A Theory of Network Games Part 1: Utility Representations

Technical Summary: A Theory of Network Games Part I: Utility Representations

Problem Statement
Network games are a popular tool for applied modeling due to their tractability, particularly when employing the linear-quadratic utility function popularized by Ballester et al. [2006]. In this standard framework, player $i$'s utility is given by $u_i(s) = b_i s_i + s_i \sum_{j \neq i} g_{ij} s_j - s_i^2/2$. While analytically convenient—yielding linear best responses and reducing equilibrium computation to solving a linear system—the substantive restrictions these preferences entail are often unclear. Researchers frequently lack a firm grasp on what assumptions are being made when adopting this form or how to judge whether real-world preferences are consistent with it. Furthermore, there is a growing consensus that the linear-quadratic framework is too restrictive for many applications, yet theoretical generalizations often lack principled foundations. This paper seeks to provide interpretable axiomatic foundations for utilities used in network games and to identify principled generalizations.

Methodology
The authors decompose the problem of utility representation into two distinct steps, utilizing axiomatic analysis and results from consumer theory (specifically additively separable utility representations).

1. Step 1: Separability and Independence Axioms. The authors first establish conditions under which preferences are separable across opponents. They formalize the ubiquitous feature of network games—bilateral strategic interactions—through two axioms:

   • Opponent Independence: Player $i$'s preference ranking over the actions of opponent $j$ is independent of the actions of any other opponent $k$.
   • Strategic Independence: If a change in opponent $j$'s action from $s_j$ to $s'_j$ shifts player $i$'s preference between two of her own actions $s_i$ and $s'_i$ in a specific direction, this shift cannot reverse direction under any other profile of opponents' actions.
     The authors distinguish between separable preferences (requiring both opponent and strategic independence) and strategically separable preferences (requiring only strategic independence). They employ technical assumptions regarding the continuity of preferences and the "richness" of action sets (e.g., connected topological spaces, existence of essential opponents) to apply theorems from Wakker [1989] and Debreu [1959].

2. Step 2: Functional Form Derivation. Assuming real-valued actions and strategic separability, the authors introduce further axioms to pin down specific functional forms:

   • Constant Rate of Substitution (CRS): The marginal effect of increasing one opponent's action on player $i$'s preferences is constant relative to another opponent's action.
   • Linear Best Replies: The optimal action for player $i$ is a linear function of opponents' actions.
   • Midpoint Indifference: A novel axiom stating that if $s_i$ is a best response to $s_{-i}$ and $s'_i$ is a best response to $s'_{-i}$, the player is indifferent between $s_i$ and $s'_i$ when opponents play the midpoint $(s_{-i} + s'_{-i})/2$.

3. General Characterization. For cases where technical richness assumptions fail (e.g., finite action sets), the authors utilize a revealed-preference approach based on "balanced sequences." They employ Motzkin's Transposition Theorem (a variant of Farkas' lemma) to characterize strategic separability without topological assumptions.

Key Contributions and Results

• Axiomatic Foundations for Separability: The paper demonstrates that bilateral strategic interactions are equivalent to having utilities that are additively separable across opponents.

  • Theorem 3: Under mild technical restrictions, a preference order over strategy profiles is separable (admitting a utility $u_i(s) = \sum_{j \neq i} g_{ij}(s_i, s_j)$) if and only if it satisfies joint independence (a combination of opponent and strategic independence).
  • Theorem 4: Under stronger technical assumptions (joint continuity, connected intervals, joint solvability), a family of preference orders over a player's own actions is strategically separable if and only if it satisfies strategic independence alone. This result covers many common generalizations where preferences depend on linear aggregates of neighbors' actions (e.g., Belhaj et al. [2014]), which are not fully separable but are strategically separable.

• Derivation of Canonical Forms:

  • Theorem 1: If preferences are strategically separable and exhibit a constant rate of substitution between opponents, they are strategically equivalent to a utility linear in opponent actions: $u_i(s) = b(s_i) + \gamma(s_i) \sum_{j \neq i} g_{ij} s_j$. This class includes the linear-quadratic form as well as non-linear generalizations.
  • Theorem 2: If, in addition to the above, preferences satisfy linear best replies and midpoint indifference, they are strategically equivalent to the classic linear-quadratic utility: $u_i(s) = b s_i + s_i \sum_{j \neq i} g_{ij} s_j - s_i^2/2$.

• Characterization via Balanced Sequences:

  • Theorem 5 & 6: For finite action sets (and generally via lexicographic representations), preferences are strategically separable if and only if they admit no "balanced sequence." A balanced sequence is a collection of strategy profile pairs where the focal player strictly prefers one set of profiles over another, yet the frequency of every action pair $(s_i, s_j)$ is identical in both sets. This provides a precise revealed-preference test for separability that does not rely on topological assumptions.

Significance and Claims
The authors claim their findings offer three primary contributions to the literature:

1. Interpretable Foundations: The paper provides clear, interpretable axioms for the utility functions used in network games. It clarifies exactly what is assumed when writing down a linear-quadratic utility or its generalizations, moving beyond mere tractability to substantive preference restrictions.
2. Principled Generalizations: By identifying the specific axioms (strategic independence, CRS, midpoint indifference) that lead to specific functional forms, the paper offers a template for constructing principled generalizations of the linear-quadratic framework. This allows researchers to relax key assumptions (e.g., linearity) while maintaining a rigorous theoretical basis.
3. Revealed-Preference Tests: The results provide concrete tests for whether observed or modeled preferences are consistent with these assumptions. The "balanced sequence" condition, in particular, offers a necessary and sufficient condition for strategic separability that can be checked without assuming specific functional forms or topological properties.

The authors note that while strategic equivalence justifies using these simplified utilities for equilibrium computation and comparative statics, it does not necessarily justify their use for welfare analysis, as different utility representations can yield different welfare rankings despite identical strategic incentives. The paper concludes by positioning itself as "Part I" of a larger project, with "Part II" intended to explore conditions for potential games and substantive predictions independent of specific utility functions.