Magic: The Gathering, Integer Linear Programming and Arbitrage

Context and Motivation

Magic: The Gathering® is a collectible card game created in 1993, and iswidely regarded as the first trading card game ever produced. To this day,it remains wildly popular, with millions of players and billions of cardsprinted. WhileWizards of the Coast, the company that creates Magic, doesnot like admitting it, a thriving secondary market exists for Magic cards,with hundreds of online retailers offering to buy and sell cards. In fact,most Magic players acquire cards by purchasing individual cards (knownas "singles") from retailers, rather than gambling on booster packs fromthe manufacturer that they hope will contain cards that they seek.

Online retailers publish prices of cards, along with the quantity available,in sell-lists. Likewise, retailers publish prices and quantities of cardsto buy in their buy-lists. Of course, the price at which a retailer will purchasea card is always less than the price at which they'll sell that card.

Besides offering cash for cards, most retailers also offer to buy cards inexchange for store credit, which can be used to purchase singles from theirshop. The amount of store credit is always greater than the amount of cashthat they offer for purchasing a card.

There are hundreds of online Magic retailers in the secondary market, allof them with buy-lists and sell-lists for cards, with some buy-lists offeringto exchange cards for trade credit. Surely there must be some inefficiencieshere!