Rational market making with probabilistic knowledge

A market maker sets prices over time for wagers that pay out contingent on the future state of the world. The market maker has knowledge of the probability of realizing each state of the world, and of how the price of a bet affects the probability that traders will accept it. We compare the optimal policy for risk-neutral (expected utility maximizing) and Kelly criterion (expected log-utility maximizing) market makers. Computing the optimal policy for a risk-neutral market maker is relatively simple, while computing the optimal policy for a Kelly criterion market maker is challenging, requiring advanced techniques adapted from the computational economics literature to run efficiently. We show that while a riskneutral market maker has an optimal policy that does not depend on the market maker’s state, a Kelly criterion market maker’s optimal policy has an intricate dependence on both time and state. Counterintuitively, a Kelly criterion market maker may offer bets that are myopically irrational with respect to the market maker’s beliefs for the entire trading period. In contrast, a risk-neutral market maker never offers a myopically irrational bet.