1. Three Rival Specifications

The determination of a FRAND royalty for a licensor holding a subset of standard-essential patents (SEPs) can be expressed in three alternative specifications. All three begin with an aggregate royalty Ragg and a patent share Q = NL / Ntotal. They differ in the structure imposed on the relationship between the royalty and the observable variables.

1.1 The Additive Model (Bottom-Up

The additive model constructs the licensor’s royalty from the bottom up by summing individual patent values:

RL = \sum_{j=1}^{NL} v(j)

where v(j) is the per-patent royalty value of the j-th SEP held by licensor L. Under the equal-value assumption v(j) = v for all j, this collapses to:

RL = NL \times v

The per-patent value v is not estimated—it is derived from the aggregate: v = Ragg / Ntotal. To my knowledge, this formula has no published derivation and no identified originator. It is often attributed to Swanson & Baumol (2005), but their paper proposes the Efficient Component Pricing Rule (ECPR) adapted to FRAND—not this decomposition. The model’s critical assumption is that all patents contribute equally. The coefficient on NL is mechanically fixed—it is neither estimated nor tested and carries no standard error. Thus, relative reliability cannot be assessed.

1.2 The Multiplicative Model (Top-Down)

The multiplicative model starts from the aggregate royalty and allocates it proportionally by patent share:

RL = R_{\text{agg}} \times Q

where Q = NL / Ntotal. Under the equal-value assumption, the additive and multiplicative models yield the same result: NL × (Ragg / Ntotal) = Ragg × (NL / Ntotal).

The distinction is rhetorical (bottom-up vs. top-down), not algebraic. The multiplicative structure does not escape the equal-value assumption. No additional information is extracted; no heterogeneity is accommodated; no testable restriction is imposed. Both are identities with zero degrees of freedom. The sample size equals the number of parameters.

1.3 The Regression Model

The structural FRAND royalty equation:

Y_i = a + b_1 Q_i + b_2 C_i + b_3 D_i + u_i

where Yi is the royalty payment by implementer i, and:

The regression model differs from its rivals in three fundamental respects. First, it estimates the coefficients b1, b2, and b3 from observed data—they are not assumed. Second, it produces standard errors and confidence intervals, allowing the arm’s-length (or reasonable) range to be defined as b̂1 ± SE(b̂1) (the 68% confidence interval at t* = 1) when the sample size is large. Third, it includes an intercept a whose significance is testable: if a ≠ 0, the additive and multiplicative models are biased by the ratio a / H, where H is the harmonic mean of the base variable.

2. The Contrast in Tabular Form

3. Data Gathering Feasibility

3.1 Variable Q: Patent Share

The patent share, Qi = NL / Ntotal, is the most tractable variable in the specification. Its numerator and denominator can be constructed from ETSI IPR declarations, patent pool composition data, and court-ordered essentiality determinations. The feasibility of Q is identical across all three models. No model has an advantage or disadvantage in constructing the patent share. The regression model uses the same Q as the additive and multiplicative models—but estimates its coefficient rather than assuming it.

3.2 Variable C: Patent Quality or Contribution

The factor C is intended to capture heterogeneity among patents that Q alone does not reflect. ꞁꞁ Candidate operationalizations include forward citation counts, essentiality rate, claim breadth, and standard contribution records. The additive and multiplicative models do not use C—they assume it away. The regression model requires C as a regressor, but it can be omitted from estimation if the variable is unavailable, yielding the reduced specification Yi = a + b1 Qi + ui. Omitting C does not destroy the model—it reduces it to a bivariate regression that still estimates b1 and tests the intercept. The bias from omitting C is absorbed into the error term and reflected in wider confidence intervals, which is the honest result. ꞁꞁꞁ

3.3 Variable D: Categorical Distinctions

The factor D captures discrete heterogeneity: technology domain (radio access vs. core network), blocking vs. non-blocking status, and the geographic scope of patent family coverage. Candidate operationalizations include technology category indicators, geographic coverage, and patent expiration profile. As with C, the additive and multiplicative models ignore D. The regression model includes D as a regressor when data are available and omits it when data are not—with the same honest consequence: wider confidence intervals reflecting greater uncertainty. In the end, it is unlikely that differences in royalty can be measured with such granularity.

3.4 The Dependent Variable Y: Observed Royalties

The most consequential data requirement is the dependent variable: observed royalty payments Yi across implementers i = 1 to N. Sources include patent pool royalty schedules, judicially determined FRAND rates, and comparable license agreements disclosed in litigation, SEC filings, or regulatory proceedings.

The additive and multiplicative models also require Ragg—the aggregate royalty—which is itself a contested input. The regression model does not require a separate aggregate royalty input: it estimates the relationship between Y and Q directly from cross-sectional data, and the aggregate emerges as the sum of fitted values rather than as a presupposed input. ꞁV

4. Defensibility Under Evidentiary Standards

4.1 The Additive and Multiplicative Models

Both additive and multiplicative models are accounting identities. They decompose a total (Ragg) into parts and, by construction, impose the equal-value assumption. Under Daubert or equivalent evidentiary standards, an expert offering this model in court is offering an assumption, not a finding. The model cannot be wrong because it makes no testable prediction.

The suppressed intercept (a = 0) is an untested restriction. When the true intercept is non-zero—as it almost certainly is whenever licensors incur fixed licensing costs, whenever royalty schedules include lump-sum components, or whenever small-portfolio holders receive minimum payments—the ratio-based PLI is biased by a / H. This bias affects all quartiles of the distribution and cannot be detected within the model because the model has no mechanism for detecting it.

4.2 The Regression Model

The regression model is falsifiable and yields testable predictions with quantifiable uncertainty. The intercept test (H0: a = 0) directly assesses whether the additive and multiplicative models are biased. If the null hypothesis is rejected, the rival models are shown to be biased—not merely assumed to be biased but demonstrated by the data.

The arm’s length range is the 68% confidence interval around the slope coefficient: b̂1 ± SE(b̂1) at t* = 1. The interval is narrow when the data are informative and wide when they are not—an honest reflection of evidentiary or probative strength. By contrast, the additive and multiplicative models produce a point estimate regardless of data quality, conveying false precision.

5. The Reduced Specification

When C and D are unavailable—as they often are—the regression model reduces to:

Y_i = a + b_1 Q_i + u_i

This bivariate regression uses the same data as the additive model (royalties and patent shares) but extracts more information: it estimates b1 and a rather than assuming them and produces a confidence interval rather than a point estimate. The omission of C and D is reflected in the residual variance and the width of the confidence interval—the model is honest about what it does not know.

The reduced specification is superior to the additive model: it nests the additive model as the restricted case a = 0 and tests that restriction empirically. If the restriction is not rejected, the regression confirms the additive model. If it is rejected, the regression demonstrates the additive model’s bias. Either way, the regression is the more informative analysis.

6. Conclusion

The additive and multiplicative FRAND royalty models are identities with zero degrees of freedom. They assume their conclusions. The regression model estimates its conclusions from observed data, tests the assumptions embedded in the rival models, and reports the uncertainty of its results. It requires the same data as the additive model when reduced to the bivariate specification Y = a + b1Q + u, and it extracts more information from that data.

The data-gathering burden for the regression model is not materially greater than that for the additive model. Both require patent share data (Q) and royalty observations (Y or Ragg). The regression’s additional variables (C, D) improve precision when available but can be omitted without destroying the model. The aggregate royalty—the most contentious input in top-down analysis—is not a presupposed input to the regression; it emerges as a fitted value.

An identity dressed up as a model is not improved by judicial adoption. A model that can be tested, falsified, and shown to be biased or unbiased by the data is the defensible alternative. The regression model is the more reliable alternative.

ꞁ The European Telecommunications Standards Institute (ETSI) — the Sophia Antipolis–based SSO that administers the IPR declarations for 3GPP standards (LTE, 5G). When SEP holders declare patents as potentially essential to an ETSI standard, they commit to license on FRAND terms. The ETSI IPR database is the primary public source for counting declared-essential patents per licensor, which is why it appears in the paper as the data source for constructing Q.

ꞁꞁ Patents within the same standard are not equally valuable. Some cover core functionality that every implementer must use (e.g., the OFDM waveform in LTE), while others cover optional features or narrow implementation choices. The equal-value assumption behind the additive and multiplicative models treats every declared SEP as interchangeable — one patent, one share of the royalty pie. Heterogeneity is the reality that this assumption suppresses: differences in technological importance, claim breadth, forward citation rates, whether a patent is blocking or design-around-able, how many standard releases it reads on, and how many jurisdictions it’s granted in. The variable C in your regression specification is the mechanism for letting the data reflect these differences rather than assuming them away. When C is omitted, that heterogeneity doesn’t vanish — it migrates into the error term and widens the confidence interval, which is at least honest about the omission.

ꞁꞁꞁ The omitted variable bias can be calculated. See Jan Kmenta, Elements of Econometrics (2nd edition), Macmillan, 1986 (Section 10-4: Specification Errors).

ꞁV External comparables from databases, such as best-in-class RoyaltyStat, contain the royalty rate, and not the royalty level Y. Thus, internal comparables are the most likely data source. The overall royalty rate can be tested using data from RoyaltyStat.