Price of information in games of chance: A statistical physics approach

Information in the form of data, which can be stored and transferred between users, can be viewed as an intangible commodity, which can be traded in exchange for money. Determining the fair price at which a string of data should be traded is an important and open problem in many settings. In this work we develop a statistical physics framework that allows one to determine analytically the fair price of information exchanged between players in a game of chance. For definiteness, we consider a game where N players bet on the binary outcome of a stochastic process and share the entry fees pot if successful. We assume that one player holds information about past outcomes of the game, which they may either use exclusively to improve their betting strategy or offer to sell to another player. We find a sharp transition as the number of players N is tuned across a critical value, between a phase where the transaction is always profitable for the seller and one where it may not be. In both phases, different regimes are possible, depending on the "quality" of information being put up for sale: we observe symbiotic regimes, where both parties collude effectively to rig the game in their favor, competitive regimes, where the transaction is unappealing to the data holder as it overly favors a competitor for scarce resources, and even prey-predator regimes, where an exploitative data holder could be giving away bad-quality data to undercut a competitor. Our analytical framework can be generalized to more complex settings and constitutes a flexible tool to address the rich and timely problem of pricing information in games of chance.