Source-linked AI summary
The Role of Design Complexity in Technology Improvement
James McNerney, J. Doyne Farmer, Sid Redner, Jessika E. Trancik
TL;DR
The paper examines how design structure and the difficulty of improving individual components shape technology-improvement trajectories. It models interacting components and finds that mean cost follows a power law governed by design complexity and component-improvement difficulty.
Problem
The paper asks how a technology’s component interactions and component-improvement difficulty determine its improvement trajectory.
Method
The model represents production designs as interacting components whose costs change through repeated improvements, analyzing trajectories under fixed and variable connectivity.
Results
The mean cost follows a power law with exponent α = 1/(d∗γ), where d∗ is design complexity and γ is component-improvement difficulty.
Takeaways & Limitations
Constant-connectivity designs tend toward steady improvement, whereas heterogeneous designs are more likely to improve in fits and starts; design details determine the rate.
Takeaways & Limitations
Comparisons with real data rely on proxies for innovation effort, whose measurement remains debated.
Abstract
from arXiv · showhide
We study a simple model for the evolution of the cost (or more generally the performance) of a technology or production process. The technology can be decomposed into $n$ components, each of which interacts with a cluster of $d-1$ other, dependent components. Innovation occurs through a series of trial-and-error events, each of which consists of randomly changing the cost of each component in a cluster, and accepting the changes only if the total cost of the entire cluster is lowered. We show that the relationship between the cost of the whole technology and the number of innovation attempts is asymptotically a power law, matching the functional form often observed for empirical data. The exponent $α$ of the power law depends on the intrinsic difficulty of finding better components, and on what we term the {\it design complexity}: The more complex the design, the slower the rate of improvement. Letting $d$ as defined above be the connectivity, in the special case in which the connectivity is constant, the design complexity is simply the connectivity. When the connectivity varies, bottlenecks can arise in which a few components limit progress. In this case the design complexity is more complicated, depending on the details of the design. The number of bottlenecks also determines whether progress is steady, or whether there are periods of stasis punctuated by occasional large changes. Our model connects the engineering properties of a design to historical studies of technology improvement.
The Model
The model represents a technology as interacting cost-bearing components connected through a design structure matrix. Innovation repeatedly proposes cost changes for a component’s dependent cluster and accepts them only when the cluster’s total cost decreases.
- The Model: A technology contains n components with additive total cost c = c_1 + c_2 + ··· + c_n.Each component can be replaced by a new implementation with a different cost.
- The Model: Components interact when changing one component affects the costs of dependent components.These dependencies form a directed network represented by a design structure matrix (DSM).
- The Model: A DSM records whether changing component j affects dependent component i, with diagonal links representing each component’s effect on its own cost.The matrices can depict randomly generated fixed-out-degree designs or an automobile brake system.
- The Model: With constant connectivity, each component affects exactly d components: itself and d − 1 others.The dependencies are chosen randomly with equal probability.
- The Model: Each innovation attempt selects a component, redraws costs for its dependent set from f, and accepts the set only if its total cost falls.Otherwise, the proposed cost set is rejected.
- The Model: For low component costs, f(c_i) follows a power form whose exponent γ measures the difficulty of reducing individual component costs.Higher γ indicates greater difficulty.
Independent Components
For independent components, repeated innovation attempts reduce cost through extreme-value search. The resulting asymptotic trajectory is a power law, with c(t) scaling as 1/t in the uniform γ = 1 case.
- Independent Components: The independent-component case treats the cost after t innovation steps as the minimum of t independent, identically distributed random variables.The paper presents this as the basis for the single-component analysis and its extension to independent components.
- Independent Components: For uniform proposals, a reduction succeeds with probability c, and the average accepted new cost is c/2.The resulting continuous-time derivation gives the correct scaling but underestimates the amplitude.
- Independent Components: c(t) ∼ 1/t as t → ∞.The power law arises because reductions become proportionally larger while successful reductions become increasingly difficult to obtain.
Interacting Components, Fixed Out-Degree
With fixed out-degree, interacting components produce a power-law improvement whose exponent depends on both component-improvement difficulty and design connectivity. Simulations support the predicted asymptotic scaling, while the approximation overestimates average cost amplitude.
- Interacting Components, Fixed Out-Degree: The interacting model computes expected total-cost changes by summing accepted cluster changes over proposals with c′ < c.The resulting integral leads to the analytical solution for the fixed-out-degree case.
- Interacting Components, Fixed Out-Degree: The theory correctly predicts the simulations’ asymptotic power-law scaling and predicts an initial downward concavity whose duration increases with n.The concavity prediction is accurate for small d but underestimates it for large d.
- Interacting Components, Fixed Out-Degree: The asymptotic solution consistently exceeds the simulated average cost because of approximations in deriving Eq. [5].Alternative distribution-based calculations recover the correct amplitude only in the limit t → ∞.
- Interacting Components, Fixed Out-Degree: α = 1/(γd), linking the performance-curve exponent to connectivity d and the difficulty parameter γ.Here d measures design complexity for constant connectivity.
- Interacting Components, Fixed Out-Degree: If γd = 1, α = 1 and the progress ratio is 50%; if γd = 3, α = 1/3 and the progress ratio is approximately 80%.The latter value is described as common in empirical performance curves.
Interacting Components, Variable Out-Degree
Variable out-degree creates bottlenecks because components differ in how readily they improve, making the slowest-improving component determine the technology’s overall rate. The design complexity therefore depends on the interaction structure rather than connectivity alone.
- Interacting Components, Variable Out-Degree: Variable out-degree allows slowly improving components to become bottlenecks that hinder the overall rate of improvement.A component’s improvement rate depends on the clusters of modifying components that affect it.
- Interacting Components, Variable Out-Degree: A component can improve either when its own cluster is selected or when another modifying component’s cluster is selected.These are the two innovation processes represented by the dependent and modifying-component clusters.
- Interacting Components, Variable Out-Degree: Higher out-degree makes a component less likely to improve when selected because the probability of lowering the cluster’s total cost decreases with connectivity.The trial changes must reduce the combined cost of the affected components.
- Interacting Components, Variable Out-Degree: The overall improvement rate is determined by the slowest-improving component, while each component’s rate is governed by the most likely modifying component to improve it.This yields a design complexity based on the interaction structure and its bottleneck components.
- Interacting Components, Variable Out-Degree: For sufficiently large t, α = 1/(d*γ) predicts the mean rate of improvement, and constant out-degree reduces design complexity to d.The special-case reduction connects the variable-connectivity definition to the fixed-connectivity model.
Fluctuations
Fluctuations around the mean depend on the design’s bottleneck structure: homogeneous designs tend toward steadier improvement, while heterogeneous designs can produce erratic, punctuated progress.
- Fluctuations: The model links fluctuation behavior to the properties of the design structure matrix.The analysis addresses variation around mean improvement, not only the average rate.
- Fluctuations: Four components with the same worst improvement rate share the long-run cost, keeping fluctuations relatively small.Their comparable contributions average the total cost across multiple bottlenecks.
- Fluctuations: A single slowest-improving component can dominate costs, producing rare dramatic improvements and large downward steps.In the illustrated case, component 7 improves slowly because it is rarely selected.
- Fluctuations: All quantiles eventually approach slope 1/d∗, but lower quantiles can converge much later than upper quantiles.Successive bottlenecks temporarily influence improvement before the slowest-improving component becomes dominant.
- Fluctuations: Greater variability in out-degree is associated with improvement in fits and starts, whereas constant out-degree is associated with steadier progress.The model interprets high trajectory variability as a role for historical contingency and punctuated equilibria.
- Fluctuations: Fluctuations decrease as the number of worst bottlenecks increases.This number is all components in the constant out-degree case, but depends on DSM details when out-degree varies.
Testing the Model Predictions
The model proposes testing whether design complexity predicts historical improvement rates using design structure matrices, while noting that γ and innovation effort are difficult to measure.
- Testing the Model Predictions: Design complexity can be computed from a design structure matrix and compared with the historical exponent α of a technology.The model makes this relationship a testable empirical prediction.
- Testing the Model Predictions: The proposed comparison is complicated because α depends on γ, the inherent difficulty of improving individual components.γ also reflects the innovation problem and the collective effectiveness of inventors, and is problematic to measure independently.
- Testing the Model Predictions: The model predicts design complexity should be inversely proportional to the estimated historical α, subject to assumptions about γ.Cross-technology studies could assume constant γ or that variations in γ average out.
- Testing the Model Predictions: Empirical testing requires a measurable proxy for attempted improvements, but the best proxy for innovation effort remains debated.Candidates include cumulative production, cumulative investment, installed capacity, R&D expenditure, and time.
Possible Extensions to the Model
The model could be extended to allow designs to change over time and to represent innovation attempts that use knowledge of prior improvements and current designs.
- Possible Extensions to the Model: A more realistic model could let the design network described by the DSM evolve through time.One proposed extension adds an evolutionary model for generating new DSMs.
- Possible Extensions to the Model: If design complexity d∗ changes over time, the theory predicts that the power-law exponent α changes as well.Historical sequences of design matrices could be used to compare predicted and observed α values across periods.
- Possible Extensions to the Model: The current learning model assumes innovation attempts are random and ignore the history of previous improvements and knowledge of the technology.An intelligent designer could do better by drawing on scientific, engineering, and design knowledge.
Discussion
The model predicts technology improvement as a power law whose exponent depends on design complexity and the difficulty of improving components.
- The mean cost trajectory follows a power law with exponent α = 1/(d∗γ).Here, d∗ is design complexity and γ measures the intrinsic difficulty of improving individual components.
- For constant connectivity, design complexity equals the connectivity; with variable connectivity, it can depend on the design matrix.
- The framework applies to any additive technology performance measure, not only cost.
- The model predicts variation in technological improvement trajectories in addition to their mean behavior.