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Axiomatic Consensus Theory in Group Choice and Biomathematics

William H. E. Day and F. R. McMorris
Publication Date: 
Number of Pages: 
Frontiers in Applied Mathematics 39
[Reviewed by
Donald G. Saari
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It takes a wild stretch of imagination to relate the infamous 2000 Florida presidential election, your right to make personal decisions, the ultimate choice of a fine single-malt scotch, the ability to reach consensus, analyzing DNA structure [Waterman], and finding classifications within the field of biology. Yet, if it can be done, be confident that mathematics must play a central unifying role. Although the book under review does not discuss these pragmatic concerns, all of them, and many others, are direct applications and consequences of the mathematics discussed.

What does the book cover? Day and McMorris, both dominant players in this research area, describe ways in which a wide variety of different kinds of information can be aggregated, and there are unexpected consequences. Clearly, it is not possible to address all forms of information, but the kinds Day and McMorris can handle is surprisingly inclusive including where the information, such as the DNA structure, or data from psychology, or opinion polls, can be described with complete transitive rankings, or partial orders, or trees, or hierarchies, or lattices, or... What adds to the surprise is that although some of the approaches they describe seem to be reasonable, they prove that these methods do not do what we think they do. After describing what cannot be done, Day and McMorris then describe what can be done. But rather than outlining the contents of the book, it is more valuable to provide a sense of what is going on.

A way to set the stage for the first three chapters is to remind you of Arrow's famous theorem — the first topic Day and McMorris describe. In his 1951 PhD thesis, which contributed to his 1972 Nobel Prize in Economics, Arrow characterized all decision rules that possess some obviously desired properties. A convenient way to describe his result to mathematicians is with the structure of a domain, range, and a mapping. I will call them, respectively, the "inputs," "outputs," and "decision or voting rule."

For inputs, let each person have a complete, transitive ranking of the candidates: there is no restriction on what ranking a voter selects. This condition makes sense; to have rational outcomes, then the inputs, the voters' preferences, should be rational (i.e., transitive). The output is the societal ranking of the candidates: we require it to be transitive. All that remains is to specify the mapping, or decision rule. But rather than being specific, Arrow considers a class of voting rules that satisfy the following two rather innocuous appearing properties.

The first condition is called the Pareto condition: think of it as an unanimity condition. Namely, if everyone ranks some pair of candidates in the same way, then that ranking should be the pair's societal ranking. To introduce the second condition, why should a pair's societal ranking be determined only in the very special setting of unanimity? Why not always rank each pair in terms of what the voters think of these two alternatives?

To illustrate the kind of problems we want to avoid, imagine a department forced to select only one of the three candidates, Barbara, Connie, and Donna, for a tenure track position where the top two choices are Barbara and Connie. Suppose that after much discussion, the personnel committee announces that Barbara will be offered the position in accordance with its ranking of Barbara > Connie > Donna. But imagine the departmental outcry if the committee admits that the ranking of Barbara and Connie was so close that if a couple of committee members had a better opinion of Donna, then Connie would have been top-ranked and selected. What is going on? What does Donna have to do with the relative ranking of Barbara and Connie and the ultimate choice of Barbara? To avoid these kinds of troublesome outcomes, Arrow imposed the condition of binary independence. This condition requires the societal ranking of each pair to be strictly determined by how the voters rank this particular pair. In other words, when determining the societal ranking for a particular pair, all other information is irrelevant.

Arrow then characterized all possible decision rules for three or more candidates that satisfy these conditions: there is only one, a dictatorship! Namely, there is precisely one voter who is endowed with the powerful role where the societal ranking must always agree with his personal ranking: this dictator need not be benevolent because the societal outcome is totally independent of what the other voters want.

A dictator! Outside of the influence imposed by some departmental chairs, rarely in our society or daily life do we use, or even consider, dictatorial decision rules. As an immediate illustration, the widely used plurality vote is not dictatorial. What we learn from Arrow's Theorem about the plurality vote or any other non-dictatorial rule, then, is that there must exist situations where this rule violates one of the posed conditions. This observation brings us to the 2000 presidential elections in Florida: Bush won the election, but if fewer voters liked Nader, Gore would have won. In other words, in a head-to-head election Gore would have been the strong winner, which means that our plurality vote violates binary independence. For references and further reading about Arrow's Theorem, let me suggest Arrow's seminal book and another one, Saari, [Decisions], that for obvious reasons I particularly like. In [Decisions], I show that Arrow's theorem admits a far more benign interpretation than given above and that we can avoid Arrow's "no method is fair" implication. (While [Decisions] is written in a manner to make the new results available to an audience beyond the mathematics community, the math reader probably will detect from the central arguments that Arrow's result occurs because a Z2 x Z2 x Z2 orbit differs from an S3 orbit.)

A bit later I will weave Arrow's Theorem back into the discussion about the book under review, but for now, let's turn to scotch. Not any scotch, but fine, smooth single-malt scotch! An argument that must rage among connoisseurs, whether professionals or amateurs, is to determine which is the better one: this deep thirst for the truth requires perpetual experimentation and careful tasting. To put a cork on all of this fun, maybe an answer can be found from mathematics and existing opinions. In a paper that received considerable attention, even from the popular press (for obvious reasons), Lapointe and Legendre in [Scotch] used consensus theory to develop a tree structure ranking for the different single-malts in terms of quality and other variables. Different connoisseurs had different opinions: opinions that, rather than expressed in the form of a transitive preference ranking, had to be captured with a tree structure. The emergence of tree structures is easy to understand just by imagining a school teacher asking students to line up alphabetically based on the last name and according to height. To handle conflicting data, Lapointe and Legendre used consensus theory to find the "consensus" tree structure. The point is that if this theory, which is indicated next, can handle scotch, it can be used to address a wide range of other topics where we need to combine data in a manner to have rankings and comparisons, and to make decisions.

Indeed, this is what consensus theory tries to do. To make a connection with the earlier "input-output-decision rule" story, rather than restricting attention to data satisfying the powerful transitivity structure, consensus theory allows the data, and outcomes, to have more general structures that can be expressed in tree formats. In this manner, the topics that can be addressed extend beyond voting to include issues from biology, chemistry, molecular structures, psychology, etc. — even scotch. The demands of these different areas from the social, biological, and behavioral sciences quickly lead to settings where inputs and outputs need to be described in terms of a variety of rooted and unrooted trees.

To indicate what is going on, think of voting as where each voter has a particular tree structure of preferences; namely, our personal transitive ranking of the candidates. If we all agree, then the outcome is obvious: the societal "tree ranking" is the unanimous ranking. The purpose of voting is to handle those settings where the data fails to agree; it is to find the appropriate tree structure — a transitive ranking of the candidates — that best fits all of the input data. The goal of "consensus theory" is to do the same for far more general settings than allowed by transitivity. If the data for a particular discipline must be expressed in a tree structure, then different data sets probably define different trees. Just as voting searches for the "best fit" for the societal ranking, in the more general settings coming from biology and elsewhere, the objective is to make sense out of the data by finding an appropriate tree structure that best fits the data. This is the goal and theme of "consensus theory" where, rather than voting, clever ways are developed to find a "consensus" out of conflicting data expressed in a tree form. While this topic clearly is of importance and value, I doubt if more than a couple of handfuls of the readers of this review know much about this area.

A natural question about consensus theory just jumps out and nearly attacks us. If the highly structured setting of "transitive preferences" allows an impossibility theorem, how do we know whether we can do what we want to do in this deep, more unstructured forest where data is expressed in terms of trees? This "can we do what we want to do?" issue has consumed a portion of the consensus theory community for decades. In what I found to be a satisfying manner, Day and McMorris explore the existence of consensus rules that satisfy different sets of clearly desired properties, but the news they report can be discouraging.

These difficulties bring us right back to Arrow's Theorem. This is because many of the negative results in this area can be, and are, described as extending Arrow's seminal result from transitive rankings to different classes of trees. That is, rather than transitive rankings, the spaces of inputs and outputs now have the structure of certain kinds of trees. The classes of decision rules are replaced with classes of consensus rules expressed in terms of desired properties. Some mimic the above Pareto, or unanimity, condition and the binary independence requirement, but other properties differ. A frequent conclusion, which Day and McMorris describe in detail, is a negative Arrovian conclusion. But to avoid leaving us with a negative sense of frustration, the authors also explore interesting ways to obtain positive conclusions.

We now arrive at the final issue: is this book for you? Who should read this book to learn more about this intriguing topic? If you are an expert, then, without question, get a copy for your personal library! This is because Day and McMorris put together, in one place and in a manner that allows us to compare the relationship among results, a long overdue exposition of these negative and positive assertions. But notice how my "If you are an expert" qualification suggests a "For members only" readership. Unfortunately, this is the case. In my opinion, if you are not an expert, or have frequent access to an expert, then to get anything out of this book, you will need to exercise extreme will power and dedication to plow through the (maybe overly) abstract representation that is essentially devoid of explanatory examples and applications. For the newcomer to this area, this book might confuse and frustrate rather than enlighten and encourage. But if you know something about this field, then, yes, this book was intended for you.

This book's limitation on who can read it with value added probably was a necessary compromise so that the authors could include the considerable amount of information in one place. But this restriction to "experts only" should not continue. Indeed, I have had the good fortune to hear both Day and McMorris give talks about how these tools can be used to further our understanding of a surprisingly wide range of topics going from libraries to biology to anywhere where "this is better (stronger, before, etc) than that" makes sense. Based on their lectures, I suspect that they could write a bang-up book introducing the "rest of us" to this field: an introduction complete with many examples and a strong dose of intuition. If Day and McMorris are not interested, I hope that somebody will write such a book. This is too nice of a topic to keep hidden.


[Arrow] Arrow, K. Individual Values and Social Choice, Wiley, 1951 (2nd ed. 1963).

[Decisions] Saari, D. G., Decisions and Elections; Explaining the Unexpected, Cambridge University Press, 2001.

[Scotch] Lapointe, F-J, and P. Legendre, "Classification of pure malt scotch whiskies." Appl. Statist., 43 (1994), 237-257.

[Waterman] Waterman, M. S., "Consensus patterns in sequences." In Mathematical Methods for DNA Sequences, ed. Wateman, M. S., CRC Press, 1989.

Donald G. Saari is Distinguished Professor of Mathematics and Economics and Director of the Institute for Mathematical Behavioral Sciences and of the Center for Decision Analysis at the University of California at Irvine.

List of Figures; List of Tables; Foreword; Preface; Chapter 1: Achieving Consensus; Chapter 2: Axiomatics in Group Choice; Chapter 3: Impossibilities in Bioconsensus; Chapter 4: Possibilities in Bioconsensus; Chapter 5: General Models of Consensus; Chapter 6: Beyond Consensus; Quick References; Bibliography; Index.