The Pie Model for Sharing Equity

I have noticed that many entrepreneurs, leaders, and managers have trouble trading off and determining the ideal level of sharing on a particular project. The question becomes how much control should they give up in order to include others in their endeavor.

In the case of an entrepreneur, it’s necessary to understand the relinquishing equity in order to gain the capital or expertise to sustain growth. My experience is that many entrepreneurs start out thinking that they want to maintain 100% of the equity for as long as possible and have a hard time agreeing to sell or otherwise parting with equity and diluting their ownership stake.  Their fear is they will lose control and/or lose a big part of the value if the enterprise is ultimately successful

Managers are conventionally concerned with complicating decision authority on projects if others are included. If the project is successful then the credit will be diluted and spread across others.

Basically, as individuals that like to optimize our decisions, we should be willing to include others and dilute our equity, control, or authority, if we believe that the resulting endeavor will be bigger, more profitable, or more successful.

I subscribe to a fairly simple model that helps people trade-off the psychological concerns and help make the decision appear to be more objective or analytical.

I start with the natural premise that there is a pie (money, credit, etc.) we will be spitting up. At the end of the activity everyone wants the most pie they can possibly get. It’a pretty common expression to seek the largest piece of pie in any deal.  So our goal is to develop a working symbolic model that helps us make decisions about splitting up a pie.

So from basic geometry, we know that the area of a pie can be represented as:

Areapie = ᴨ ∙ Radius2pie

where Areapie is the total area of pie available as a reward for this project. We would say the “area of a piece of pie is pi times the radius of the pie squared.

So now if we are going to split the pie into slices that are shared with others, we still want to maximize the pie we eat. In fact, our goal is that if we offer pieces of pie to others that we still get more pie than we had originally before we offered someone a piece. Our plan of action is that we won’t give up pieces of the pie unless we are relatively confident that the slice of pie we are left with will actually be larger than the original total pie since we expect the pie to grow.

Turns out, also from high school geometry, the area of a piece of pie (sector) is equal to:

Areaslice = ᴨ ∙ Radius2pie ∙ Angleslice

We would say that the area of a piece of pie is equal to pi times the radius squared times the angle.

So if we are going to give out pieces of pie to others we need to know that the area of our new slice is bigger than our old slice, even if the angle is smaller. This is because we expect that the pie radius will grow. In fact, we hope the growth of the pie will grow very rapidly.

The good news is that a pie grows by radius-squared much faster than by the angle. In fact, intuitively things growing at radius-squared are much better than things growing like the angle.

One way of verbalizing the model is:

In order to entice others to help us grow our endeavor, we are willing to relinquish a proportion of ownership (angle or linear reduction in pie) as long as we are convinced that the resulting growth (radius-squared growth in pie) will more than offset the reduction in proportion.

That looks a lot like math.  Let’s restate it as:

It’s much better to make the pie radius bigger rather than it is protecting the angle of pie ownership. 

Of course, projects are not necessarily pies, but this gives a symbolic way to understand why we bring partners into projects. We would be willing to give up some angle in our pie to get that radius growing and we should analyze it accordingly.

Here are two anecdotes to illustrate my point:

Anecdote 1 — NASA program manager model

Anecdote 2 — Legal Services for Start-Ups

Example 1 — One of the most critical examples is when the founders/owners of early-stage businesses need to decide if they should dilute their equity and raise more capital.  Typically there is ‘sticker shock’ when the first term sheets arrive.  The implied valuation of the company is less than anticipated meaning that to raise a fixed amount of capital, the ownership team will need to give up more of the company (larger slice of the pie).  This can be hard to swallow and new entrepreneurs are reluctant to give up any stake, especially if control is likely to be threatened in this or future rounds.

Hopefully, the Pie Model presented above will be helpful for founders/owners/managers into trading off whether to take the round of funding or not.  One should approach it analytically and not just dismiss the funds due to a principle of not sharing control or just accept it because one is hungry for a capital infusion.

One should truly attempt to understand:  Will the funds end up helping us to grow the pie more than we could without the funding and will the net result be that we will end up with more pie?  As they say, 100% of nothing is much smaller than 25% of something.  Very few businesses are able to grow without raising funds so some compromise is needed.

In fact, most VCs (Venture Capitalists) I know usually try to understand the enterprise risk from 4 vantage points prior to investing: technical, market, execution, and fundraising.  Fund raising risk for future rounds is actually an important consideration.  If the company is unable (steady progress toward critical milestones in unlikely) or unwilling to raise additional funds, if and when they are needed ( to fund new opportunities, unexpected delays, changes in markets, etc.) then the company gets a big black mark.  Access to funds is needed and in most cases, loans from banks will not be able to provide the growth capital.

So investors like to believe the management team understands that accepting dilution to sustain the growth can be a proper course of action.  If the management team is pre-disposed toward avoiding dilution at all costs, this can be a danger signal and may prevent sophisticated investors from participating.

Advanced Pie Model — Engineers would say that we are trying to make sure that radius squared growth has to be more than the angle we give up.  Or for those who desire even more rigorous math model, with a little algebra, we can write the following ratio equation:

(RadiusResulting/RadiusInitial)2 > (AngleInitial/AngleAfter Sharing)


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