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Question

Introduction

I'm creating humongous city in a renaissance-esque fantasy setting. This city is governed by a council/senate with 51 seats. Division of these seats is based on ownership of stocks in 5 different merchant guilds in the city.

Question

How would one design a division algorithm that gives the illusion of a fair allocation of seats/stock, but is ultimately unfair in that it grants more seats to the super-rich and none/very few to the not-so-rich.

Notes

One person can hold many seats

- ...but never a majority of them, unless they own ALL stocks.

Each merchant guild has exactly 1000 stocks, which means 5000 stocks in total.

- Only stocks are divided by merchant guild. The seats does not necessarily go to specific guild stocks.

- One can own stocks from multiple guilds.

Anyone can buy stocks

Owning a few stocks does not guarantee a seat (obviously)

It should be possible to mathematically figure out the system, but complex enough that most people can't do it.

This is a ridiculously unfair system as it literally states that you need lots of money to be allowed influence over the government.

Seats are not voted on by the stock-holders. You either have enough to get a seat or you don't.

Answers from comment questions

Seats can not be empty. All seats are divided among the stock-holders and you cannot decline a seat, when you're given one.

Welcome to Worldbuilding SE. I'm too lazy to write up an answer right now, but you might look up gerrymandering in the USA. If someone wants to take this and run with it for the points I won't be offended. — 4 hours ago
"Unfair" is relative. You have to define "fair" in order to know what "unfair" looks like. To my European social democratic eye your system looks suspiciously like extremely unfair rampant capitalism o the sort we have revolutions to get rid of. :-) — 4 hours ago
Your last remark about the mathematical difficulty is quite confusing. I don't see your goal here. — 4 hours ago
@StephenG First of all, bold of you to assume that I'm not (I'm Swedish) ;). Secondly: Oh, it is very unfair from a classist point-of-view, since you can literally buy votes in this system. Third: you make an excellent point, I'll revise the question. — 4 hours ago
@pojo-guy, gerrymandering doesn't apply here as there's no voting. — 4 hours ago

score 9 · Answer 1 · 2018-10-23 14:06:49Z

So there are 51 seats, meaning each guild gets 10, and there's one left for the ruler or something.

Each guild has 1000 shares of stock, meaning that if you own 100 shares then you should have a seat.

Looks simple. But there won't be clean lots of 100. Someone might own 125, while another owns 70, and you have a handful of merchants with 1 share each. Now, a person who owns 70 might go out and look for smaller shareholders with similar beliefs to support their claim on a seat, essentially a proxy position, but if the shares get split up a lot, with a lot of poorer people with only 1 share then that might be hard, so what would probably happen is that the guy with 125 would just vote the second seat so that at least all the votes are accounted for.

The rich people who own enough stock to already own one seat would be in place to claim the empty seats, while the less rich would be unlikely to get any say at all, and the normal people who can't even afford a single share would be even less likely.

Also, 5000 shares is pretty limited for any decent size city, meaning that they won't be cheap, and you'd only be able to buy one if someone else was willing to sell, which would drive the price up if someone was bidding to get enough for a full seat.

Example:

Bill has 125 shares.

Ted has 120 shares.

Sherryl, Pam, Nancy, Sam, Pat, and Rodge have 105 each.

Don has 70.

the rest are split up to varying amounts among merchants and gentry.

Everyone above 100 has enough stock to legitimately claim one seat, and because because Bill and Ted have more than the others and they are on the council already they claim an extra seat each.

If Don wants a seat he has to go out and talk to the people who own the remaining 55 shares, and see if they will either sell them (assuming he can afford more) or let him borrow them as a proxy. If he can get over the 100 share threshold then he'd be able to take the extra seat that Ted is claiming. Now Ted might not like this, and so can go to the people loaning Don their shares and make them a better offer, give them sweetheart deals or tax breaks, offer to pay more for their shares, or in some cases use extortion (which would be dangerous since these are still wealthy enough people to own stock at all).

If he can get Don back bellow 100, then Don would lose the seat and Ted could claim it back.

On the other side, if Bill can muster enough support as proxy, maybe by recruiting Don and a few others, he could get enough borrowed stock to claim the extra seat that Ted has, giving himself 3 votes.

There can't be any empty seats. All seats will be allocated to someone. — 4 hours ago
Ah, and a person with 126 would get their second seat before a person with 124, etc, etc? So first give 1 seat to everyone with 100+, then divide the rest on by one from the top down? And once someone below 100 manages to buy up to 100+, they are given the extra seat held by the lowest 100+ owner? — 4 hours ago
This is where you send around a few "friendly lads" to either "encourage" small shareholders to sell to you outright, or else. Ah, or else obtain promises to vote their proxy with your relatively weakly claimed seat, thus making a stronger position. Or else! Violence & bribery ~~ Demobocracy in action! — 4 hours ago
Basically, if I understand right, you'd have senators "in sede forte" positions. Those with at least 100 shares. They have a guaranteed seat; a strong position to be in. Then you'd have senators "in sede tenente" positions. Those with a majority of a position (51 to 99 shares) who would be able to command a seat and withstand a coup by all but a very powerful coalition of small shareholders. Lastly, you'd have senators "in sede debile" positions. These are, like your example, those with perhaps 25 to 49 shares. (cont) — 4 hours ago
@GunnarSÃ¶dergren Yeah, exactly. I added an example, but that's essentially what you said. — 3 hours ago

score 7 · Answer 2 · 2018-10-23 14:01:29Z

You wanted complicated maths and loophole abuse? Lets do this.

We're going to use the Log function. Specifically, the inverse log.

By Calculating the inverse log of the number of seats you want, divided by 22.03, then rounding that number to the next whole number of stocks, we reach the threshold of stocks required to own that number of seats.

If you're wondering why 22.03, it's because using that number forces you to need exactly 1000 stocks for 10 seats.

This gives us the following thresholds:

Num Seats. Num Stocks

1

1

1

3

7

19

50

136

368

1000

Now, at first glance you'd think that any schmuck with a stock would have a seat, but that's why I say it's a Threshold. You need at least that many stocks. Seats are actually determined in order of most owned stocks. So, if you own the most stocks at 501, you have met the threshold for 9 seats. The next guy, even if he owns the other 499, can only claim the remaining 1 seat.

The 51st seat is reserved for the guy who owns the most stocks across all 5 merchant guilds.

Now, if you want to work out how many stocks are worth, you need to perform the following maths (lets say we have 50 seats, for simplicity, and we want to find out X, the number of seats):

50 = invLn(X)/22.05

50*22.05 = invLn(X)

1101.5 = invLn(X)

Ln(1105.5) = X

7.008... = X

So, we can have 7 seats with 50 stocks. Assuming no-one owns more stock than us anyway.

So, you could theoretically own 46 seats, whilst only holding 2,505 stocks total. Barely half of them.

((If someone better at formatting can help with my Seats -> Stocks table, I'd appreciate it))

Woah! Love it! Could you elaborate a bit more on how the calculation is made? Add an example, perhaps? — 4 hours ago
Logarithms weren't invented until the 17th century. (If I understand right!) And thus will be an anachronism for the setting. — 3 hours ago
It's renaissance-esque, which google says is 1300 Ã¢Â€Â“ 1600. That touches on the 17th Century, and fun anachronisms are a staple of fantasy. — 3 hours ago
Anachronisms when it comes to mathematics are not a problem. It's renaissance-esque, not actual renaissance. — 3 hours ago

score 2 · Answer 3 · 2018-10-23 15:51:38Z

Let's say you need one hundred shares to get a seat, since that is the fairest (or fairest-looking) allocation. Then you want to minimize the number of people with 100 shares.

So, in the name of democratic accountability and fairly-shared ownership, you want to encourage each guild's thousand shares to be spread as evenly as possible (read: in the smallest lots possible) among the guild's members.

These shares should pay a dividend, or allow voting in internal guild matters, or have some other benefit so the owners will be resistant to selling their share(s) - this inflates the cost of buying up lots of small holdings to combine into the required hundred.

You could add a transaction tax ("to discourage speculators") to further reduce liquidity. Offer a discount to people already with large holdings (say, >100 shares) if you need an even bigger bias towards consolidating ownership in large holdings.

Now, that should leave us with fewer than ten seat-eligible shareholders per guild, so you need a way to allocate the remaining seats. A popular real-world system is to provide a proxy: but have the guild choose who controls any un-filled seats ("on behalf of the shareholders").

To further concentrate power, just have the proxy voter be the largest shareholder in that guild. If you want to disguise that concentration, you can use the (also real-world) non-executive directorship trick. Nominate the largest shareholder of a different guild to be the proxy voter. Maybe allow all guild shareholders to vote on which other guild's controlling shareholder should be chosen as the proxy, but don't allow anyone else to be nominated.

There's nothing stating that one must have 100 stocks to get a seat. That would, of course, be the fairest way to do it: 1 seat for every 100 stocks-ish, but I'm looking for an unfair system. — 2 hours ago
The point is that you can make it look fair ("anyone can have a seat for only 100 shares!"), while tilting the playing field so that it's practically impossible to actually achieve. — 2 hours ago

Joe C 111 · Answer 4 · 2018-10-23 16:49:35Z

It sounds to me that you want to delve into the maths behind Ã¢Â€Âœproportional representationÃ¢Â€Â - a principle of electoral theory that states that the number of seats won by each political party in an election should be as proportional as possible to the votes they received. Except in this case, weÃ¢Â€Â™re replacing votes with stocks; parties with guilds; and proportionality with favouritism for the rich!

For obvious reasons, most work in this field has been dedicated to getting proportional results, not the unproportional results that we want. However, some systems have proved to be less proportional than their designers originally hoped.

What comes to mind for me in particular is the DÃ¢Â€Â™Hondt quotient system. Under the DÃ¢Â€Â™Hondt system, first, give a seat to the guild with the most stocks. Then divide that guildÃ¢Â€Â™s stocks by 2. LetÃ¢Â€Â™s say that same guild still has the most stocks - so give them a second seat, and divide their original stock by 3. And so on.

Another way of thinking about it is you give each seat one by one - you could think of this as Ã¢Â€ÂœroundsÃ¢Â€Â. In each Ã¢Â€ÂœroundÃ¢Â€Â, you give the seat to the guild with the highest Ã¢Â€ÂœscoreÃ¢Â€Â, where their Ã¢Â€ÂœscoreÃ¢Â€Â is the number of stocks they hold divided by one more than the number of seats that guild has already won.

You can always look this up - IÃ¢Â€Â™m sure someone else has explained this much better than me!

Once grasped, the system intuitively seems fair, but in practice tends to give more seats to guilds with more stocks.

EDIT: HereÃ¢Â€Â™s the BBCÃ¢Â€Â™s explanation: http://news.bbc.co.uk/1/hi/northern_ireland/91150.stm.

score 0 · Answer 5 · 2018-10-23 13:54:36Z

Right now your system is quite resilient, yes you can buy the whole system but that's expensive.

Current system:

~98 stocks per seat.

Fill any remaining unclaimed seats with the largest remaining stockholders.

There's not a lot of room to exploit this system other than the direct purchase of stocks. The only thing more resilient would be:

Largest 51 stockholders.

We need to create some weaknesses to get our crowbar into the cracks

10 seats per guild.

1 seat appointed by first past the post election for an upstanding member of society to break any tied votes.

Now you have the opportunity to elect the pirate king and get a free extra seat, however you still have to be the largest player in the game. So lets add an extra rule:

Minimum holding of 100 stocks (total across all guilds) to hold a seat

Now if I hold 1 stock there's a seat empty somewhere and that can't be. It needs to be offered to someone who holds enough stocks to hold a seat, which basically means someone who already holds a seat is getting an extra one cheap. The balance of the system is now off kilter, the small stock holders are affecting the major players by giving them a boost.

The more stocks held by stockholders holding <100 stocks, the more valuable each stock over that 100 threshold is.

If there are enough small players then a small group of large stockholders can control the whole system.

Roger 4865 · Answer 6 · 2018-10-23 16:51:16Z

This is a great question, because it doesn't look that hard, but it keeps getting harder the longer you look at it.

The first hard part is the "seemingly fair" part -- it is a requirement that the system provides "the illusion of a fair allocation of seats/stock".

In my opinion this rules out a simple open market in stock, as that hardly seems fair at all.

Let me offer up an alternate system based on a series of modest proposals:

Firstly, that all 5000 shares will be distributed among the populace by means of a fair lottery.

Sure, that seems fair -- but is it? What about those bumpkin farmers that just show up to sell their weekly vegetables? Surely we don't want to add them to the lottery on the off chance they happened to be in town on census day.

Fine, fine -- we'll add a residency requirement. You'll actually have to live in town for a year and a day. But now we've got shady landlords claiming that all these people are renting a closet in their tenements! That amounts to a bunch of foreigners buying tickets into our government lottery, and that will not do.

Alright then, we'll institute a requirement that someone needs to actually be a landowner within the city (along with being a resident.) That sort of seems fair. If someone doesn't even have a house in town, they hardly have any real interest in the good government of the city. They could just pack up their bindle and move on to the next town tomorrow. (Such a requirement is not exactly unheard of in the real world -- to pick one example, a senator, today, in Canada is required to own at least $4,000 of land in the province they represent.)

While we're at it, it seems a bit unfair that a family of ten brothers, each holding a 400-square-foot villa of their own, get ten chances at the lottery, while their father with a single 4000-square-foot warehouse gets only one. So we'll give people lottery tickets based on square footage somehow.

Once the lottery happens, the shares are attached to the land parcel, not the owner -- if the land changes hands, so does the share.

This takes us, through a series of relatively-reasonable steps, to a situation where rich people who own lots of land are going to end up with most of the seats in government.

To fulfill the "no one can own a majority of seats" requirement, maybe we can make that the trigger for another round of the lottery. Which means that a super-rich person can trigger it at will, which they probably see as a feature.

This system also (I believe) fulfills the requirement of "more seats to the super-rich" without circularly-defining the super-rich as someone who owns lots of seats, which appeals to me.

There's probably a few more details and tweaks to be sorted out, but I think that gets us pointed in the right direction.

score 9 · Answer 7 · 2018-10-23 14:06:49Z

So there are 51 seats, meaning each guild gets 10, and there's one left for the ruler or something.

Each guild has 1000 shares of stock, meaning that if you own 100 shares then you should have a seat.

Looks simple. But there won't be clean lots of 100. Someone might own 125, while another owns 70, and you have a handful of merchants with 1 share each. Now, a person who owns 70 might go out and look for smaller shareholders with similar beliefs to support their claim on a seat, essentially a proxy position, but if the shares get split up a lot, with a lot of poorer people with only 1 share then that might be hard, so what would probably happen is that the guy with 125 would just vote the second seat so that at least all the votes are accounted for.

The rich people who own enough stock to already own one seat would be in place to claim the empty seats, while the less rich would be unlikely to get any say at all, and the normal people who can't even afford a single share would be even less likely.

Also, 5000 shares is pretty limited for any decent size city, meaning that they won't be cheap, and you'd only be able to buy one if someone else was willing to sell, which would drive the price up if someone was bidding to get enough for a full seat.

Example:

Bill has 125 shares.

Ted has 120 shares.

Sherryl, Pam, Nancy, Sam, Pat, and Rodge have 105 each.

Don has 70.

the rest are split up to varying amounts among merchants and gentry.

Everyone above 100 has enough stock to legitimately claim one seat, and because because Bill and Ted have more than the others and they are on the council already they claim an extra seat each.

If Don wants a seat he has to go out and talk to the people who own the remaining 55 shares, and see if they will either sell them (assuming he can afford more) or let him borrow them as a proxy. If he can get over the 100 share threshold then he'd be able to take the extra seat that Ted is claiming. Now Ted might not like this, and so can go to the people loaning Don their shares and make them a better offer, give them sweetheart deals or tax breaks, offer to pay more for their shares, or in some cases use extortion (which would be dangerous since these are still wealthy enough people to own stock at all).

If he can get Don back bellow 100, then Don would lose the seat and Ted could claim it back.

On the other side, if Bill can muster enough support as proxy, maybe by recruiting Don and a few others, he could get enough borrowed stock to claim the extra seat that Ted has, giving himself 3 votes.

There can't be any empty seats. All seats will be allocated to someone. — 4 hours ago
Ah, and a person with 126 would get their second seat before a person with 124, etc, etc? So first give 1 seat to everyone with 100+, then divide the rest on by one from the top down? And once someone below 100 manages to buy up to 100+, they are given the extra seat held by the lowest 100+ owner? — 4 hours ago
This is where you send around a few "friendly lads" to either "encourage" small shareholders to sell to you outright, or else. Ah, or else obtain promises to vote their proxy with your relatively weakly claimed seat, thus making a stronger position. Or else! Violence & bribery ~~ Demobocracy in action! — 4 hours ago
Basically, if I understand right, you'd have senators "in sede forte" positions. Those with at least 100 shares. They have a guaranteed seat; a strong position to be in. Then you'd have senators "in sede tenente" positions. Those with a majority of a position (51 to 99 shares) who would be able to command a seat and withstand a coup by all but a very powerful coalition of small shareholders. Lastly, you'd have senators "in sede debile" positions. These are, like your example, those with perhaps 25 to 49 shares. (cont) — 4 hours ago
@GunnarSÃ¶dergren Yeah, exactly. I added an example, but that's essentially what you said. — 3 hours ago

score 7 · Answer 8 · 2018-10-23 14:01:29Z

You wanted complicated maths and loophole abuse? Lets do this.

We're going to use the Log function. Specifically, the inverse log.

By Calculating the inverse log of the number of seats you want, divided by 22.03, then rounding that number to the next whole number of stocks, we reach the threshold of stocks required to own that number of seats.

If you're wondering why 22.03, it's because using that number forces you to need exactly 1000 stocks for 10 seats.

This gives us the following thresholds:

Num Seats. Num Stocks

1

1

1

3

7

19

50

136

368

1000

Now, at first glance you'd think that any schmuck with a stock would have a seat, but that's why I say it's a Threshold. You need at least that many stocks. Seats are actually determined in order of most owned stocks. So, if you own the most stocks at 501, you have met the threshold for 9 seats. The next guy, even if he owns the other 499, can only claim the remaining 1 seat.

The 51st seat is reserved for the guy who owns the most stocks across all 5 merchant guilds.

Now, if you want to work out how many stocks are worth, you need to perform the following maths (lets say we have 50 seats, for simplicity, and we want to find out X, the number of seats):

50 = invLn(X)/22.05

50*22.05 = invLn(X)

1101.5 = invLn(X)

Ln(1105.5) = X

7.008... = X

So, we can have 7 seats with 50 stocks. Assuming no-one owns more stock than us anyway.

So, you could theoretically own 46 seats, whilst only holding 2,505 stocks total. Barely half of them.

((If someone better at formatting can help with my Seats -> Stocks table, I'd appreciate it))

Woah! Love it! Could you elaborate a bit more on how the calculation is made? Add an example, perhaps? — 4 hours ago
Logarithms weren't invented until the 17th century. (If I understand right!) And thus will be an anachronism for the setting. — 3 hours ago
It's renaissance-esque, which google says is 1300 Ã¢Â€Â“ 1600. That touches on the 17th Century, and fun anachronisms are a staple of fantasy. — 3 hours ago
Anachronisms when it comes to mathematics are not a problem. It's renaissance-esque, not actual renaissance. — 3 hours ago

score 2 · Answer 9 · 2018-10-23 15:51:38Z

Let's say you need one hundred shares to get a seat, since that is the fairest (or fairest-looking) allocation. Then you want to minimize the number of people with 100 shares.

So, in the name of democratic accountability and fairly-shared ownership, you want to encourage each guild's thousand shares to be spread as evenly as possible (read: in the smallest lots possible) among the guild's members.

These shares should pay a dividend, or allow voting in internal guild matters, or have some other benefit so the owners will be resistant to selling their share(s) - this inflates the cost of buying up lots of small holdings to combine into the required hundred.

You could add a transaction tax ("to discourage speculators") to further reduce liquidity. Offer a discount to people already with large holdings (say, >100 shares) if you need an even bigger bias towards consolidating ownership in large holdings.

Now, that should leave us with fewer than ten seat-eligible shareholders per guild, so you need a way to allocate the remaining seats. A popular real-world system is to provide a proxy: but have the guild choose who controls any un-filled seats ("on behalf of the shareholders").

To further concentrate power, just have the proxy voter be the largest shareholder in that guild. If you want to disguise that concentration, you can use the (also real-world) non-executive directorship trick. Nominate the largest shareholder of a different guild to be the proxy voter. Maybe allow all guild shareholders to vote on which other guild's controlling shareholder should be chosen as the proxy, but don't allow anyone else to be nominated.

There's nothing stating that one must have 100 stocks to get a seat. That would, of course, be the fairest way to do it: 1 seat for every 100 stocks-ish, but I'm looking for an unfair system. — 2 hours ago
The point is that you can make it look fair ("anyone can have a seat for only 100 shares!"), while tilting the playing field so that it's practically impossible to actually achieve. — 2 hours ago

Joe C 111 · Answer 10 · 2018-10-23 16:49:35Z

It sounds to me that you want to delve into the maths behind Ã¢Â€Âœproportional representationÃ¢Â€Â - a principle of electoral theory that states that the number of seats won by each political party in an election should be as proportional as possible to the votes they received. Except in this case, weÃ¢Â€Â™re replacing votes with stocks; parties with guilds; and proportionality with favouritism for the rich!

For obvious reasons, most work in this field has been dedicated to getting proportional results, not the unproportional results that we want. However, some systems have proved to be less proportional than their designers originally hoped.

What comes to mind for me in particular is the DÃ¢Â€Â™Hondt quotient system. Under the DÃ¢Â€Â™Hondt system, first, give a seat to the guild with the most stocks. Then divide that guildÃ¢Â€Â™s stocks by 2. LetÃ¢Â€Â™s say that same guild still has the most stocks - so give them a second seat, and divide their original stock by 3. And so on.

Another way of thinking about it is you give each seat one by one - you could think of this as Ã¢Â€ÂœroundsÃ¢Â€Â. In each Ã¢Â€ÂœroundÃ¢Â€Â, you give the seat to the guild with the highest Ã¢Â€ÂœscoreÃ¢Â€Â, where their Ã¢Â€ÂœscoreÃ¢Â€Â is the number of stocks they hold divided by one more than the number of seats that guild has already won.

You can always look this up - IÃ¢Â€Â™m sure someone else has explained this much better than me!

Once grasped, the system intuitively seems fair, but in practice tends to give more seats to guilds with more stocks.

EDIT: HereÃ¢Â€Â™s the BBCÃ¢Â€Â™s explanation: http://news.bbc.co.uk/1/hi/northern_ireland/91150.stm.

score 0 · Answer 11 · 2018-10-23 13:54:36Z

Right now your system is quite resilient, yes you can buy the whole system but that's expensive.

Current system:

~98 stocks per seat.

Fill any remaining unclaimed seats with the largest remaining stockholders.

There's not a lot of room to exploit this system other than the direct purchase of stocks. The only thing more resilient would be:

Largest 51 stockholders.

We need to create some weaknesses to get our crowbar into the cracks

10 seats per guild.

1 seat appointed by first past the post election for an upstanding member of society to break any tied votes.

Now you have the opportunity to elect the pirate king and get a free extra seat, however you still have to be the largest player in the game. So lets add an extra rule:

Minimum holding of 100 stocks (total across all guilds) to hold a seat

Now if I hold 1 stock there's a seat empty somewhere and that can't be. It needs to be offered to someone who holds enough stocks to hold a seat, which basically means someone who already holds a seat is getting an extra one cheap. The balance of the system is now off kilter, the small stock holders are affecting the major players by giving them a boost.

The more stocks held by stockholders holding <100 stocks, the more valuable each stock over that 100 threshold is.

If there are enough small players then a small group of large stockholders can control the whole system.

Roger 4865 · Answer 12 · 2018-10-23 16:51:16Z