Modern technology has created many new opportunities for buying and selling goods and services - whether it is trading stock shares or airplane tickets. However, as trading intensified, it became apparent that the design of the trading mechanism (e.g., stock exchanges), itself, plays a very important role. In particular, certain features of the trading mechanism, such as, e.g., the trading frequency, may affect the stability and efficiency of a market. A notorious example of the latter is the existence of so-called “flash crashes” and “flash rallies” – fast and significant changes in price, which cannot be explained by any fundamental economic reasons. In this talk, I will present several results unified by the common purpose of developing a rigorous modeling framework for studying the interaction between market participants on a micro level. The game-theoretic nature of the proposed models makes them particularly useful for analyzing the potential effects of introducing new trading mechanisms, for which no statistical data exists. In particular, I will use such models to describe the behavior of a market with limited trading frequency and to explain how exactly the occurrence of flash crashes is related to high trading frequency. Another application I will consider is related to “tick size” – the smallest price increment allowed in the market – and the potential consequences of changing it. On the mathematical side, this work contributes to the study of asymptotic properties of general Itô processes, fixed point problems with discontinuities, and control-stopping games.