The Pricing War Continues: On Competitive Multi-Item Pricing

TL;DR

This paper analyzes a multi-item pricing game with strategic vendors and a buyer with submodular valuation, revealing the existence conditions of pure Nash equilibria and providing bounds on the price of anarchy.

Contribution

It demonstrates that pure Nash equilibria may not always exist in multi-item pricing games and establishes tight bounds on the price of anarchy, extending previous single-item results.

Findings

Pure Nash equilibria may not exist in multi-item vendor games.

The price of anarchy can be logarithmic in the general case.

Efficient pure Nash equilibria always exist for certain submodular functions.

Abstract

We study a game with \emph{strategic} vendors who own multiple items and a single buyer with a submodular valuation function. The goal of the vendors is to maximize their revenue via pricing of the items, given that the buyer will buy the set of items that maximizes his net payoff. We show this game may not always have a pure Nash equilibrium, in contrast to previous results for the special case where each vendor owns a single item. We do so by relating our game to an intermediate, discrete game in which the vendors only choose the available items, and their prices are set exogenously afterwards. We further make use of the intermediate game to provide tight bounds on the price of anarchy for the subset games that have pure Nash equilibria; we find that the optimal PoA reached in the previous special cases does not hold, but only a logarithmic one. Finally, we show that for a special case of submodular functions, efficient pure Nash equilibria always exist.