Donald Gene Saari (born March 1940) is an American mathematician, a Distinguished Professor of Mathematics and Economics and former director of the Institute for Mathematical Behavioral Sciences at theUniversity of California, Irvine. His research interests include then-body problem, theBorda count voting system, and application of mathematics to thesocial sciences.

Saari has been widely quoted as an expert invoting systems^[1] andlottery odds.^[2] He is opposed to the use of theCondorcet criterion in evaluating voting systems,^[3] and amongpositional voting schemes he favors using theBorda count overplurality voting, because it reduces the frequency of paradoxical outcomes (which however cannot be avoided entirely in ranking systems because ofArrow's impossibility theorem).^[4] 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.^[5] 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.^[3] However, other voting theorists such asSteven 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 bytactical voting.^[4][6] Saari also applies similar methods to a different problem in political science, theapportionment of seats to electoral districts in proportion to their populations.^[3] He has written several books on the mathematics of voting.^{[S94][S95a][S01a][S01b][S08]}

Ineconomics, Saari has shown that naturalprice mechanisms that set the rate of change of the price of a commodity proportional to its excess demand can lead tochaotic behavior rather than converging to aneconomic 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]}

Incelestial mechanics, Saari's work on then-body problem "revived the singularity theory" ofHenri Poincaré andPaul Painlevé, and provedLittlewood's conjecture that the initial conditions leading to collisions havemeasure zero.^[7] He also formulated the "Saari conjecture", that when a solution to the Newtoniann-body problem has an unchangingmoment of inertia relative to itscenter of mass, its bodies must be in relative equilibrium.^[8] More controversially, Saari has taken the position that anomalies in therotation speeds of galaxies, discovered byVera 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 simplified mathematical models of galaxies as systems of large numbers of bodies arranged symmetrically on circular shells could be made to formcentral configurations that rotate as arigid body rather than with the outer bodies rotating at the speed predicted by the total mass interior to them. According to his theories, neitherdark 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 ongravitational lenses and irregularities in thecosmic microwave background.^[9] 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: areductionist 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.^[10]

Saari is also known for having some discussion withTheodore J. Kaczynski in 1978, prior to the mail bombings that led to Kaczynski's 1996 arrest.^[11]

Saari grew up in aFinnish Americancopper mining community in theUpper Peninsula of Michigan, the son of twolabor organizers there. Frequently in trouble for talking in his classes, he spent hisdetention time in private mathematics lessons with a local algebra teacher, Bill Brotherton. He was accepted to anIvy 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.^[12] While attending Michigan Tech, Saari joined the Beta chapter ofTheta Tau Professional Engineering Fraternity.

He received his Bachelor of Science in mathematics in 1962 from Michigan Tech, and his Master of Science and PhD in mathematics fromPurdue University in 1964 and 1967, respectively.^[13]At Purdue, he began working with his doctoral advisor,Harry Pollard, because of a shared interest inpedagogy, but soon picked up Pollard's interests in celestial mechanics and wrote his doctoral dissertation on then-body problem.^[12]

After taking a temporary position atYale University, he was hired atNorthwestern University byRalph P. Boas Jr., who had also been doing similar work in celestial mechanics.^[12]From 1968 to 2000, he served as assistant, associate, and full professor of mathematics at Northwestern, and eventually became Pancoe Professor of Mathematics there.^[14]He was led tomathematical economics by discovering the high caliber of the economics students enrolling in his courses infunctional analysis,^[12] and added a second position as Professor of Economics.^[14]He then moved to theUniversity of California, Irvine at the invitation ofR. Duncan Luce, who had founded the Institute for Mathematical Behavioral Sciences (IMBS) in theUCI School of Social Sciences in 1989.^[12] At UC Irvine, he took over the directorship of the IMBS in 2003, and stepped down as director in 2017.^[15] He is a trustee of theMathematical Sciences Research Institute.^[16]

He was editor in chief of theBulletin of the American Mathematical Society from 1998 to 2005,^[17] and published a book on the early history of the journal.^[S03]

• In 1985, with John B. Urenko, Saari received aLester Randolph Ford Award for a paper they wrote demonstrating thatNewton's method for the approximation of theroot of a polynomial can exhibitchaotic behavior when its initial conditions are poorly chosen.^[SU]
• He has honorary doctorates fromPurdue University (1989), theUniversity of Caen Normandy (1998),Michigan Technological University (1999), and theUniversity of Turku (2009).^[14]
• He received in 1995 theChauvenet Prize for another of his papers, relating the history of then-body problem and showing how to usespinors to eliminate some of the singularities arising in this problem.^[S90]
• In 1999, he and Fabrice Valognes won theAllendoerfer Award for their work on the geometry of voting schemes.^[SV]
• In 1999, a conference oncelestial mechanics was held at Northwestern in honor of his 60th birthday.^[7]
• In 2001 he was elected to theUnited States National Academy of Sciences,^[18] and in 2004 he was named aFellow of theAmerican Academy of Arts and Sciences.^[19]
• He gave thePacific Institute for the Mathematical Sciences Distinguished Chair Lecture at theUniversity of Victoria in 2002, speaking on the title "Mathematical Social Sciences, an oxymoron?".^[20]
• He was elected as an external member of theFinnish Academy of Science and Letters in 2009,^[21] and in the same year he became a fellow of theSociety for Industrial and Applied Mathematics "for contributions to dynamics, voting, and economics".^[22]
• In 2012 he became one of the inaugural fellows of theAmerican Mathematical Society.^[23]
• In 2018 he was elected as a foreign member of theRussian Academy of Sciences.^[24]
• Asteroid9177 Donsaari, discovered byEleanor Helin atPalomar Observatory in 1990, was named in his honor. The officialnaming citation was published by theMinor Planet Center on 5 February 2020 (M.P.C. 121135).^[25]

• Home page

• 1940 births
• Living people
• People from Houghton, Michigan
• American people of Finnish descent
• Economists from California
• American game theorists
• Voting theorists
• 20th-century American mathematicians
• 21st-century American mathematicians
• University of California, Irvine faculty
• Michigan Technological University alumni
• Fellows of the American Mathematical Society
• Fellows of the Society for Industrial and Applied Mathematics
• Members of the United States National Academy of Sciences
• Santa Fe Institute people
• Foreign members of the Russian Academy of Sciences
• Economists from Michigan
• 21st-century American economists

• CS1 Finnish-language sources (fi)
• Articles with short description
• Short description is different from Wikidata
• Articles with hCards
• Pages using infobox person with deprecated parameters