Translating Extensive Form Games to Open Games with Agency

We show open games cover extensive form games with both perfect and imperfect information. Doing so forces us to address two current weaknesses in open games: the lack of a notion of player and their agency within open games, and the lack of choice operators. Using the former we construct the latter, and these choice operators subsume previous proposed operators for open games, thereby making progress towards a core, canonical and ergonomic calculus of game operators. Collectively these innovations increase the level of compositionality of open games, and demonstrate their expressiveness. The game on the left has three players, each of them making one decision. The strategy profile ( L , L , L ) is a Nash equilibrium of this game, which yields the utility ( 1 , 3 , 1 ) . The game on the right has only two players, with p 1 making two decisions. In this second game, (( L , L ) , L ) is not a Nash equilibrium because p 1 can change strategy to (( R , R ) , L ) and get a better reward. In the first game, even though p 1 and p 3 always get the same reward, they are different players and so cannot similarly coordinate changes to their strategies.