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A Theory of Network Games Part 1: Utility Representations

Joseph Root, Evan Sadler

Abstract

We demonstrate that a ubiquitous feature of network games, bilateral strategic interactions, is equivalent to having player utilities that are additively separable across opponents. We distinguish two formal notions of bilateral strategic interactions. Opponent independence means that player i's preferences over opponent j's action do not depend on what other opponents do. Strategic independence means that how opponent j's choice influences i's preference between any two actions does not depend on what other opponents do. If i's preferences jointly satisfy both conditions, then we can represent her preferences over strategy profiles using an additively separable utility. If i's preferences satisfy only strategic independence, then we can still represent her preferences over just her own actions using an additively separable utility. Common utilities based on a linear aggregate of opponent actions satisfy strategic independence and are therefore strategically equivalent to additively separable utilities--in fact, we can assume a utility that is linear in opponent actions.

A Theory of Network Games Part 1: Utility Representations

Abstract

We demonstrate that a ubiquitous feature of network games, bilateral strategic interactions, is equivalent to having player utilities that are additively separable across opponents. We distinguish two formal notions of bilateral strategic interactions. Opponent independence means that player i's preferences over opponent j's action do not depend on what other opponents do. Strategic independence means that how opponent j's choice influences i's preference between any two actions does not depend on what other opponents do. If i's preferences jointly satisfy both conditions, then we can represent her preferences over strategy profiles using an additively separable utility. If i's preferences satisfy only strategic independence, then we can still represent her preferences over just her own actions using an additively separable utility. Common utilities based on a linear aggregate of opponent actions satisfy strategic independence and are therefore strategically equivalent to additively separable utilities--in fact, we can assume a utility that is linear in opponent actions.
Paper Structure (28 sections, 17 theorems, 179 equations)

This paper contains 28 sections, 17 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Additively Separable Utilities
  3. Separable Preferences
  4. Proof Preliminaries
  5. Proof of Theorem \ref{['theo:main1']}
  6. Discussion
  7. Strategically Separable Preferences
  8. Technical Assumptions
  9. The Result
  10. Local Aggregates
  11. Constant Rate of Substitution
  12. Beyond Strategic Independence
  13. Proof of Theorem \ref{['theo:cyclic']}
  14. Discussion
  15. Final Remarks
  16. ...and 13 more sections

Key Result

Theorem 1

Suppose Assumptions as:separable1 and as:separable2 hold for action sets $\{S_i\}_{i=0}^n$ and the preference order $\succeq_0$. The order $\succeq_0$ is separable if and only if it satisfies joint independence. Moreover, the representation of $\succeq_0$ via a utility of the form is unique up to a positive affine transformation.

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Lemma 1: Wakker, 1989, Theorem III.4.1
  • Example 1
  • Definition 4
  • Definition 5
  • Theorem 2
  • Lemma 2
  • ...and 22 more