From: John Conover <john@email.johncon.com>
Subject: Re: The California Recall Polls
Date: 26 Aug 2003 20:27:09 -0000
Note: this analysis originally appeared in the NdustriX and NtropiX mailing lists-and generated substantial interest; more so than I anticipated. There is a new formal version in Appendix III of Quantitative Analysis of Non-Linear High Entropy Economic Systems Addendum which is one of the mathematical and numerical methods series at the NdustriX site, (and analyzes many more things that you might be interested in, too.) You might want to skip over to Appendix III since it is better written and has graphical content; or you can read the original, below. (The poll data was from the Los Angeles Times, August 24, 2003, by Susan Pinkus, Times Poll Director.)
The polls have shifted[footnote]; about 50% favor recalling Davis. If
the recall succeeds-as of today-Bustamante would have about a 35%
favorability against Schwarzenneger's 22%. There are 42 days left
until the recall vote.
And, how do things stand as of today?
Davis:
Davis has a 50% chance of being recalled. What are the chances of
Davis maintaining, (at least,) 50%? 50%, of course, since there is
an equal likelihood of it going either way in the next 42 days,
(making Davis-astonishingly-the front runner since if he defeats
the recall, the other's poll numbers are irrelevant.)
Bustamante:
Bustamante is in a poll "bubble," that started about ten days
ago. What are the chances of him being able to make it "stick" to,
(at least,) October 7? Its erf (1 / sqrt (52)), which is about 1 /
sqrt (52), or about 14%. The deviation of Bustamante's poll
numbers, 42 days from now, is 0.02 * sqrt (42) = 13%, or there is
an 16% chance that at the time of the election, Bustamante's poll
will be above 48%, and a 16% chance that it will be below a 22%
favorability.
Schwarzenneger:
Schwarzenneger's numbers haven't moved much. However, there is a
16% chance that Schwarzenneger's favorability would be above 22% +
13% = 35%, and a 16% chance that it would be below 22% - 16% = 6%
by the time of the election.
So, what's the chances of Schwarzenneger beating Bustamante on
election day, (if Davis loses?) Fortunately, its kind of easy to
estimate; Bustamante has a 50% chance of a favorability that is at
least as good as it is today, (35%,) and Schwarzenneger has a 16%
chance, (one standard deviation,) of the same favorability, so it is
0.5 * 0.16 = 8%.
The other contenders have substantially less of a chance, (McClintock
now has 12% since Simon has dropped out; to get to Bustamante's 35%,
would be 35 - 12 / 13 = 1.8 standard deviations, or about 3.6%, and
McClintock would only have half of that since there is a 50% chance
that Bustamante will have more than a 35% favorability at the time of
the election.)
What's the chances of the Democrats, (i.e., Davis or Bustamante
winning, and/or, all Republicans losing?) 0.5 + 0.35 = 85%.
So how do the chances stack up as of today?
Davis: 50%
Bustamante: 0.5 * 0.35 = 18%
Schwarzenneger: 0.5 * 0.08 = 4%
McClintock: 0.5 * 0.018 = 1%
All Others, (less than a 1% chance, each,) combined: 27%
John
Footnote:
There is cause for concern in the accuracy of the poll numbers.
Bustamante's favorability was 18% on August 11-15 days ago. What's the
chances of a move to 35% in 15 days? The standard deviation of the
chance is 0.02 * sqrt (15) = 7.7%, and 35 - 18 / 7.7 is 2.2 standard
deviations, which is a chance of 1.4%. Not a very likely occurrence,
(not impossible, but not very likely, either.) Such things are a
hallmark signature of leptokurtosis in the daily movements of the
favorability numbers-meaning that there are "fat tails" in the
distribution of the movements. This implies the volatility is a factor
of 2.2 what would be measured by the deviation of the daily movements,
(the metric of volatility is the deviation of the movements, where
volatility is interpreted as risk, or uncertainty; similar to the
classic definition of Black-Scholes-Merton's "implied volatility.")
Most political economists argue that leptokurtosis in social systems
is the rule, and not the exception. An interpretation is that things
go steadily along-with a reasonable amount of uncertainty-and then,
spontaneously there is a jump of many standard deviations. The jumps
tend to cluster, and are characteristic of social change, (for
example, during insurgence/rebellion/revolt, jumps in the range of
orders-of-magnitude many standard deviations are common; yet-using the
mathematics that are used by pollsters-such occurrences have a
probability that is indistinguishable from zero.)
There is probably something else that is going on besides the recall
of the California Governor. At least that's what the poll numbers tend
to indicate.
John Conover writes:
>
> The polls of today indicate that Schwarzenneger and Davis are about
> tied, with 25% of the vote, each. In the last 6 days, each moved about
> 5%, (Bustamante is slightly ahead, at the expense of Schwarzenneger.)
> What's the chances of that happening?
>
> The deviation of such a happening is 5 / 2 * sqrt (6) = 1.0. In other
> words, there is a 68% chance that such a movement in 6 days would be
> less than 5%, and 32%, more, (if we consider it a two person zero sum
> game; if not-considering the dark-horses-its the single sided tail of
> the distribution, at 84% chance.)
>
> What's the chances of Bustamante's good fortune continuing longer?
> erf (1 / sqrt (7)) which is about 1 / sqrt (7), or about 38%,
> (ignoring the chances of the dark-horses.)
>
> There are 51 calendar days to the vote, (pending several legal
> decisions, of course,) and Bustamante has won the poll lottery of the
> first 6 days of the campaign. (Schwarzenneger lost 13% of a chance of
> being governor, Bustamante won 13%; now either has a 50/50 chance,
> ignoring the dark horses.)
>
> John
>
> John Conover writes:
> >
> > Several Polls came out today. 59% of Californians are for bouncing
> > Gray Davis, and if that sticks, Schwarzenneger has 31% of the vote,
> > Bustamante 18%, with all of the other 150, or so, contenders well
> > below 10%, each. There are 57 calendar days to the vote.
> >
> > Polls are very fractal, and if a percentage point that is gained by
> > one opponent is at the expense of another, (e.g., a zero sum game,)
> > then the deviation of poll movements will be about 2% per
> > day. Fractals are a random process, so the deviation of the polls at
> > the end of 57 days would be 0.02 * sqrt (57) = 15%.
> >
> > So, there is a 16% chance, (one single sided deviation,) that Davis
> > can move his fractal poll from 59% to 59 - 15 = 44% for the recall by
> > October 7. (And, 9 / 15 = 0.6 deviations, which is a probability of
> > 27% that he can move it to a 50/50, and defeat the recall.)
> >
> > Schwarzenneger's chance of winning against Bustamante is 31 / (18 +
> > 31) = 63%, and Bustamante's winning against Schwarzenneger is 18 / (18
> > + 31) = 37%, (the Gambler's Ultimate Ruin is a good way to determine
> > such things; what it says is that each will be required to gamble on
> > strategies, but the success of the strategies are uncertain.) The next
> > candidate has 6% of the vote, so each of the other candidates will
> > have about 6 / 100 = 6% chance of passing Schwarzenneger and
> > Bustamante, (and all the others,) winning the Gubernatorial race.
> >
> > However, most of the remaining 150 candidates are not serious
> > contenders-about 6 actually have a chance. And that means that the
> > chances of any one of the 6 winning the race is 1 - (1 - 0.06)^6 =
> > 31%.
> >
> > So:
> >
> > Chances of Davis defeating the recall: 27%
> >
> > Chances of Schwarzenneger beating Bustamante: 63%
> >
> > Chances of Bustamante beating Schwarzenneger: 37%
> >
> > Chances of any dark horse beating all the other dark horses and
> > both Schwarzenneger and Bustamante: 31%
> >
> > John
> >
> > BTW, note there are no long calls. The best odds are on
> > Schwarzenneger, but for Schwarzenneger to win requires two things to
> > happen: Davis to be recalled, and defeating Bustamante, or a chance of
> > (1 - 0.27) * 0.63 = 46%, (assuming all the others have virtually no
> > chance of winning.)
> >
--
John Conover, john@email.johncon.com, http://www.johncon.com/