Decisions and Elections : Explaining the Unexpected

Rating:
Summary: Mathematician Bangs on the Pulpit: Circular Thinking Exposed
Review: In my humble opinion, although the copy editors should have been a little more attentive to some glaring typographical errors, this is a very important book, an important contribution to the mathematical development of the social sciences. Saari shows how one fundamental insight involving the subtle loss of available information when a whole is broken down into parts leads to surprising resolutions to a broad spectrum of mind boggling problems, dilemma's and paradoxes. Fundamentally, this book is all about recognizing cyclic thinking for what it is, and straightening it out.

Reading this book reminded me of another book I reviewed, titled _A Darwinian Left_, by Princeton University philosopher Peter Singer, in which he issued a call for "the development of a field of social research that shows the way towards a more cooperative society" (pg. 47). Singer should be pleased with Saari's book, as it makes a fundamental theoretical contribution along that line, and shows how to apply it.

The single most memorable part of Saari's book, to my mind (as something of a community activist), is Saari's analysis of the logic of a noise ordinance in Keweenaw County, Michigan. He uses this ordinance to illustrate how individual and societal rights can be logically consistent after all, in spite of a Theorem by another Nobel Laureate, Amartya Sen, which asserts something to the contrary. Another stand out, in my mind, is Saari's explanation of how the well known "Prisoner's Dilemma" is resolved by a slightly revised version of the Golden Rule: "Do unto others as they did to you."

Saari shows how Kenneth Arrow's seminal "Impossibility Theorem," which is often interpreted as a proof that can be no such thing as a fair and consistent voting method when there are more than 2 candidates, is based on fairness criteria which are inconsistent with one another. I learned in logic class that you can resolve a dilemma by eliminating an internal inconsistency of the argument. Saari does just that with Arrow's Theorem.

"Obviously," writes Saari, "whenever the actual conditions defining our decision procedures differ from what we intended, then unexpected conclusions and paradoxes can occur. This point, although obvious, is sufficiently important that I repeat it often enough to resemble a preacher banging on the pulpit." (pg. 26).

Many introductory math textbooks draw too strong a conclusion from Arrow's Theorem, and claim that it proves that a fair and consistent voting method is an impossibility when there are more than 2 candidates. To the contrary, his theorem only proves that there is no method which can satisfy all of his fairness criteria. In other words, Arrow proved that his criteria are inconsistent with one another. In particular, Saari shows that Arrow's "Binary Independence" criterion is inconsistent with non-cyclic outcomes. We must remember that "fairness" is not a strictly objective thing. It necessarily involves an evaluative judgment, and is based on questionable intuitions. Arrow's Theorem may be interpreted as providing a good reason to subject his fairness criteria to further scrutiny, to try to understand why his particular criteria are inconsistent with each other, and to come up with more satisfactory results with other elementary fairness criteria or axioms. Saari interprets and scrutinizes Arrow's Theorem in exactly this way, and comes up with more satisfying results. Among other things, he finds that, if Arrow's "binary independence" condition is slightly modified so as to require a procedure to pay attention to the strength of a voters preferences (he calls his version the "intensity of binary independence" condition), then the Borda Count procedure solves the problem and satisfies the theorem.

Now, I am no professional voting theorist, but I have studied this subject and his work in some depth, and I think Saari has made a very important contribution to voting theory. At least two other ground breaking voting theorists, Amartya Sen and Kenneth Arrow, have received Nobel Prizes in Economics for their contributions. It seems to me that Saari should the next.

Rating:
Summary: Cuts through the hype on voting methods & paradoxes
Review: This books cuts through much of the hype surrounding voting methods and paradoxes. Saari explains the primacy of the Borda Count when voters rank their choices. This will make the book controversial among devotees of other methods, such as Condorcet Voting, Instant Runoff Voting, and Approval Voting.

For example, Saari fully explains the paradoxes of Condorcet or pairwise tallies of voter rankings (A is ranked over B in the final tally if over half the voters rank A over B). This is due to the fact that a pairwise tally ignores the strength or intensity of a voter's ranking of A over B (is A ranked just above B? or is A ranked first and B last? etc.). Saari shows how to modify a controversial hypothesis of Arrow's Theorem in a reasonable way so that good voting methods, such as the Borda Count, do exist. However most other voting methods, such as those mentioned above, fail one or more of the modified Arrow hypotheses, and Saari shows how to construct voting paradoxes out of these failures.

Mathematician Saari has investigated other kinds of "social choice" paradox as well. However the quality of the exposition is somewhat uneven - sometimes there are brilliant illustrations of various phenomena, sometimes obtuse verbiage (I am a mathematician myself). This is an "undergraduate level" book which should accessible to a larger audience. I can recommend that Saari find a good expositer co-author to better reach that large audience.