Desert Bus donation formula analysis

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MarkVI
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Desert Bus donation formula analysis

Postby MarkVI » 23 Sep 2009, 14:54

For those who are interested, I decided to put up a short summary of how to compute the total donations required to achieve k hours of game play. Conversely, I've derived the formula to compute how many hours of Desert Bus would have to be played given C dollars of total donations.

The basic idea: Hour 1 costs $1 and subsequent hours cost 1.07 times the previous hour.

Let H(k) be the cost of hour k. Then
H(k) = (1.07)^(k-1).

Moreover, the total cost C(k) of hours 1 through k will be
C(k) = H(1) + H(2) + ... + H(k) = \sum_{j=1}^{k} H(j) = \sum_{j=1}^{k} (1.07)^(j-1) = \sum_{j=0}^{k-1} (1.07)^j.

This simplifies to:
C(k) = (1.07^(k) - 1)*(100/7)


Now to find the total hours achieved for a given donation amount C. It turns out that
k = ln( 7C/100 + 1) / ln(1.07) = log_{1.07}(7C/100 + 1)
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Re: Desert Bus donation formula analysis

Postby Lyinginbedmon » 23 Sep 2009, 15:10

This is cool, but I must ask:

How bored were you?
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Re: Desert Bus donation formula analysis

Postby Jillers » 23 Sep 2009, 15:11

Nope, still hate math.
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Re: Desert Bus donation formula analysis

Postby Master Gunner » 23 Sep 2009, 15:43

Lyinginbedmon wrote:This is cool, but I must ask:

How bored were you?


Probably not that bored, the formulas are standard, you just need to plug in the numbers that apply here. I did that last year to calculate for the last few hours. Crusher did it the first year to come up with his list of the first 120 hours.

As well, last year somebody came up with a few charts to show how different increases would work out. Its not hard, and the formulas are easy to look up.
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Re: Desert Bus donation formula analysis

Postby steelfox » 23 Sep 2009, 16:55

I thought the formula was:

Give money until they all go mad from driving... Also ask for ridiculous things.
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Re: Desert Bus donation formula analysis

Postby Master Gunner » 23 Sep 2009, 17:15

Well that too.
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Re: Desert Bus donation formula analysis

Postby Arius » 23 Sep 2009, 17:40

steelfox wrote:I thought the formula was:

Give money until they all go mad from driving... Also ask for ridiculous things.

Don't forget to compensate for when they have time away from DB, gotta make sure to break their willpower when they get back. Otherwise they resist.
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Re: Desert Bus donation formula analysis

Postby Evil Jim » 23 Sep 2009, 20:21

Isn't it, each hour costs seven or eight percent more than the previous one, with the first hour costing a dollar?
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Re: Desert Bus donation formula analysis

Postby MarkVI » 23 Sep 2009, 22:02

I didn't have to look anything up other than the 1.07 multiplier. Everything else is easy to derive from scratch in 5 min.
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Re: Desert Bus donation formula analysis

Postby Graham » 23 Sep 2009, 23:29

Evil Jim wrote:Isn't it, each hour costs seven or eight percent more than the previous one, with the first hour costing a dollar?

That's what multiplying by 1.07 will do—a 7% increase.
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Re: Desert Bus donation formula analysis

Postby Evil Jim » 24 Sep 2009, 00:45

Ah, whups. I skipped over the equations in the first post.
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Re: Desert Bus donation formula analysis

Postby Lord Chrusher » 27 Sep 2009, 06:39

Master Gunner wrote:
Lyinginbedmon wrote:This is cool, but I must ask:

How bored were you?


Probably not that bored, the formulas are standard, you just need to plug in the numbers that apply here. I did that last year to calculate for the last few hours. Crusher did it the first year to come up with his list of the first 120 hours.

As well, last year somebody came up with a few charts to show how different increases would work out. Its not hard, and the formulas are easy to look up.


I will have to post a bigger table for this year.
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Re: Desert Bus donation formula analysis

Postby Lord Chrusher » 28 Sep 2009, 00:59

Ok here is this year's version of the donation table. It goes for seven days - over a million dollars would have to be raised for Desert Bus for it to go that long, roughly the total amount of money Child's Play raised in total last year.

Code: Select all

Hours    Total    To Next Hour
0           0.00        1.00
1           1.00        1.07
2           2.07        1.14
3           3.21        1.23
4           4.44        1.31
5           5.75        1.40
6           7.15        1.50
7           8.65        1.61
8          10.26        1.72
9          11.98        1.84
10         13.82        1.97
11         15.78        2.10
12         17.89        2.25
13         20.14        2.41
14         22.55        2.58
15         25.13        2.76
16         27.89        2.95
17         30.84        3.16
18         34.00        3.38
19         37.38        3.62
20         41.00        3.87
21         44.87        4.14
22         49.01        4.43
23         53.44        4.74
24         58.18        5.07
25         63.25        5.43
26         68.68        5.81
27         74.48        6.21
28         80.70        6.65
29         87.35        7.11
30         94.46        7.61
31        102.07        8.15
32        110.22        8.72
33        118.93        9.33
34        128.26        9.98
35        138.24       10.68
36        148.91       11.42
37        160.34       12.22
38        172.56       13.08
39        185.64       13.99
40        199.64       14.97
41        214.61       16.02
42        230.63       17.14
43        247.78       18.34
44        266.12       19.63
45        285.75       21.00
46        306.75       22.47
47        329.22       24.05
48        353.27       25.73
49        379.00       27.53
50        406.53       29.46
51        435.99       31.52
52        467.50       33.73
53        501.23       36.09
54        537.32       38.61
55        575.93       41.32
56        617.24       44.21
57        661.45       47.30
58        708.75       50.61
59        759.36       54.16
60        813.52       57.95
61        871.47       62.00
62        933.47       66.34
63        999.81       70.99
64       1070.80       75.96
65       1146.76       81.27
66       1228.03       86.96
67       1314.99       93.05
68       1408.04       99.56
69       1507.60      106.53
70       1614.13      113.99
71       1728.12      121.97
72       1850.09      130.51
73       1980.60      139.64
74       2120.24      149.42
75       2269.66      159.88
76       2429.53      171.07
77       2600.60      183.04
78       2783.64      195.85
79       2979.50      209.56
80       3189.06      224.23
81       3413.30      239.93
82       3653.23      256.73
83       3909.95      274.70
84       4184.65      293.93
85       4478.58      314.50
86       4793.08      336.52
87       5129.59      360.07
88       5489.66      385.28
89       5874.94      412.25
90       6287.19      441.10
91       6728.29      471.98
92       7200.27      505.02
93       7705.29      540.37
94       8245.66      578.20
95       8823.85      618.67
96       9442.52      661.98
97      10104.50      708.31
98      10812.81      757.90
99      11570.71      810.95
100     12381.66      867.72
101     13249.38      928.46
102     14177.83      993.45
103     15171.28     1062.99
104     16234.27     1137.40
105     17371.67     1217.02
106     18588.69     1302.21
107     19890.90     1393.36
108     21284.26     1490.90
109     22775.16     1595.26
110     24370.42     1706.93
111     26077.35     1826.41
112     27903.76     1954.26
113     29858.03     2091.06
114     31949.09     2237.44
115     34186.52     2394.06
116     36580.58     2561.64
117     39142.22     2740.96
118     41883.18     2932.82
119     44816.00     3138.12
120     47954.12     3357.79
121     51311.91     3592.83
122     54904.74     3844.33
123     58749.07     4113.44
124     62862.51     4401.38
125     67263.88     4709.47
126     71973.36     5039.13
127     77012.49     5391.87
128     82404.37     5769.31
129     88173.67     6173.16
130     94346.83     6605.28
131    100952.11     7067.65
132    108019.75     7562.38
133    115582.14     8091.75
134    123673.89     8658.17
135    132332.06     9264.24
136    141596.30     9912.74
137    151509.04    10606.63
138    162115.68    11349.10
139    173464.77    12143.53
140    185608.31    12993.58
141    198601.89    13903.13
142    212505.02    14876.35
143    227381.37    15917.70
144    243299.07    17031.93
145    260331.00    18224.17
146    278555.17    19499.86
147    298055.04    20864.85
148    318919.89    22325.39
149    341245.28    23888.17
150    365133.45    25560.34
151    390693.79    27349.57
152    418043.36    29264.04
153    447307.39    31312.52
154    478619.91    33504.39
155    512124.30    35849.70
156    547974.01    38359.18
157    586333.19    41044.32
158    627377.51    43917.43
159    671294.93    46991.65
160    718286.58    50281.06
161    768567.64    53800.73
162    822368.37    57566.79
163    879935.16    61596.46
164    941531.62    65908.21
165   1007439.84    70521.79
166   1077961.62    75458.31
167   1153419.94    80740.40
168   1234160.33    86392.22
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Iqy
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Re: Desert Bus donation formula analysis

Postby Iqy » 28 Feb 2013, 11:22

What would that mean if someone donated a billion dollars? I make it about 230 years.
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Re: Desert Bus donation formula analysis

Postby Jamfalcon » 28 Feb 2013, 11:27

It would mean a lot of entertainment.
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Re: Desert Bus donation formula analysis

Postby MarkVI » 28 Feb 2013, 12:26

Not too bad actually. It would be just 267 hours.

And to make the team bus even for 1 year (8760 hours) would cost in the range of 3.6*10^(258) dollars (many orders of magnitude more than the number of elementary particles in the universe)
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Re: Desert Bus donation formula analysis

Postby JackSlack » 28 Feb 2013, 13:50

Now a better question would be: What's the best formula for predicting a new year's DBfH donation total? I remember Fugiman and I had a go at it last year, only to way, way undershoot.
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Re: Desert Bus donation formula analysis

Postby Master Gunner » 28 Feb 2013, 14:12

I have in the past tried it with software that uses genetic algorithms to try and derive the best equation to describe a data set. The problem is that there anywhere near enough data points to get a reliable result. It was a few years ago the last time I tried it, and the range between the best-fitting equations was around $100,000.
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Re: Desert Bus donation formula analysis

Postby Lord Hosk » 28 Feb 2013, 19:47

Lord Chrusher made this graph.

Image

He predicted based on the power law that they would raise $536 917.

Based on a linear progression taking DB5 as a aberration in the math I predicted $422,550. I was off by less than one desert buck.

Desert Bus 6 total $443,630

I will predict right now that DB7 will raise $512,560 and run for 155 hours They will not have purchased the 155th hour yet but will run over because "they are so close".
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Re: Desert Bus donation formula analysis

Postby Lord Chrusher » 02 Mar 2013, 06:18

I ran my analysis again with the DBFH total and I got this:

Image

The linear model predicts that they will raise $522 450 (155 hours).

The power law model predicts that they will raise $599 864 (157 hours).
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Re: Desert Bus donation formula analysis

Postby the amativeness » 02 Mar 2013, 13:24

Now, random thought:

If you subtract 100,000 from both DB5 and DB6, you end up with what looks like a perfectly straight line. Based on this line, what is the DB7 prediction (adding back in that 100K that we removed)?

Image

y=66.0008(7)+-52.8888 (+100)

Based on these numbers, I predict $509,116.60. This means that they will bus for 154 hours, and possibly push to 155.
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Re: Desert Bus donation formula analysis

Postby Lord Chrusher » 02 Mar 2013, 16:31

<GIF of Kate shaking her head>Nope</GIF>

When you are making projections you can not just modify data to fit a trend.
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Re: Desert Bus donation formula analysis

Postby Keab42 » 04 Mar 2013, 07:13

Unless your study is designed to show that homoeopathy actually works.
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Re: Desert Bus donation formula analysis

Postby Dave-O_Boy » 04 Mar 2013, 07:41

MarkVI wrote:Not too bad actually. It would be just 267 hours.

And to make the team bus even for 1 year (8760 hours) would cost in the range of 3.6*10^(258) dollars (many orders of magnitude more than the number of elementary particles in the universe)

So you're saying it's probably an unrealistic goal for next year then?
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Re: Desert Bus donation formula analysis

Postby Yaxley » 04 Mar 2013, 08:00

Lord Chrusher wrote:<GIF of Kate shaking her head>Nope</GIF>

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I didn't have that saved, and for some reason I tried to google it. I got as far as "kate nop" before Google suggested "kate stark nope gif". I'm a little scared that it knew that.

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