Selling Enhanced Attempts

Achieving a milestone often requires numerous attempts: a student takes and retakes an exam or a video-game player takes numerous attempts to complete a challenging puzzle. Attempters may be willing to pay for enhancements that increase their chances. A testing agency may bundle an initial attempt with an option to retake the exam. Players may be induced to purchase a booster (e.g., extra time or moves). Typically, if the milestone is passed in the next attempt without relying on the enhancement, it is wasted. In some cases, it can be purchased later at a higher price. This paper asks the question of how to optimally deploy and price enhancements. Selling enhanced attempts is a novel form of advance selling, a classical topic in operations management, marketing, and information systems. We contrast our results with findings in the classical literature in advance selling, uncovering nuances not present in standard settings. We model the selling of enhanced attempts as an a stochastic extensive form game, which we study analytically and numerically. We show how the distribution of skill among agents (i.e., their inherent ability to pass the milestone), and the inherent randomness of the milestone, influence enhancement selling strategies. For instance, we show that if the proportion of highly skilled agents is either sufficiently large or sufficiently small, firms should adopt pure advance selling strategies with no market. Our results yield implications for games companies, including the suggestion that they should only offer advance purchases of enhancements in games with a sufficiently high level of randomness, and if they are committed to offering both advance and spot purchases, might earn additional revenue by reducing the overall randomness in their design.