We study liquidity provision by competitive high-frequency trading firms (HFTs) in a dynamic trading model with private information. Liquidity providers face adverse selection risk from trading with privately informed investors and from trading with other HFTs that engage in latency arbitrage upon public information. The impact of the two different sources of risk depends on the details of
the market design. We determine equilibrium transaction costs in continuous limit order book (CLOB) markets and under frequent batch auctions (FBA). In the absence of informed trading, FBA dominates CLOB just as in Budish et al. (2015). Surprisingly, this result does no longer hold with privately informed investors. We show that FBA allows liquidity providers to charge markups and
earn profits – even under risk neutrality and perfect competition. A slight variation of the FBA design removes the inefficiency by allowing traders to submit orders conditional on auction excess demand.