Structure of voting simple games with several levels of approval (2002)

In modeling the type of voting intended to register collective approval or
disapproval of a proposal, the literature on voting systems has confined itself
almost exclusively to the mathematical structures known as simple voting
games. By their nature, simple games allow for exactly two possible levels of
approval in the �input� (voters must choose between �yes� and �no�) and two
levels in the �output� (the proposal either passes or does not pass). Yet in
many of the real voting systems that have been modeled by these games, such
as that of the United Nations Security Council, abstention plays a key role as
a middle level of voter approval.
One factor that may have retarded the study of voting games with abstention
is the absence of a completely satisfactory definition of weighed voting in this
context. In a weighted simple game with no abstentions allowed, each voter is
assigned a numerical weight, and a proposal is approved if the total weight cast
in favor meets or exceeds some preset threshold. There are two tempting ways
to modify the definition in order to account for abstention: collective approval
can require that the ratio of total weight cast in favor to total weight cast
against meet or exceed a preset threshold, or that the difference between these
totals meet or exceed such a threshold. These two approaches are distinct,
though each reduces to the standard definition when no voters abstain. With
abstentions allowed, the UN Security Council voting system fails to be weighted
under either definition.
We argue in favor of a third definition of weighted voting with abstention,
which is strictly more general than either of the two mentioned above. The
UN Security Council system is weighted under this new definition. The new
definition, generalizes to the structures we call (j, k) games, in which there
are j ordered levels of input approval and k ordered levels of output approval.
While it is the (3, 2) games and (3, 3) games that are probably most relevant to
legislative voting, these are not empty generalizations. Our examples of grading
systems suggest that level structures with larger j and k values arise naturally
as models of certain types of decision making.
The study of the weightedness of (j, k) simple voting games is solved by
means of our �grade trade robust� notion. However two possible extensions
of (j, k) weighted games arise in a natural form. We also focus our interest in
developing them.
The first extension involves preference relations among voters intending to
measure, what it means for one voter to have more influence than another. 
Those games for which the associated �desirability� relationship is total will
be called �linear�. There are several subclasses of (j, k) weighted voting games
that deserve be considered, and we will show that each one of these subclasses
may be extended by means of an appropriate linear desirability order.
A second natural extension is that of considering (j, k) simple games defined
as intersections of (j, k) weighted games. It is of interest to ask how efficiently
this can be done for a given (j, k) simple game and whether each (j, k) simple
game admits a finite intersection. We check the affirmative answer to the second
question, providing a notion of �dimension� for (j, k) simple games.