Pricing Public Goods for Private Sale

TL;DR

This paper investigates how a seller should price a good that provides benefits to others around the buyer, analyzing equilibrium strategies and proposing approximately revenue-maximizing pricing mechanisms for various network structures.

Contribution

It introduces a pricing framework for locally public goods in networks, providing approximately optimal solutions for fully public and certain networked cases.

Findings

Proposes a near revenue-optimal pricing mechanism for fully public goods with regular distributions.

Develops a pricing method that maximizes revenue in worst-case equilibria for d-regular networks.

Identifies challenges in extending results to general networks and distributions.

Abstract

We consider the pricing problem faced by a seller who assigns a price to a good that confers its benefits not only to its buyers, but also to other individuals around them. For example, a snow-blower is potentially useful not only to the household that buys it, but also to others on the same street. Given that the seller is constrained to selling such a (locally) public good via individual private sales, how should he set his prices given the distribution of values held by the agents? We study this problem as a two-stage game. In the first stage, the seller chooses and announces a price for the product. In the second stage, the agents (each having a private value for the good) decide simultaneously whether or not they will buy the product. In the resulting game, which can exhibit a multiplicity of equilibria, agents must strategize about whether they will themselves purchase the good to receive its benefits. In the case of a fully public good (where all agents benefit whenever any agent purchases), we describe a pricing mechanism that is approximately revenue-optimal (up to a constant factor) when values are drawn from a regular distribution. We then study settings in which the good is only "locally" public: agents are arranged in a network and share benefits only with their neighbors. We describe a pricing method that approximately maximizes revenue, in the worst case over equilibria of agent behavior, for any $d$-regular network. Finally, we show that approximately optimal prices can be found for general networks in the special case that private values are drawn from a uniform distribution. We also discuss some barriers to extending these results to general networks and regular distributions.