A Mathematical Analysis of Why Ratios Are Bad, and a Better Solution to the Problem of Unwanted Leechers

In the file-sharing community, balance is everything. If everyone contributes, the system works, and everyone is happy. Community generosity leads to high-speed transfers and high availability.

But not everyone is generous. Some people are downright grinchy when it comes to sharing. And that really pisses people off. So people make private communities to keep out the grinches. But how do you decide who to let in?

The primary method of weeding out the trash for most torrent sites has almost universally been the invocation of a minimum ratio requirement. The reason is obvious: everyone should share the responsibility of uploading. The problem with ratios is that they don't really solve the problem, and are actually contrary to the idea of file-sharing.

In fact, ratio systems can actually create problems and invite leeching. On this page I'm going show mathematically why it is that ratios are a bad idea.

Then I'm going to talk about alternative solutions that actually keep out the bad guys.

1. It is impossible for all users to maintain a ratio over 1.

It constantly amazes me how many sites require a ratio over 1 because doing so is a mathematical impossibility.

For the mathematically inclined, here's a quick proof:

• Let there be N users ranging from 1 to N.
• Let u_i be the number of bytes uploaded by user i.
• Let d_i be the number of bytes downloaded by user i.

• Let's assume the 1st user has a ratio greater than 1.

By definition d_1 is not negative, so the inequality is preserved.  

Multiplying both sides by -1 causes the inequality to flip.  

• Note that the sum of all uploads has to equal the sum of all downloads. Call that alpha.

We can always add the same value to both sides without altering the inequality.  

Now substitute the summations appropriate to each side for the two alphas.  

• We will now assume that every other user other than the Nth user has a ratio of exactly 1, so their download total must equal their upload total.

We can always subtract the same value from both sides without altering the inequality.  

Again, simple substitution.  

• We can repeat this process for all j=3..N-1 finally arriving at:

The ratio of the Nth user MUST be less than 1.

The math really isn't necessary to see this, though. For every byte someone uploads, someone has to be downloading that byte. The only way everyone could have a ratio greater than 1 is if every person uploaded more than they downloaded -- but that's impossible because someone has to be the goat that downloads those extra bytes, and that person has to download more than they upload.

There is a common misconception that you can get around this by "uploading something new". This is not true: someone else has to download what you upload.

The clear implication of this is that if you run a site and require a ratio of at least 1, someone is going to get banned.

2. The best case scenario when requiring a ratio of 1 results in a deadlock under which no one can download.

Let's say everyone of the people you get on your site is a good guy and doesn't want to break the rules. To make it really easy and obvious, lets say there are just two users, user A and user B.

Let's pretend there's a grace period of 1 day after which you must have a ratio of 1.

On the same day User A and User B joins. User A uploads a 1GB file, and User B downloads it. ratio_A={1 \over 0}=\infty and ratio_B={0 \over 1}=0

Immediately thereafter, user B uploads a 2GB file to get a better ratio, and user A downloads it. ratio_A={1 \over 2}=.5 and ratio_B={2 \over 1}=2

Now the grace period is about to end, and user A starts uploading a 2GB file to correct his ratio. But at 1 GB, user B shuts down his download because he's hit his ratio limit.... ratio_A={2 \over 2}=1 and ratio_B={2 \over 2}=1

Now the grace period has expired. Both users are "good users" and won't break the ratio rules, so neither one will download from the other. They've deadlocked!

3. Only leechers or frustrated good users who give up on the tracker can break the deadlock

In order for our deadlocked users to get a higher ratio, a 3rd user has to join and download during his grace period. That results in an increase in ratio for our two initial users, but also results in the new user having a 0 ratio, or at least one much less than 1. Since our initial 2 users are dead set on maintaining their ratio requirement, it is completely impossible for the third user to ever have a ratio greater than 1. He either gives up trying to upload while he is still below the grace-limit, or he just leeches and gets banned.

4. Leeched bytes behave similarly to money

Each byte the leecher downloads raises the ratio of the user he downloads from by one byte. That enables that user to "spend" that byte downloading from someone else, who in turn can do the same. Thus, leeched bytes become a form a currency. The total added currency on the server is given by:

where users N+1..M represent those leechers which are no longer active on the server, for whatever reason. "Leechers" always have a download byte count greater than their upload byte count, and so always produce currency.

5. Generous over-seeders get other people banned.

The irony of a "good user" is that good users try to over-seed. It's not unusual to hear of someone with a ratio much greater than 1. What's not so obvious is that over-seeders are generating upload credits for themselves that they don't intend to use and usually end up hording this "upload currency" making it unavailable for use by others. (i.e., the total is always positive.)

where users M+1..L represent those generous seeders (hoarders) who do not intend to use their credit for whatever reason, even if they are still active. "Over generous seeders" always have an upload byte count greater than their download byte count, and so always remove currency.

We can now calculate the total currency on the server as:

noting that retired leechers N+1..M contribute currency, and all hoarding overseeders M+1..L remove currency.

If the currency ever becomes negative, someone's ratio has fallen below 1 and that person will be banned unless a generous user frees up byte currency by downloading something or the ratio requirement is lowered below 1.

How badly can a generous user effect other users?

Consider the situation where half of your users are "generous over-seeders" and each of them maintains a Group 1 ratio of x. Now assume the other half of your users all have the Group 2 ratio y. In this specific case y={1 \over x} which is the all familiar inverse graph. The yellow area show that the more "generous" the over-seeders are (x), the lower everyone else's ratio (y) becomes.

As the 2nd groups ratio falls, users become desperate for more ratio and try to download "popular" torrents, even though they are not really interested in them. This doesn't fix the overall ratio problem and wastes bandwidth. The entire system becomes a pyramid scheme of users trying to obtain credits.

5. Ratios below 1 and grace settings generate "byte currency" but promote leeching

By lowering the ratio, you are instituting "forced deflation," effectively creating new "byte currency", enabling the economy to function. However, the lower the ratio and grace limits are, the more leechers are encouraged to take resources from your server, the very thing you are trying to prevent. Ironically, when people come to your tracker and download without uploading, they inject extra "byte currency" into the economy. A large number of leechers can thus become the most prominent generator of "currency" on the tracker.

When the ratio requirement is lowered, the currency on the tracker becomes:

Notice that this permits active users to create currency by causing uploads to be worth more (ratio requirement is <1), but does not actually prevent the drain of currency from the economy. Since active users can now create currency, the system will either tend toward infinite currency or 0 currency, depending on whether more currency is being created or being drained.

6. If "Byte currency" is depleted, ratio requirements must be lowered.

As generous over-seeder rack up an ever higher ratio and users retire from the tracker, they take unused "byte currency" out of the economy, causing "inflation by attrition". (Remember, only active users contribute currency.) If activity by active users is not sufficient to counter-balance this, the ratio must be lowered or more free-leech must be offered in order to to counter this inflation with forced deflation. If the ratio requirement is low enough and the grace settings high enough, leechers abusing the system can give the appearance of semi-stability by regenerating lost liquidity. This keeps the server running, but defeats the purpose of the ratio requirement by letting in leechers.

The more currency is in circulation, the more downloading takes place and the slower the network becomes. The less currency is in circulation, the less downloading takes place and the faster the network becomes.

A "network activity score" can be created by defining it to be the "currency per active user". The higher the activity score, the slower the network will be and the more transfer will take place. The lower the activity score, the less data will be transfered and the faster the network will be.

Ratio Requirements Under-Reward 'Good but Obscure Content' Seeders

The definition of a "good user" is one who:

• Seeds what he has for long period of times
• Does not overly restrict his upload bandwidth
• Only downloads what he wants (instead of downloading with the mere intent to seed)

The definition of a "bad user" is one who:

• Downloads without care
• Does not seed, or
• Seeds, but restricts his bandwidth so much as to make his seeding worthless

Note that if the server consists of only "good users", it is quite possible for many, even most, users to have a poor ratio. Ratio's are not a good indicator of whether a user is a "good user" or a "bad user". Obviously users with high ratios are desirable, but what may not be so obvious is that "good users" with low ratios can also be highly desirable. These users are the users often responsible for the seeding obscure material not desired by everyone. When these users are forced off the server they take their esoteric offerings with them.

In an ideal environment, it would be possible to have no active downloads occurring because no one wants anything, while still having a large number of seeders and substantial download credit available to all users. Obviously if no one is downloading, no user's ratio is being increased. Under a ratio system it's possible for a new user to join, offer a good but unpopular seed, and not be rewarded for it either because few download it from him or because no one who wants it is able to download it from him.

Better Solutions

The problem with ratio systems is that ratio requirements reward the wrong thing. You don't really want to reward uploading, because ratios only reward the early possession of popular material. When ratios are dropped too far, the system becomes stable, but only because leechers provide that stability. What you really want to do is reward sharing, even if someone doesn't take advantage of it. You want over-seeders to be the boon to the community that they should be and to make life difficult for the true leechers that you don't want.

A much better system would be to give credit for making upload bandwidth available, even if it isn't used. For example, you might offer 1KB of upload credit for every second a user makes 5KB/s of upload bandwidth available. Unfortunately, no clients reliably report the true available upload bandwidth, so this would require a lot of programming to work right.

Another system would be to reward users every hour with a fixed number of KB for each torrent they seed, up to a set maximum. Under this system you don't worry about what a user's upload bandwidth is, but instead simply reward them for seeding torrents. This isn't quite as good, but can be implemented today.

A practical "compromise" alternative would be to combine ratios with rewards based upon the above systems. This has already been implemented in the form of "bonus points" at HDBits.org.

You can have "free leech" days where leeching does not affect your net download count. (Of course, this promotes leeching.)

You can create "byte currency" by selling it for cash. (This also promotes leeching, but at least you get paid!)

Finally, one very reasonable method is to be very strict about who can join and do away with ratios all together.

Conclusion

The "enemy" is not actually leeching, but leechers. Leeching by good users is fine, and should be promoted in-so-far as it does not adversely impact network speed. The true enemy is the leecher, who not only downloads more than he uploads, but also will not contribute upload bandwidth when it is necessary to do so. Thus, ratio requirements are fundamentally attacking the wrong issue.

Alternative systems based on seeding instead of uploading discourage leeching, maximize the seed:leecher ratio and therefore network speed, and allow people to become instantly accepted simply by seeding even if they have not uploaded much. They reduce waste by eliminating the treadmill act of downloading with the intent to seed for ratio. Finally, they promote the ideal of high-speed file sharing which comes as a result of generosity without simultaneously killing the network.

-- ThoughtKeeper - 22 Feb 2007

Comments

• Holy cow. Comments. Never so many on anything I have written before. Where did you all come from?! - ThoughtKeeper

I have tried sites that require a ratio of 'seeding' but the problem I have after three days I am banned because the upload is not enough. If I try to upload a file I am stopped because my ratio is not enough to allow uploading. Can't download, can't upload. I avoid those that require ratios.

-- TWikiGuest - 24 Feb 2007

So you create your own upload. How hard is that? That means that you don't download, just upload, so you CREATE credits. How's that for math

• You can't create credits just by uploading (in a 1:1 ratio system) -- someone has to balance the equation by downloading that torrent from you. How's that for math? - ThoughtKeeper

-- TWikiGuest - 25 Feb 2007

Your insight isn't new i myself (staffer of the Azureus project) have argued many times that ratios should be weighted by the seeds:peers ratio of a torrent, this way leeching on a swarm in need (few seeds, many peers) would be punished, uploading on such a swarm would be rewarded.

• No, the insight isn't that new, but perhaps it's the first time anyone has sat down and fully described the problem. I was just tired of listening to all the people who keep saying, "only bad users have a ratio below 1. If they'd only '...' then they'd be fine." -ThoughtKeeper

Overzealous uploaders on the other hand wouldn't receive much credit for seeding an overseeded torrent, because the seeds:peers weighting would reduce the credit they get from seeding an overseeding torrent. Together with some "safety margin" above and below a weighted 1:1 ratio that should do the job, everything beyond those margins (above AND below) should result in a ban/other limitations.

• The problem with banning over-seeders is that they are the ones who make your network fast. Of course that would keep the network running, but it would be a slow network. -ThoughtKeeper

And your math proofs are a bit of overkill imo, a bit knowledge about bittorrent and "mathematical intuition" should be enough

• I agree, but so many people have said it wasn't true, I thought I'd just prove it anyway. It's not like it was difficult. -ThoughtKeeper

-- TWikiGuest - 25 Feb 2007

I'm not sure about this...I can easily keep over 1.0 ratio in lots of private trackers, and my upstream bandwidth is nothing to write home about (30kB/s).

• Being a very good user who uploads new content helps ensure you will be one of the people "on top of the dog heap", but that does not mean there are not people on the bottom of it. - ThoughtKeeper

-- TWikiGuest - 25 Feb 2007

No overseeders, goodbye fast speeds!

• That's true, but the point isn't to get rid of over-seeders. The point is to reward people simply for the act of seeding, which actually encourages over-seeding. - ThoughtKeeper

-- TWikiGuest - 25 Feb 2007

A simpler solution is simply to set the ratio requirement at a lower value - e.g. 0.6 or thereabouts.

• Unfortunately lowering the ratio too far encourages leechers, who become mostly responsible for keeping the server running. Ack! But yes, this is better than a high ratio. -ThoughtKeeper

This is suggested to me by actual experience of various torrent sites. At sites where there is high and strict ratio requirement, it's a desperate struggle to maintain membership, and members hardly dare to download anything for fear of not getting enough leechers when they then need to seed it back. Membership at such sites is an experience of anxiety rather than enjoyment.

On the other hand, sites with more lenient rules have a much better performance than one would expect based on a cynical idea that everyone will "hit and run" if not prevented. There are some premature deaths of torrents, but in general there are enough seeders to answer demand.

Another way of looking at it is this. Obviously if everyone always seeded to 1.0, no torrent would ever die. But this is not ideal. Allowing for torrents to die out after interest falls below a certain level, and making an additional allowance for the inevitable minority of "hit and run" free riders, and carrying out the mathematical analysis accordingly, one would come up with an "optimal ratio" between zero and one. Set that as the requirement and a site will prosper.

-- [element 76] at austin.rr.com

-- TWikiGuest - 25 Feb 2007

this is a very interesting read, but a few things crossed my mind, maybe you can address them as well.

You didnt really factor for sites that allow new members to join, or for the attrition rates of a site. As the membership numbers will always go up and down, this allows for new people to join and impact the numbers quite a bit as well. Perhaps another way of wording it would be, everyone cannot have a ratio of 1, at the exact same point in time?

• Well, over 1, anyway. Yes, I could do a further analysis. It would take the form of a differential equation, probably that of the hunter-prey equation. I don't think anyone is interested, but I'll think about it. - ThoughtKeeper

-- TWikiGuest - 25 Feb 2007

everyone can keep a ratio over 1 if they upload new stuff, instead of just leeching what's already on the tracker.

• No, they can't. That's the point of the proof in point 1. I keep hearing this over and over again, and it's WRONG. In order to upload something new, someone else has to download it. That is, someone has to sacrifice their ratio to download your upload. The net effect is that its impossible for everyone to maintain a ratio of 1 without injecting additional liquidity through some other means. - ThoughtKeeper

That way, by forcing people to have a fair sharing ratio of 1, you also make sure that the tracker grows.

Isn't that the point?

• No, the point is to ensure that everyone participating is WILLING to seed and DOES seed, even if no one takes advantage of it, so the bandwidth is available if a 'good user' wants it. - ThoughtKeeper

-- TWikiGuest - 25 Feb 2007

I've run into the problem you mentioned, that someone in a closed >1 ratio MainP2P system will be always banned (aka musical p2p chairs), but in a slightly different way.

There's a private torrent site I wanted to join. They required all users to maintain a transfer ratio greater than 1:1 in order to maintain their membership. But they also started everyone off with zero upload bytes. So you had to seed something in order to earn credits before you could download anything.

I tried to seed content to the membership, but no one ever downloaded what I had to offer.

Two things happened. Either everyone already downloaded had what I was seeding, or no one had it (including me) except for one seeder. But because I hadn't uploaded anything yet, I wasn't able to download from that one seeder, so I never was able to upload anything. A catch-22.

I eventually gave up trying to join that private site and only used public sites, even though I was more than willing to always maintain a >1 transfer ratio. I always do that when I use public sites.

So strictly enforcing a >1 transfer ratio had the well known chilling effect. Private sites have no idea how many non leechers have given up trying to join their anti-leecher site.

-- TWikiGuest - 25 Feb 2007

I totally agree with this article. I am SysOp on a private torrent site (not sure if it was ok to name it) and we appreciate the problems outlined in this article. This is why we run 2 separate bonus systems and also give all new members a 2GB upload credit safety buffer on joining. Our minimum global ratio is set at 0.7 We have a similar system as mentioned where we reward members who seed by giving them bonus points. For every hour they seed, regardless of any peer activity we award a bonus point. Points can be accumulated and exchanged for upload credit. We also run a smaller system which is designed to increase ratio and also boost our irc channel numbers. This script simply awards members who join our irc channel 15mb every 15 minutes. The combination of these 2 scripts means there is absolutely no reason not to maintain a good ratio. We also have 'free-leech' torrents (at the uploaders discretion) where only upload is counted towards your ratio, again an excellent ratio-builder. We obviously also have many other scripts in place to watch for hit and run, cheating and any other unfair activity. We continuously ask our community for feedback and suggestions and I am glad to say that we get nothing but good comments.

-- TWikiGuest - 25 Feb 2007

As a footnote to my last post (above ^^^) we feel that by running a relaxed site members are not afraid to download or live in fear of getting banned, which is reflected by our excellent seeder to leecher ratio (1211%)

-- TWikiGuest - 25 Feb 2007

-- TWikiGuest - 26 Feb 2007

One tracker I use has a good work-around for this. It has what are called "seed points", and if you are seeding a torrent that has a 20:1 seed:peer ratio or more (or no leechers at all), and also if your ratio for that torrent is under 0.800, it rewards you with points each hour you seed that file. If your points reach the "cost" of the torrent in question (determined by filesize and how much you had left to upload at the time the points requirements were met), then you'd automatically be levelled out. Your upload stat for that torrent would automatically be raised to give you a share ratio of 1.000, and your account credited with that upload amount. So even with your actual upload ratio for that torrent at 0.550 because no one else wanted it anymore (perhaps old or unpopular), your account gets credit for it because you were sharing it.

-- TWikiGuest - 26 Feb 2007

What I don't understand is what is to stop a new member from simply providing new content before downloading anything? It's easy to keep your ratio above the minimum if you simply create a new torrent to upload if your ratio starts to get low.

Another way to monitor this is to put a cap on your download speeds to more closely match your uploads.

Admittedly this requires a large user base and can be difficult for trackers just starting up and trying to gain users.

-- TWikiGuest - 26 Feb 2007

Many trackers don't let members upload. This is done to keep the upload speed high and stop fake torrents being posted. Most uploaders on private sites have high speed seedservers and scene access.

-- TWikiGuest - 26 Feb 2007

Some private sites asked you to download like $20 and so on to get a free 200 GB upload credit. The money goes into keeping up the servers and maintance. That usually helps out.

-- TWikiGuest - 27 Feb 2007

Wow. I was unclear about ratios before...but now, I am a bit more informed. Thanx. But tell me something. Where is the balance for someone that looks for obscure and esoteric torrents? Can a person download what he wants, and then seed it for a month...with no limits on his bandwidth? I would like to have a simple formula to use...and then stick with it. Ratios give me a headache..heh. I try and see myself as a customer at a trading post...if I have nothing to trade, then I will get nothing. But if I keep a supply of what people want and need, then I should be able to get a few things I would like to have. Does that make sense?

-- TWikiGuest - 28 Feb 2007

requiring users rather than being open to all is 100% pointless in the first place.

-- TWikiGuest - 01 Mar 2007

Beautifully written, lovely maths, nice graphics. But if people want to leech, they're going to. The counter to that is a seed requirement.

When man is altruistic, your approach will be important. Until then... private trackers remain the bastion of those tired of a slow, leech-filled swarm.

-- TWikiGuest - 02 Mar 2007

You say that "They reduce waste by eliminating the treadmill act of downloading with the intent to seed for ratio". The implication being that downloading stuff just to seed it is wasteful.

Take an extreme example. Imagine I have insane upload bandwidth. I download something I don't want then seed it back at a high rate. The throughput on that torrent is thus increased. Everybody gets the file sooner. Is this actually a bad thing?

-- TWikiGuest - 02 Mar 2007

"Holy cow. Comments. Never so many on anything I have written before. Where did you all come from?! -"

StumbleUpon? .

-- TWikiGuest - 15 Mar 2007

Just read this post on a 'stumbleUpon' recommendation (hence all the reurns?) - one thought: have ytou thought about publishing this scientifically? I believe the scientific community has journals running p2p math and studies as a hot subject right now. Take care...

-- TWikiGuest - 15 Mar 2007

I just started reading this page, you're conclussion is probably right (I didn't reach it yet) but you made a small mistake in your fourth formula: (alpha - u1 < alpha - d1). This is only right if u1=d1 and u1 is not d1 since your first formula says : u1/d1 > 1. Your fourth formula should be (alpha - u1 < alpha - dpersonthatdownloadsfromperson1)

-- TWikiGuest - 19 Mar 2007

'Only downloads what he wants (instead of downloading with the mere intent to seed)'

What is wrong with downloading with the intent to seed? Those who download what they want are inclined to be leechers. Those who download with the intent to seed are by definition the seeders. Those people are not only responsible for the wide availability of fast torrents but are the backbone of the content distribution network.

-- TWikiGuest - 19 Mar 2007

this is way to complicating the simple truth is without seeding more than 1:1 there will be very little in the way of downloading How can anyone come and say i get banned after 3 days is just not acceptable and you will need to stick to the public trackers that don't record your stats as you will never be welcome or allowed to be part of any of the private trackers so yes join up today download multi gig torrents and then whine you can not upload enough

this is how most people use their credit cards for the first couple of month and then you go bankrupt go figure hehe

-- TWikiGuest - 20 Mar 2007

Very good article! Yet, I think because of ration problems people are more interested to find/create and upload torrents that don't already exist in the tracker thus creating themselves more upload ration, but more importantly new content to the site.

-- TWikiGuest - 27 Mar 2007

I too have thought about this when i was bored. i was like wait every user can't have a 1 or over ratio, thanks for doing the math.

-- TWikiGuest - 11 Apr 2007

No, you are an idiot. Clearly you've never actually been an admin at a ratio-enforced torrent site, as you have no idea what you are talking about.

Firstly, all the nice mathematics you've done to show us that everyone's ratio cannot be > 1 is great, but I think everyone already knew this. In a closed system, yes, the average MUST be 1.

HOWEVER, as anyone will tell you, bittorrent sites aren't closed systems, and there are large injections and leakages of credit.

-Cheaters. -Inaccurate stats reporting. -New members joining the site, downloading, and then getting banned. -Established members letting their accounts lapse. -Periods of free download/increased upload. -Staff giving free credit to people. -People donating to buy credit.

All these things essentially generate extra upload credit which is then distributed between the users, sort of like a government issuing more currency. Everyone's ratio gets boosted, and so everyone CAN enjoy a ratio above 1.

More things to consider -

You are also assuming that the only content people can upload is either content they download from the tracker, or their own torrents they upload, which is clearly not the case. The vast majority of trackers deal with the same sort of content, and so a user struggling to upload can just download it from somewhere else and upload it at the tracker in question.

Your "better solution" of offering credit based on willingness to upload is also pretty badly flawed. There is no way to accurately determine how much capacity someones line has at the current time. For instance, I could have all my torrents seeding at your tracker with nothing being uploaded, while having my line maxed out uploading something else, or more simply, I could claim I had a 100mbit line, and gain credits for the "unused" 99.5mbit.

You also seem to assume that all users are well-meaning, and genuinely want to increase their ratio at the site. I know from experience that there are many people who will simply take all they can during the 2-week grace period (or however long it is), knowing full well that they will lose their account, and then just signup again (or acquire another invitation) once this happens.

I think you have made some decent points, but this article is a gross over-simplification. Just look at the popularity of sites that enforce such a scheme, and I think we can see that the system works well, if not perfectly.

-- TWikiGuest - 14 Apr 2007

Very good insights, but wasted I'm afraid. Yes, it's impossible for EVERYONE to maintain a 1.0 ratio. Smart people already know that. If you're at the bottom, though shit. Upload something, etc etc. As the old adage goes, you don't have to be faster than the bear...just faster than the fat kid.

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-- TWikiGuest - 05 Oct 2007