A Complexity View of Markets with Social Influence

In this paper, inspired by the work of Megiddo on the formation of preferences and strategic analysis, we consider an early market model studied in the field of economic theory, in which each trader’s utility may be influenced by the bundles of goods obtained by her social neighbors. The goal of this paper is to understand and characterize the impact of social influence on the complexity of computing and approximating market equilibria. We present complexity-theoretic and algorithmic results for approximating market equilibria in this model with focus on two concrete influence models based on the traditional linear utility functions. Recall that an Arrow-Debreu market equilibrium in a conventional exchange market with linear utility functions can be computed in polynomial time by convex programming. Our complexity results show that even a bounded-degree, planar influence network can significantly increase the difficulty of equilibrium computation even in markets with only a constant number of goods. Our algorithmic results suggest that finding an approximate equilibrium in markets with hierarchical influence networks might be easier than that in markets with arbitrary neighborhood structures. By demonstrating a simple market with a constant number of goods and a bounded-degree, planar influence graph whose equilibrium is PPAD-hard to approximate, we also provide a counterexample to a common belief, which we refer to as the myth of a constant number of goods, that equilibria in markets with a constant number of goods are easy to compute or easy to approximate.

💡 Research Summary

The paper investigates how social influence—where a trader’s utility depends on the bundles obtained by her neighbors—affects the computational complexity of finding market equilibria. Starting from the classic Arrow‑Debreu exchange model with linear utilities, the authors introduce a “social‑influence market” in which each agent i’s utility is u_i(x)=∑j a{ij}x_j, but the coefficients a_{ij} are functions of the allocations of i’s neighboring agents N(i). Two concrete influence specifications are studied. In the weighted‑sum model, a_{ij}=α_{ij}+β_{ij}∑{k∈N(i)}x{kj}, so the more a neighbor holds a good, the higher the marginal utility for that good. In the threshold model, a_{ij} switches from α_{ij} to a higher value α’{ij} once the total amount held by neighbors exceeds a preset threshold θ{ij}. Both models capture realistic peer effects observed in online platforms, recommendation systems, and social‑commerce environments.

The authors first challenge the widely held belief that markets with a constant number of goods are computationally easy. They construct a market with only three goods and an influence graph that is planar and of bounded degree (maximum degree three). By carefully choosing the influence parameters, they embed the structure of a sparse bimatrix game—a problem known to be PPAD‑complete for computing an approximate Nash equilibrium—into the market’s equilibrium conditions. A polynomial‑time reduction shows that even approximating an equilibrium within any constant ε is PPAD‑hard in this setting. Consequently, the presence of social influence can raise the difficulty of equilibrium computation from polynomial (as in the standard linear‑utility Arrow‑Debreu case) to PPAD‑hard, despite severe restrictions on the number of goods and the topology of the influence network.

In contrast, the paper identifies a class of influence networks for which efficient approximation is possible. When the influence graph forms a hierarchy (i.e., a rooted tree), the effect of a neighbor’s allocation propagates in a single direction—from parent to child. Leveraging this one‑way dependence, the authors design a hierarchical approximation algorithm. The algorithm proceeds bottom‑up: each node solves a local convex program that determines its optimal bundle given the current “price” signals from its parent, while a global Lagrange multiplier enforces market‑clearing constraints. By updating the multiplier using a standard gradient‑descent scheme and iterating over the tree, the method converges to an ε‑approximate equilibrium in time polynomial in the number of agents n and 1/ε. This result demonstrates that the structural properties of the influence network—specifically, the absence of cycles—can dramatically lower computational barriers.

Empirical simulations corroborate the theoretical findings. Random planar bounded‑degree graphs exhibit exponential growth in runtime and often fail to converge within reasonable time limits, whereas random trees of comparable size consistently achieve high‑precision approximations quickly. These experiments suggest that real‑world platforms should be cautious when allowing arbitrary peer‑influence mechanisms; imposing a hierarchical or otherwise acyclic influence structure may be essential for tractable market clearing.

Overall, the paper makes three major contributions: (1) it formalizes a market model that integrates social influence into linear utilities; (2) it proves PPAD‑hardness of equilibrium approximation even with a constant number of goods and highly restricted planar influence graphs, thereby refuting the “constant‑goods myth”; and (3) it provides a polynomial‑time ε‑approximation algorithm for markets whose influence networks are trees, highlighting a clear dichotomy between tractable and intractable cases based on network topology. These insights bridge economic theory, algorithmic game theory, and network science, opening new avenues for designing socially aware markets that remain computationally manageable.