Games in which one player acts first, followed by both players acting Games in which one player acts first, followed by another player acting are referred to as Stackelberg games.
Does anyone know of a class of games where one player acts first, followed by both players acting simultaneously?

Stackelberg games are only a very small subset of sequential games with perfect information, where players (not necessarily just two) move one after another. 
The type of games you asked for -- one player acts first, followed by both players acting simultaneously -- falls under the set of sequential games with imperfect information. In the context of industrial organization, there are variants of "market entry games" such as the Spence-Dixit entry-deterrence game, that fit your description.
