choice theorists and political scientists use voting-terminology differently.

In this talk I will state the Gibbard-Satterthwaite Theorem in purely mathematical terms; the statement will refer to sets, relations, and functions. I will give an overview of the framework in which Gibbard originally presented the theorem; this framework features voters, preferences, strategies, and game forms. I will then use these two tools — the purely mathematical theorem, Gibbard’s framework — to build an interdisciplinary method for applying the Gibbard-Satterthwaite Theorem.

Speaker: Benjamin Hadjibeyli (ENS de Lyon)
Topic: Geometry of distance-rationalization
When: 2:30-3:30, Tuesday 27 August
Where: Room 5115, OGGB
Abstract: Representing voting rules in the unit simplex by considering only the distribution of voter preferences is a classical approach to voting theory, for example in the books of Donald Saari. However, it has not yet been applied to the distance-rationalization framework. We aim to analyse general properties of distance-rationalizable voting rules by looking at the geometry of their consensus and metric under this representation. This leads to interesting geometric questions involving metric spaces.

Slides are available.

Speaker: Mark Wilson (Computer Science)
Topic: “Distance rationalization of voting rules”
When: 2:30-3:30, Tuesday 20 August
Where: Room 5115, OGGB
Abstract:
A promising unifying framework for social choice involves the concept of measuring how far a preference profile is from an acknowledged consensus, with respect to some distance measure. This has been actively studied recently, particularly by Elkind, Faliszewski, and Slinko.

This is an introductory talk, giving basic definitions, examples, and results, to set the scene for next week’s talk.

Slides are available.

Speaker: Patrick Girard (Philosophy)
Topic: “Belief revision and the limit assumption: Tension between static belief and belief dynamics”
When: 2:30-3:30, Tuesday 13 August
Where: Room 5115, OGGB
Abstract:
…so there’s this assumption called the limit assumption which basically says that doxastic orders are well-founded. If you only consider beliefs as being static, the assumption is philosophically implausible. However, when you do belief change, than it becomes crucial for a lot of doxastic operations. Without it, you can’t be sure that revising a belief set returns a belief set. Which considerations is more important? Static or dynamic? I will try and explain what that all means.

Speaker:     Arkadii Slinko
Affiliation: University of Auckland
Title:       Clone Structures
Date:        Tuesday, 4 Jun 2013
Time:        14:00
Location:    303-412

In Economics, a set of linear orders is normally interpreted as a set of opinions of agents about objects in C.  Cloning candidates (products) is one of the most sophisticated tools of manipulation of elections (consumer surveys). Unfortunately most common voting rules are vulnerable to this method of manipulation. So clones do matter.

Mathematically, a subset of C which is ranked consecutively (though possibly in different order) in all linear orders is called a clone set. All clone sets for a given family of linear orders form the clone structure. In this talk I will formalise and study properties of  clone structures. In particular, I will give an axiomatic characterisation of clone structures, define the composition of those, classify irreducible ones, and show that it is sufficient to have only three linear orders to realise any clone structure.

This is a joint work with Piotr Faliszewski (Krakow) and Edith Elkind (Oxford).

All welcome!

Speaker: Arkadii Slinko
Affiliation: The University of Auckland
Title: Secret sharing schemes 2 (elementary introduction)
Date: Tuesday, 21 May 2013
Time: 4:00 pm
Location: Room 6115, Owen Glenn Building

This is a continuation of my talk on 7 May 2013.

This time I will first introduce two large classes of ideal access structures, namely, conjunctive and disjunctive hierarchical access structures. They are characterised by the fact that users are divided into classes so that users within each class are equivalent but users belonging to different classes have different status with respect to the activity. For example, the UN Security Council with its permanent and non-permanent members is a conjunctive hierarchical access structure (to the passage of a resolution).

The main part of the talk will be focused on the connection between ideal secret sharing schemes and matroids. The theorem of Brickel and Davenport (1991) which describes this connection plays a central role in the theory of secret sharing. A short introduction to matroids will be given, no prior knowledge of matroids will be necessary.

Speaker: Professor Bettina Klaus
Affiliation: University of Lausanne
Date: Friday 3 May 2013
Time: 12pm
Venue: Room 317, Level 3, Owen G Glenn Building

Abstract: In college admissions and student placements at public schools, the admission decision can be thought of as assigning indivisible objects with capacity constraints to a set of students such that each student receives at most one object and monetary compensations are not allowed. In these important market design problems, the agent-proposing deferred-acceptance (DA-)mechanism with responsive strict priorities performs well and economists have successfully implemented DA-mechanisms or slight variants thereof. We show that almost all real-life mechanisms used in such environments – including the large classes of priority mechanisms and linear programming mechanisms – satisfy a set of simple and intuitive properties. Once we add strategy-proofness to these properties, DA-mechanisms are the only ones surviving. In market design problems that are based on weak priorities (like school choice), generally multiple tie-breaking (MTB) procedures are used and then a mechanism is implemented with the obtained strict priorities. By adding stability with respect to the weak priorities, we establish the first normative foundation for MTB-DA-mechanisms that are used in NYC.

This is a joint Department of Economics/CMSS seminar.

Speaker: Arkadii Slinko
Affiliation: The University of Auckland
Title: Secret sharing schemes (an elementary introduction)
Date: Tuesday, 7 May 2013
Time: 4:00 pm
Location: Room 6115, Owen Glenn Building

Certain cryptographic keys, such as missile launch codes, numbered bank accounts and the secret decoding exponent in an RSA public key cryptosystem, are so important that they present a dilemma. If too many copies are distributed, one may be leaked. If too few, they might all be lost or accidentally destroyed. Secret sharing schemes invented by Shamir (1979) and Blakley (1979) address this problem, and allow arbitrarily high levels of confidentiality and reliability to be achieved. A secret sharing scheme `divides’ the secret S into `shares’ – one for every user – in such a way that S can be easily reconstructable by any authorised subset of users, but an unauthorised subset of users can extract absolutely no information about S. A secret sharing scheme, for example, can secure a secret over multiple servers and it remains recoverable despite multiple server failures.

Secret sharing schemes are a sort of cooperative games where the information and not money is being distributed among players. The set of authorised coalitions of a secret sharing scheme is a simple game so there is a rich connection to the theory of games.

In my talk I will give an elementary introduction to secret sharing.