Hello everyone, My 25 Feb posting on Pij definitions, prepared in so-called 'rich text', was judged by the latest EM-digest to be in an irreproducible format. In hopes of getting a posting which WILL reproduce in an EM-digest, the present posting restates the earlier one, but in (what I believe to be) 'plain text' format. Joe Weinstein Long Beach CA USA THE TWO Pij DEFINITIONS ARE DIFFERENT PREAMBLE AND ABSTRACT. Fellow EM-listers: Most of us on this list, like me, have other lives - both at work and elsewhere - and often lack the time to prepare intelligent postings - even when we�ve got new ideas or insights to share, or would like to rebut ill-considered criticisms of our prior postings. Even when others� postings get really silly or confusing, time constraints keep most of us helpless spectators. That�s been my situation for the past six weeks. Most frustrating has been Mike and Richard�s recent furious exchange as to whether two different definitions of Pij - a probability value defined for an impending election - really must amount to the same thing. Would-be conclusions have been obfuscated by various irrelevant asides, including dubious pronouncements and basic queries about math and logic conventions and about probabilities. Unfortunately, I lack time to answer the queries (but thanks to Anthony Simmons for a good show on the logic part) - despite the fact that I did do my math Ph.D. dissertation on logic, and now for a living (as an applied statistician) am always computing probabilities. But here in this posting I will do my best to set the record straight on the original Pij question. Namely, I give an example and use it to illustrate conclusively and derive numerically that the two Pij definitions can refer to distinctly DIFFERENT probability values - which thereby yield different �strategic voting�. I first review, restate and give names to the two definitions at issue. Next, I give an illustrative (if in some respects fanciful) example of an impending election. Then I derive and calculate the numerical probability values. Finally, I use the example�s situation and numbers to illustrate why (despite both authors� confusing asides) Richard�s main argument is correct and Mike�s main counter is erroneous. DEFINITIONS AT ISSUE. The two definitions are reviewed in a 17 Feb. posting by Richard,which begins by noting that earlier Mike wrote: > >> > Pij is the probability that i & j will be the 2 frontrunners. > > > > > >Slight correction/clarification: The precise meaning of Pij is usually > >taken to mean the probability, given a tie exists for first place, that > >i and j will be involved in that tie. > >Different wording, same thing. If i & j are the 2 highest vote-getters, >and there's a tie for 1st place, then it's between them. If they >aren't the 2 highest vote-getters, and there's a tie for 1st place, then >it isn't between them. Richard then comments: "They aren't the same. It's easy to think they'd be proportional, but keep in mind Bayes' Theorem for conditional probabilities..." My side comment: Indeed, as we shall see, the two definitions of Pij aren�t the same at all. However - as we shall also see - in order to understand and explain the difference, it is unnecessarily arcane to appeal to such devices as conditional probability or Bayes' Theorem. Richard�s extended comment may have been prompted by the following two facts, true but unneeded here: (1) one way the two definitions of Pij differ is that the first-defined probability is �unconditional� but the second is �conditional�; and (2) in some situations (not really here) Bayes' Theorem aids computations with conditional probabilities. Let�s restate and briefly name the two probability definitions (or essentially similar ones) at issue. Let i and j be any two candidates in an impending election. (1) �lead�: Pij = probability that i and j will be the two leading candidates (i.e. each will get more votes than any of the candidates other than i and j). (2) �tie�: Pij = probability, given that precisely two candidates tie for first place, that i and j will be those candidates. In some postings, the �lead� definition was ascribed to Mike; and the �tie� definition was ascribed to Bart Ingles (who in turn ascribed it to a published reference). Note that, for outcomes of a given election, we are discussing PRIOR probability values, NOT POST-facto values. After the fact, an election has but one actual outcome and its data, so post-facto outcome probabilities must be 0 or 1; but an impending election has a set of possible outcomes, each with its data - so prior probabilities need not all be 0 or 1. EXAMPLE. This example has two versions, one for lone-mark (�plurality�) voting and one for unrestricted pass-fail (�approval�) voting. In the state of Catatonia, in order to avoid a steep fine, each potential voter appears at the local polling place by 9 AM on Election Day, and remains there until 9:15 AM. Prior to Election Day, each voter has received one ballot for each race, and may mark it before entering the polling place. No marking of ballots is allowed within the polling place. Upon exit, each voter may - but need not - cast any or all of the voter�s ballots received for the various races. There are two political parties: the Hots and the Colds. Cold voters always cast their ballots. In order to decide whether to cast ballots, all Hots listen to the result - broadcast in all polling places at 9:05 AM - of the latest State Lottery random drawing of six digits (000000 through 999999). If the drawn number is 900000 or more, all Hots cast all their ballots; if not, they cast no ballots. Hot and Cold voters of both genders are very partisan. On each ballot received, each such voter will mark only suitable candidates, i.e. candidates of the same party and gender; conversely, so long as at least one suitable candidate exists, each such voter will mark at least one. Further, for every married couple (and Catatonia allows marriage only between opposite genders), if either spouse is Hot, then both are Hot; and if either is Cold, then both are Cold. The electorate comprises a grand total of T voters, split as follows: 60% of all voters are married Hots. 34% of all voters are married Colds. 4% of all voters are single female Colds. 2% of all voters are single Independents. The example�s specific election contest is for state governor, between four candidates: #1 is a female Hot; #2 is a male Hot; #3 is a female Cold; and #4 is a male Cold. PROBABILITY VALUES. It is easy to see that the partisan (i.e., Hot or Cold) 98% of the electorate will produce just two outcomes. These outcomes differ only according as the Hots do or do not cast ballots. Apart from the Independent voters� ballots, candidates #3 and #4 will always get respectively 21% x T and 17% x T votes. Suppose - as will occur with probability 10% - the Hots do cast ballots. Then, candidates #1 and #2 are the leaders - each with at least 30% x T votes. (Meanwhile, # 3 and #4 each have at most 23% x T votes). Depending on how the Independents vote, #1 and #2 may tie for first place. Suppose - as will occur with probability 90% - the Hots do not cast ballots. Then candidates #3 and #4 are the leaders and cannot tie: #3 has at least 21% x T votes and #4 has at most 19% x T votes. (Meanwhile #1 and #2 each have at most 2% x T votes). Using the �lead� definition: P12 = 10% and P34 = 90%. That is, probability is 10% that the two Hot candidates #1 and #2 will be the leaders, and is 90% that the Cold candidates #3 and #4 will be the leaders. However, using the �tie� definition, P12=100% and P34=0%. That is, considering just all cases with a first-place tie: in all of these cases the tie is between Hot candidates #1 and #2. INTERPRETATIONS AND CONSEQUENCES. The above example and numbers illustrate that each of the defined probabilities may be interpreted simply as a fraction (valued between 0 and 1 inclusive). Thus,with the �lead� definition, Pij is just the fraction of all possible outcomes which have candidates #i and #j as the leaders. With the �tie� definition, Pij is the fraction - of those outcomes which have a first-place tie - where the first-place tie is between candidates #i and #j. The two fractions have different denominators. For the �lead� definition, the denominator counts all possible outcomes, whereas the �tie� definition�s denominator counts only certain of the outcomes - namely those which meet the extra condition of having a first-place tie. For this reason, usual terminology calls the �lead� probability Pij �unconditional� and the �tie� probability Pij �conditional�. What do the two definitions� Pij values tell us about possible �strategic� voting on the part of the only voters - namely the Independents - who are open to such voting? Clearly, the only way an Independent voter can have any winner-determining effect is to break a first-place tie (among all other voters). Here, this situation can possibly occur only if her (or his) ballot is marked for exactly one of the two candidates - namely the Hot candidates #1 and #2 - who could possibly be in a first-place tie. The �tie� probability values (P12=100% and Pij=0% for all other choices of i and j with i<j) directly reflect this reality; but the �lead� probability values do not. CORRECT AND INCORRECT ARGUMENTS. A 20 February posting by Richard quoted and critiqued the crux of Mike�s argument that necessarily the lead and tie definitions must yield the same Pij value. Richard wrote: "MIKE OSSIPOFF wrote: ... >But when I showed you why those 2 probabilities can't be different, >I asked you which part of my argument you disagree with. You didn't >answer ... You didn't show they can't be different. Instead you wrote this: >1. We agree that if i & j are the 2 top vote-getters in the election, >then, if there's a tie for 1st place, it will be between them. > >2. We agree that if i & j are not the 2 biggest vote-getters in the >election, then, if there is a tie for 1st place, it will not be between >them. > >3. That means that either both of the following 2 statements are true, >or neither of them are true: > >a) i & j are the 2 biggest vote-getters in the election. >b) If there's a tie for 1st place, it will be between i & j. > >4. Since, after the election, those 2 statements are either both true >or both false, then the probability, at a time before the election >that a) will be true after the election must be the same as the >probability, >at that same time before the election, that b) will be >true after the election." As Richard goes on to comment, Mike�s argument is subtly erroneous. Namely, despite the truth of Mike�s statements 1 and 2, statement 3 in fact can in some cases be false: it does not follow from 1 and 2. Detailed reasons for this failure of derivation are worth noting. Statement 1 asserts that �if a) then b)�. Statement 2 is intended to assert that �if not-a), then not-b)�: then statement 3 WOULD follow from statements 1 and 2. However, what statement 2) actually asserts is in effect that �if not-a) then not-c)�, where c) is not quite equivalent to b). Namely, b) asserts that �if there is a first-place tie, then #1 and #2 tie for first place�, whereas c) asserts that �there is a first-place tie and (moreover) #1 and #2 tie for first place� - or, in effect, simply that �#1 and #2 tie for first place�. In fact, contra 3, statement b) can be true and meanwhile statement a) can be false. For instance, to simplify and concretize Richard�s argument, take our example with i=#1 and j=#2. For ALL our election outcomes, #1 and #2 are the only possible first-place-tied candidates, so for ALL outcomes statement b) is true: IF there is a first-place tie (at all), THEN #1 and #2 tie for first place. On the other hand, for the 90% of our example�s outcomes where Hots do not cast ballots, statement a) is false, because the leaders are not #1 and #2. To be sure, some people may be confused by the idea that, for our example, b) is always true (i.e. is true for all possible election outcomes). Well, b) is a conditional statement of the form �if X then Y�. Here X (�there is a first-place tie�) is sometimes true and sometimes false, but anyhow Y (�#1 and #2 tie for first place�) happens always to be true whenever X is true - which is all that is required for truth of a statement of form �if X then Y�. [As Anthony Simmons has noted, this approach to truth of if-then statements not only very well describes ordinary usage but also is a precise convention of mathematics and logic.] THAT�S ALL!! THANKS FOR YOUR HEED!! Joe Weinstein Long Beach CA 90807 USA _________________________________________________________________ Get your FREE download of MSN Explorer at http://explorer.msn.com
