  Game theory relies heavily on the availability of cardinal utility functions,
but in fields such as matching markets, only ordinal preferences are typically
elicited. The literature focuses on mechanisms with simple dominant strategies,
but many real-world applications lack dominant strategies, making the intensity
of preferences between outcomes important for determining strategies. Even
though precise information about cardinal utilities is not available, some data
about the likelihood of utility functions is often accessible. We propose to
use Bayesian games to formalize uncertainty about the decision-makers'
utilities by viewing them as a collection of normal-form games. Instead of
searching for the Bayes-Nash equilibrium, we study how uncertainty in utilities
is reflected in uncertainty of strategic play. To do this, we introduce a novel
solution concept called $\alpha$-Rank-collections, which extends $\alpha$-Rank
to Bayesian games. This allows us to analyze strategic play in, for example,
non-strategyproof matching markets, for which appropriate solution concepts are
currently lacking. $\alpha$-Rank-collections characterize the expected
probability of encountering a certain strategy profile under replicator
dynamics in the long run, rather than predicting a specific equilibrium
strategy profile. We experimentally evaluate $\alpha$-Rank-collections using
instances of the Boston mechanism, finding that our solution concept provides
more nuanced predictions compared to Bayes-Nash equilibria. Additionally, we
prove that $\alpha$-Rank-collections are invariant to positive affine
transformations, a standard property for a solution concept, and are efficient
to approximate.
