Fund I/O can be extremely useful in funding projects with a social or environmental impact: producing vaccines, for example. One challenge of vaccine production is getting the market for a new vaccine to scale to the point where the vaccine becomes affordable. Because prospective buyers expect prices to drop, many will wait with their purchase, which may even prevent the market from reaching the required size. Fund I/O offers an original solution to this problem, by giving buyers incentives to buy now. This can provide a viable alternative to difficult-to-negotiate advance market commitments.

Fund I/O can be used not only for digital goods, but for anything that has large fixed costs and low marginal costs. This includes physical goods that have a large social impact, such as vaccines. Here is how this might work.

Suppose a vaccine for a certain disease has already been developed. Now it needs to be produced at scale to reach millions of people world wide. The problem is vaccine production requires a substantial investment to get going. This means, if a supplier were to produce just 100,000 doses of the vaccine, each dose might be prohibitively expensive. But at a scale of 10 million doses, each dose could be very cheap. As long as the market is small, the price for the vaccine will be very high so that only few people will be able to afford it. But once the market grows, the price will drop, and many more people will be able to afford the vaccine.

This creates a chicken and egg problem: To get the price down, many people would need to buy the vaccine. But for many people to buy the vaccine, the price would need to drop. So how can we get the market started?

One approach to getting the market started are advance market commitments. A couple of large buyers of the vaccine, such as governments or charitable donors, make a joint commitment to buying a large number of doses at a specified price. The idea is that by guaranteeing a certain demand for the vaccine in advance, the market can be jump-started at a volume attractive to suppliers and at a price attractive to buyers.

There is a catch, however: Because, the price for the vaccine will drop over time, buyers have an incentive to wait. Those who start buying early will pay a premium. It you wait, you pay less, and the later you get in, the more you save. Even if everyone wants to buy the vaccine, those who manage to start buying last will pay the least. Early adopters effectively subsidize late entrants.

You can imagine that this leads to quite a lot of maneuvering over who gets to be the last. Especially because the market for governments buying vaccines has several features that exacerbate the problem: First, very large sums of money are at stake. Second, buyers (such a governments) are often frugal, by necessity. Third, implicitly subsidizing other parties may be unpopular, even if it is for a good cause such as getting more vaccines to more people. And finally, the only benefits of entering early are measured in terms of impact (saving lives) and not profit (return on investment), which often makes spending money harder instead of easier - unfortunately. Advance market commitments can help a lot in this setting: by forming a coalition of first movers, the financial disadvantage to buying early is reduced considerably. But the incentive to get in later remains, and this makes it often very hard to establish such a coalition.

Here Fund I/O can help. In fact, Fund I/O was designed to solve exactly this chicken-and-egg problem. Using Fund I/O, buyers do not have an incentive to wait. To achieve this, all you need to do is apply the refund mechanism used in the second phase of the Fund I/O model. Buyers can purchase immediately and still benefit from economies of scale as the market matures. Through the refund mechanism, they can rest assured they will benefit from decreasing prices in just the same way as a late entrant.

This solution makes jump-starting markets much easier, by offering a smooth transition from a low volume/high price market to a high volume/low price market. Moreover, it does not require complicated negotiations aimed at getting buyers to commit to making purchases now, even though it would be in their financial interest to wait. The Fund I/O mechanism guarantees the same financial benefits as an advance market commitment involving all potential buyers, present and future, but it can be set in motion by just a few interested parties right away. Agreements are not made at a negotiating table but on a platform of exchange that incentivises all participants to reveal their true valuation for the vaccine.

Of course there are many more details to be addressed here, both as far as the theoretical model and practical issues are concerned. I will deal with these as I go along. In the next post, I’ll start with a detailed example of this use-case of Fund I/O in action.

I think we need more crowdfunding projects for open source software! And here is one, for example: Yorba, the guys behind Shotwell, are currently running an Indigogo campaign to fund the development of their open source email client Geary. As of this writing, they have raised about \$50,000 of their \$100,000 goal and the rush of the last 24 hours has just begun. So, please head over there now and consider making a dontation. You can always come back here to read on!

Why Funding Open Source Software is Difficult on Kickstarter

Crowdfunding has taken off, for physical goods as well as for software-related products like games. So why do we see so few open source project take advantage of crowdfunding? In fact, Kickstarter is based on the Street Performer Protocol (SPP), a method for raising funds that was originally intended as a model for financing open source software, and that had some early successes such as making Blender open source. Could it be that rewards-based crowdfunding as on Kickstarter simply does not work as well for open source software?

There is one subtle difference between open source and closed source projects on Kickstarter, and this difference may be crucial: The successful game projects on Kickstarter have no intention of giving away their core product for free. This makes pledges pre-sales instead of donations. Open source projects on the other hand announce right away that their product is going to be free. Does this influence backer behavior?

It is certainly true that many crowdfunding campaigns are successful because some altruistic backers pledge huge amounts of money to the project - much more than they would need to pledge to get a copy of the product for themselves. But projects rarely can be funded through such altruistic contributions alone. Take a look at Shrouds of the Avatar, for example: the large majority of backers pledged the minimum amount required to receive a copy of the game. These backers are obviously price conscious and don’t pledge more than they have to. I will call these backers rational, in the economic sense. This means that they are self-interested and act to maximize their own welfare.

This leads me to the following working hypothesis:

Crowdfunding projects need contributions from both rational and altruistic backers to raise large amounts of money.

This would explain why open source projects appear (so far) to be less successful than other creative projects at raising money via crowdfunding: Because open source software is (going to be) free, rational backers don’t pledge anything.

In fact, there is research in the economics literature on the optimal behavior of an entirely rational backer who is faced with the decision of backing an open source project. The amount he or she is willing to pledge is determined by the probability of their pledge being pivotal and getting the project funded. Backers correctly observe that this probability decreases with the total number of backers interested in a project. A consequence is that - if all backers were rational - the amount of money open source projects can expect to raise grows only proportional to the square root of the number of people interested in the project. In contrast, the amount a closed source project can raise grows linearly in the number of people interested in the project. This can quickly lead to a difference of several orders of magnitude!

(Now, before anyone gets a wrong impression: I do not think that people behave entirely rationally or that they should behave rationally. It is vital for our society that people act out of altruistic motives. And it is great that crowdfunding has found a way to get altruism to scale, so that we can make great things happen. I think we should look for more ways in which we can integrate altruism into our economic activities. All I am saying is that there is a reason that our entire economic system is built on the assumption that people tend to behave rationally. Our goal should be to find economic models that incorporate both altruistic and rational motives.)

Fund I/O as a Model for Funding Open Source Software

Fund I/O is a new crowdfunding model based on the Average Cost Threshold Protocol (ACTP). (If you are not yet familiar with Fund I/O, you may want to check out this example before continuing here.) When I developed the ACTP, one of my goals was to find a crowdfunding model that would work for open source software. And in many ways Fund I/O achieves precisely that:

• Most importantly, Fund I/O gets rational backers to reveal how much they would truly be willing to pay for the software.
• This allows open source project to raise as much money as closed source projects.
• Moreover a project can raise more money on Fund I/O than using the Kickstarter model, because it lets the market determine the optimal price of the product.
• At the same time, Fund I/O still allows altruistic backers to dontate as much as they want to the project. (This requires Fund I/O to be used with the option of allowing each backer to set their own minimum price level.)
• Finally, Fund I/O provides a clear path towards the relase of the software product under an open source license.

Now, you may have noticed that I did not quite say Fund I/O “works” for open source software. In fact, I do not think the Fund I/O model gives me everything I would personally want from an open source funding model. However, I do think it works much better than anything that is currently out there. So let me first explain in what way it does work for open source software. In the next section I will then go on to explain the limitations I see.

Fund I/O works great for open source projects if your project meets the following assumptions:

• Your goal is to finance the creation of version 1.0 - no more.
• You are willing to release your software under a closed license at first.

Then you just run through the three phases of the Fund I/O mechanism (with individual minimal price levels) and at the end:

• you have completed the software you wanted to write,
• you have released the software and its source code under an open source license, free for everyone to use and modify,
• your costs were covered right from the start and
• you even made a certain profit that you can determine in advance.

The release under an open source license happens if and only if there is enough altruism among your customers to finance the costs of production through voluntary donations. However, this absolutely not the same as simply using the SPP. First of all, your project gets made if there is enough rational interest in your software. It just won’t be open sourced unless there is enough altruistic interest. Second, Fund I/O achieves a smooth transition from the state where there are few users who would need to donate very large amounts to the state where there are many users who need to donate just a couple of dollars. This lessens the burden on any individual donor considerably. Moreover, the donation is a sunk cost for the customer: donating means forgoing future refunds instead of paying money now. All of these factors make it much easier to gather the required donations, than using the SPP, even if Fund I/O is used with individual minimal price levels. If a fixed minimal price level is used for everyone, then gathering the required donations gets easier still.

Caveats

Now, I believe the above could work great for developing, say, an open source game. But for open source software I see a couple of caveats.

First of all, the user base of the software has to be willing to use closed source software during the second phase of the protocol. If a substantial number of users will never run anything but free software, then you are of course out of luck. I do not think this will be a problem; it was not a problem for Light Table during its crowdfunding campaign.

Second, it has to be reasonable to make the software non-free during the second phase. This means that the software is for sale and according to the license users are not allowed to pass copies of the software along to their friends. This is a stronger restriction than just not releasing the source code and may turn off more of the free software crowd. More importantly, this limits viral marketing effects during the crowdfunding and sales phases. People have to buy the product to try it out. (Even though a demo could certainly be offered.)

Of course the whole point of Fund I/O is that because of the refund mechanism, there is less of a disincentive to buying than in a classical sales context. In particular, if a customer sets their minimal price level to zero, then they get a full refund in case the software is released as open source. But in a world where many people expect application software (as opposed to games) to be free, this may be a hurdle. I’d be optimistic and say that the compromise of making software non-free for a limited amount of time, according to a transparent mechanism, in order to cover the costs of development is convincing enough to assuage these concerns.

Finally, and most importantly from my point of view, software requires maintenance. Application software needs to be developed continually. A game may be “done” after the 7th patch or so, but application software never is. If the only goal is to fund the creation of a version 1.0, then this is not a problem. But what if you are looking for a way to finance the development of open source software throughout its entire lifecycle?

An obvious modification to Fund I/O for financing several consecutive versions would be to run a new campaign for each major version. The mechanism could easily be adjusted to make a part of the revenues from the version 2.0 sales refunds for version 1.0. And once 1.0 is free, buyers/backers of subsequent versions could stop paying refunds to version 1.0 owners. The problems I see with this are therefore not theoretical, but practical in nature. The company developing the software would have to maintain several versions of the software side-by-side, they would have to backport security patches and provide support, and they would have to finance these activities from their version 2.0 revenues. While all of these concerns appear in commercial (and open source) software development anyway, they are exacerbated by keeping different versions under different licenses around - just because of the funding scheme.

Ultimately, it will depend on the particular project which of these considerations are relevant, and which variant of Fund I/O makes the best fit. Personally, I view Fund I/O as a toolbox that can be tailored to fit many different needs.

Conclusion

I hope to have convinced you that crowdfunding open source software is possible at much bigger scale than we have seen so far. What we have to do to make this happen is to find a crowdfunding model that takes the backers’ diverse patterns of economic behavior into account. I think Fund I/O is a step in the right direction, but can certainly be improved upon with regard to funding open source software.

What do you think would be the challenges when applying Fund I/O to open source software? How would you modify Fund I/O to make it more applicable to open source software? Comments and questions are welcome!

Throughout my conversations about Fund I/O over the past couple of weeks, one thing has become clear: Fund I/O implies a radically different way of thinking about business.

What is the Fund I/O business model? Fund I/O is a business model for the production of goods with substantial economies of scale. In a nutshell:

• Financing is provided through presales made during a crowdfunding campaign.
• The producer commits in advance to a pricing scheme that passes savings due to economies of scale on to customers directly, by lowering prices as fast as possible.
• Early buyers are guaranteed to get a refund when the price drops, so that they have no incentive to wait with their purchase.

If you are looking for details, good places to start are fund-io.com and this example.

What makes Fund I/O different from the conventional way of doing business? Suppose you run a company and want to produce a product that benefits from economies of scale. Here is what Fund I/O means to you:

• You finance your product entirely through presales. You do not have take on a loan, you do not have to pay interest, and you do not have to maximize profit for your investors. This drastically reduces your risk and your cost of capital. The presales appear as a non-interest bearing liability on your balance sheet, and the primary interest of your financiers is to receive a great product, not to get paid.
• You place an absolute limit on the profit you can make off a product. This limit can be chosen arbitrarily, but by choosing a refund scheme you are committing to an absolute limit on your profits. Note that because you used presales from customers to finance production you are making a profit without having invested any capital yourself, so from that point of view your return on investment is infinite, even if your maximum profits are not!
• You obtain a tremendous amount of market information: By committing to a refund scheme, you give your customers an incentive to reveal exactly how much they are willing to pay for your product. This translates into higher sales, more revenue and faster growth.

Through the combination of crowdfunding, a clear price reduction scheme with a limit on profits and a refund mechanism, Fund I/O manages to align the interests of key stakeholders and it provides them with a mechnism to communicate truthfully about value: It resolves the fundamental conflict of interest between investors who want to maximize their profits and customers who want to minimize what they have to pay.

What makes Fund I/O different from other crowdfunding schemes? There are basically two different crowdfunding models out there: reward-based crowdfunding and equity crowdfunding.

Equity crowdfunding is not that different from the conventional business model. Financiers are still interested in return on investment. The key difference is that investors often have another stake in the business in addition to their profit interests. For example, they may be not only investors but customers at the same time, or they may have an interest in the social impact of the business. But still the business is obliged to maximize profits on behalf of its investors, which gives customers incentives to understate their valuation of the products the business is offering.

Reward-based crowdfunding makes customers the financiers of a product, aligning the interests of two key stakeholders. But it still does not provide a mechanism to communicate about value. In reward-based crowdfunding, creators have to fix the price a priori and often a majority of backers pledge just this price and no more. The success of reward-based crowdfunding hinges on the generosity of backers who self-select for price targeting by willingly paying a large premium for the product. This type of altruism (whether motivated by appreciation of the project, the thrill of being part of something great or the enjoyment of receiving secondary merchandise) is great, if it suffices to finance the project. Fund I/O can be tweaked to captialize on this type of gift economy as well, but the main contribution of Fund I/O lies elsewhere: Fund I/O makes crowdfunding work in cases where the capital requirements of a project are large or where the potential customers are price sensitive. Fund I/O provides the mechanism that allows producers and customers to join forces in order to create great products, even if all parties involved need to avoid paying more than they have to.

What problem does Fund I/O solve? From the point of view of a business, Fund I/O reduces risk, aligns interests of stakeholders, provides a wealth of market information and generates more sales. Moreover Fund I/O provides financing alternatives to debt and equity and conventional crowdfunding. This is particularly useful in cases where debt and equity are too expensive, too risky or simply unavailable and when altruistic contributions made through a rewards-based crowdfunding campaign do not suffice.

However, this is just one angle on the question: “What problem does Fund I/O solve?” Over the next days, I will post other answers, taking the point of view of a customer, or examining the applicability of Fund I/O to non-profit projects aimed at social impact. One very technical answer is already written up: Fund I/O provides a practical implementation of an incentive compatible, individually rational mechanism for the private provision of excludable public goods that is asymptotically optimal.

Fair crowdfunding has a new home: Fund I/O.

As you may have seen, I have been blogging recently about a new crowdfunding mechanism for public goods, the Average Cost Threshold Protocol. Since the last post on the topic, I have spent quite some time discussing this idea with people, and have decided to give this idea into a project dubbed Fund I/O with an own website fund-io.com and an own Twitter account.

To get updates about his project and idea, you can subscribe to a (very-low volume) mailing list on the project website, or just follow this blog. I’ll keep posting whatever I have to say on the subject of fair crowdfunding on this blog. (If and when Fund I/O gets its own blog, I will make an announcement here.)

Finally, in case you are wondering, Fund I/O is not a startup. (Not yet, anyway.) The current goal of Fund I/O is to try some variant of the ACTP in practice. If you have got just the right project in need of a revolutionary funding mechansim, let me know!

There are many beautiful geometric interpretations of the GCD and visualizations of the Euclidean algorithm out there, see for example here, here and here. I have written about this myself some time ago. But I find that some of the pictures become much clearer when lifted into the third dimension.

[Note: I am using Sage Cells to produce the pictures in this post. This has the advantage that you, dear reader, can interact with the pictures by modifying the Sage code. However, this will probably not run everywhere. It certainly won’t work in most RSS readers…]

Here is the graph of the GCD! First a little static preview…

… and now a proper interactive graph. Be sure to rotate the image to get a 3d impression. (Requires Java…)

What precisely are we seeing here? The gcd is a function $\mathrm{gcd}:\mathbb{N}^2\rightarrow\mathbb{N}$. Its graph is a (countable) set of points in $\mathbb{N}^3$. This graph is precisely the set of integer points contained in one of the (countably many) lines indicated in the above picture. The value of the gcd is given by the vertical axis.

Now, what are those lines? Suppose $a$ and $b$ are two natural numbers that are relatively prime, i.e., $\gcd(a,b)=1$. Then we know that $\gcd(ka,kb)=k$ for any positive integer $k$. So all (positive) integer points on the line through the origin and $(a,b,1)$ are in the graph of the gcd. Conversely, for any numbers $x,y$ with $gcd(x,y)=z$, we know that $\gcd(\frac{x}{z},\frac{y}{z})=1$, that is, the point $(x,y,z)$ lies on the line through the origin and $(\frac{x}{z},\frac{y}{z},1)$. So by drawing all these lines, we can be sure we hit all the integer points in the graph of the gcd.

What I find very beautiful about this is that when looking at this picture from the right angle, you can see the binary tree structure of this set of lines. It is precisely this binary tree that we walk along when running the Euclidean algorithm. The lines in the above picture are color coded by their depth in this “Euclidean” binary tree.

The version of the Euclidean algorithm I like best this one. Suppose you want to find $\gcd(a,b)=1$. If $a>b$, compute $\gcd(a-b,b)$. If $a<b$, compute $\gcd(a,b-a)$. If $a=b$, we are done and the number $a=b$ is the gcd. At each point in the algorithm we make a choice, depending on whether $a>b$ or $a<b$. If we keep track of these choices by writing down a 1 in the former case and a 0 in the latter, then, for any pair (a,b), we get a word of 0s and 1s that describes a path through a binary tree. As we walk from one pair $(a,b)$ to the next in this fashion, we trace out a path in the plane: For each 1 we take a horizontal step and for each 0 we take a vertical step. This path can look as follows.

Now comes the twist. We can turn the Euclidean algorithm upside down. Instead of walking from the point $(a,b)$ to a point of the form $(g,g)$ and thus finding the gcd, we can also start from the point $(1,1)$ and walk to a point of the form $(\frac{a}{g},\frac{b}{g})$ to find the gcd $g$. (Why $(\frac{a}{g},\frac{b}{g})$? Well, $(\frac{a}{g},\frac{b}{g})$ is the primitive lattice vector in the direction of $(a,b)$. In terms of the graph at the top, $(\frac{a}{g},\frac{b}{g},1)$ is the starting point of the ray through $(a,b,\gcd(a,b))$.)

How do we get there? We start out by fixing the lattice basis with “left” basis vector $(0,1)$ and “right” basis vector $(1,0)$. Their sum $(1,1)$, which I call the “center”, is a primitive lattice vector. Now we ask ourselves: is the point $(a,b)$ to the left or to the right of the line through the center? If it is to the left of that line, we continue with the basis $(0,1)$ and $(1,1)$. If it is to the right of that line, we continue with the basis $(1,1)$ and $(1,0)$ and recurse: We take the sum of our two new basis vectors as the new center and ask if $(a,b)$ lies to the left or to the right of the line through the center, and continue accordingly. If $(a,b)$ lies on the line through the center, we are done: the gcd of $a$ and $b$ is the factor by which I have to multiply the center to get to $(a,b)$.

As we follow this algorithm, we draw the path from one center to the next, until we reach $(\frac{a}{g},\frac{b}{g})$. Then we get the following picture. It looks almost like a straight line from $(1,1)$ to our goal, but it is only straight whenever we take two consecutive left turns or two consecutive right turns. Whenever we alternate between right and left, we have a proper angle, it only gets difficult to see this angle the farther out we get. If we write down a sequence of 1s and 0s, corresponding to the left and right turns we take, we get the exact same sequence as with the standard Euclidean algorithm.

The obvious next step is to draw the complete binary trees given by the above two visualizations of the Euclidean algorithm (up to a certain maximum depth). Drawing the first tree gives a big mess, because many of the edges of the tree overlap. You can clean up the picture by removing the lines and drawing only the points: In the code below comment the first definition of thisdrawing and uncomment the second. This then gives you a picture of all primitive lattice points (pairs of relatively prime numbers) in the plane at depth at most maxdepth in the tree.

The second variant of the Euclidean algorithm gives a much nicer picture, revealing the fractal nature of the set of primitive lattice points. The nodes of the tree, i.e., the points drawn in the picture are the starting points of the rays plotted in the 3d graph of the gcd given at the beginning of this post. In this way, this drawing of the tree gives the precise structure of the binary tree intuitively visible in the three-dimensional graph. One thing you may want to try is to increase the maxdepth of the tree to 10. (Warning: this may take much longer to render!) Note how far out from the origin some of the points at depths 10 are.

The reader may have observed that the binary tree described in this post gives us a very nice way of enumerating the rational numbers. Calkin and Wilf made this observation in the paper Recounting the Rationals, whence it is sometimes called the Calkin-Wilf tree, even though Euclid tree might be a better name as it is given by the Euclidean algorithm itself. As we have seen above, a look through the lens of lattice geometry tells us how to draw this tree in the plane and how to find it in the graph of the GCD.

Can fair crowdfunding resolve the conflict over copyright? Or, as Cory Doctorow put it yesterday:

Does the Average Cost Threshold Protocol incentivize content creators to “sue the Internet”?

Very short answer: The incentive for creators to sue the Internet is very small (much smaller than usual) but not zero.

Short answer: The Average Cost Threshold Protocol (ACTP) drastically reduces the incentive of content creators to sue people, as creators’ costs are covered from the start. Moreover it drastically reduces the incentive for people to copy content outside of what is allowed by the protocol, because buyer’s payments are used to lower the price (for themselves and others) and to eventually release the content under a copyleft license.

That said, there is a “sales phase” in the protocol where such incentives exist, even if they are limited. However, this drawback is outweighed by the protocol’s key advantages:

• Creators can expect to raise significantly more money than with funding models that employ only copyleft licenses.
• Creators and users agree from the outset on a clear path towards release of the content under a copyleft license.

Moreover the protocol can be modified so that creators have no incentive whatsoever to sue the Internet - however, in this case creators also lack an economic incentive to deliver a great product.

Long answer: Here are the key reasons why creators have very little incentive to sue the internet, if they choose to adopt the ACTP.

• The protocol guarantees that creators’ costs are covered before content is ever released. This drastically reduces the creators’ financial risk, which in turn reduces the threat unlicensed copies pose to creators.
• Costs are covered by users, not investors. This frees creators from having to worry about paying back loans or maximizing their return on equity. Thus the creators’ economic goal is to maximize user value not shareholder value.
• The ACTP provides a clear path towards releasing the content under a copyleft license. By adopting the ACTP, creators subscribe willingly to this goal.
• Creators’ profits are maximized if and only if the content is released under a copyleft license, at the end of ACTP. This means that creators’ have an incentive to work towards the goal of releasing the content for free.

That being said, the ACTP does have a “sales phase” where the content is available only under a “closed” copyright license and people have to buy access. In this sales phase, users have an economic incentive to obtain unlicensed copies.

• Creators receive a share of the revenues earned during the sales phase as profit. They thus have an incentive to sue people who avoid paying for access in the sales phase.
• However, if there are enough buyers to “free” the content and cause its release under a copyleft license, then the creators do not care how many people made unlicensed copied during the sales phase: their profits in this case are fixed from the start.
• The ACTP could be modified so that creators do not receive any profits during the sales phase. All revenues from sales could go towards lowering the price by refunding previous buyers/backers. In this case creators do not have any incentive whatsoever to sue the Internet, at any point in time. However, I do not think that this is a good idea. Because in this case, creators’ do not receive any revenue after the initial fundraising campaign and thus they have no economic incentive to make their product shine and promote it accordingly.

Moreover, the ACTP provides incentives for users to actively participate in the sales phase of the protocol by paying, instead of by making or obtaining unlicensed copies:

• In the short term, the more people pay, the faster the price will drop. As a buyer this means you yourself will get a larger refund and at the same time you will enable other people to afford access.
• In the long run, the more people pay, the sooner will the content be released freely, under a copyleft license.

The sales phase thus becomes a cooperative game among users, where they can help each other by lowering the price in the short term and ultimately “freeing” the content for everyone. This of course has the drawback that it might cause cooperative users (who buy) to sue uncooperative users (who make or obtain unlicensed copies). But the amount of refund money cooperative users stand to loose from unlicensed copies is very limited, so I hope this is not going to be an issue. Users who obtain unlicensed copies hurt themselves by making it less likely that the content is eventually going to be officially released under a copyleft license.

But why should we have a “sales phase” at all? Would it not be better to release the content under a copyleft license right away? The reason is a phenomenon that has been well-studied in the economics literature: By restricting access to content during the sales phase, it is possible to raise significantly larger amounts of funds for the creation of digital content than would otherwise be possible. In economics jargon: It is significantly easier to raise funds for the private provision of a public good with use exclusions than for the private provision of a pure public good without use exclusions. The amount of money we can raise for content that is going to published under a copyleft license from the start grows on the order of $\sqrt{n}$ where $n$ is the number of people in the economy. If we introduce use exclusions, on the other hand, the amount of money grows on the order of $n$. This is a difference of several orders of magnitude, with a huge potential impact in practice! (I will write more about these results in a future post. In the meantime I recommend this post and these two articles.) The ACTP aims to strike a balance between the conflicting goals of publishing free content and raising funds by employing use exclusions in an intermediate phase and providing a clear path towards release under a copyleft license.

Bottom-line: The ACTP does a far better job of aligning incentives than any other mechanism for funding digital content that I know. I think it goes a long way towards resolving the economic conflict over copyright and I can’t wait to put it into practice and try it out.

We live in a wonderful age! So much that we value - music, movies, books, games, software, designs, research, art, ideas - can be stored in digital form. Once these digital goods are created, they could in principle be made available to everyone at virtually no extra cost. So why don’t we? Why do we fight about copyright instead?

The reason is a conflict of economic interests. On the one hand we need to fund the creation of digital goods. On the other hand, our creation can do the most good if it is made available to literally everyone. But how can we convince people to pay for a digital good, if they know that eventually everybody can download a copy for free? What we need is a new idea that addresses both of these issues: the public welfare and the profitability of creating digital goods.

In this regard crowdfunding platforms are very promising, as they can in principle be used to fund digital goods that are then made freely available to everybody. However, a closer look reveals that in practice the digital goods financed by crowdfunding campaigns are often sold like apples - as if they could not be copied at all. Here is one idea how we can do better:

The Average Cost Threshold Protocol is a fair crowdfunding mechanism that takes economic interests of both users and creators into account.

How it works is best explained by way of a concrete example. Suppose a company wants to raise \$1,000,000 to finance the production of a computer game. They start a crowdfunding campaign on a website implementing the Average Cost Threshold Protocol. The website starts collecting pledges until a deadline is reached. Suppose on the day of the deadline the pledge look as in picture below. In particular, we find that there are 20,000 people who pledged \$50 each or more. If all of them pay exactly \$50, then the company’s costs will be covered and everybody will have paid the same amount. Moreover \$50 is the lowest price that covers costs: for example 24,000 people would be willing to pay \$40, but that yields only \$960,000. This lowest price now becomes the threshold price: everybody who pledged at least \$50 pays exactly \$50 and they become backers, who will receive a copy of the game once it is finished. All others do not pay anything and do not get a copy. The principle is simple: the costs of production are distributed equally among all those who get access. The threshold price is chosen such that it provides access to the largest number of people at the lowest possible price such that the costs of production are covered.

Now that the game is released, more people have heard about it and want to purchase access. Typically, the company would simply sell copies and keep the revenues as profit. But here comes the twist! Instead of paying a fixed price, interested buyers submit pledges on what they would be willing to pay for the game. Suppose there are another 10,000 people who each pledge \$40 or more. Then we drop the price to \$40 dollars and charge each newcomer \$40 for a copy of the game. The revenues of \$400,000 are split 50-50. One half goes as profit to the company. The other half goes to the original backers, who each get a refund of \$10. Following this principle the price keeps dropping the more people buy the game and thanks to the refunds everybody who gets access is guaranteed to pay the same price no matter how much they pledge or when they pledge.

The more people buy the game, the lower the price for everyone. And the lower the price, the more people can afford to buy the game! What if this virtuous cycle takes off and the game becomes really popular? At this point another feature of the mechanism kicks in: At the very beginning of the crowdfunding campaign a price of freedom of \$10 is announced. This means that as soon as the threshold price drops to the price of freedom of \$10, the game is made available to everyone, the whole world, no matter if they paid or not. This includes the release of the game under a copyleft license as well as the publication of all source code and assets, to allow people to freely modify the game. In the example, the price of freedom of \$10 is reached if the game becomes so popular that at least 180,000 people buy the game. In this scenario, the company has not only covered its costs, but on top of that it also made a profit of \$800,000. (This amounts to an infinite (!) return on investment as the customers contributed all the funding for the project!) The public gains a game they are free to use and modify. And the backers and buyers of the project spent only \$10 each, both to gain early access to the game and to make all of this happen.

We can even go one step further and allow every customer to set their own price of freedom, that is, the price below which they do not care to receive refunds. With this small modification, it is entirely rational for people to pledge exactly as much as they are really willing to pay for the game. This is a truly amazing property that most other methods of raising funds do not satisfy. In a retail sales context, on Kickstarter, in you-name-your-price mechanisms like the Humble Bundle, customers always have an incentive to understate how much the product is worth to them. Here, even entirely self-interested customers have an incentive to tell the truth, at every point during the process. This is not the end of the story, though! The Average Cost Threshold Protocol has a host of variations and wonderful theoretical properties.

As you can see, there is a lot of room to improve upon existing methods for funding digital goods. In a world where digital goods make up an ever increasing percentage of the global economic output this can have a profound impact on our economic lives. With new ideas and experiments we can come up with business models that can raise large amounts of money to finance the creation of high-quality digital goods and at the same time make sure that as many people as possible get access. In particular, we can provide a clear path towards releasing digital goods under a copyleft license. This way, everybody can benefit from the economic power of copying!

Christopher Smy asked for a geometric proof of the identity $$1^3+\ldots+n^3 = (1+\ldots+n)^2.$$ Because I like that sort of thing, I couldn’t resist drawing a picture:

If you like that sort of thing, check out this video or these lecture notes.

A while ago, I started a series of posts on finding new ways to fund public goods. Crowd-funding platforms, which have been enormously successful in 2012, promise a new way of financing creative projects and many of them employ the Street Performer Protocol, which was originally invented with a view towards financing public goods such as open source software projects. However, a closer look at the way the Street Performer Protocol is used on Kickstarter reveals that it is rarely employed to finance goods published under a copyleft license. Rather pledges are treated as pre-orders for goods sold as if they were classic private goods. I think we can do better than that!

In this post, I want to describe a different way to fund the private provision of public goods: the Average Cost Threshold Protocol.

Before we get started, let me clarify what I mean by the term public good. The term does not have the egalitarian meaning of a common good that is shared by everyone and that all members of a community are entitled to. Rather, a public good is any good that has a specific economic property: If a public good is provided to one person, it could also be provided to everybody else at no additional cost. Note that I am not saying the public good actually is provided to everyone. It may very well be that some people are excluded from using the public good. Also, I am not saying that nobody has to pay anything for the public good.

Practically all digital goods are natural public goods. This includes movies, music, ebooks, online lectures, scientific articles, software and computer games. These can, in principle, be copied at virtually no cost. Often, legal or technical restrictions are placed on copying these goods, as producers want to raise money for the provision of these goods by selling these goods as private goods (or by selling advertising tied to these goods). This post is about finding better ways to fund the creation of such digital goods. As my running example of a public good, I am going to use a computer game. Not because games are the most important type of public good, but simply because they are widely popular, require no maintenance after their release (or after, say, the tenth patch) and because a number of computer games have been very successful on Kickstarter, which shows that the audience is open to experimenting with new funding mechanisms.

The Theory of Average Cost Pricing of Public Goods

A very natural variation of threshold pledge systems like the Street Performer Protocol is a fixed fee mechanism with average cost prices. In this section I will present the theory behind this mechanism, before turning to its practical implementation (and a practical example!) in the next section.

The basic idea is very straightforward:

1. Everybody interested in a given project makes a pledge, saying they are willing to contribute a certain amount of money to funding the project. If somebody is not interested in contributing that just don’t pledge anything.
2. After all pledges have been made, we distribute the costs of the project equally among as many people as possible: Everybody pays exactly the average cost of the project, as long as they pledged at least as much. More precisely, we figure out the smallest price $p$ such that if everybody who pledged at least $p$ pays exactly the price $p$, then the costs of the project are covered exactly. Nobody pays more than $p$ and if somebody pledged less than $p$ they don’t have to pay anything.
3. If there is no such price $p$, for example because people pledged too little, then the project is not funded and nobody pays anything!
4. In case the project is funded, the money raised is used to create the public good. But only those people who paid get access to the public good! (Here we assume that the public good in question is excludable.)

Points 1 and 3 are very similar to the Street Performer Protocol (SPP) and what happens on Kickstarter. Point 2 is crucially different, as in the SPP and on Kickstarter everybody pays what they pledged and not the price $p$. Point 4 is what happens in many projects on Kickstarter, as I observed in my last post, but it is very different from the idea behind the SPP, which was intended to fund pure public goods without use exclusions. There is another vital difference to what happens on Kickstarter that will become clear in the next section.

The fixed fee mechanism with average cost prices has crucial theoretic advantages:

• Everybody has an incentive to pledge exactly what the project is actually worth to them. The technical term for this incentive compatibility. Note that the SPP is not incentive compatible: there, people have an incentive to understate the value the project has for them, in order to pay less. (This is called “free riding” in the public goods literature.)
• Nobody is forced to contribute anything if they don’t want to. In other words, no matter what happens, after we have used this mechanism, everybody will be at least as well-off as before. This is called individual rationality. Many mechanisms for funding public goods, both practical ones, such as taxation by the government, and theoretical ones, such as the Groves-Clarke mechanism, are not individually rational. But for the private provision of public goods in which people participate voluntarily, individual rationality is essential.
• If the fundraising is successful, we raise exactly as much money as is needed to fund the project. That is, the mechanism not only covers costs but moreover it balances the budget.

Of course this mechanism also has a key disadvantage: We exclude people form using the public good even though the public good could be provided to them at no extra cost. In technical terms we say the mechanism is inefficient. This, however, is unavoidable as there are theoretical results such as the Myerson-Satterthwaite Theorem which says, roughly, that there does not exist a mechanism for the provision of public goods that is incentive compatible, individually rational and efficient. This result tells us that the first best solution of giving everybody access to the public is impossible to attain. The good news is, though, that in face this impossibility result, the fixed fee mechanism with average cost prices is the best possible alternative:

• According to a result by Peter Norman the above mechanism is (asymptotically, in a large economy) the most efficient one among all mechanisms that are incentive compatible, individually rational and guarantee budget balance.
• If the cost of funding the public good is small in comparison to the total size of the economy and the number of people interested in the public good, then the average cost will approach zero. This allows us to get arbitrarily close to giving everyone access to the public good. See also this paper by Martin Hellwig.

Regarding the history of the fixed fee mechanism with average cost prices, I want to mention that average cost prices have been studied for a very long time in the context of monopoly economics and a number of authors have examined fixed fee mechanisms in the context of public goods. However, as far as I was able to find out, the paper by Peter Norman is the first instance where this exact mechanism has been studied in a public good setting.

Of course there is a lot more to say about these results and I plan to write more about the technical details in the future. But today, I want to talk about how this mechanism could work in practice and present a practical implementation which I dub the Average Cost Threshold Protocol.

The Average Cost Threshold Protocol in Practice

Suppose a company wants to create a computer game and they need money to cover their costs, which total, say, $1 million. They decide to use the above mechanism to raise the funds and so they start a project on a website like Kickstarter which provides all the necessary infrastructure. The project is open to receive pledges for 30 days and at the end of that period the pledges are tallied to see if the project can be funded according to the above rule. (Strictly speaking, setting a deadline is not necessary, but given that many crowd-funding projects raise a large part of their funds the days immediately before the deadline, it seems like a good idea to make use of this psychological effect.)

Many people chip in and pledge various amounts. There are 20,000 people who pledge \$50 or more. So setting a price of \$50 would exactly cover the costs of the game. However, there are only 23,000 people who pledge \$40 or more, so setting a price of \$40 would raise only \$0.92 million which does not cover the costs of the game. Let us assume that \$50 is the lowest price that covers the costs of the game.

Thus, the price is fixed at \$50 dollars. Everybody who pledged \$50 or more now has to pay exactly \$50 dollars. The people who pay these \$50 are now called backers. The total amount raised is \$1 million which covers the costs of the game. This money is used to create the game, and once it is finished, every backer receives a copy.

So far so good. But what about all the other people who would like to play the game? In all likelihood, there are many more people out there who would be happy to pay \$50 to get a copy! Maybe they heard about the project only after the fundraising ended, so they did not have a chance to become a backer. Or maybe they wanted to wait and see how the game turned out before committing to the purchase. Or maybe they did not have \$50 to spare back then, but they do now. Whatever their reasons, it makes perfect sense to provide these people with the public good - we just have to find a way that is consistent with our mechanism.

How do games companies currently do it on Kickstarter? Well, they just sell copies of the game. And the companies keep the revenues from these sales for their own profit. There is nothing wrong with creators making a profit from their work. The problem here is that this breaks our mechanism! Suppose there are another 20,000 people (let’s call them buyers as opposed to backers) who pay \$50 for the game and these revenues are the profits of the company. Now there are a total of 40,000 people (buyers + backers) who have paid \$50 each, which amounts to a total of \$2 million dollars. However distributing the total cost of the game (\$1 million) among 40,000 people would lead to an average cost of just \$25! So, in this scenario, if the company decides to sell the game for profit after it is finished, people would pay twice the average cost, which is very different from what our mechanism specifies.

So let’s suppose we take our mechanism seriously. What would need to happen with the profits from selling the game? Suppose we have 20,000 backers plus 20,000 buyers. As we observed above, distributing the costs of \$1 million equally among those 40,000 people would lead to an average cost of just \$25. But the backers already paid \$50! So here is what needs to happen according to the mechanism: The buyers need to pay just \$25 each. And these revenues need to be given to the backers instead of the company. This makes sure that everybody pays exactly the average cost of the public good.

But now the price of the good has dropped by 50% and the product is gaining ever more public attention. So now there are, say, 40,000 additional people out there who would be willing to pay \$25 for the game. Here is what the mechanism tells us to do: Charge each of the 40,000 newcomers \$12.50 and give the proceeds to the 40,000 people who purchased the game first. This way, everybody just paid \$12.50.

You can now see where this is going:

• As more and more people buy the public good, the average cost of the good drops.
• The price of the good drops, making the good available to more and more people.
• In this process, early buyers are refunded for the high price they paid in the beginning.
• In particular, because of the transfers from late buyers to early buyers, nobody has an incentive to wait with their purchase until prices drop.

All we need for this to work is for the website that coordinates this process to manage these transfer payments. Of course actually distributing money among many different bank accounts whenever a single purchase is made would incur way too much transaction costs. But the website could simply keep track of purchases and how revenue needs to be redistributed and allow customers to withdraw funds every once in a while. The transaction costs could be passed on to backers/buyers directly or could they could simply be financed from the interest the website earns from keeping the payments for some time.

It is also important to note that the infrastructure for pledging available on the website should still be used for sales, even after the product is finished. This way, if the current price for the game is still \$40, people who would be willing to pay \$20 for the product could submit this “bid” on the website. If enough people pledge \$20, the price of the product will actually drop and they will get a copy. The important thing is that people have an incentive to reveal the true price they are willing to pay! This is in contrast to a classical sales context, where customers have an incentive to understate their true valuation to get the company to lower their price.

The Benefits in Summary

The Average Cost Threshold Protocol is a practical mechanism for funding public goods that allow use exclusions. It is an implementation of the well-known fixed fee mechanism with average cost prices, and thus it enjoys many desirable properties:

• It is in everybody’s own best interest to pledge exactly what they are willing to pay. This is true even in hindsight!
• The protocol makes everybody better off. In particular, nobody is forced to pay anything.
• The costs of the project are guaranteed to be covered for the company.
• Among all possible mechanisms with these properties this one is optimal, that is, it creates the most value for society as a whole.

Beyond these theoretical merits, the practical protocol suggested above has a number of additional benefits:

• Prices to the public good are lowered continuously, in order to provide the widest possible audience with access to the public good as quickly as possible.
• At all times throughout the process, everybody has paid exactly the same price for access to the public good. So nobody has an incentive to wait with their purchase for prices to drop.
• In the sales-phase, it is in the interest of customers to bid what they would be willing to pay, even before prices drop. This allows the mechanism to lower prices more quickly, via optimal price targeting.
• Everybody benefits from getting more people to buy access. Even today, backers of crowd-funding projects often provide significant marketing services to their projects through word of mouth, long after fundraising has ended. Using this protocol, backers are not only personally rewarded for this engagement, but they also help to give more people access at lower prices.

Of course this is not the end of the story! There are a number of variants of this mechanism that are worth exploring.

Variations of the Protocol

There are a number of aspects of the basic version of the Average Cost Threshold Protocol presented above which can be improved further.

1) Most importantly, the public good provided by the protocol is still subject to use exclusions. We would really like to actually provide a pure public good that everybody has access to, no matter if they paid anything or not. But if we modified the mechanism so that people know the good is eventually going to be provided for free, the economic incentive to contribute something would disappear. Remember, this is the essence of the Myerson-Satterthwaite impossibility theorem!

Nonetheless, there are a number of ways the protocol could be modified to fund the creation of a pure public good without use exclusions. For example, we could set a “reserve price” of, say, \$5 dollars. If the price of the good falls below \$5, the good becomes a pure public good available to everyone for free. Now, of course, everyone who did pay \$5 has paid \$5 too much, which would destroy incentive compatibility in a strict sense. But as \$5 is a relatively small loss, backers who care about the project may very well be ready to accept this loss and gain the warm glow-effect of having made the public good available for everyone. Instead of fixing a common reserve price of \$5 for everyone, backers might also set their own individual reserve price when buying the product. (This of course would require the redistribution scheme to be adjusted.)

A completely different option would be to set a fixed “expiration date” of the use exclusions, for example, three years after the release of the finished product. Buyers would then purchase early access to the product, which is a common business model already today. The difference is that this early access would come with a guarantee that the product will become a pure public good eventually.

Of course such modifications would ruin some of the nice game theoretical properties of the mechanism. But these theorems hinge on the assumption that all backers are entirely rational anyway. And humans are not entirely rational, they are also benevolent and they tend to be tremendously enthusiastic about creative projects they like. So there is room enough for such small changes to work, even if they don’t fit into the rational framework.

2) Companies or creative individuals funding a project using the “vanilla” Average Cost Threshold Protocol as defined above enjoy the tremendous benefit of having their costs covered in advance by payments very similar to pre-purchases. They are not funded through equity and they have no liabilities, which means they have no investors that they need to satisfy through profits and they have no debt that they need to pay off. But that does not mean that all is well.

First of all, after the project is funded and the company received the payment to cover their costs, they are not going to receive any further payments, whatsoever. This means that they have no further economic incentives to make the project succeed. They may have incentives in terms of their creative ambition, their reputation as a company, their individual careers or simply their personal integrity. But the economic incentives to make the product shine, to market it well, to finish it on time and on budget or even to complete the project all - they are all gone. This is clearly not in the interest of anybody! Therefore it is a good idea to allow the company to make some profits in order to create the corresponding economic incentives.

Moreover, no matter how accurately the company projected the costs of the project at the outset, the actual development may run over budget. Projects often (always?) do. So despite the fact that the initial fundraising is expected to cover costs in advance, the company still faces financial risks. To make these financial risks worthwhile to the company, it stands to reason that the company is allowed to make some profit.

Fortunately it is straightforward to allow the company to make a profit and at the same time allow the public to enjoy decreasing prices. The rule is simple: Half of the payment a new buyer makes goes to the previous backers, the other half is profit for the company. An example. Initially, 20,000 backers paid \$50 each to raise \$1 million. Now, 10,000 additional buyers want access. According to the original protocol, everybody would have to pay \$33.33 now. But instead, we ask the newcomers to pay \$40. Half of that is the profit of the company, which amounts to \$200,000 in total. The other half goes towards refunding the original 20,000 backers so that everybody paid just \$40 in total. In this way, prices will decrease steadily with an increasing number of buyers. But still, the company stands to make an unlimited profit from creating a great product! This amounts to a reasonable compromise between the interests of the company and the social goal of giving access to as many people as possible.

The great thing about this variation is that it preserves many of the nice game-theoretic properties of the original mechanism. In particular, this modified mechanism is still incentive compatible and individually rational. It still covers costs by producing a budget surplus instead of a balanced budget. It is less efficient than before, because fewer people get access for the same amount of money. But still, as the number of buyers grows, the price goes to zero, enabling everybody to afford access if the public good becomes popular enough.

Of course, nobody says that revenues always have to be split 50-50 among the company and its backers. Any other ratio would do. The ratio could change over time. Or each backer could choose their own ratio (similar to what the Humble Bundle is doing), leading to a democratic vote on how revenues should be split. Also, this idea can be combined with the first variation presented above to create a protocol that produces both profits for the company and a public good without use exclusions, provided the good becomes popular enough. In this case the profits for the company are bounded and it is not entirely rational for buyers to reveal their true valuation, but still this promises to be an excellent compromise.

3) From a game theoretic perspective it is important (though not indispensable) that everybody pays the same price. However, from a practical perspective that may not be desirable. Some backers may want to be charged more than other backers. Maybe because they want to show how much they value the project. Maybe the company decides to offer rewards for backers who pay much. Most importantly, there is a real possibility that projects cannot be funded without backers who self-select to pay a very large premium on the average cost. The public interest in the project may not be broad enough to get the costs covered on an average cost pricing basis, but the interest may well be deep enough to cover costs if some people are allowed to pay more.

A special case is the money the company itself puts into the development. Companies and creators running crowd-funding projects often put a significant amount of personal wealth into their projects. Usually, these are funds that do not show up during the fundraising campaign. But an effective mechanism for funding public goods should explicitly incorporate a facility to account for this common practice.

Again there are several ways to allow backers to pay more than the average cost during the initial fundraising campaign. (Note that backers can always pledge as much as they want, but in the original protocol, they will never pay more than the average cost.) One way to accommodate this is via the variable reserve price mechanism mentioned above. Backers who want to pay a lot could simply set their personal reserve price to exactly the same amount as their pledge. Then, they could get charged the entire amount if the fundraising campaign is successful.

However, the above variation would also imply that these backers are not refunded anything. This may well be in the interest of very enthusiastic backers, but it would not fit the needs of the company who wants to recoup the large investment it made into the project in this way. Also there may be backers who are willing to make a very large payment, if the project can’t be funded otherwise, but would like to be refunded if the project turns out to be widely popular in the long run. To accommodate these interests, one could allow backers to specify that they want to be charged more during the fundraising campaign. Later on, the revenues earned from sales of the product could be used to refund backers in proportion to the payment they made during fundraising. This way large backers could eventually recoup their investment if the project is widely successful.

While it may generally not be rational for backers to make such large payments, the presence of this option does not change the fact that for the average backer the protocol remains incentive compatible and individually rational. As before, this variation can also be combined with the variations 1) and 2) mentioned above.

Conclusion

The Average Cost Threshold Protocol and its variations promise to yield an effective mechanism for the private funding of public goods. Even if this particular mechanism is not the ultimate answer, it shows that there is a lot of room out there for improving upon existing crowd-funding mechanisms in this regard. I hope that more people apply their creativity to invent new ways of making the private provision of public goods attractive. In a world where public goods make up an increasing share of the global economic output, such a mechanism could change the way we do business and interact with each other’s creative work.

My latest math paper has hit the arXiv. It is called Enumerating Colorings, Tensions and Flows in Cell Complexes and it is joint work with Matthias Beck, Jeremy Martin and Logan Godkin. It has been great working with all of them. I am particularly pleased that this is my first joint paper with Matthias, the great guy who brought me to San Francisco!

This paper is a continuation of the work that Matthias Beck and Thomas Zaslavsky started on using inside-out polytopes to prove reciprocity results for counting polynomials and that Raman Sanyal and I adapted to the case of modular counting functions. In the present work we generalize all of this considerably, by proving a whole collection of combinatorial reciprocity theorems for flow, tension and chromatic quasipolynomials defined on cell complexes, i.e., in terms of arbitrary integer matrices. The fact that the Ehrhart theory methods developed for the graph case do generalize to cell complexes is a testament to the remarkable power of the geometric approach to combinatorics. Head over here to hear the full story!