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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.
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closed as primarily opinion-based by Mołot, Renan, JohnWDailey, Vincent, kingledion Oct 23 '18 at 20:33

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – James Oct 25 '18 at 19:45
  • $\begingroup$ i don't think the answers given bear out the criticism of their being opinion-based. this is essentially an algorithm question, and people with some interesting responses have weighed in. $\endgroup$ – theRiley Oct 27 '18 at 7:36

10 Answers 10

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You wanted complicated maths and loophole abuse? Let's 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 shares, we reach the threshold of shares 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 shares for 10 seats.

This gives us the following thresholds:

enter image description here

Now, at first glance you'd think that any schmuck with a share would have a seat, but that's why I say it's a Threshold. You need at least that many shares. Seats are actually determined in order of most owned shares. So, if you own the most shares 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 much shares 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 shares. Assuming no-one owns more stock than us anyway.

So, you could theoretically own 46 seats, whilst only holding 2,505 shares in total. That is barely half of them.

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

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  • $\begingroup$ Woah! Love it! Could you elaborate a bit more on how the calculation is made? Add an example, perhaps? $\endgroup$ – Gunnar Södergren Oct 23 '18 at 13:52
  • $\begingroup$ Logarithms weren't invented until the 17th century. (If I understand right!) And thus will be an anachronism for the setting. $\endgroup$ – elemtilas Oct 23 '18 at 14:01
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    $\begingroup$ It's renaissance-esque, which google says is 1300 – 1600. That touches on the 17th Century, and fun anachronisms are a staple of fantasy. $\endgroup$ – Kyyshak Oct 23 '18 at 14:03
  • $\begingroup$ Anachronisms when it comes to mathematics are not a problem. It's renaissance-esque, not actual renaissance. $\endgroup$ – Gunnar Södergren Oct 23 '18 at 14:07
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    $\begingroup$ This seems really unfair (and unstable), as you only need about 1/50th of all stocks (and be the guy with most stocks) to always have the majority of seats. All the small scale owners will quickly be sold out in a race to have most stocks, and the first to cross the 500-line wins. $\endgroup$ – Karl Oct 23 '18 at 20:00
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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.

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  • $\begingroup$ There can't be any empty seats. All seats will be allocated to someone. $\endgroup$ – Gunnar Södergren Oct 23 '18 at 13:29
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    $\begingroup$ 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? $\endgroup$ – Gunnar Södergren Oct 23 '18 at 13:39
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    $\begingroup$ 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! $\endgroup$ – elemtilas Oct 23 '18 at 13:40
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    $\begingroup$ 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) $\endgroup$ – elemtilas Oct 23 '18 at 13:49
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    $\begingroup$ @GunnarSödergren Yeah, exactly. I added an example, but that's essentially what you said. $\endgroup$ – AndyD273 Oct 23 '18 at 14:35
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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.

We can do this entirely with real-world shareholder capitalism.

  1. Discourage owners of small holdings from selling - this inflates the cost of buying up lots of small holdings to combine into the required hundred.

    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).

  2. Discourage potential purchasers from buying - same motivation as above through a different mechanism.

    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, hopefully those leave us with fewer than ten seat-eligible shareholders per guild, so we need a way to allocate the remaining seats.

  1. Provide a proxy to vote "on behalf of the owners" of any blocks smaller than one hundred
  2. Have the Guild itself nominate the proxy

    • either have the proxy voter be the largest shareholder in that guild
    • or, if you want to disguise the concentration of power, you can use a variant of the (also real-world) non-executive directorship:

      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.


Now we should have lots of small fragmented share holdings, owners who want to keep their shares, and tax disincentives for anyone thinking of acquiring a seat-eligible shareholding.

If all this isn't enough, there are another couple of tricks we can borrow from the real world:

  • Class B shares:

    In an effort to share the Guild's wealth more widely with its members (who could object?), the Guild is offering an exciting new class of shares. Each costs only one hundredth of a single normal share, and offers a proportional fraction of the dividend! More members can own a share of our profits than ever before!

    Obviously the only way for the Guild to issue these without diluting existing shareholders is to buy back one share to issue one hundred new shares. The new shares however have no voting rights, so the original class A share is added to those controlled by the Guild's proxy.

  • Nominee accounts:

    Guild shares were originally issued to Guild members, but our success sees non-members wanting to invest, and we feel that Guild members with small shareholdings would benefit from a secondary market so they can cash out without having to sell their share back to the Guild.

    Behold, a new marketplace where Guild members can sell shares freely to non-members!

    Obviously we don't want non-members acquiring a vote in our internal affairs, even though we're happy to share our dividend payments. So, shares owned by non-members will be held on their behalf by the Guild. They're free to buy as many shares as they want, and to sell them back to both members and non-members, but voting rights will be exercised on their behalf by the Guild proxy.

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  • $\begingroup$ 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. $\endgroup$ – Gunnar Södergren Oct 23 '18 at 15:45
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    $\begingroup$ 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. $\endgroup$ – Useless Oct 23 '18 at 15:50
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It sounds to me that you want to delve into the maths behind “proportional representation” - 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, we’re replacing votes with stocks; parties with shareholders; 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 D’Hondt quotient system. Under the D’Hondt system, first, give a seat to the shareholder with the most stocks. Then divide that shareholder’s stocks by 2. Let’s say that same shareholder 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 “rounds”. In each “round”, you give the seat to the shareholder with the highest “score”, where their “score” is the number of stocks they hold divided by one more than the number of seats that shareholder has already won.

You can always look this up - I’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 shareholders with more stocks.

EDIT: Here’s the BBC’s explanation: http://news.bbc.co.uk/1/hi/northern_ireland/91150.stm.

ANOTHER EDIT: As Bobson points out I misunderstood the setup slightly, replaced “guild” with “shareholder” in every occurrence.

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    $\begingroup$ This is basically what I was going to suggest. One correction: You don't give the seats to the guild with the highest score, you give it to the shareholder with the highest score. $\endgroup$ – Bobson Oct 23 '18 at 18:58
  • $\begingroup$ Bobson, you’re quite right, I misunderstood the setup slightly. I’ll just correct it now. $\endgroup$ – Joe C Oct 24 '18 at 21:17
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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.

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Similar to the answer Joe C added, this sounds like you'd be looking for a proportional representation system, similar to the one the US uses.

The formula here is $A_n=\frac{V}{\sqrt{s(s+1)}}$, where $V$ is the number of shares a given shareholder has, and $s$ is the number of seats already allocated to that shareholder. You calculate the $A_n$ value for every shareholder, and then allocate a seat to the one with the highest value. Then $s$ goes up and you repeat.

There's a worked example on the Wikipedia page linked to above, and you can find instructions for building a spreadsheet to calculate it here. I took the provided google spreadsheet and changed it to calculate seats for 10 people: Three rich, one middle class, and six poor.

Here are all the numbers of seats and the shares I used to calculate them: enter image description here

You can see that the rich have notably more voting power than the shares they have, and the poor are very disenfranchised.

This is what the calculation looks like in process: seats ordered by priority


  • One person can hold many seats

Yep.

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

It doesn't quite manage this - a second shareholder needs 134 shares to get one seat.

  • 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.

These are all covered by simply looking at the total number of stocks, rather than what guilds they're from.

  • Anyone can buy stocks

It's likely that with a constrained supply, the demand will cause the price to continue rising so that only the rich will be able to afford a share (see: Taxi medallions), but that's outside the scope of what you're looking for.

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

Yep.

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

Yep. I doubt anyone in the real world works these numbers by hand, as there's a ton of calculation involved. It's certainly possible, but not easy.

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

Large concentrations of shares are disproportionally favored, but not by so much that it's blatantly obvious. (The US gets around this problem by assigning one Representative to each state before doing this math, which shifts it to disproprortionally favor the small states instead.)

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  • $\begingroup$ Is "s" the number of seats already allocated to the person currently being calculated for or the total number of seats already allocated? $\endgroup$ – Gunnar Södergren Oct 25 '18 at 10:45
  • $\begingroup$ @GunnarSödergren Oops - it’s seats for the person. Edited to fix that. $\endgroup$ – Bobson Oct 29 '18 at 11:24
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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.

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Perhaps each stockholder with a seat chooses the number of a stock they do not own each "election cycle", and whoever holds that stock gets the seat. Of course, anyone with 100 or more stock gets a seat automatically (and 2 seats if they have 200, and so on).

This is fair in theory - each council member must give up their seat every few years (depending on the period picked, say 4). However, in practice there could be a couple wealthy people with ~150 stock each, and the rest distributed among poorer people. Then those people could trade seats and have nothing change, despite them only owning 30% of the stock.

This could be exacerbated by a portion of the stock being tied to property, so that half (or some other percent) of the stock is given to landlords (even moderately wealthy ones), as then it would be harder for someone else to collect 100 stock for their automatic seat. If you have to buy someone's business to get his stock, it's a lot less likely you can get him to sell.

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Disclaimer: Any resemblances to modern-day political election systems are purely in your mind

Shares are allocated based on population affiliated with each guild in the city. As you have already pointed out, no single guild (no matter how populous) can have a majority, so there are minimum representation share quotas (say a minimum of 3 shares per 250 must be allocated to each guild). Within the guild, resident members who can prove they are residents via stamped documentation (which carries a stamp tax and administration fee) can vote for representatives to take the seats that their shares allocate their guild. In light of fairness, there are regulations restricting individuals from abusing personal wealth to essentially buy votes, but merchants within each guild can of course campaign on behalf of their favored seat occupant. As there are a finite number of seats for each guild, each seat is assigned a subset of the guilds voting population from which it is to be filled (though you can campaign for a seat outside of your residential guild subdivision). All seats are filled with a simple majority vote, discarding all non-majority votes for that subdivision.

This system allows the ultra-wealthy to promise kickbacks to special interest groups and merchants in exchange for their support (all off the books of course). This also allows current seat holders to re-divide their guilds' subdivisions to skew support in their favor and the favor of their constituents. All the while, each individual in the city is promised an equal voice and told how valuable their vote is, while the mentioned fees for voting ID documentation keep the lowest earners from every having a voice. Oh, and for good measure, the voting should take place during a time where it inconveniences those who can barely afford to live.

This is just scraping the surface of the potential abuses of a system of course, but it should be a good starting point with ample real-world examples to fill it out.

EDIT: If you want to make it even more ridiculous, divide the 50 seats out evenly, and force the election to be for representatives to vote on the occupant of the last seat. For this last seat to hold power, each guild should aim to polarize their seat occupants to always vote with each other on every issue, thereby locking issues to guild-lines, and making the single seat the deciding factor on almost every issue that has the city divided. violators should be shunned by the propaganda machine and made an example of, to ensure loyalty in the future.

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  • $\begingroup$ I must repeat, this system in no way resembles a political system currently in place in the real world. $\endgroup$ – GOATNine Oct 23 '18 at 18:01
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These are shares in a guild, not in a company - so tie them to Companies in the guilds, with some restrictions on stock ownership:

  • Each Company can control at most 1 seat, and elect their representative to sit in it.
  • Seats are allocated to the 50 companies with the most stock, with some form of run-off election in case of a draw (e.g. 10 companies with 6 stock each competing for the last 4 seats) voted on by stockholders not in the running for a seat

Now, every Butcher, Baker and Candlestick Maker in the city can buy stocks. So long as they own at least 1 stock, they have a small chance to win a seat, and a larger chance to vote on who gets a seat.

The problem is, of course, conglomerates: Merchant Moneybags owns a shipping company, a warehouse & storage company, and a wholesaler. And possibly an exotic goods bazaar a jewellery store, and a clothing shop too. Oh, half of them may only offer goods and services to each other, but those are 6 different companies.

He doesn't even need to own the most stocks, either, so long as he can either keep the most companies in the Top 50 - or the most single-stock companies to control the tie-breaker elections and wring concessions from everyone on the cut-off line. Anyone trying to stockpile as many stocks as possible in 1 company actually works in his favour - as it reduces the number of stocks needed to get into the Top 50 companies

As for "how can one person run that many companies"? Well, maybe he doesn't - he just happens to be the Wealthy Patron for a large number of people who do run companies, who agree to give him shares in their company.

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