<HTML><BODY style="word-wrap: break-word; -khtml-nbsp-mode: space; -khtml-line-break: after-white-space; "><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">Dear Elections List,</SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">Some comments about apportionment.</SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">1. The major contribution of E.V. Huntington</SPAN></FONT><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"> </SPAN></FONT><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">to the study of apportionment methods was to call attention to fairness questions with regard to moving one seat assigned to some state to another state. This led him to study "divisor methods" and their properties. Among several contributions of Balinski and Young was to call attention to problems created by ties. Ties (using a particular method) can occur not only when two states have the same population but also when they have different populations but the house size is a particular value.</SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">Consider the example of where Webster is applied to a house size h = 4 for two states with populations of 300 and 500, and each state should have, as usual, at least one seat. Since the total population is 800 and 4 seats are to be apportioned, the ideal district size should be 200 people. Thus, the state with population 300 is entitled to 300/200 = 1.5 seats which using the Webster rounding rule means 2 seats while the state with 500 people is entitled to 500/200 = 2.5 seats which using the Webster rounding rule means 3 seats. Thus, we have not assigned the proper number of seats. However, if one changes the divisor 200 up or down a little bit Webster will not assign exactly 4 seats. Thus, for a house size of 4 with these populations the "Webster method" must have a prevision for how to choose to give the extra seat to either the 300 state or the 500 state. It is not difficult to find examples and house sizes where the Webster method may have many simultaneous ties similar to the example just provided. For house sizes of 3 of 5 the situation is straightforward.</SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">From a mathematical point of view, therefore, one can not describe a method by saying "by trial and error" find a "value" of some parameter that distributes exactly h seats; no such number may exist. (Find a fraction a/b (b not zero, a and b integers) whose square is 2. No such fraction exists.) One must show that a solution exists or describe the situations where one gets into trouble and resolve what to do in those cases. Huntington did this, and Balinski and Young did it even more carefully.</SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">2. Huntington, Balinski, and Young have a very precise meaning for what they call a "divisor method." There are definitely infinitely many divisor methods in the technical sense and many methods that are described in some very different way from, say, the rounding rule approach to divisor methods, turn out to be a divisor method. On the other hand, just because one does some "divisions" that does not mean that one has produced a divisor method in the sense that Huntington, Balinski, and Young use this term. Furthermore, suppose that one describes some apportionment and it really is a divisor method in the sense of Huntington, Balinski, and Young, then it is subject to conclusions that were discovered by Huntington. Balinski and Young discuss the idea of "stability" with regards to transfer of a seat between two states using one of the sixteen different ways of expressing the ways that state i can be better off than than state j with respect to inequalities involving the population of the states and the number of seats they are assigned. (See page 101 and 102 of Fair Representation by Balinski and Young, where they explain Huntington's ideas in a fairly nontechnical way.) Quoting Balinski and Young:</SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">"Not every measure of inequality gives stable apportionments: for some measures there exist problems for which every apportionment can be improved upon by some transfer. Huntington showed that, except for four such "unworkable" measures, all others resulted in the methods of either Adams, Dean, Hill, Webster or Jefferson."</SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">What this means in practice is that if one develops a method which is actually a divisor method, however, it is described, if it is not one of the 5 "classical divisor methods" it has the unattractive property of not behaving nicely for having transfers between pairs of states which is "stable." Now if one wants to one can say that issues of transfer equity between pairs of states does not matter, just as one can say that violation of quota does not matter or that violation of population monotonicity does not matter. There are different views on what make methods reasonable and fair.</SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">3. Whereas in America the practical debates about how to apportion the House of Representatives has turned on discussions of measuring inequities between pairs of states, in Europe's attempts to cope with apportionment problems this approach did not surface. Rather, there was a tradition of using methods of finding a "global" discrete optimal answer. For a detailed look at this point of view you should consult:</SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">Evaluation and Optimization of Electoral Systems, by Pietro Grilli di Cortona, Cecilia Manzi, Aline Pennisi, Federica Ricca, and Bruno Simeone, SIAM Monographs on Discrete Mathematics and Its Applications, Philadelphia, 1999. (All of the authors of this book teach at universities or work in Rome, Italy.)</SPAN></FONT><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"> </SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">Perhaps not surprisingly, the 5 classical methods also show up in this very different approach involving "global optimization."</SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">4. The word "biased" means many things in common parlance and it also can be given many technical definitions. To be "proportional' a method should give states with increasingly large populations more seats than what it gives to smaller states. A method which does not do so has a form of "bias." There are different ways to try to measure "bias" in failing to be "proportional." On the other hand if a method tends to give groups of larger states more than their fair share as compared with groups of smaller states (using some definition of small and larger) then this is also a form of "bias." Again there are different approaches to measuring such bias. Bias of methods from different perspectives can be looked at in an abstract theoretical way or it can be looked at as an empirical question for data which has been used in the historical context.</SPAN></FONT><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"> </SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">5. For many years I have been teaching a fairness and equity course both as part of my college's program in Humanities, in our Liberal Studies Program, Honors Program, and mathematics courses.</SPAN></FONT><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"> </SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><FONT class="Apple-style-span" face="Helvetica"><A href="http://www.york.cuny.edu/~malk/courses/fairness-outline.html">http://www.york.cuny.edu/~malk/courses/fairness-outline.html</A></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><FONT class="Apple-style-span" face="Helvetica"><A href="http://www.york.cuny.edu/~malk/courses.html">http://www.york.cuny.edu/~malk/courses.html</A></FONT><FONT class="Apple-style-span" face="Helvetica"></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR class="khtml-block-placeholder"></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">Often, at the start of the course, students ask me what insights mathematics can have into these questions. By the end of the course they typically have some answers. For many fairness situations there are so many properties of fairness that one would like a "fair method" to have in solving the problem that mathematicians have been able to prove that there is no method that obeys all of the desired fairness conditions. One reason there is so much acrimony about fairness questions is that there are groups of people who say conditions X, Y, and Z are the essential ones and others who say conditions A, B, and C are the essential ones. Since there are no methods that obey X, Y, Z, A, B, and C these groups endless argue even though often there are methods that they might reach a consensus on as being better than the methods currently in use.</SPAN></FONT><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"> </SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">Another thing mathematics can do is for each "reasonable" method Z find a list of axioms that are satisfied by method Z but no other method. When one has such axioms for each method that one might think is appealing one can see more clearly what one gains and loses by adopting one method rather than another.</SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">Cheers,</SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" face="Lucida Bright" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;">Joe</SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Lucida Bright; min-height: 15px; "><BR></DIV><BR><DIV> <SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 16px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 16px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 16px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 16px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 16px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><DIV><BR class="khtml-block-placeholder"></DIV><DIV>------------------------------------------------</DIV><DIV>Joseph Malkevitch</DIV><DIV>Department of Mathematics</DIV><DIV>York College (CUNY)</DIV><DIV>Jamaica, New York 11451</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Phone: 718-262-2551 (Voicemail available)</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>My new email is:</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV><A href="mailto:malkevitch@york.cuny.edu">malkevitch@york.cuny.edu</A></DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>web page:</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV><A href="http://www.york.cuny.edu/~malk">http://www.york.cuny.edu/~malk</A></DIV><DIV><BR class="khtml-block-placeholder"></DIV><BR class="Apple-interchange-newline"></SPAN></SPAN></SPAN></SPAN></SPAN> </DIV><BR></BODY></HTML>