Equilibria of Chinese Auctions

TL;DR

This paper analyzes the existence of pure Nash equilibria in Chinese auctions, revealing conditions under which equilibria exist or can be stabilized, with insights into both continuous and discrete bidding scenarios.

Contribution

It provides the first comprehensive analysis of pure Nash equilibria in Chinese auctions, including existence conditions and stabilization strategies for asymmetric and symmetric cases.

Findings

Pure Nash equilibria may not exist in asymmetric continuous games with zero valuations.

The auctioneer can stabilize the game by placing tickets in each basket when some valuations are zero.

Pure Nash equilibria are guaranteed to exist when all valuations are strictly positive, with symmetric strategies under symmetry.

Abstract

Chinese auctions are a combination between a raffle and an auction and are held in practice at charity events or festivals. In a Chinese auction, multiple players compete for several items by buying tickets, which can be used to win the items. In front of each item there is a basket, and the players can bid by placing tickets in the basket(s) corresponding to the item(s) they are trying to win. After all the players have placed their tickets, a ticket is drawn at random from each basket and the item is given to the owner of the winning ticket. While a player is never guaranteed to win an item, they can improve their chances of getting it by increasing the number of tickets for that item. In this paper we investigate the existence of pure Nash equilibria in both the continuous and discrete settings. When the players have continuous budgets, we show that a pure Nash equilibrium may not exist for asymmetric games when some valuations are zero. In that case we prove that the auctioneer can stabilize the game by placing his own ticket in each basket. On the other hand, when all the valuations are strictly positive, a pure Nash equilibrium is guaranteed to exist, and the equilibrium strategies are symmetric when both valuations and budgets are symmetric. We also study Chinese auctions with discrete budgets, for which we give both existence results and counterexamples. While the literature on rent-seeking contests traditionally focuses on continuous costly tickets, the discrete variant is very natural and more closely models the version of the auction held in practice.