Specifying nodes as sets of actions

The nodes of an extensive-form game are commonly specified as sequences of actions. Rubinstein calls such nodes histories. We find that this sequential notation is superfluous in the sense that nodes can also be specified as sets of actions. The only cost of doing so is to rule out games with absent-minded agents. Our set-theoretic analysis accommodates general finite-horizon games with arbitrarily large action spaces and arbitrarily configured information sets. One application is Streufert (2012), which specifies nodes as sets in order to formulate and prove new results about Kreps-Wilson consistency.