How to rebalance founder equity

by Ymir Vigfusson

You and a number of your coder friends have come up with a genius idea. It’s so brilliant that you decide to turn it into a startup, Flugeldufel Inc. You can taste the excitement. You grow your team, heavy on the technical side of course because those are the people in your network. You dream of all the good you will do with your inevitable riches. Life is grand.

Cara putting in more work than Alice and Bob on Flugeldufel Inc.

One of the many things to do when forming a company is to define the relative ownership. Through misplaced conflict avoidance, you have decided to divide the startup up into equal pieces. Everybody is contributing equally to the success of the company, right? You had been lucky enough to receive a grant or angel investment that let you and your team spend several months on just creating a prototype for the course-altering technology that will propel Flugeldufel. There were other more urgent conversations than equity so you punt on it. Life is still grand.

Now it’s a few months into the venture. There are mounting tensions on the team. When you probe the people, you uncover people are no longer fine with how the equity is allotted. Alice and Bob, for instance, have clearly put in less effort than Cara and Deatrik, according to Cara anyway, and Deatrik and Eve feel that at least Bob should definitely have a smaller stake in the enterprise than they should, anything different would be “totally unfair”. The situation has festered for a while, and is now heating up and getting more complicated. Expectations are misaligned.

Does this sound familiar?

I advised a startup where this equity allocation problem came up. A stellar technical crew had been building a stellar prototype of a technology. There was, however, mounting apprehension over the equity allocation that was slowly engulfing the group with mistrust and other cardinal sins from the Five Dysfunctions of a Team. Rather than issuing new stock and allocating to employees according to some well-defined metrics, as is custom in established companies, it was clear that the equity division needed to be changed retroactively.

Some of the headache could have been averted by learning to Slice the Pie. The Slicing the Pie model attaches a monetary market value on each of the at-risk contributions made by the co-founders and early employees. This includes tangible contributions, such as capital investment or below-market salary, and intangible contributions, such as the use of someone’s social network. These contributions are all converted into hypothetical monetary amounts. The equity of the company is then divided in proportion to the accumulated dollar amount contributed by each employee so far, whether directly or indirectly. For example, if Xander took a $1,000/month salary cut for a year relative to his market rate to be part of the startup, while Zola contributed $6,000 up-front capital to help get the company off the ground, they would receive an equal share of the company.

A big downside of the model is that the market value of individuals may not be available or accurate. Some people are, through training or circumstance, simply far more productive engineers. Some people are more productive on particular projects or on different teams---a standard low-risk steady-salary job may not unleash their potential. Some people are young and don’t have a job history to prove their worth. Some markets are small and inefficient. So should you slice the pie?

Slicing the pie

The EigenEquity method

Returning to Flugeldufel Inc., the crux of our problem is now to discover what equity allocation would make everybody happy. Or, when push comes to shove, the least unhappy. Our goal is to let the team itself make the determination of how much individuals have contributed and then distribute equity proportionally. This needs to be done carefully to avoid fueling potential conflicts.

The core idea behind EigenEquity is that the relative contributions of a team member should mostly be judged by other high-contributing team members. This is a recursive definition—Alice’s contribution score now depends on the contribution score of Bob and others, but Bob’s contribution score is in turn dependent on Alice’s contribution score!

This problem has been studied in other settings, notably to rank the importance of web sites for search queries. In that scenario, a web site A that hyperlinks to other web site B is thought to have “voted” for B, and so if A is important through votes of its own, B gets some of that importance. The full algorithm, which must take care of idiosyncrasies, such as pages with no outgoing or incoming hyperlinks, is the celebrated PageRank.

EigenEquity solves the recursive equity refitting problem by considering it as a simple case of PageRank. At first, a trusted outsider, call them Frodo, asks each person in the company to independently submit an idea what they really think would be the fair allocation of equity. This information is private and will not be shared with the other co-founders. Each such relative allocation is a distribution meaning it sums to 100%. We place as a separate row in a table. Here is an example where Cara, for instance, feels Alice should receive 5% but Bob 10%.

The clashes between Alice and Bob, and Cara and Dietrik are evident in the table. Notably, four of the five co-founders found themselves as being the most deserving of equity in the company. This is probably usual.

Diving in to EigenEquity

The next step is to compute the first left eigenvector of this table. Say what?

Rather than focusing on the calculations, let’s rather get an intuitive understanding. There is an equivalent way of thinking about PageRank called the random surfer model.

imagine we knocked on the door of any person at Flugeldufel Inc., say Cara. She will turn around and send us to another office, but that new office is chosen at random with probability proportional to her votes above. So 5% of the time, she will send us over to Alice. 10% of the time she will send us to see Bob. We’ll also stay in her office 35% of the time, and so forth.

Once we are referred to another office, say Dietrik’s, the same process will take place. He will choose another office for us to visit at random with probability proportional to his own votes. So we would next wind back up in Cara’s office with 35% probability, for example. We can continue this process, roving from one office to the next using the person’s votes to determine the probability of which one to visit next. If we did for a very long time, what proportion of our visits would we spend in each office?

This question has a surprising answer. In math terminology, the process of bouncing between offices at random is called a Markov chain. The proportion we will spend in each office is called a limiting distribution of that random process. It turns out that the limiting distribution of a Markov chain, if it exists, is exactly the first left eigenvector of the matrix behind our table above.

In PageRank, a node is important if it's linked to by important nodes

Let’s get some more intuition. If we represent the matrix as $M$, where for instance $M_{\text{Cara},\text{Eve}} = 0.15$, then the element of the matrix $M^2_{\text{Cara},\text{Eve}}$ where $M \cdot M = M^2$ describes how likely it is for the surfer to wind up in Eva’s office after starting at Cara in exactly two steps. We can continue multiplying by $M$ and consider $M^3, M^4,$ and so forth. What do we expect to happen when we consider the distribution of offices after making 10,000 steps as a random surfer? The result from math is that there is a pattern, so long as the matrix $M$is “nice”. Specifically, if there is a limit $\lim_{n \to \infty} M^n = \vec e \vec \pi$ as the random surfer takes more and more steps, where $\vec \pi$ is a strictly positive probability row vector and $\vec e$ is a column vector of all ones $(1,1,\ldots,1)^T$, then $\vec \pi$ is precisely the importance vector we sought. It turns out that the common row vector $\vec \pi$ is exactly the left eigenvector of $M$ with eigenvalue 1, meaning $\vec \pi M = \vec \pi$. Moreover, it doesn’t even matter in what office we started, the proportion of time we will have spent in each office at Flugeldufel will be the same down the line.

The “nice” matrices we mentioned above have a technical bent. The results do not hold if the matrices over time have periodicity as the random surfer takes steps (like $M=$ $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$), or if they have coalitions that are not willing to concede power to others (an irreducible matrix). To overcome this restriction, Sergei Brin and Larry Page in their PageRank work were inspired to rank web sites by considering the limiting behavior of a random surfer, as above, but one who would occasionally surf to a completely random web site. By adding the random jumps, the augmented vote matrix $M$ for PageRank becomes automatically aperiodic and irreducible and thus has a unique limiting distribution which matches the first left eigenvector of $M$.

What about self-interest?

You may have noticed that each co-founder also rank themselves. This creates an odd incentive to give oneself a disproportionately large slice of the pie. However, a part of the EigenEquity process is to simply disregard one ranking of oneself, and dividing the remainder proportionally to others for whom one is voting. However, it is illustrative to see how far off the ultimate allocation to a person (which was determined by others) is from what the person wanted. This can help understand how well aligned co-founders are with their own expectations. In our case, every single person allotted themselves a larger slice of the pie than what they ultimately received.

Beyond self-interest among person, just like spam in the web hyperlink graph, a coalition of two or more people who point to one another can skew the picture. But then there are deeper team issues in the start-up that go beyond what EigenEquity is geared to solve. In those cases, you simply need to have the difficult conversations.

Using EigenEquity in practice

EigenEquity is simple to use.

$ git clone https://github.com/SimBioSysLab/EigenEquity.git

For the example above, we first encode the matrix into a tab-separated file votes.csv where each row is a vote.

Alice Bob Cara Dietrik Eve

0.4 0.3 0.1 0.1 0.1

0.3 0.45 0.1 0.05 0.1

0.05 0.1 0.35 0.35 0.15

0.05 0.1 0.35 0.4 0.1

0.2 0.1 0.1 0.2 0.4

Next, we simply run the program.

$ python3 eigenequity.py votes.csv

Alpha value of EigenEquity: 1.0

Alice should receive: 19.68% (Wanted 40.0%)

Bob should receive: 23.34% (Wanted 45.0%)

Cara should receive: 19.12% (Wanted 35.0%)

Dietrik should receive: 22.24% (Wanted 40.0%)

Eve should receive: 15.62% (Wanted 40.0%)

Conclusion

Hopefully, EigenEquity can help guide you towards good and fair allocation for a start-up equity, even when expectations and reality diverged early in the journey. If you do use it, let us know! We’d love to hear your story.

Slicing the Startup Pie at the 11th Hour