Axiomatics in Bioconsensus
Bioconsensus is a rapidly evolving field in which consensus rules,
having been developed for use in social choice theory, are adapted for
use in areas such as systematic biology and evolutionary or molecular
biology. Given a set 
of objects of interest, a basic problem of
bioconsensus is to identify an appropriate consensus rule
that, when presented a 
-tuple 
of objects
of , returns from a unique consensus object that in some
sense best represents ; if is the set of all hierarchical
classifications or all partitions of a set, then the problem concerns
finding a consensus of classifications. Solving this problem requires
evaluating consensus rules in terms of the basic properties (axioms)
that they satisfy. Sometimes one can obtain an impossibility result
by listing a set of reasonable axioms that any consensus rule on
should satisfy, then proving that no such rule can exist. Such a
contradiction encourages one to explore how to weaken the axioms while
maintaining the contradiction, or how to alter the axioms so as to
remove the contradiction. Such analyses may yield characterizations of
consensus rules in terms of axioms with which users may assess the
appropriateness of the rules for given applications. Although the
axiomatic approach may seem to be abstract and purely technical, its
practical and concrete aspects enable one to discover what is
realizable and what is not.
This short course explores how axiomatics can be used to investigate
consensus rules in classification, clustering, and systematic biology.
I describe how axiomatics can establish the nonexistence of suitable
consensus rules and the existence of unique rules or families of
rules. The rules discussed include majority, plurality, and median
rules; the objects on which they act include partitions of a set,
hierarchical structures, unrooted trees, and sequences. The course
is introductory and assumes only that participants have a certain
familiarity with mathematical reasoning, a certain capacity for
abstract thought, and a certain interest in how mathematics is used to
model aspects of biological problems.
Related Reading
W. H. E. DAY AND F. R. MCMORRIS, Axiomatic Consensus Theory
in Group Choice and Biomathematics. Society for Industrial and
Applied Mathematics, Philadelphia, Pennsylvania, 2003.
M. A. STEEL, A. W. M. DRESS, AND S. BSOCKER, Simple but
fundamental limitations on supertree and consensus tree methods,
Systematic Biology, 49(2000), pp. 363-368.