Interdependence Framework

Formalizing Structural Dependencies for Strategic Coopetition (TR-1)

This document provides a comprehensive treatment of the interdependence formalization from Technical Report 1 (TR-2025-01), explaining how structural dependencies from conceptual models translate to quantitative game-theoretic analysis.

Executive Summary

For Practitioners: Interdependence captures why actors must consider partner outcomes even while competing. When your success depends on your partner’s success, you have rational incentive to care about their performance, not from altruism, but from structural necessity.

For Researchers: We formalize interdependence through translation from i* dependency networks to an interdependence matrix $\mathbf{D}$, where $D_{ij} \in [0,1]$ quantifies the structural coupling of actor $i$’s outcomes to actor $j$’s actions. This enables game-theoretic equilibrium analysis with dependency-augmented utility functions.

Conceptual Foundation

The Interdependence Problem

Classical game theory assumes purely self-interested payoffs: each actor maximizes their own returns without intrinsic concern for others. This fails to capture a fundamental aspect of real organizational relationships:

Structural Dependency Creates Rational Concern for Partner Outcomes

When Actor A depends on Actor B for critical resources, capabilities, or goal achievement:

Example: A startup developing for iOS cannot succeed if Apple’s App Store fails. The startup rationally cares about Apple’s platform health, not from goodwill, but because their business depends on it.

Distinguishing Interdependence from Altruism

Concept Source Motivation Varies With
Interdependence Structural coupling Instrumental self-interest Dependency structure
Altruism Psychological preference Concern for others’ welfare Personal values
Reciprocity Behavioral response Conditional cooperation Partner behavior

Interdependence is structural: it emerges from how goals and capabilities connect, not from psychological disposition. Two actors with identical personalities will have different interdependence coefficients based on their positions in the dependency network.

The i* Framework Foundation

Strategic Dependency Modeling

The i* framework (Yu, 1995) provides the conceptual basis for interdependence analysis through its representation of strategic actors and dependencies.

Core Elements:

Element Symbol Definition
Actor Circle Intentional entity with goals
Depender i Actor who depends
Dependee j Actor who is depended upon
Dependum d Object of dependency (goal, task, resource)

Dependency Relationship: When Actor i depends on Actor j for dependum d, this creates a structural constraint: i cannot fully achieve certain goals without j’s successful performance in delivering d.

Example: Platform Ecosystem

Platform Dependencies

Asymmetry: The developer critically depends on the platform (D = 0.80), while the platform moderately depends on any single developer (D = 0.35). This asymmetry has profound strategic implications.

Mathematical Formalization

The Interdependence Matrix

Definition: The interdependence matrix $\mathbf{D}$ is an $N \times N$ matrix where element $D_{ij} \in [0,1]$ quantifies the structural dependency of actor $i$ on actor $j$.

Interdependence Heatmap

Equation 1: Interdependence Coefficient

\[\Large D_{ij} = \frac{\sum_{d \in \mathcal{D}_i} w_d \cdot \text{Dep}(i,j,d) \cdot \text{crit}(i,j,d)}{\sum_{d \in \mathcal{D}_i} w_d}\]

Components:

Component Symbol Range Meaning
Importance Weight $w_d$ $\mathbb{R}^+$ Strategic priority of goal $d$ for actor $i$
Dependency Indicator $\text{Dep}(i,j,d)$ ${0, 1}$ Does $i$ depend on $j$ for $d$?
Criticality Factor $\text{crit}(i,j,d)$ $[0, 1]$ How critical is $j$ for achieving $d$?

Computing Each Component

Importance Weights (w_d)

Purpose: Quantify the strategic priority of each goal/dependum for the actor.

Elicitation Methods: 1. Analytic Hierarchy Process (AHP):

  1. Direct Assessment:
    • Allocate 100 points across goals
    • Simple, fast, intuitive
    • May lack consistency guarantees
  2. Goal Criticality Analysis:
    • Rate goals on urgency, impact, stakeholder priority
    • Composite scoring
    • Comprehensive but complex

Example:

Developer Goals:
  - Revenue generation: w = 0.50 (highest priority)
  - User acquisition: w = 0.30
  - Technical capability: w = 0.20

Σw = 1.00 (normalized)

Dependency Indicator (Dep(i,j,d))

Purpose: Binary flag indicating whether a dependency relationship exists.

Determination: Direct from i* model analysis

Example:

Developer depends on Platform for:
  - API access: Dep = 1 (critical dependency)
  - App distribution: Dep = 1 (critical dependency)
  - Marketing support: Dep = 0 (developer does own marketing)

Criticality Factor (crit(i,j,d))

Purpose: Quantify how essential actor j is for achieving dependum d.

Calculation Rules:

Scenario crit(i,j,d) Rationale
Sole provider, no alternatives 1.0 Complete criticality
n equal alternatives 1/n Distributed criticality
Preferred but substitutable [1/n, 1] Partial lock-in
Fully substitutable 0.1-0.3 Minimal switching cost

Example:

Developer criticality for Platform API:
  - Only one platform available: crit = 1.0
  - Two platforms (iOS, Android): crit = 0.5 per platform
  - Developer prefers iOS but could switch: crit = 0.65 for iOS

Worked Example: Computing D_ij

Scenario: Developer’s dependency on Platform

Goal (d) w_d Dep crit w × Dep × crit
Revenue 0.50 1 0.90 0.450
Users 0.30 1 0.85 0.255
Tech 0.20 1 0.60 0.120
Total 1.00     0.825 
D_developer,platform = 0.825 / 1.00 = 0.825

Interpretation: The developer’s outcomes are 82.5% dependent on the platform’s performance. This is a high-dependency relationship creating significant strategic exposure.

Properties of the Interdependence Matrix

Structural Properties

Property Description Implication
Bounded $D_{ij} \in [0, 1]$ Normalized interpretation
Asymmetric $D_{ij} \neq D_{ji}$ generally Power imbalances
Zero diagonal $D_{ii} = 0$ No self-dependency
Non-negative $D_{ij} \geq 0$ Dependencies cannot be negative

Interpretive Guidelines

$D_{ij}$ Value Interpretation Strategic Implications
0.00 No dependency Independent operation
0.01 - 0.20 Minimal dependency Low strategic coupling
0.20 - 0.40 Moderate dependency Some coordination needed
0.40 - 0.60 Significant dependency Strategic partnership
0.60 - 0.80 High dependency Critical relationship
0.80 - 1.00 Near-complete dependency Survival depends on partner

Dependency Patterns

Mutual High Dependency ($D_{ij} \approx D_{ji}$, both high):

Asymmetric Dependency ($D_{ij} \gg D_{ji}$):

Low Mutual Dependency ($D_{ij} \approx D_{ji} \approx 0$):

Integration with Utility Functions

The Integrated Utility Function

The interdependence matrix enters the utility function as weights on partner payoffs:

Equation 13 (TR-1):

\[\Large U_i(\mathbf{a}) = \pi_i(\mathbf{a}) + \sum_{j \neq i} D_{ij} \cdot \pi_j(\mathbf{a})\]

Components:

Term Meaning
$\pi_i(\mathbf{a})$ Actor $i$’s private payoff
$D_{ij} \cdot \pi_j(\mathbf{a})$ Dependency-weighted concern for $j$’s payoff
$\sum_{j \neq i} D_{ij} \cdot \pi_j(\mathbf{a})$ Total structural concern for all partners

Why This Works

The integrated utility captures rational self-interest in the presence of structural coupling: 1. When $D_{ij} = 0$: $U_i = \pi_i$ (pure self-interest)

  1. When $D_{ij} > 0$: $U_i$ includes weighted partner payoffs
  2. Higher $D_{ij}$: More weight on partner’s success

This is not altruism, it’s recognizing that when your outcomes depend on your partner’s performance, maximizing your utility requires considering their payoffs.

Coopetitive Equilibrium

Definition: A Coopetitive Equilibrium is a Nash Equilibrium where each actor maximizes the integrated utility function:

\[\Large \mathbf{a}^* \text{ is Coopetitive Equilibrium if: } a_i^* \in \arg\max_{a_i} U_i(a_i, \mathbf{a}_{-i}^*) \text{ for all } i\]

Key Insight: Coopetitive Equilibrium generally produces more cooperative outcomes than standard Nash Equilibrium because the integrated utility includes positive spillovers through interdependence terms.

Translation Framework

Step-by-Step Process

Step 1: Elicit the i* Dependency Network

Step 2: Quantify Importance Weights

Step 3: Assess Criticality Factors

Step 4: Compute Interdependence Matrix

Step 5: Validate and Refine

Iterative Refinement

The translation process is inherently iterative, quantification translates qualitative dependencies into the $\mathbf{D}$ matrix, while equilibrium analysis reveals gaps in the conceptual model.

Validation: Samsung-Sony S-LCD Case Study

Case Background

Joint Venture: Samsung and Sony formed S-LCD Corporation (2004-2011) to manufacture LCD panels.

Key Dynamics:

Interdependence Analysis

Estimated D Matrix:

           Samsung  Sony
Samsung  [  0.00   0.45 ]
Sony     [  0.40   0.00 ]

Interpretation:

Validation Results

Metric Model Prediction Historical Data Match
Cooperation level 41% increase 15-50% range ✓
Relationship duration 7-8 years 8 years ✓
Eventual dissolution Predicted Occurred 2012 ✓

Validation Score: 58/60 criteria matched (96.7%)

Practical Applications

For Requirements Engineering

For Alliance Management

For MARL Research

Common Pitfalls

Pitfall 1: Confusing Dependency with Preference

Wrong: “We like working with them, so D is high” Right: D reflects structural necessity, not preference

Pitfall 2: Assuming Symmetry

Wrong: “If I depend on them, they depend on me equally” Right: $D_{ij}$ and $D_{ji}$ are independent; asymmetry is common

Pitfall 3: Static Analysis

Wrong: “Compute D once and use forever” Right: D should be updated as relationships evolve

Pitfall 4: Over-Precision

Wrong: “$D_{ij} = 0.4237$ exactly” Right: $D$ estimates have uncertainty; use sensitivity analysis

Further Reading

Primary Source

Background

Technical Reports