Behind the Design of Stockholders

[Judge Sam]

There was a lot that went into the Stockholder's voting twist. It actually didn't start out with anything to do with stocks or a business setting that was a flavor that was added on later.


One of the rounds that was always included in the bunch was something I called Trusted Weighted Vote. The more people trust you, the more your vote counts! I thought that was a really cool idea since all the other voting processes were like 1 person = 1 vote. And who wouldn't want to give the trusted cits more voting power??


Later on this was merged to the really cool idea of a shareholder's meeting at a company. People vote with stocks depending on what % of the company they own. There's a board of directors who manages it. (thought of including this in some way but ended up not doing it.) The first problem was what to call it... Wikipedia gives so many names. Should it be stockholder, stakeholder, or shareholder?


So that worked awesomely especially with the Union theme. For the record here were the #s of votes:


1. 501 stocks
2. 440 stocks
3. 400 stocks
4. 380 stocks
5. 360 stocks
6. 340 stocks
7. 320 stocks
8. 300 stocks
9. 260 stocks
10. 200 stocks
11. 100 stocks


A good bit of really interesting math went into figuring out these numbers. In previous essays I would just handwave over it but I actually want to go through this!


But let's start at the beginning of the twist cause that's kinda like the end.


So at the chat before I threw out ONE HUNDRED VOTES which got everyone excited haha. I hadn't really intended to like completely deceive y'all on that so much. What I was going for was that first I told you that you all had 2 votes, then I think that one person would get 100. The thing with the two votes and then the upgrade to a 100 was that it implied that the number of votes could change a lot. (and to further the logic, would change again as more people got more votes)

But that didn't really get through with the way I presented it and ended up kinda silly on my part. Another part which sucked was that that was 2 out of 4 days in the episode spent worrying about something that didn't come to fruition rather than spending that time strategizing about the possibilities with the real, interesting voting twist which was unfortunate. I wish y'all had more time to think about it.


So first I had you make up a trust list. For awhile it was going to be a Spy list - rank people who you thought were a Spy. Actually wait... before all this it might not have even been a trust or a spy list. I think I considered having a ranking based on a challenge, or a group decision, I'm not sure. Anyway, okay so before the game I was debating Spy List vs. Trust List. I had done Spy lists before and usually they affect the game a lot. I was thinking the spy list would affect things too much at this late point in the game and that's not what I really wanted. Plus, Borda count was going to be originally rank everyone, and that was going to give out tons of info too.


Spy list had it's advantages but I'm really glad I shifted it to trust list. It's kinda the same, kinda not, and has a little bit less of an impact which I liked.


So anyway usually with these things I tell you to rank everyone and I assign this unique, cool point system. But this time I wanted to try something different! I told you to rank your top three trusted and your bottom three trusted. I would then combine them to make a master list.


This may be a little mathy but stick with me haha I find this really interesting. Now most people when combining lists would just go like okay, ranked #1, gets 1 point, ranked #2 gets 2 points, #3 3 points and tally them all up. Person with least points overall is #1, second least #2, etc.

But these point values are so arbitrary. Why are we assigning a point value for second place that is double of first place, while assigning a point value for tenth place that is 10/9ths of 9th place? (10 points for 10th place, 9 points for 9th place). Why are the magnitudes that way and would the result end up changing if we did some other intuitive way, say ranking them 1, 2, 4, 8, 16 points?


Yes it would actually, it would change quite a bit! Each way you change it ends up in different results. So what's the best way to change it? How do you combine several lists into one master list?


Actually if anyone can point out a mathematically more efficient method I'd love to see it. I'm sure there is one for combining several lists into one master list. But here's what I thought. First, the 3 points, 2 points, 1 point system I just thought wouldn't work. Why is most trusted 3 times as valuable as 3rd more trusted? That seems a little high to me, I mean they are 3rd most trusted lol. It's a little weird to make the 1st most trusted worth 3 times as much as the 3rd most trusted, but make the 3rd most trusted only 1 point more than 4th and 5th most trusted.


But how much more valuable should most trusted be? I didn't know, but I wanted to find out. So here's what I did, I assigned variable point levels to your rankings.

1st most trusted: 'a' points
2nd most trusted: 'b' points
3rd most trusted: 'c' points

1st least trusted: '-a' points
2nd least trusted: '-b' points
3rd least trusted: '-c' points

definition: a > b > c

So rather than making a definitive point list, I decided to keep them variable. That way I could try out every different possible point list while I was comparing them. And if someone was always higher in every single point list than someone else, then they would be ranked higher for sure!


And then I tried to rank all of you on that which was tons of fun. Here's actually what y'all had:

So for Jenya, who got 3a + b - c, that would mean 3 people ranked her 1st most trusted, 1 person ranked her 2nd most trusted, and 1 person ranked her 3rd least trusted.


Jenya: 3a + b - c
Kirsten: 3b
Jason: 3a + 2b
Roxy: -3a + b
Khaled: -a - b - 2c
Boris: -2a - 2c
Pete: 2a + c
Tiberius: -a + b + 2c
Mirela: -a - 3b
Levi: b - 2c
Gretchen: -2a - 3b - 2c
Georgia: -4b + 2c


Okay so rank all of those into a list from greatest to least! That's the fun part haha.

To do that, I first started out with my definition a > b > c which is self-evident. I also found out I needed to add in other assumptions. For example, I added in that b + c > a. This means that the 'worth points' of you voting for 2nd most trusted and 3rd most trusted added together should be more than you voting for 1st most trusted. By the way, the 3, 2, 1 system does not meet this assumption since 2 + 1 = 3.


I'm not going to go through every single calculation haha but with these specific values it wasn't as difficult as it could be, which was nice. There's some quick things you notice with this list. First, some of the values like 2a + c are always going to be positive numbers, and some of the values like -4b + 2c are negative. (because b > c.) Turns out all of them are one or the other so you can divide it into two tiers which makes it easier. The top half only has four people and it's pretty straight forward to rank these:

3a + b - c
3b
3a + 2b
2a + c

for example let's look at 3a + b - c and 2a + c . Which is more?

3a + b - c > 2a + c because
---
a > c and b > c [definition]
a + b > c + c [adding the two]
a + b > 2c
2a + a + b > 2a + 2c [adding 2a to both sides]
2a + a + b - c > 2a + 2c - c [subtracting c from both sides]
---
3a + b - c > 2a + c [what we wanted to show]


The negative ones were a bit more fun cause some of them were pretty close. For example, I don't think it is immediately obvious how the rankings of these should go:

-a - 3b
-3a + b
-2a - 2c
- a - b - 2c


The way I did it was via comparison of all the expressions. If one equation is bigger than another and that another is bigger than a third, then the first one is bigger than the third one. By fitting them all in bigger or smaller than one another it all lines up.


Of course like I said these relations depend on what point system you end up using. You could change the order if you decided that a = 1000 b = 950 and c = 500 for example which fits all our current assumptions. You need to make a few more expression assumptions to rank them properly. The closest assumption I ended up making was between:


3a + 2c and 5b. Which is larger?


Maybe you could argue it either way. But my opinion is that the 'gap' between 1st and 2nd most trusted is slightly larger than the gap between 2nd and 3rd most trusted, or a - b > b - c . Why? Well think of it this way. Your absolute least trusted comes to mind right away. It takes a lot for that person to get the world's worst award. But now think about ranking a 9th least trusted and a 10th least trusted. It's kinda the same especially when there's so many people, it's hard to distinguish. The difference between 10th and 9th and maybe say, 5th and 4th is a different value there.

Following that logic to the end is how I got that the 'gap' between 1st and 2nd most trusted is slightly larger than the gap between 2nd and 3rd most trusted, or a - b > b - c .

Let's try and show that 3a + 2c > 5b using our new assumption.
---
a - b > b - c [assumed]
a + c > 2b [rearranging of terms]
2.5a + 2.5b > 5b [multiply by 2.5]
a > c [definition]
a - c > 0 [subtract c from both sides]
.5a - .5c > 0 [multiply by .5]
---
3a + 2c > 5b [add .5a - .5c > 0 to 2.5a + 2.5c > 5b]


So anyway, surprisingly, that all worked out! If you are in lala land with all these variables and would want an exact number system which fits all the criteria I decided upon, a simple one is:

a = 10 points for most trusted
b = 7 points for 2nd most trusted
c = 5 points for 3rd most trusted

which sounds fair I think.


Yeah so that's part one haha. We haven't even gotten to the actual vote part yet, which is the interesting part. Part two up soon!

[Ellis]

W00t, an opportunity to indulge my inner maths-geek. Actually, my outer math-geek as well.

So to find A, B, and C, did you average all the totals, add to them all so that the lowest was 1, then multiply by 100? Or something along those lines?

[Judge Sam]

Not exactly sure what you mean... if you mean how did I translate the results from part one into the # of votes each person got to cast then that's answered in this part two.


So how did I come up with all those 100, 200, 260 vote numbers? This was actually probably the most difficult calculation I came up with. As you can imagine, you could make it any collection of numbers and it would still work. There's a whole bunch of sets of votes that might provide for an interesting Exile. Here's some examples in a situation with 5 voters:

100, 110, 120, 130, 140 votes
100, 300, 500, 700, 900 votes
100, 300, 400, 500, 700 votes
1/5, 1/4, 1/3, 1/2, 1 votes
0, 1, 2, 3, 4 votes
1, 1, 1, 2, 2, 3 votes


Each one might produce a different outcome. And there's tons of possibilities. So how did I pick one and get the numbers to where I wanted it? Well I started off with a few guiding principles and properties I wanted the vote to have. For example, I wanted everyone's vote to matter somewhat. This wasn't a kingmaker or a council so it shouldn't be like the non-trusted people have no say at all. That'd be kinda boring for them. So the smaller numbers shouldn't be so small that they have no chance of affecting things.


But there's another corollary to that that was very important to me. Not only do the smaller numbers have to be within range, but the bigger numbers can't be too big. As I'm sure you know, it takes 50% + 1 votes to win a election. Now normally those 50% + 1 votes are cast by 50% + 1 of the people. But in stockholders that's not the case. Just three or four people could cast a majority of the votes.


And that would lock out people just as effectively as the non-trusteds not having enough votes would. If three or four people all came together as a group, and decided it would be them that makes the call - automatically everyone else's vote doesn't matter. Cause collectively those three have amongst themselves 50% + 1 of the votes. They could choose who goes amongst themselves.


Yeah I know the top three or four aren't always going to do that since they have different opinions but as you'll see later that was a huge factor in what I was thinking. Trying to factor that in caused a lot of issues I had to solve haha.


So I had to figure out a few things. First was scale. How spread out would the votes be? If it was 100, 110, 120, 130, 140 then that's not very spread out. Maybe think about how that vote system would work out. That might as well just be 1 vote each cause then the #s don't really matter except if it's a 2-2 tie. I was trying some ranges like between 100-301 and 100-801 to see what worked best... it changed quite a bit. I had to factor in how much more the max vote was worth than the least vote, in this case 3x to 8x. But I also had to factor in how much more the max vote was worth than the average vote, since tons more people would have around the average score.


The final factor was how the points should be distributed. Like if I added up the rank points and 1st, 2nd, and 3rd most trusted were all really close, should I factor that into the points? Should I base it on the 'points' of the trust rankings? Like maybe have it 400, 390, 380 since the points between the top 3 trusted were all really close. Or should I just fit it to a set point scale regardless of the gaps in between.


I decided to go with the set point scale as it would be easier to work with.


Another way this factors in is what the 'line graph' should look like. It's easier to do with pictures. Should the points just go straight through like a diagonal line? Or should the point values clump in the middle and have high/low values on the end?





Should it be just a straight line like that green one? Or maybe it should curve off like the red one so the high values aren't as high?





Or maybe it should look like this? To visualize this, think of points equally spaced along the graph (those tic marks at the bottom of the image are good markers). The y value (the vertical one) would be how many points they get. Starting off on the left the first person doesn't get many points, but it quickly skyrockets for the second. Then we hit a plateau point where we're going horizontally so these people around here all have the same points. Then we skyrocket again so the most trusted people get lots more points.


That graph above is actually y= (6/5)(x-6)^3 + 250 lol which is one I was considering. Looking at my graphs I've tried, that, a y = x line, y = x^(3/5), and a piecemeal one where it kinda looks like a connect the dots because each segment has a different steepness.


So after all these tries, formulas, and all these number values like 100 votes, 173.2, 217.6, 240.4 etc, why didn't any of these work? Why couldn't I find an equation that would do what I wanted it to do?


Well here's where a problem I talked about earlier came in. Remember, I didn't want a group of the Top X trusted people to just come together themselves and decide since they had 50% +1 of the votes and block out everyone else. That'd suck. But in all these formulas the top 3 or 4 people kept having a majority of the votes. It was all about them. I couldn't get out of it.


At this point, it was locked in that there would be 11 voters. (before the game I had worked with this where it could be at a time between 9 and 15 voters.) So with 3 out 11 deciding themselves, that'd be kinda dumb. But would 4 out of 11 be okay? Remember that in a normal vote 6 out of 11 could come together and decide. And if 4 out of 11 wasn't okay, that means it only be a little different than a normal vote.


But I decided to not allow 4 out of 11 to come together to decide, even though that would be unlikely, just to be on the safe side. Maybe you wouldn't have made it this strict. It was a tough choice mostly because of the small number of people (11). If there were 15 it'd probably be easier.



Okay so I'm trying to get the sum of the highest 4 numbers to be less than half of the total sum and I'm hitting brick wall after brick wall. I try all sorts of things to smoosh it down, from whittling the top 4 numbers down to basically nothing, to boosting the lower numbers up so high but none of these are working with any of the formulas.


So then I go back to an assumption I had made from the beginning and I realize that it's a terrible assumption and will always cause whatever I do to fail. It's gotta go.


The assumption was, it's a bit difficult to type out, but like the assumption was that the range of numbers should be balanced around the middle. Like it should be 1,2,3,4,5 and not 1,2,3,6,10. 3 is in the middle, so each side should be like even. In the graphs above, if you cut them down the middle it should kind of look the same on both sides.


Once I threw that pretty similarity out I was able to bump up the bottom numbers enough to get it to work.


That's why it goes like 100 to 200 to 260 at the bottom but at the top it goes like 400 to 440 to 501.


Once I threw that pretty similarity out I was able to bump up the bottom numbers enough to get it to work. What I liked was that I didn't have to smoosh the top numbers to make them indistinguishable either and it still worked. I played around with some numbers using this new way of doing things and it really worked out nicely. I picked the prettiest numbers so it would be easy to grasp (at some points I had like 100, 133.33333, 166.666666 lol). Luckily it all fell together perfectly at this point!

As a reminder, here they are:

1. 501 stocks
2. 440 stocks
3. 400 stocks
4. 380 stocks
5. 360 stocks
6. 340 stocks
7. 320 stocks
8. 300 stocks
9. 260 stocks
10. 200 stocks
11. 100 stocks

Sum: 3601.
50% +1 = 1801
Sum of Top 4: 501 + 440 + 400 + 380 = 1721. Yay!

Overall I really like it but maybe some of the middle values are too close. 3rd having 400 stocks and 8th having 300 stocks isn't too much of a difference. As a result, these players votes have less tilting the result impact and the higher ones have more and the lower ones have less. Maybe I could have spread that out a bit. Or maybe I should have gone from 100 - 401 stocks instead.


The actual vote itself turned out to be pretty cool it was:

KHALED (1180): 360 stocks from Levi, 380 stocks from Kirsten, 440 stocks from Jenya
GEORGIA (801): 300 stocks from Khaled, 501 stocks from Jason
GRETCHEN (760): 100 stocks from Mirela, 260 stocks from Boris, 400 stocks from Pete
MIRELA (660): 320 stocks from Roxy, 340 stocks from Georgia
LEVI (200): 200 stocks from Gretchen



which was pretty spread out but shows you the power of the system which turned out well I think. Even though Khaled and Gretchen both got 3 votes, Khaled's were from more trusted people. And Georgia even went a little bit above Gretchen with just 2 votes because those votes were from more trusted people while Gretchen's were from 2 of the 3 lowest. And still, Mirela with two votes was under Gretchen with 3 votes so it still does show that a bit.



Despite the difficulties, this was a really fun round to work out and another one I enjoyed.

[Ellis]

Sept 9, 2009 2:07:26 GMT -5 Judge Sam said:

That graph above is actually y= (6/5)(x-6)^3 + 250

Well, in all fairness, you could probably tell that just by looking at it.

Very very interesting though, takes me back to my days of programming computer games with random-entry opponents and things like that.