Optimal Pricing for Linear-Quadratic Games with Nonlinear Interaction Between Agents

TL;DR

This work studies a linear-quadratic network game where peer effects are mediated by a strictly concave interaction function $f$. The authors establish existence and uniqueness of Nash Equilibr under Lipschitz and curvature conditions, and reformulate the monopolist's bilevel pricing problem into a single-level, strongly concave optimization in the agents’ action vector $\mathbf{x}$, enabling efficient computation via projected gradient methods. The optimal prices satisfy $\mathbf{p}^* = \mathbf{a} + \mathbf{G} f(\mathbf{x}^*) - \bm{\mathcal{B}}\mathbf{x}^*$, with the NE corresponding to $\mathbf{p}^*$ being the unique maximizer of the objective $J(\mathbf{x})$. A key finding is that network-aware pricing strictly improves revenue over network-agnostic strategies when $f$ is strictly concave, and the paper provides a concrete bound on the price of information (PoI). Numerical experiments on star and preferential attachment graphs illustrate that asymmetry can maximize PoI and that the lower bounds capture the observed revenue gaps, validating the theoretical results and highlighting practical implications for pricing interventions in networks with diminishing peer effects.

Abstract

This paper studies a class of network games with linear-quadratic payoffs and externalities exerted through a strictly concave interaction function. This class of game is motivated by the diminishing marginal effects with peer influences. We analyze the optimal pricing strategy for this class of network game. First, we prove the existence of a unique Nash Equilibrium (NE). Second, we study the optimal pricing strategy of a monopolist selling a divisible good to agents. We show that the optimal pricing strategy, found by solving a bilevel optimization problem, is strictly better when the monopolist knows the network structure as opposed to the best strategy agnostic to network structure. Numerical experiments demonstrate that in most cases, the maximum revenue is achieved with an asymmetric network. These results contrast with the previously studied case of linear interaction function, where a network-independent price is proven optimal with symmetric networks. Lastly, we describe an efficient algorithm to find the optimal pricing strategy.

Figures (2)

• Figure 1: (Left) Star graph with $a = 1; b = 1; n = 10$. Maximum PoI achieved by $f_1(\cdot)$ is $\frac{1}{0.989}$ at $\alpha = 0.41$, $f_2(\cdot)$ is $\frac{1}{0.995}$ at $\alpha = 0.32$; (Right) PA graph with $a = 1; b = 2; n = 100$. Maximum PoI achieved by $f_1(\cdot)$ is $\frac{1}{0.999}$ at $\alpha = 0.44$, $f_2(\cdot)$ is $\frac{1}{0.998}$ at $\alpha = 0.32$.
• Figure 2: (Left) Comparing the actual ratio $\frac{J({\bm x}^*)}{J({\bm x}_0)}-1$ to the lower bound in Theorem \ref{['the:the2']} with $a=2,b =\bar{g}$; (Right) Comparing the ratio with uniform prices $\frac{J(\bar{x}^\star {\bm 1} )}{J(\bar{x}_0 {\bm 1} )}-1$ to the lower bound in Theorem \ref{['the:the3']} with $a=2,b =1$.

Theorems & Definitions (14)

• Lemma 1
• Corollary 1
• Lemma 2
• Proposition 1
• Theorem 1
• Corollary 2
• Theorem 2
• proof
• proof
• proof