WASHINGTON—Wall Street’s top regulator will study potential new rules related to two recent episodes of stock-market turbulence — the GameStop frenzy among small investors and the implosion of Archegos Capital Management.

In testimony prepared for the House Financial Services Committee, Securities and Exchange Commission Chairman Gary Gensler says applications that “gamify” trading—by using appealing visual graphics to reward a user’s decision to trade—might encourage frequent trading that results in worse outcomes for investors. Some Democratic lawmakers have blamed gamification for the boom in retail trading that helped drive the rise in GameStop shares.