  Recent years have seen a surge of artificial currency-based mechanisms in
contexts where monetary instruments are deemed unfair or inappropriate, e.g.,
in allocating food donations to food banks, course seats to students, and, more
recently, even for traffic congestion management. Yet the applicability of
these mechanisms remains limited in repeated auction settings, as it is
challenging for users to learn how to bid an artificial currency that has no
value outside the auctions. Indeed, users must jointly learn the value of the
currency in addition to how to spend it optimally. In this work, we study the
problem of learning to bid in two prominent classes of artificial currency
auctions: those in which currency, which users spend to obtain public
resources, is only issued at the beginning of a finite period; and those where,
in addition to the initial currency endowment, currency payments are
redistributed to users at each time step. In the latter class, the currency has
been referred to as karma, since users do not only spend karma to obtain public
resources but also gain karma for yielding them. In both classes, we propose a
simple learning strategy, called adaptive karma pacing, and show that this
strategy a) is asymptotically optimal for a single user bidding against
competing bids drawn from a stationary distribution; b) leads to convergent
learning dynamics when all users adopt it; and c) constitutes an approximate
Nash equilibrium as the number of users grows. Our results require a novel
analysis in comparison to adaptive pacing strategies in monetary auctions,
since we depart from the classical assumption that the currency has known value
outside the auctions, and moreover consider that the currency is both spent and
gained in the class of auctions with redistribution.
