Title: 

Gacha: A Simple Mechanism to Screen a Budget-Constrained Buyer

Abstract: 

A lottery mechanism that allows repeated purchases until the buyer wins a designated item is widely used for both digital goods in Gacha games (e.g.,  Genshin Impact) and physical goods such as collectible toys (e.g., Labubu). We study this mechanism---referred to as a Gacha mechanism---in what is arguably the simplest possible setting: a seller offering one single item to a budget-constrained buyer. While the optimal mechanism in this setting typically involves designing a large (potentially infinite) menu of lotteries that can only be purchased once, the Gacha mechanism requires only a selling price and a winning probability, making it far more practical to implement. We show that the Gacha mechanism is particularly effective at screening along the budget dimension. While the posted price mechanism can perform arbitrarily poorly compared to the optimal mechanism when the buyer’s valuation is public, the Gacha mechanism achieves at least 63.2% of the optimal revenue. It becomes asymptotically optimal as the valuation grows large. Moreover, when the seller has almost no information about the buyer’s budget distribution, 63.2% is also the max-min revenue ratio guarantee, which can be achieved by a Gacha mechanism with vanishing winning probability. We also show that the Gacha mechanism is less effective at screening along the valuation dimension. When the valuation is private and the budget is public, the optimal Gacha mechanism reduces to a posted-price mechanism that achieves at least 50% of the optimal revenue. When both valuation and budget are private and independent, and the valuation follows a monotone hazard rate distribution, the Gacha mechanism guarantees at least 23% of the optimal revenue. Finally, we explore two extensions of the Gacha mechanism---one that includes the option of direct purchase, and another that incorporates a pity system.

Bio: 

Zhenyu Hu is a Dean’s Chair Associate Professor at the Department of Analytics & Operations at the National University of Singapore. He obtained his PhD in industrial engineering from the University of Illinois at Urbana-Champaign and a bachelor in mathematics from Sun Yat-sen University. His research focuses on dynamic pricing and revenue management, supply chain management, and mechanism and information design. He currently serves as an associate editor for M&SOM.