$α$-Rank-Collections: Analyzing Expected Strategic Behavior with Uncertain Utilities

Game theory largely rests on the availability of cardinal utility functions. In contrast, only ordinal preferences are elicited in fields such as matching under preferences. The literature focuses on mechanisms with simple dominant strategies. However, many real-world applications do not have dominant strategies, so intensities between preferences matter when participants determine their strategies. Even though precise information about cardinal utilities is unavailable, some data about the likelihood of utility functions is typically accessible. We propose to use Bayesian games to formalize uncertainty about decision-makers utilities by viewing them as a collection of normal-form games where uncertainty about types persist in all game stages. Instead of searching for the Bayes-Nash equilibrium, we consider the question of how uncertainty in utilities is reflected in uncertainty of strategic play. We introduce $\alpha$-Rank-collections as a solution concept that extends $\alpha$-Rank, a new solution concept for normal-form games, to Bayesian games. This allows us to analyze the strategic play in, for example, (non-strategyproof) matching markets, for which we do not have appropriate solution concepts so far. $\alpha$-Rank-collections characterize a range of strategy-profiles emerging from replicator dynamics of the game rather than equilibrium point. We prove that $\alpha$-Rank-collections are invariant to positive affine transformations, and that they are efficient to approximate. An instance of the Boston mechanism is used to illustrate the new solution concept.