Analysis of the 2000 Florida presidential balloting with program code and graphical output by Roger Luebeck © 2000, 2017----------------
Analysis of the 2000 Florida presidential balloting with program code and graphical output by Roger Luebeck © 2000, 2017 |
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Balloting This iteration of a balloting paradigm, using a random number generator, generated surprising patterns amidst chaos. It is very different from a coin toss paradigm, in that the voting population decreases as the balloting progresses. That is a very important distinction, as this algorithm was designed to determine the margin of error in the predicted outcome. I developed this algorithm in November 2000 as part of a DOS program I wrote to simulate the Florida presidential balloting, in order to predict who would win the recount - Bush or Gore. The program predicted that if a standard for counting ballots was adopted that accounted for the difference in undervote between "Marksense" counties and "Optical" counties, then Gore would have an 80 percent chance of winning, with the most likely margin of victory being approximately 350 votes. (I had requested undervote data from all 67 Florida counties and promptly received the data from all but two small counties.) One year later, such a recount was performed, and Gore actually won by about 350 votes, using the standard whereby "[pin] pricks" counted as votes. It was the only standard that ever made any sense. [Pin] pricks don't appear without the voter taking some action, as emphasized by the president of the company that manufactured the Marksense machines, and as verified by their own testing. In fact, all other standards for interpreting ballots produced results that fantastically failed to account for the difference in undervote between Marksense counties and Optical counties. Anyhow.... Even though my program incorporated the BASIC random number generator, distinct patterns always appeared in the graphical output. This is an example of applying an algorithm to randomness. Without the algorithm, no number of iterations of XY plots, with Y being assigned randomly, will produce any pattern. The pattern that is generated matches that which is generated by three other very different algorithms: logistic map a repeating line drawing routine used to color a plane in a CAD program simple trig They are shown on other pages of this website. (VisualBASIC and DOS BASIC program code displayed further down on page) The following graphs were generated using a slightly modified version of the DOS program using VisualBASIC: |
Output: First 20 percent Amplitude: .5 X increment: .04 Votes: 20000 Trials: 500 |
Output: First 50 percent
Amplitude: .5
X increment: .04
Votes: 20000
Trials: 500
Output: First 80 percent
Amplitude: .5
X increment: .04
Votes: 20000
Trials: 500
More amazement: When I shrunk the images, even more
accentuated and previously undetected patterns appeared.
The greater the shrinkage, the more revealing are the
newly seen patterns. The following three images are
1/2 size, 1/4 size and 1/6 size (then re-enlarged) of
the "50 percent" image:
Output: First 50 percent
Amplitude: .5
X increment: .04
Votes: 20000
Trials: 500
Strangely enough, I discovered this identical pattern
when I graphed the logistic difference equation in a
circular manner for a given value of the parameter r.
Those images are displayed on the logistic difference
equation page of this website.
Just as surprising, this very same pattern showed up on
one of my 3D-CAD images, due to some odd combination of
perspective angle and the ratio of cross-hatch lines used
to color a plane. I've never seen that pattern or any
other pattern appearing on any of my thousands of CAD
images. Here it is:
Here is an image from my original DOS BASIC program, which
I used to determine the margin of error in my prediction:
By varying the scale and the percent of output, one can
adjust the manner in which the patterns appear:
Shrinking the above image gradually revealed some
grouping of the Y values. See next two images:
Here is my original code for the VisualBASIC version.
Private Sub voterunbtn_Click()
Set viewport.Picture = LoadPicture("")
DONE$ = "done"
per = Val(perbox.Text)
amp = Val(ampbox.Text)
inc = Val(incbox.Text)
iter = Val(iterbox.Text)
trials = Val(trialsbox.Text)
iterper = Int(iter * (per / 100))
For total = 1 To trials
Randomize Timer
X = 0: G = 0: A = 0
For C = 1 To iterper
RN = Rnd(1)
If RN >= 0.5 Then G = G + 1
A = A + 1
A2 = A / 2
Y = (A2 - G) * (iter / A)
X = X + ((100 * inc) / per)
viewport.PSet (X, Y / amp), RGB(0, 0, 0)
Next C
Next total
finalnum.Text = DONE$
End Sub
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Here is my program code for the original voting algorithm
in QuickBASIC for DOS:
SCREEN 12
CLEAR
CLS
KEY(10) ON
ON KEY(10) GOSUB 300
5 :
VIEW (1, 1)-(639, 218)
CLS
KEY(9) ON
ON KEY(9) GOSUB 400
COLOR 14
LOCATE 28, 5: PRINT "Exit - F10"
COLOR 13
LOCATE 20, 46: PRINT "Current total ballots: "; TT
LOCATE 22, 46: PRINT "Current scale: "; S
LOCATE 24, 46: PRINT "Current percent: "; PER
COLOR 11
LOCATE 20, 3: INPUT "Input total ballots"; T
LOCATE 22, 3: INPUT "Scale (multiple of 1.5)"; S
6 LOCATE 24, 3: INPUT "Percent used for final
projection"; PER
IF PER = 10 OR PER = 20 OR PER = 40
OR PER = 80 OR PER = 99 THEN 7
LOCATE 22, 20: PRINT " ": GOTO 6
7 LOCATE 26, 3: INPUT "Number of trials"; TRI
VIEW (1, 1)-(639, 479)
CLS
TT = T
TPER = INT(TT * PER / 100)
TINC = INT(TPER / 1000)
YSCALE = TT / 750
VIEW (2, 218)-(600, 470), , 14
WINDOW (0, -YSCALE)-(1100, YSCALE)
COL = 1
ROW = 7
COLOR 13
LOCATE 1, 1: PRINT "Start - press spacebar"
LOCATE 2, 1: PRINT "Exit - F10 Restart - F9"
COLOR 14
LOCATE 1, 34: PRINT "Total ballots: "; TT
LOCATE 2, 34: PRINT "Scale: "; S
LOCATE 3, 34: PRINT "Trials to run: "; TRI
YLINE = INT(.9 * YSCALE)
COLOR 8
LINE (100, YLINE)-(100, -YLINE)
LINE (200, YLINE)-(200, -YLINE)
LINE (300, YLINE)-(300, -YLINE)
LINE (400, YLINE)-(400, -YLINE)
LINE (500, YLINE)-(500, -YLINE)
LINE (600, YLINE)-(600, -YLINE)
LINE (700, YLINE)-(700, -YLINE)
LINE (800, YLINE)-(800, -YLINE)
LINE (900, YLINE)-(900, -YLINE)
LINE (990, YLINE)-(990, -YLINE)
L6 = INT(.75 * YSCALE)
L5 = INT(.625 * YSCALE)
L4 = INT(.5 * YSCALE)
L3 = INT(.375 * YSCALE)
L2 = INT(.25 * YSCALE)
L1 = INT(.125 * YSCALE)
COLOR 7
LINE (60, L6)-(990, L6)
COLOR 8
LINE (60, L5)-(990, L5)
LINE (60, L4)-(990, L4)
COLOR 7
LINE (60, L3)-(990, L3)
COLOR 8
LINE (60, L2)-(990, L2)
LINE (60, L1)-(990, L1)
COLOR 7
LINE (60, 0)-(990, 0)
COLOR 8
LINE (60, -L1)-(990, -L1)
LINE (60, -L2)-(990, -L2)
COLOR 7
LINE (60, -L3)-(990, -L3)
COLOR 8
LINE (60, -L4)-(990, -L4)
LINE (60, -L5)-(990, -L5)
COLOR 7
LINE (60, -L6)-(990, -L6)
LOCATE 16, 67: PRINT INT(S * TT / 1000); (.1) * (S)
LOCATE 19, 67: PRINT INT(S * TT / 2000); (.05) * (S)
LOCATE 22, 67: PRINT " 0"
LOCATE 25, 67: PRINT INT(S * TT / 2000); (.05) * (S)
LOCATE 28, 67: PRINT INT(S * TT / 1000); (.1) * (S)
COLOR 15
LOCATE 28, 14: PRINT "Variation of
projection for A from actual A"
COLOR 15
IF PER = 10 THEN 11
IF PER = 20 THEN 12
IF PER = 40 THEN 14
IF PER = 80 THEN 18
IF PER = 99 THEN 20
11 LOCATE 14, 14: PRINT "2%"
LOCATE 14, 27: PRINT "4%"
LOCATE 14, 41: PRINT "6%"
LOCATE 14, 54: PRINT "8%"
LOCATE 14, 68: PRINT "10%"
GOTO 30
12 LOCATE 14, 14: PRINT "4%"
LOCATE 14, 27: PRINT "8%"
LOCATE 14, 41: PRINT "12%"
LOCATE 14, 54: PRINT "16%"
LOCATE 14, 68: PRINT "20%"
GOTO 30
14 LOCATE 14, 14: PRINT "8%"
LOCATE 14, 27: PRINT "16%"
LOCATE 14, 41: PRINT "24%"
LOCATE 14, 54: PRINT "32%"
LOCATE 14, 68: PRINT "40%"
GOTO 30
18 LOCATE 14, 14: PRINT "16%"
LOCATE 14, 27: PRINT "32%"
LOCATE 14, 41: PRINT "48%"
LOCATE 14, 54: PRINT "64%"
LOCATE 14, 68: PRINT "80%"
GOTO 30
20 LOCATE 14, 14: PRINT "20%"
LOCATE 14, 27: PRINT "40%"
LOCATE 14, 41: PRINT "60%"
LOCATE 14, 54: PRINT "80%"
LOCATE 14, 68: PRINT "99%"
30 :
COLOR 13
LOCATE 13, 2: PRINT "Press spacebar to continue"
40 I$ = INKEY$
IF I$ = "" THEN 40
LOCATE 13, 2: PRINT " "
FOR VOTE = 1 TO TRI
RANDOMIZE TIMER
LOCATE 12, 5: PRINT "Number of trials :"; VOTE
50 B = 0: G = 0: C = 0: X = 0
INCRE = 0: ACT = 0: VARIA = 0
BP = TT / 2: GP = TT / 2
FOR C = 1 TO TPER
V1 = RND(1)
IF V1 < .5 THEN B = B + 1
IF V1 >= .5 THEN G = G + 1
ACT = ACT + 1
A = ACT / 2
100 INCRE = INCRE + 1
IF INCRE = TINC THEN
INCRE = 0
COLOR 10
VARIA = INT((A - G) * (TT / ACT))
X = X + 1
PSET (X, VARIA / S)
END IF
150 :
NEXT C
T = TT
NEXT VOTE
200 I$ = INKEY$
IF I$ = "" THEN 200
COLOR 12
LINE (100, YLINE)-(100, -YLINE)
LINE (200, YLINE)-(200, -YLINE)
LINE (300, YLINE)-(300, -YLINE)
LINE (400, YLINE)-(400, -YLINE)
LINE (500, YLINE)-(500, -YLINE)
LINE (600, YLINE)-(600, -YLINE)
LINE (700, YLINE)-(700, -YLINE)
LINE (800, YLINE)-(800, -YLINE)
LINE (900, YLINE)-(900, -YLINE)
LINE (990, YLINE)-(990, -YLINE)
COLOR 12
LINE (60, L6)-(990, L6)
COLOR 12
LINE (60, L5)-(990, L5)
LINE (60, L4)-(990, L4)
COLOR 12
LINE (60, L3)-(990, L3)
COLOR 12
LINE (60, L2)-(990, L2)
LINE (60, L1)-(990, L1)
COLOR 12
LINE (60, 0)-(990, 0)
COLOR 12
LINE (60, -L1)-(990, -L1)
LINE (60, -L2)-(990, -L2)
COLOR 12
LINE (60, -L3)-(990, -L3)
COLOR 12
LINE (60, -L4)-(990, -L4)
LINE (60, -L5)-(990, -L5)
COLOR 12
LINE (60, -L6)-(990, -L6)
250 I$ = INKEY$
IF I$ = "" THEN 250
GOTO 5
300 KEY(10) OFF: KEY(9) OFF
END
400 KEY(9) OFF: GOTO 5
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