METROPLEX AGANT SEMINAR
(Algebraic Geometry, Algebra and Number Theory Seminar)
a joint venture of UNT, UTA and TCU
Date/Time/Room: Fri April 28, 2006, at 4:00 pm in
PKH1 487 at UTA
Refreshments will be served at 3:45 pm.
Speaker:
Irinel Dragan,
Department of Mathematics, UTA
``"On Quasi-cores, the Shapley Value and
Semivalues''
Abstract:
The presentation will start by an example of a cooperative game, in
order to make clear the meaning of the concept of Solution, and Core
solution. Quasi-cores have been introduced by L. S. Shapley and M. Shubik
(1966), in connection with market games. A comprehensive survey on the
Core, by Y.Kannai (1992), where some quasi-cores are discussed, can be
found in the Handbook of Game Theory Vol.1, Chapter 12. In this talk,
we define and interpret more general quasi-cores and some byproducts,
and necessary and sufficient conditions for the nonemptiness of the
quasi-cores, which are derived from the Bondareva-Shapley theorem for
the nonemptiness of the Core. The quasi-cores will be used to solve the
problem of the appurtenance of a Semivalue to the Core.
The problem of the appurtenance of the Shapley value to the Core was
solved by E. Inarra and J. M. Usategui (1993), and different forms of
the results were given by J. Marin-Solano and C. Rafels (1996) and Y.
Izawa and W. Takahashi (1998). The Core and Quasi-cores are given by
mixed linear systems of inequalities defined on the linear space of
cooperative n-person games with a given set of players. The Shapley
value is a linear operator with a set of properties defined by a system
of axioms on the same space. The Core is considered a good solution in
economics, due to its properties; the Shapley value has also its
properties given by the axioms. Clearly, the appurtenance to the Core
is providing an element of the Core with more properties than the usual
ones, and sufficient conditions for the nonemptiness of the Core. A
family of more general linear operators called the Semivalues would
provide also solutions with less properties, but an opportunity to choose
in the family, where the Shapley value is a member. Therefore, the
appurtenance of the Semivalues to the Core is an important problem.
In this talk, we give necessary and sufficient conditions for this event
to happen. In order to derive these conditions, we need several
intermediate steps: conditions for the appurtenance of the Shapley value
to a symmetric quasi-core, the connection between Semivalues and the
Shapley value and the Average per capita formulas for Semivalues and the
efficient normalization of a Semivalue, which give these linear operators
as functions of averages, instead of as values of the characteristic
function of the game.
Please mention/email this talk to any interested parties
and to your students if you are teaching an advanced undergraduate algebra
class or a graduate algebra class.
1Driving & Parking (Revised Jan 12, 2006)
If you are travelling from out of town, then a map of UTA may be found at
http://www.uta.edu/maps/. Note that
PKH is the triangular building in the central-campus region of the map.
Click on central campus to zoom in, & PKH is in the SE corner of that
submap, next to the "garage" & is on Nedderman Drive between College St
(aka Planatarium Place) & West St.
West St is one-way northbound between W Mitchell St & 3rd St. Parking
is available in the nearby (East & SE of PKH) parking lots with a UTA
permit (but many people park there late on Fridays with no permit & are
not caught). There are a few metered parking spots on the East side of
Pickard Hall on West St. Visitor (free), & metered, parking is available
by Davis Hall (marked DH in SW part of the submap). A parking garage
(expensive) is nextdoor to PKH (use West St to get to it). Note that if
heading south on Cooper, then no left turn is allowed to Nedderman Dr.
Recommended route from UNT: eventually head south on Hwy 360 (or Industrial
Blvd = Collins), West on Abram, south (left) on Center, West (right) on
W Mitchell. Then use map. Alternatively, get to I30 heading East, & follow
directions from TCU given below. From TCU: east on I30, exit Cooper, South
on Cooper, left (East) on W Mitchell, then use map. (Note that Nedderman
Drive between West St & Pecan St is permanently closed.)
Note that PKH 487 is on the 4th floor of PKH; turn right out of elevator
on 4th floor.
Questions, etc on arrangements/directions? Contact Michaela Vancliff.