From: John Conover <john@email.johncon.com>
Subject: Re: CNET.com - News - E-Business - Latest dot-com bomb: TheMan.com
Date: 11 Nov 2000 08:09:52 -0000
While we are on the subject, what happens if everyone defects in a
democratic vote, (if you voted for the lesser of two evils on Tuesday,
you are guilty of it,) and does what is called insincere voting?
The chances are very high that the vote will go to a 50/50 deadlock,
with everyone getting a candidate that represents their worst ranking
of priorities.
That's what Kenneth Arrow's so called Impossibility Theorem, (its a
game-theoretic phenomena,) is all about-democratic process can lead to
very undemocratic results. (The theorem was the result of an
investigation into the so-called social welfare function-he was
attempting to use linear/mathematical programming to optimize the
economic welfare function; and ended up proving it was impossible.)
Insincere voting is when one does not vote for one's preference in the
ordering of priorities, (like welfare, for example-usually because one
realizes that one can not pull it off with enough votes,) and uses the
vote to exclude the WORST ordering of priorities by voting for
something else.
As a pledgarized, (from "Archimedes Revenge," Paul Hoffman, Fawcett
Crest, New York, 1989, ISBN 0-449-21750-7, pp. 221-225,):
In 1951, Kenneth Arrow, an American economist, astounded
mathematicians and economist alike with a convincing demonstration
that any conceivable democratic voting system can yield
undemocratic results. Arrow's unsettling game-theoretic
demonstration drew immediate comment in academic circles the world
over.
One year later, in 1952, Paul Samuelson, later the winter of the
Nobel Memorial prize in Economic Sciences, put it his way: "The
search of the great minds of recorded history for the perfect
democracy, it turns out, is the search for a chimera, for logical
self-contradiction. Now, scholars all over the world-in
mathematics, politics, philosophy, and economics- are trying to
salvage what can be salvaged from Arrow's devastating discovery
that is to mathematical politics what Kurt Godel's 1931
impossibility-of-proving-consistency theorem is to mathematical
logic."
Arrow's demonstration, called the impossibility theorem, (since it
showed, in effect, that perfect democracy is impossible,) helped
earn him the Nobel Prize in economics in 1972. Today, the fallout
from Arrow's "devastating discovery," one of the earliest and most
astonishing results of game theory, is still being felt.
The undemocratic paradoxes inherent in democratic voting are best
explained by an example. Consider three friends, Ronald, Clara,
and Herb, who, after a hard day of work, have a craving for fast
food. They are determined to dine together at one of three
eateries, McDonald's, Burger King, or Wendy's, but they cannot
agree on which. Ronald, who longs for a McD.L.T., served in the
nifty partitioned container that keeps the greasy hamburger from
flooding the crisp, farm-fresh veggies, wants to go to McDonald's;
of the other two restaurants, he favors Burger King over
Wendy's. Eager to go where the beef is, Clara prefers Wendy's to
McDonald's, and McDonald's to Burger King. Herb, dreaming of a
double Whopper with cheese, likes Burger King best and McDonald's
least.
Ronald Clara Herb
-------------- -------------- --------------
1. McDonald's 1. Wendy's 1. Burger King
2. Burger King 2. McDonald's 2. Wendy's
3. Wendy's 3. Burger King 3. McDonald's
FAST-FOOD PREFERENCES
The three friends decide to settle the matter by voting first
between McDonald's and Wendy's and then between the winner of that
vote and Burger King. If Ronald, Clara, and Herb each vote their
real preference, they'll end up at Burger King, (with Wendy's the
runner-up.)
McDonald's \ Wendy's \
(Ronald) \ (Clara) \
\ \
vs. > vs. > Burger King
/ /
Wendy's / Burger King /
(Clara, Herb) / (Ronald, Herb) /
Since Burger King is Clara's last choice, she will not be happy.
If Clara votes on the first ballot not for her real preference,
Wendy's, but for her second choice, McDonald's, she ensures that
McDonald's will win the first ballot as well as the second. It is
paradoxical that Clara ultimately achieves a preferable result by
initially violating her own preference.
McDonald's \ McDonald's \
(Ronald, Clara) \ (Ronald, Clara) \
\ \
vs. > vs. > McDonald's
/ /
Wendy's / Burger King /
(Herb) / (Herb) /
Moreover, even if Ronald and Herb get wind of Clara's strategy,
they cannot effectively interfere. Herb is outraged because
Clara's crafty voting has made his third-choice restaurant the
winner, whereas "honest" voting on her part would have made his
first choice the winner. Herb tries to persuade Ronald to conspire
with him in some insincere voting of their own, but Ronald wants
no part of it, because he cannot possibly improve his own
position: Clara's voting has made Ronald's the first-choice
restaurant the winner.
A change in the voting sequence cannot eliminate the possibility
of crafty voting. All it would do is give someone other than Clara
the opportunity to be insincere. Suppose the three friends vote
first between Burger King and Wendy's, with the winner set against
McDonald's: if they all vote "honestly," they'll end up at
McDonald's, leaving Herb Disappointed.
Burger King \ Burger King \
(Ronald, Herb) \ (Herb) \
\ \
vs. > vs. > McDonald's
/ /
Wendy's / McDonald's /
(Clara) / (Ronald, Clara) /
If Herb's shrewd enough to foresee this, he'll cast a sly first
vote that will steer them ultimately to Wendy's.
Burger King \ Wendy's \
(Ronald) \ (Clara, Herb) \
\ \
vs. > vs. > McDonald's
/ /
Wendy's / McDonald's /
(Clara, Herb) / (Ronald) /
The other possible voting sequence-first between McDonald's and
Burger King and then between the winner of that vote and
Wendy's-is no better.
Burger King \ McDonald's \
(Herb) \ (Ronald) \
\ \
vs. > vs. > Wendy's
/ /
McDonald's / Wendy's /
(Ronald, Clara) / (Clara, Herb) /
It simply gives Ronald the opportunity for shrewd voting.
Burger King \ Burger King \
(Herb, Ronald) \ (Herb, Ronald) \
\ \
vs. > vs. > Burger King
/ /
McDonald's / Wendy's /
Clara) / (Clara) /
Although the predicament of the three would-be diners is
fictitious, it is not contrived. The possibility of crafty voting
may arise in any majority-rule voting in which a series of ballots
re cast to select a single winner from three or more
alternatives. This happens in the House of Representatives when an
amendment to a bill is introduced. First the House votes on the
amendment. If it passes, a second and final vote is taken between
the amended bill and the option of no bill at all. If the
amendment is defeated, the second vote is between the original
bill and no bill.
...
As early as the eighteenth century, the French mathematician
Jean-Antoine-Nicolas Caritat, marque de Condorcet, identified a
fundamental voting paradox. He discovered that society often has
preferences that, if held by an individual, would be dismissed as
irrational. Consider again our three hungry friends. Ronald
prefers McDonald's to Burger King, and Burger King to
Wendy's. Given those preferences, he would be irrational to prefer
Wendy's to McDonald's. Yet these are precisely the preferences of
our friends as a group! In a vote of all three friends, they
prefer McDonald's to Burger King, Burger King to Wendy's, and
Wendy's to McDonald's. Could it be that from a mathematical point
of view democracy is inherently irrational?
which is an interesting tautology.
John
BTW, insincere voting is very common in politics. See "Mathematical
Applications of Political Science," William Riker, University of
Rochester. He does a game-theoretic analysis of a 1956 House vote on a
bill calling for federal aid for school construction; complete with
empiricals and interviews of the members of Congress. Insincere voting
not only happens, it is an operational agenda.
John Conover writes:
> BTW, for the record, I stand corrected. My statement:
>
> "Its interesting because the popular and electoral votes may
> differ for the first time in history, (a very small chance, but
> not insignificant.) ..."
>
> is not correct-the electoral and popular votes for President were not
> the same on 3, (or 4, depending on which historian is telling the
> story,) previous occasions. The last was 1888, (but other historians
> are claiming W. Wilson's election was the last time.)
>
> However, it is kind of interesting the way things turned out. Gore,
> (et al, or whoever,) did sling mud, G. W. Bush did not, (at least not
> as much and as bad,) and it looks like a reasonably high probability
> that Gore will take the popular vote-coming from behind-and loose the
> electorate, and the nomination for Presidency, to G. W. Bush, by
> playing a nearly optimal game, under the circumstances.
>
> John
>
> BTW, you can arrive at the same conclusion from a game-theoretic
> perspective, too. You will find the solution, (simplex tableu,)
> identical to the well known non-iterated, zero-sum, prisoner's dilemma
> game, where the optimal strategy is always to play a defection
> strategy, (a la linear algebra/simplex.)
>
> If the game is iterated, the resultant time series will have fractal
> characteristics, and the last game will have a go-for-broke
> optimization, (sling all the mud that can be slung.)
>
> The trouble is that power is a zero-sum game, (winner takes all,)
> which makes the defection strategy for the the only, or last,
> iteration of the game the optimal solution.
>
> If the game was positive-sum, (i.e., all candidates got some power for
> running,) then a mixed mode solution would be best, i.e., sling mud
> sometimes, but not always, and do so randomly, (at least where one's
> opponents wouldn't be able to figure out when one is going to do it
> next,) would be the optimal solution. (That's the theory of the
> parliamentarian system-which has other zero-sum optimal defection
> solutions.)
>
> The game-theoretic solutions are interesting because it can be shown,
> re: Kenneth Arrow/Impossibility Theorem, that there is no perfect
> political system where cooperation is optimal. It is impossible to
> remove the zero-sum nature of politics in a social system of more than
> 3 people. All one can hope to do is shuffle around the zero-sum
> problem via structural implementation-but it can not be eliminated.
>
> Slinging mud, (and other assorted political defection strategies,) are
> completely rational-and have been around since the beginning of
> civilization, (see the Gilgamesh, circa 3K BC, the Summarian God of
> Wisdom's describing the gift to Holy Ianna of Uruk.)
>
> John Conover writes:
> >
> > And what should Gore do with a 22% chance of winning? What is his
> > optimal strategy?
> >
> > He should use a Greedy algorithm, (which has characteristics of what
> > the mathematicians call a "Devils Stair Case",) over the next four
> > days, which means he bets half of 47 - 43 = 6 / 2, or 3% of his
> > tracking poll, every day, until he is in the lead.
> >
> > This means Gore should risk PO'ing 3% of his constituents for a chance
> > at gaining 3% of Bush's constituents, every day until the end of the
> > election, or until he is ahead of Bush, (i.e., sling some mud.)
> >
> > And what would Gore's probability of winning be?
> >
> > let r = (1 - p) / p = (1 - (0.43 / 3)) / (0.43 / 3) = 5.98
> >
> > P = (r^0.43 - 1) / (r^0.47 - 1) = 0.89
> >
> > remarkably, almost 90%!
> >
> > That is, unless G. W. Bush does the same thing, in which case it moves
> > Gore's chances back to a 22%.
> >
> > Bottom line, mud slinging is not only the way of politics-its optimal.
> >
> > And, in the current state of affairs, Gore has absolutely nothing to
> > lose by slinging mud, (with only a 22% of winning if he doesn't.)
> >
> > John
> >
> > BTW, tsinvest does not use a Greedy algorithm, or Devils Stair Case.
> > If p < 0.5, it simply refuses to invest in the stock-and if no stocks
> > have p > 0.5, it will withdraw from the market, (although you can
> > override this if you are foolish-its the -D option.)
> >
> > John Conover writes:
> > > That's about a 4 in 5 chance of winning. But it can be wrong 1 chance
> > > in 5, too. What this means is that you play this game many times, you
> > > will win 4 out of 5 times.
> > >
> > > So, you wouldn't want to bet your nest egg on it, (you stand a 20%
> > > chance of losing everything on the first game, if you do, and you
> > > can't play any longer.) But you have to bet something, otherwise you
> > > can't make anything, (another of mathematics most profound insights.)
> > >
> > > The optimum lies in between. And the magic optimum is when F = 2P - 1,
> > > where P is the probability of a win, (0.78 on Bush in this case,) and
> > > F is the fraction of your nest egg to wager, (or about 56% in this
> > > case.)
> > >
> > > Based only on the popular vote, of course-and I don't know any bookies
> > > that taking bets based only on the popular vote.
> > >
> > > John
> > >
> > > BTW, the -d option to tsinvest controls how the program does this
> > > methodology; the -d1 is what was outlined here.
> > >
> > > John Conover writes:
> > > >
> > > > So, Bush has a 0.84 * 0.93 chance of winning, or about 78%,
> > > > (considering only the popular vote,) based on the accuracy of the
> > > > tracking polls, and the ability of Gore to move them.
> > > >
--
John Conover, john@email.johncon.com, http://www.johncon.com/