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In 1970, the mathematical economist Amartya Sen published a troubling paper, "The Impossibility of a Paretian Liberal," which seemed to prove that no completely rational system of making social choices could accommodate individual rights. Born in India, Dr. Sen has recently resigned a professorship of economics and philosophy at Harvard to become Master of Trinity College at Cambridge University in England. In the fall of 1998 he became the first Asian to be awarded the Nobel Prize in Economics.
Famous for his work on the economics of poverty and inequality, Amartya Sen was the last person anyone would expect to establish a result so at odds with the ideals of a democratic society. For a generation, economists and mathematicians have struggled to make some sense of his unsettling theorem.
This year, another mathematical economist, Donald G. Saari of Northwestern University, has discovered how to escape the consequences of Sen's Theorem. This exit route is described in a highly readable article, "Are Individual Rights Possible?" in the April 1997 issue of Mathematics Magazine, one of the journals of the Mathematical Association of America (MAA).
Among other strong points, the article is a good introduction to how mathematicians use axioms to launch an investigation of a mathematical field. An axiom, of course, is an explicit assumption that we make at the beginning of an investigation. Everything else should then follow from the axioms and the use of accepted mathematical procedures.
Sen's axioms are quite reasonable. He supposes that society is confronted with some number k of alternatives (with k > 2 to eliminate trivial cases). Then he assumes three axioms:
It is the third axiom that incorporates individual rights. It says there are certain choices over which individuals have autonomy. Only you can decide what shoes to wear tomorrow, what newspaper to read, what flag to fly over your doorstep. Once you have decided, society must agree with your decision.
Sen then set out to see what procedures for making societal decisions were compatible with the axioms. Of course, he wanted society's choices to be transitive as well: if society chooses a over b and b over c, then it should choose a over c. What he proved was that there is no such procedure: no procedure can satisfy axioms 1-3 and be transitive in all cases.
Situations in which transitivity fails, in which a is preferred to b, b to c, and c to a, are called cyclic. It is perfectly possible for society to have cyclic preferences among three alternatives. For example, in 1992 it was possible that American voters as a group preferred Clinton to Bush, Bush to Perot, and Perot to Clinton, meaning that in two-man races Clinton would beat Bush, Bush would beat Perot, and Perot would beat Clinton (we're not arguing that this actually happened, only that it seemed possible at the time).However, society needed to choose among these three alternatives and a particular procedure called a Presidential Election, involving lots of ballots and something called the Electoral College, was used to make the choice. Sen's Theorem troubled economists and political scientists because it seemed to say that no such procedure can ever be completely rational and allow for individual rights at the same time.
Saari found a way out of this dilemma--several ways, in fact--by backing away from the problem. Instead of seeking a procedure, he decided to find out what procedures were permitted by each of the axioms. He discovered that every procedure compatible with axioms 2 and 3 allows individuals to have non-transitive preferences. In other words, Sen got in trouble because axioms 2 and 3 conflict with axiom 1!
In fact, we can get out of trouble by fixing either end of the conflict:
Social choice, in particular election procedures and methods of fair division and apportionment, is an attractive application of mathematics to real life accessible to students in both middle and high schools. Most discrete mathematics texts, such as the popular Discrete Mathematics through Applications (W. H. Freeman and Co.) now used in many high schools, contain one or more chapters on social choice, and these chapters generally require no mathematics beyond about grade 6. Obviously, this area also provides a good way to integrate mathematics with other subjects, especially social studies.
Most new mathematical theorems are beyond the ken of most of us. But not all: Saari's work reminds us that when we go to apply mathematics we find there is a lot to be learned even about very elementary things.
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First posted September 1, 1997; updated and revised November 4, 1998. Features remain online as long as they remain current; they may be updated if new information becomes available.
Copyright © 1998, Center for Mathematics and Science Education. Teachers have permission to duplicate this page for use in teaching their own classes. All other rights reserved. You are welcome to link to this page, but do not copy its contents. "Claris Works" is a trademark of the Claris Corporation.
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