Optimal Pricing Strategy for Linear-Quadratic Games with Diminishing Marginal Peer Influence

Core Concepts

The optimal pricing strategy for a monopolist selling a divisible good to agents in a network game with linear-quadratic payoffs and strictly concave peer interaction function is strictly better than a network-agnostic pricing strategy, and the price of information increases with the curvature of the interaction function.

Abstract

This paper studies the optimal pricing problem for a class of network games with linear-quadratic payoffs and externalities exerted through a strictly concave interaction function. The authors make the following key contributions:

They prove the existence and uniqueness of a Nash Equilibrium (NE) for this class of games under mild conditions on the interaction function and the network structure.

They analyze the optimal pricing strategy for a monopolist selling a divisible good to the agents. They show that the optimal pricing strategy, found by solving a bilevel optimization problem, is strictly better when the monopolist knows the network structure compared to the best strategy agnostic to network structure.

They provide a lower bound on the "price of information" (PoI), defined as the ratio between the total revenues under the optimal and network-agnostic pricing strategies. They show that the PoI increases with the curvature of the interaction function.

They describe an efficient algorithm to find the optimal pricing strategy by reformulating the bilevel optimization problem as a strongly concave optimization problem.

The results contrast with the previously studied case of linear interaction function, where a network-independent price is proven optimal with symmetric networks. The authors demonstrate through numerical experiments that the optimal pricing strategy can exploit the asymmetry in the network structure to achieve higher revenues compared to the network-agnostic strategy.

Stats

The paper does not provide any specific numerical data or metrics to support the key claims. The analysis is primarily theoretical, with numerical experiments used to illustrate the theoretical findings.

Quotes

"We show that the optimal pricing problem is equivalent to a strongly concave optimization problem through reparameterizing the latter by the NE."
"Our results suggest that the price of information (PoI) obeys a nuanced structure in the nonlinear interaction setting."

Key Insights Distilled From

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

by Jiamin Cai,C... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.01047.pdf

Deeper Inquiries

How would the optimal pricing strategy change if the monopolist has only partial information about the network structure

If the monopolist has only partial information about the network structure, the optimal pricing strategy may be suboptimal compared to having full knowledge of the network. In such a scenario, the monopolist may need to rely on statistical methods or estimation techniques to infer the network structure based on the available information. This could lead to a less precise pricing strategy, potentially resulting in lower revenue or suboptimal outcomes. The monopolist may need to implement adaptive pricing strategies that can adjust based on the partial information available, taking into account the uncertainty in the network structure.

What are the implications of the diminishing marginal peer influence on the agents' behavior and welfare in the network game

The diminishing marginal peer influence in the network game can have significant implications on the agents' behavior and welfare. When the peer influence has diminishing returns, it means that the impact of additional peers on an agent's decision decreases as the number of peers increases. This can lead to more stable equilibrium outcomes in the network game, as agents may be less responsive to changes in their peers' actions.
From a welfare perspective, diminishing marginal peer influence can affect the overall efficiency of the network. Agents may be less influenced by their peers, leading to less coordination or cooperation in the network. This can impact the overall utility or satisfaction of the agents in the network, potentially reducing the overall welfare compared to a scenario with stronger peer influence.

Can the insights from this work be extended to other types of nonlinear interaction functions beyond the strictly concave case

The insights from this work on optimal pricing strategies in network games with strictly concave interaction functions can be extended to other types of nonlinear interaction functions. By analyzing the impact of different forms of nonlinearity on the optimal pricing strategy, researchers can gain a deeper understanding of how various interaction functions affect the behavior of agents in the network.
For example, exploring convex interaction functions or other forms of concavity beyond strict concavity can provide insights into how different types of peer influence impact the optimal pricing strategy. By studying a broader range of nonlinear functions, researchers can uncover more nuanced relationships between network structure, peer influence, and optimal pricing decisions. This can lead to a more comprehensive understanding of network games and pricing strategies in complex social and economic systems.