  We study long-run market stability for repeated price competitions between
two firms, where consumer demand depends on firms' posted prices and consumers'
price expectations called reference prices. Consumers' reference prices vary
over time according to a memory-based dynamic, which is a weighted average of
all historical prices. We focus on the setting where firms are not aware of
demand functions and how reference prices are formed but have access to an
oracle that provides a measure of consumers' responsiveness to the current
posted prices. We show that if the firms run no-regret algorithms, in
particular, online mirror descent(OMD), with decreasing step sizes, the market
stabilizes in the sense that firms' prices and reference prices converge to a
stable Nash Equilibrium (SNE). Interestingly, we also show that there exist
constant step sizesunder which the market stabilizes. We further characterize
the rate of convergence to the SNE for both decreasing and constant OMD step
sizes.
