Saari has been widely quoted as an expert in voting methods and lottery odds. He is opposed to the use of the Condorcet criterion in evaluating voting systems, and among positional voting schemes he favors using the Borda count over plurality voting, because it reduces the frequency of paradoxical outcomes (which however cannot be avoided entirely because of Arrow's impossibility theorem). For instance, as he has pointed out, plurality voting can lead to situations where the election outcome would remain unchanged if all voters' preferences were reversed; this cannot happen with the Borda count. Saari has defined, as a measure of the inconsistency of a voting method, the number of different combinations of outcomes that would be possible for all subsets of a field of candidates. According to this measure, the Borda count is the least inconsistent possible positional voting scheme, while plurality voting is the most inconsistent. However, other voting theorists such as Steven Brams, while agreeing with Saari that plurality voting is a bad system, disagree with his advocacy of the Borda count, because it is too easily manipulated by tactical voting. Saari also applies similar methods to a different problem in political science, the apportionment of seats to electoral districts in proportion to their populations. He has written several books on the mathematics of voting.[S94][S95a][S01a][S01b][S08]
In economics, Saari has shown that natural price mechanisms that set the rate of change of the price of a commodity proportional to its excess demand can lead to chaotic behavior rather than converging to an economic equilibrium, and has exhibited alternative price mechanisms that can be guaranteed to converge. However, as he also showed, such mechanisms require that the change in price be determined as a function of the whole system of prices and demands, rather than being reducible to a computation over pairs of commodities.[SS][S85][S95b]
In celestial mechanics, Saari's work on the n-body problem "revived the singularity theory" of Henri Poincaré and Paul Painlevé, and proved Littlewood's conjecture that the initial conditions leading to collisions have measure zero. He also formulated the "Saari conjecture", that when a solution to the Newtonian n-body problem has an unchanging moment of inertia relative to its center of mass, its bodies must be in relative equilibrium. More controversially, Saari has taken the position that anomalies in the rotation speeds of galaxies, discovered by Vera Rubin, can be explained by considering more carefully the pairwise gravitational interactions of individual stars instead of approximating the gravitational effects of a galaxy on a star by treating the rest of the galaxy as a continuous mass distribution (or, as Saari calls it, "star soup"). In support of this hypothesis, Saari showed that central configurations (simplified mathematical models of galaxies as systems of large numbers of bodies arranged symmetrically on circular shells) could be made to rotate as a rigid body rather than with the outer bodies rotating at the speed predicted by the total mass interior to them. According to his theories, neither dark matter nor modifications to the laws of gravitational force are needed to explain galactic rotation speeds. However, his results do not rule out the existence of dark matter, as they do not address other evidence for dark matter based on gravitational lenses and irregularities in the cosmic microwave background. His works in this area include two more books.[SX][S05]
Overviewing his work in these diverse areas, Saari has argued that his contributions to them are strongly related. In his view, Arrow's impossibility theorem in voting theory, the failure of simple pricing mechanisms, and the failure of previous analysis to explain the speeds of galactic rotation stem from the same cause: a reductionist approach that divides a complex problem (a multi-candidate election, a market, or a rotating galaxy) into multiple simpler subproblems (two-candidate elections for the Condorcet criterion, two-commodity markets, or the interactions between individual stars and the aggregate mass of the rest of the galaxy) but, in the process, loses information about the initial problem making it impossible to combine the subproblem solutions into an accurate solution to the whole problem.[S15] Saari credits some of his research success to a strategy of mulling over research problems on long road trips, without access to pencil or paper.
Saari grew up in a Finnish American copper mining community in the Upper Peninsula of Michigan, the son of two labor organizers there. Frequently in trouble for talking in his classes, he spent his detention time in private mathematics lessons with a local algebra teacher, Bill Brotherton. He was accepted to an Ivy League university, but his family could only afford to send him to the local state university, Michigan Technological University, which gave him a full scholarship. He majored in mathematics there, his third choice after previously trying chemistry and electrical engineering.
He received his Bachelor of Science in Mathematics in 1962 from Michigan Tech, and his Master of Science and PhD in Mathematics from Purdue University in 1964 and 1967, respectively.
At Purdue, he began working with his doctoral advisor, Harry Pollard, because of a shared interest in pedagogy, but soon picked up Pollard's interests in celestial mechanics and wrote his doctoral dissertation on the n-body problem.
After taking a temporary position at Yale University, he was hired at Northwestern University by Ralph P. Boas Jr., who had also been doing similar work in celestial mechanics.
From 1968 to 2000, he served as assistant, associate, and full professor of mathematics at Northwestern, and eventually became Pancoe Professor of Mathematics there.
He was led to mathematical economics by discovering the high caliber of the economics students enrolling in his courses in functional analysis, and added a second position as Professor of Economics.
He then moved to the University of California, Irvine at the invitation of R. Duncan Luce, who had founded the Institute for Mathematical Behavioral Sciences (IMBS) in the UCI School of Social Sciences in 1989. At UC Irvine, he took over the directorship of the IMBS in 2003, and stepped down as director in 2017. He is a trustee of the Mathematical Sciences Research Institute.
He was editor in chief of the Bulletin of the American Mathematical Society from 1998 to 2005, and published a book on the early history of the journal.[S03]
In 1985, with John B. Urenko, Saari received a Lester Randolph Ford Award for a paper they wrote demonstrating that Newton's method for the approximation of the root of a polynomial can exhibit chaotic behavior when its initial conditions are poorly chosen.[SU] He received in 1995 the Chauvenet Prize for another of his papers, relating the history of the n-body problem and showing how to use spinors to eliminate some of the singularities arising in this problem.[S90] In 1999, he and Fabrice Valognes won the Allendoerfer Award for their work on the geometry of voting schemes.[SV]
In 2001 he was elected to the United States National Academy of Sciences,
and in 2004 he was named a Fellow of the American Academy of Arts and Sciences.
He was elected as an external member of the Finnish Academy of Science and Letters in 2009,
and in the same year he became a fellow of the Society for Industrial and Applied Mathematics "for contributions to dynamics, voting, and economics".
In 2012 he became one of the inaugural fellows of the American Mathematical Society.
In 2018 he was elected as a foreign member of the Russian Academy of Sciences.
He gave the Pacific Institute for the Mathematical Sciences Distinguished Chair Lecture at the University of Victoria in 2002, speaking on the title "Mathematical Social Sciences, an oxymoron?". He has honorary doctorates from Purdue University (1989),
the University of Caen Normandy (1998),
Michigan Technological University (1999), and the
University of Turku (2009).
In 1999, a conference on celestial mechanics was held at Northwestern in honor of his 60th birthday.