Speaker: Tatiana Gvozdeva
Affiliation: The University of Auckland
Title: Roughly weighted games with interval thresholds
Date: Friday, 30 Apr 2010
Time: 3:00 pm
Location: Room 401 (small math seminar room)

It is very well-known that not every simple game has a representation as weighted majority game. The first step in attempt to characterise non-weighted games was the introduction of the class of roughly weighted games. This concept proved to be useful. It realises a very common idea in Social Choice that sometimes the rule needs an additional tie-breaking rule that helps to decide who is the winner if the results of all candidates are on a certain ‘threshold’. We gave a criterion of rough weightedness in the previous talk at this seminar. However not all games are even roughly weighted. Hence we need a more general construction to represent all games.

In this paper we introduce and explore several concepts that generalise roughly weighted games in several directions. One idea is to make the threshold thicker, i.e. use not a number but an interval for it. A good example of this situation would be a faculty vote. If neither side controls a 2/3 majority (calculated in faculty members or their grant dollars), then the Dean would decide the outcome. We can keep weights normalised so that the lower end of the interval is fixed at 1, then the right end of the interval becomes a “resource” parameter. We show that all class of games split into the hierarchy of classes of games define by this parameter. We show that as it increases we get strictly greater descriptive power, i.e., if the resource parameter gets larger, strictly more games can be described.

A situation when the number of players n is fixed also of interest. Then there is an interval [1,s(n)], numbers from which provide us with a resource parameter for every game. There will be finitely many numbers q in this interval such that the interval [1,q] represents more n-player games that any interval [1,q’] with q'<q. We call the set of such numbers the nth spectrum. We calculate the spectrum for n<7. Also we present an upper bound for s(n).

Another hierarchy of games emerges when we restrict the number of possible values that the weight of a coalition might be when this weight is in the threshold interval.