Speaker:     Tatyana Gvozdeva
Affiliation: The University of Auckland
Title:       A new bound for simple games
Date:        Monday, 13 Jul 2009
Time:        3:00 pm
Location:    Room 401

Comparative probability orders are closely related to simple games and cancellation conditions for these orders are very similar to trading transforms for games. In this talk we exploit this similarity to obtain new examples of simple games using Fishburn’s examples of comparative probability orders. These examples give us a new lower bound on the lengths of certificate of non-weightedness for simple games, which is better than the best known one given by Taylor and Zwicker (1992). Our lower bound is linear in the number of players while the one by Taylor and Zwicker is equal to the square root of n.

Speaker:     Matthew Ryan
Affiliation: Economics Department, The University of Auckland
Title:       Mixture Sets – An Introduction
Date:        Monday, 22 Jun 2009
Time:        3:00 pm
Location:    Room 401, Science Centre

A mixture set is an abstract convex structure introduced by Herstein and Milnor (Econometrica, 1953) as a foundation for the expected utility representation theorem (representation of preferernces by a linear utility function).  Mixture sets combine a set X with a ternary relation T that maps (x,y,t) to an element of X for each x,y in X and each t in [0,1] — the t-mixture of x and y.  Herstein and Milnor consider infinite mixture sets, but the notion is well-defined even for finite X.  This raises the question of the relationship between mixture sets and abstract convex geometries (discussed by Arkadii in previous Workshops).  It appears that neither is a special case of the other. This talk will introduce mixture sets, and what is — and isn’t — known about them.

Speaker: Tatyana Gvozdeva
Affiliation: The Univesity of Auckland
Title: Roughly weighted simple games
Date: Monday, 18 May 2009
Time: 3:00 pm
Location: Room 401

In this talk we will give a necessary and sufficient condition for a simple game to have rough weights. As in the classical Taylor-Zwicker (1992) theorem this will be done in terms of trading transforms. We will give some bounds on the lengths of certificates of non-existence of rough weights and explore games with small number of players.

Speaker:     Arkadii Slinko
Affiliation: The University of Auckland
Title:       Simple games: what are the questions?
Date:        Monday, 11 May 2009
Time:        3:00 pm
Location:    Room 401

A simple game consists of a finite set of objects (players) and some subsets (coalitions) are marked as winning and the rest are therefore losing. The monotonicity condition is imposed which says that a superset of a winning coalition is winning. Simple games are used to model the distribution of power in a body of agents, say which coalitions of countries can pass a motion in the UN Security Council. A simple game also may model the access structure of a secret-sharing scheme – in this case winning coalitions are those who authorised to know the secret.

In this talk I will introduce some basic concepts and formulate a number of open questions. Most are concerned either with finding conditions under which the power of a player can be expressed by a real number or conditions under which  all players can be ranked in accordance to their power.

The concept of a trading transform will be introduced and several numerical functions which characterise the game will be introduced too. The emphasis will be on games that have extremal values of those parameters. Gabelman games will be considered in particular.

The talk will not present anything new. A week later Tatyana Gvozdeva will discuss some new results.

SPEAKER:  Matthew Ryan (Economics)

 

DAY & TIME: 11am, 30 April

 

VENUE: OGGB Room 6115

 

TITLE: “Stochastic Choice and Expected Utility”

 

DESCRIPTION/ABSTRACT:  This talk will survey some recent developments in the theory of probabilistic choice, a branch of decision theory that treats the act of choice as inherently stochastic.  Much of this theory has applications in econometrics.  For example, McFadden’s logit model is based on a famous result by R. D. Luce.  Most stochastic choice models are probabilistic extensions of models of utility over deterministic objects.  Two recent papers (one by Dagsvik in Mathematical Social Sciences in 2008, and another by Gul and Pesendorfer in Econometrica in 2006) extend these ideas to choice amongst gambles, obtaining stochastic versions of expected utility.  This talk will mostly discuss the contribution by Dagsvik.  I hope to give a follow-up talk to the MSS Group later in May, which will discuss an alternative proof of one of Dagsvik’s results.

Speaker: Reyhaneh Reyhani
Affiliation: Computer Science Department, The University of Auckland
Title: New measures of the difficulty of manipulation of voting rules
Date: Monday, 27 Apr 2009, 3:00 pm
Location: Room 401

Voting systems as a method for aggregating different opinions of group members are used extensively in different fields. Except for dictatorships, all voting systems are susceptible to strategic manipulation. From the perspective of mechanism design, it is generally regarded desirable to minimize the occurrence of strategic manipulation of voting rules.

One method for designing a safer voting system against strategic manipulation is to find rules that minimize the number of situations in which manipulation can succeed.

In this talk, we introduce new measures of manipulability of anonymous voting rules and argue for their superiority over some commonly used measures. We give a simple common framework that describes these measures and connects them to recent literature. We discuss their computation and present numerical results that allow for comparison of various common voting rules. This is joint work with Geoffrey Pritchard and Mark Wilson.

Finally, I will mention some issues that I intend to consider in future for my PhD thesis.

Speaker: Geoff Pritchard
Affiliation: The University of Auckland
Title: Impartial-culture asymptotics: a central limit theorem for manipulation of elections
Date: Monday, 30 Mar 2009
Time: 4:00 pm
Location: Room 401 (small math seminar room)

We consider the problem of manipulation of elections using positional voting rules under Impartial Culture voter behaviour. We consider both the logical possibility of coalitional manipulation, and the number of voters that must be recruited to form a manipulating coalition. It is shown that the manipulation problem may be well approximated by a very simple linear program in two variables. This permits a comparative analysis of the asymptotic (large-population) manipulability of the various rules. It is seen that the manipulation resistance of positional rules with 5 or 6 (or more) candidates is quite different from the more commonly analyzed 3- and 4-candidate cases.

This is joint work with Mark Wilson, to appear in Mathematical Socal Sciences. Slides for the talk are available.