<HTML><BODY style="word-wrap: break-word; -khtml-nbsp-mode: space; -khtml-line-break: after-white-space; ">Dear Election List,<DIV><BR class="khtml-block-placeholder"></DIV><DIV>If you look at the technical papers on apportionment, especially Balinski and Young's work, you will see that one of the most difficult problems to address is the issue of "ties." Ties not only occur when two states have identical population but also when you apply the divisor interpretation of the the "divisor" methods. Loosely speaking the reason why divisor methods violate not giving a state the integer above or below its exact quota (when its exact number of seats that should be given to it is not an integer) is that states of about the same population (not to mention exactly equal population) should be treated equally from the pairwise equity point of view. What this means is that Webster, say, will give the same number of seats to states with approximately the same population even if this means being overly generous or ungenerous to other states. Thus, loosely speaking, these methods try to treat equally situated states in an equivalent manner even if this means that "quota" (as defined above) gets violated. One way to try to get around this is to use discrete optimization methods that optimize some global criterion rather than look at pairwise equity. </DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Note that Balinski and Young have a "technical meaning" for the word "bias." </DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Regards,</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Joe</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV><BR><DIV><DIV>On Dec 10, 2006, at 5:43 PM, Warren Smith wrote:</DIV><BR class="Apple-interchange-newline"><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Actually, I claim EVERY apportionment method so far discussed is biased,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">in the sense it will, under the right circumstances, systematically always-down-round</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">one class of states and always-up-round the other.<SPAN class="Apple-converted-space"> </SPAN>(Just make the small states</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">all have exactly the right sizes and the large states all have the right<SPAN class="Apple-converted-space"> </SPAN></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">sizes, and voila, this'll happen.<SPAN class="Apple-converted-space"> </SPAN>You can make pretty much all the</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">methods prefer larger or prefer smaller states, at your whim, by setting up</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">the populations right in your contrived scenario.)</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Is there a way to get around that?<SPAN class="Apple-converted-space"> </SPAN>Yes:<SPAN class="Apple-converted-space"> </SPAN>"randomized rounding."</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">The idea would be you use</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">a random number generator as part of the input into your decision to round</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">a state up or down, and in such a way the expected net gain, was zero.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Example: 5.3<SPAN class="Apple-converted-space"> </SPAN>--><SPAN class="Apple-converted-space"> </SPAN>5 with probability 0.7 and <SPAN class="Apple-converted-space"> </SPAN>--> 6 with probability 0.3.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">(That is for absolute unbiasedness.<SPAN class="Apple-converted-space"> </SPAN>Also important is ratio-unbiasedness,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">which you can also assure by the same kind of method.)</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">OK, so, here is a possible such method: do this kind of rounding.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">If the total number of congressmen comes out wrong,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">then try again, and keep trying until it comes out right.<SPAN class="Apple-converted-space"> </SPAN>The end.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">This method seems totally unbiased.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">(Incidentally, the same idea was suggested in the 1980s for rounding</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">floating point numbers inside computers.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Biases can build up and result in large errors, and randomized rounding prevents that.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">This is a good idea but no computer hardware I know of implements it.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">The "round to even" approach is often used, which tries to get unbiasedness</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">but isn't perfect.)</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Only problem with it is, it is randomized!!</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Warren D Smith</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><A href="http://rangevoting.org">http://rangevoting.org</A></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">----</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">election-methods mailing list - see <A href="http://electorama.com/em">http://electorama.com/em</A> for list info</DIV> </BLOCKQUOTE></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></DIV></BODY></HTML>