# [EM] serious strategy problem in Condorcet but not in IRV?

Jobst Heitzig heitzig-j at web.de
Sun Apr 25 06:32:02 PDT 2004

```A week ago I suggested the following class of methods to handle the
strategic problems James described under the above subject line:

> Def. COARSE<whatever>//APPROVAL:
> 1. Coarsen the information about majorities by rounding each majority's
> magnitude s to the smallest element of {1/2, 2/3, 3/4, 4/5, ..., 1}
> which is greater or equal to s.
> 2. To these "coarsened" magnitudes, apply some magnitudes-based method
> without its tiebreaker, for example beatpath, Tideman or the river
> method. This reduces the options to a set of "acceptable" options.
> 3. Pick the most approved acceptable option.

Upon closer inspection, however, it is not at all clear what it means to
apply methods like MAM or River "without its tiebreaker" as required by
step 2, because those methods use the tiebreaker in intermediate steps
and not just at the end...

What I suggest is to define "precedence" of majorities in a way like this:

Def. PRECEDENCE OF MAJORITIES BY CLASS AND APPROVAL:
----------------------------------------------------

Let MAG(A>B)   designate the number of voters
strictly preferring A to B.
Let CLASS(A>B) designate the largest positive integer
such that mag(A>B) > (k-1)/k.
Let A>B        designate the fact that mag(A>B) > mag(B>A).
Let APP(A)     be the approval count of A
(as defined by cutoffs on the ballots, e.g.).
Let RVHR(A)    be the rank of A in some random voter hierarchy
(for example defined as in MAM).
Let SCORE(A>B) designate the "score-tuple"
(class(A>B),-app(B),mag(A>B),rvhr(B),-rvhr(A)).

Assume that A>B and C>D. Then A>B is said to PRECEDE C>D
IF SCORE(A>B) IS LEXICOGRAPHICALLY LARGER THAN SCORE(C>D).

In other words, A>B precedes C>D if either
(i)   class(A>B)>class(C>D), or
(ii)  class(A>B)=class(C>D) and app(B)<app(D), or
(iii) class(A>B)=class(C>D), app(B)=app(D), and mag(A>B)>mag(C>D), or
(iv)  app(B)=app(D), mag(A>B)=mag(C>D),
and B is below D in the r.v.h., or
(v)   B=D, mag(A>B)=mag(C>D), and A is above C in the r.v.h.

This "precedence" first looks "coarsely" at the maginitudes of the
majorities, only considering the "class" of majority (either 1="simple
majority", 2="absolute majority", or 3,4,...="qualified majorities"). It
then looks at approval counts before considering the exact magnitudes.

The formal definition of precedence as a lexicographical ordering by
score-tuples makes it easy to compare with other definitions of
precedence. Standard MAM, for example, uses the score-tuples
(mag(A>B),-mag(B>A),rvhr(B),-rvhr(A))
(class(A>B),-app(B),mag(A>B),rvhr(B),-rvhr(A)).
Because of the similarity of the construction, I am quite hopeful that
most of Steve's proofs will be adaptable to definitions of "precedence"
with different score-tuples if only the individual entries of the tuple
behave nicely (e.g., monotonic).

Other suggestions of score-tuples:
(class(A>B),-app(B),rvhr(B),-rvhr(A))
(class(A>B),-class(B>A),-app(B),app(A),mag(A>B),-mag(B>A),rvhr(B),-rvhr(A))
...

A final remark for Ernest: When using classes of majorities, it could
also be easier to present the results graphically by using a hierarchy
of colours or line-types, for example
1 gray or dotted arrows
2 black/solid
3 blue/double
4 green/triple
etc.

Jobst

```