In January 2013, my colleagues and I at Undercurrent gathered around the Milton Steinbach Professor of Management at Yale, Barry Nalebuff. Undercurrent was paused for Undercurrent University, our organization’s thrice yearly self-reflective high holiday, and Professor Nalebuff was our distinguished guest. Professor Nalebuff has a particular talent for showing parties locked in negotiation that they need one another. He helped negotiate a collective bargaining agreement for the NBA during the 2011 player lockout. He shared with us one of his most basic, and useful, concepts in negotiation. More than a year on, I’m still returning to the notes I took down. He began with an anecdote.

When he and his Honest Tea co-founder, Seth Goldman, were looking to expand their distribution footprint with Coca-Cola bottling, the team at Coke were trying to set the fulcrum to their benefit. Coke reminded the team at Honest Tea that they were massive and could bottle and distribute their product for pennies on the dollar compared to what Honest Tea could do themselves. They reiterated that Honest Tea needed Coke’s scale.

Prof. Nalebuff countered and reminded Coke that they needed Honest Tea. “Remember,” he said to us, “Coke would be unable to benefit without Honest Tea’s inefficiencies.” He reminded the negotiators at Coca-Cola that Honest Tea had a novel and growing product. Certainly the team at Honest Tea wanted it to scale, but hadn’t Coke built their incredibly efficient bottling and distribution businesses to suit opportunities like the one they had in Honest Tea? (In 2011, Coca-Cola went on to purchase Honest Tea.)

Coca-Cola and Honest Tea were better together than apart. Together they create new value that would not exist otherwise.

Central to Prof. Nalebuff’s concept is the pie. The pie is the benefit created by acting together versus to acting alone. He asserts that you begin all negotiations by splitting benefit of acting together evenly — dividing the pie. Then, you divide the costs between the parties to preserve the shared benefit. Sometimes, this means that the bigger party will share more of the costs. Tough noogies. Remember, everybody stands to gain equally — everybody gets the same size slice of pie.

Baking Up An Example

Let’s take two hypothetical companies, Abigail’s Knitwear and Bearsden Apparel. Abigail’s Knitwear is a small, up-and-coming producer of sweaters for cats. They are killing it on Etsy. Bearsden Apparel is more established, and has started to produce an exciting line of sweaters featuring images of cats on them. They primarily distribute through brick and mortar retail.

A hypothetical offering from Abigail’s Knitwear (source: https://flic.kr/p/dgNY8U)

The ownership between the firms are friendly with each other. Bearsden has approached Abigail with the prospect of co-locating and sharing the cost of purchasing automated knitting equipment. The equipment would allow each firm to increase their production efficiency. Neither firm anticipates being able to fully utilize the production capacity of the new machinery, so there is plenty of machine time for each firm to share.

An example offering from Bearsden Apparel (source: https://flic.kr/p/kHZ7Tm)

There’s a catch: Bearsden Apparel is quite a bit larger than Abigail’s Knitwear and stands to gain quantifiably more than Abigail’s. How should Abigail’s Knitwear and Bearsden Apparel divide the costs of the new equipment, particularly if they don’t quite derive the same benefit?

Should Bearsden pay more because they’d benefit more? What if Abigail’s Knitwear can’t afford the machine on their own, should they pay more to have access to the power of Bearsden’s capital? Or, should they split the cost of the machine 50/50 no matter what?

Worry no more. We can put these questions to rest by following the below method.

Step 1: Calculate Individual Benefit

In the simplest case, let’s suppose that both firms could both afford and benefit from the machinery on their own.

In our example, both firms desire to purchase a used, slightly tired Yamashita Flat Knitting machine model YFK-42. On the second-hand market, the YFK-42 costs $50,000.

Abigail’s Knitwear will be able generate an additional $100,000 annually by utilizing the machine. Bearsden, being slightly larger and with more customers, will benefit more and will generate an additional $200,000 per year.

First, we calculate their benefit of acting separately:

Abigail’s Knitwear’s individual benefit is $100K - $50K = $50K

Bearsden Apparel’s individual benefit is $200K - $50K = $150K

If we added these numbers together we’d get the individual net benefit:

$50K (Abigail’s benefit) + $150K (Bearsden’s benefit) = $200K (net benefit, acting individually)

Step 2: Calculate Combined Benefit

Next, we figure out what the benefit is of both firms acting together, the gross benefit. Above, we stated by using the new machine:

Abigail’s Knitwear will generate an additional $100K annually

Bearsden Apparael will generate an additional $200K annually

Therefore, the gross benefit is the sum of these two figures:

Abigail’s 100K + Bearsden’s $200K = $300K (gross benefit)

Less the cost of the machine, the net benefit of acting together is:

$300K (gross benefit) - $50K (cost of machine) = $250K (net benefit, acting together)

Step 3: Find the Size of the Pie

The difference between the between acting together and going it alone is the value created in partnership. Prof. Nalebuff calls this created value the size of the pie.

Our pie is calculated simply by subtraction:

$250K (net benefit, acting together) - $200K (net benefit, acting individually) = $50K (value created, a.k.a. the pie)

That means Abigail’s Knitwear and Bearsden Apparel will create $50K of additional value should they choose to divide the cost of the YFK-42 knitting machine than if they decided to individually purchase it. Sweet.

The question is, how will they agree to divide the cost of the machine?

A Flat Knitting Machine

Step 4: Divide the Pie

Negotiations on dividing the costs of partnership can often stall when the benefits derived from the partnership are unequal. That is to say, when one partner will derive more individual value. Conversations often turn to who needs who more when, in fact, both partners need each other equally.

In our example, this is could easily happen between Abigail’s Knitwear and Bearsden Apparel. By using the knitting machine, Abigail’s Knitwear will only unlock $100K vs. Bearsden Apparel’s $200K in incremental revenue. This doesn’t matter. Only the $50K created by both partners acting together matters.

By recognizing that neither partner can create the additional value without the other and agreeing to divide the created value one can fairly determine how to divide costs.

Approximately half a pie. (source: https://flic.kr/p/axkNfz)

From our calculations above, we’ve show the value we’re creating in our partnership is $50K. To come up with each partner’s benefit we divide the created value evenly by dividing the pie in half ($25K). Each partner’s individual cost is calculated by taking the smaller value between the total costs and an individual partner’s gross value.

This last bit is subtle. Why the smaller of these two values? Suppose an individual partner could not afford to go it alone (as we’ll soon see in subsequent cases). We use the maximum a partner can contibute and still break even in order to calculate their net benefit. This ensures that everybody can contibute and still receive their share of the pie.

Let’s see it in action. First, let’s divide the pie in two:

$50K (the pie) / 2 = $25K (half the pie)

Shown mathematically, first for Abigail’s Knitwear:

MIN($50K (cost of YFK-42), $100K (Abigail’s gross benefit)) - $25K (half the pie) = $25K (Abigail’s share of costs) →

$50K (cost of YFK-42) - $25K (half the pie) = $25K (Abigail’s share of costs)

And next for Bearsden Apparel:

MIN($50K (cost of YFK-42), $200K (Bearden’s gross benefit)) - $25K (half the pie) = $25K (Bearsden’s share of costs) →

$50K (cost of YFK-42) - $25K (half the pie)= $25K (Bearsden’s share of costs)

Each firm should therefore each pay $25K toward the cost of the knitting machine. In this example, it’s a 50/50 split. It may not seem revolutionary, but pause and think about it: this is like two friends deciding to split the lunch bill even though one diner ordered steak, and the other salad. The act of coming together (and presumably having a pleasant meal) is valued more than the individual cost of the dishes.

Stranger, and perhaps more illustrative still is considering the case where only one partner can afford to go it alone.

When The Small Partner Can’t Act Alone

Let’s keep the individual benefits for Abigail’s Knitwear and Bearsden Apparel the same. Abigail stands to gain $100K annually and Bearsden $200K. However, let’s consider that they want a slightly less worn knitting machine. If we increase the cost of the YDK-42 to $150K. What happens?

We proceed exactly as before. We calculate how they stand to benefit if they were to act separately. Immediately we’ll notice that Abigail’s wouldn’t act alone: the cost of the machine exceeds the benefit they’d derive! Therefore we set their individual benefit to $0.

Abigail’s Knitwear’s individual benefit is $100K — $150K = ( $50K) ∴ $0

∴ Bearsden Apparel’s individual benefit is $200K — $150K = $50K

Summing these figures, the individual net benefit is:

$0K (Abigail’s benefit) + $50K (Bearsden’s benefit) = $50K (net benefit, acting individually)

Next we calculate the net benefit of acting together. We’ll once again use the gross benefit of $300K from the two partners acting together by summing their individual benefits from our very first example:

$100K (Abigail’s individual benefit) + $200K (Bearsden’s individual benefit) = $300K (gross benefit)

(gross benefit) $300K (gross benefit) — $150K (cost of machine) = $150K (net benefit, acting together)

And use it to determine the value created, the size of the pie:

$150K (net benefit, acting together) — $50K (net benefit, acting individually) = $100K (value created, a.k.a. the pie)

Finally, we calculate the costs for each partner by dividing the pie:

$100K (the pie) / 2 = $50K (half the pie)

MIN($150K (cost of YFK-42), $100K (Abigail’s gross benefit)) - $50K (half the pie) = $50K (Abigail’s share of costs)

(Abigail’s share of costs) MIN($150K (cost of YFK-42), $200K (Bearsden’s gross benefit)) - $50K (half the pie) = $100K (Bearsden’s share of costs)

In this case the costs are shared asymmetrically. The small partner, Abigail, will pay $50K toward the knitting machine and Bearsden $100K. This may seem unintuitive at first, but this cost-sharing structure allows for each partner to have access to exactly half the value created by working together.

Further Examples

What if neither partner can afford the knitting machine on their own? Let’s consider a new YFK-42 knitting machine (vs. the used models from before) and set its cost at $250K.

Below is a table given from a Google Sheet I created called The Barry Nalebuff Pie Baker. The data from the two previous cases are given. Looking at the final row in the table, we see the cost of the machine set at $250K, yielding total size of the pie to be shared by each partner at $50K. The costs for Abigail’s (“A”) and Bearsden (“B”) are $75K and $175K, respectively.