  Migration is a major decision to leave one place and move to another, and
involves job and life changes. The migration flow of people provides relational
information across places about which is better to live by ``voting with their
feet'' (Tiebout, 1956; Douglas, 1997). From the people's votes, in a
``democratic'' process, we quantify a descriptive statistic of liveable cities
by a potential of migration flow in Combinatorial Hodge theory. As a case
study, we measure the liveability of municipalities in Japan for specific
populations such as families with small children and women of reproductive age.
Using these potentials as dependent variables, we perform a regression analysis
to identify the factors relevant to liveability. Additionally, using the
aformentioned theoretical framework, we analytically derive the expression of
the utility as a function of given flow data, which was numerically estimated
in previous studies (Douglas & Wall, 1993; Douglas, 1997; Douglas & Wall, 2000;
Wall, 2001; Nakajima & Tabuchi, 2011). The proposed method extracts a
consistent metric of interval scale from the non-transitive, pairwise
comparison between locations and provides valuable statistics for urban
planning by policymakers.
