Top-Ups 2.0

Author

Kristina Bishop, Matthew Jennejohn, and Cree Jones

1 Introduction

“Top-Ups and Telephone” discovered primary mechanisms for how legal innovations diffuse throughout the legal advisory network. We now test the theory of how diffusion happens using a family history pattern. How do firms pass down the knowledge firm by firm?

2 Methods

We assume that firms gain knowledge of how a top-up option works by doing a deal that includes a top-up option with another firm or a deal with a prior adopter of the top-up option. We classify a firm as a “parent,” meaning they pass on the knowledge of the top-up innovation to another firm in one of the following ways:

  1. Cosigner Prior Adopter: the cosigner of deal that includes a top-up option is a prior Adopter
  2. Proximate Adopter: the firm who was a prior Adopter and most recently signed a deal with the firm before it became an Adopter
  3. If the cosigner of the deal is not a prior adopter, then we use the proximate adopter as the parent firm.
  4. Multiple Adopters: all prior Adopters who a firm interacted with the firm before it became an Adopter.

We construct each of these definitions of parent.

Then, we look at the parents of each firm’s parents and so on to construct a “genealogical” relationship of how firms are connected through the adoption of the top-up option adoption.

We also construct the Cosine Similarity between the texts of each deal to test the theory that as firms become farther removed away from each other through genealogical distance, so does the text of their deal.

3 Results

Figure 1: Histogram of the Time (in years) from Proximate Adopter to Firm Adoption Date

Figure 1 shows the distribution of the time difference between the deal with the most recent Proximate Adopter. This is to show how long the influence of a deal with a previous adopter may extend. There are 6 deals less than one year, 16 deals less than three years, 21 deals less than 5 years, and 47 total deals with proximate adopters. Deal influence may exist longer than we initially thought if firms are learning from their interactions with prior Adopters.

Table 1: Parent Firm Relationships for Adopters
1 year 3 Year 5 Year All
Cosign Prior Adopter, No proximate adopter 16 16 16 16
No Cosign Prior Adopter, Proximate Adopter 3 11 13 21
Cosign Prior Adopter and Proximate Adopter 3 5 8 26
Not Cosign Prior Adoptor, No Proximate Adopter 31 23 21 13

The subsequent analysis will be carried out using the Proximate Adopter with deals recently of three years or less when the Cosigner Adopter is not present for a firm.

3.1 Descriptive Statistics

Table 2: Ancestor Relationships
N Distinct
Firm 76 76
Parent 53 31
Grandparent 24 13
Great-grandparent 10 5
Great-great-grandparent 2 2

D 130 130 61 61 130->61 58 58 61->58 285 285 61->285 104 104 58->104 84 84 285->84 125 125 285->125 10 10 26 26 10->26 28 28 10->28 77 77 10->77 119 119 10->119 139 139 10->139 52 52 26->52 66 66 26->66 18 18 28->18 25 25 28->25 112 112 77->112 132 132 77->132 69 69 84->69 29 29 52->29 31 31 66->31 60 60 66->60 115 115 25->115 27 27 29->27 2 2 3 3 51 51 3->51 167 167 3->167 215 215 3->215 45 45 215->45 32 32 8 8 32->8 63 63 8->63 44 44 9 9 44->9 15 15 44->15 92 92 44->92 94 94 44->94 198 198 44->198 85 85 9->85 50 50 141 141 50->141 73 73 152 152 73->152 86 86 136 136 137 137 246 246 49 49 246->49 64 64 246->64 16 16 49->16 196 196 85->196

Figure 2: Firm Relationships of Proximate Adopters of Less Than Three Years

(a) Parent

(b) Grandparent

(c) Great-grandparent

Figure 3: Histogram of Firm and Parent Cosine-Similarity by Generation

Figure 4: Cosine Similarity between Generations

Table 3: Cosine Similarity Statistics by Generation
Parent Grandparent Great-Grandparent
Mean 0.80 0.89 0.95
Standard Deviation 0.14 0.13 0.04
Median 0.80 0.94 0.95

3.2 Heatmaps

Figure 5: Heatmap of Cosine Similarity for All Deals with a Top-Up Option by Deal Date

Figure 6: Heatmap of Adopter Cosine Similarity by Adoption Date

Figure 7: Heatmap of family 130 (Heler Ehrman), by generation birth order

Figure 8: Heatmap of family 10 (Gibson, Dunn & Crutcher LLP), by generation birth order