Definition 3.9 A strategy for player i is a function si mapping each vertex x ∈ Vi to an element in Action(x)(equivalently, to an element in Child(x)). Vi is the set of all vertices in game tree where player i makes a move.
According to this definition, a strategy includes instructions on how to behave at each vertex in the game tree, including vertices that previous actions by the player preclude from being reached. For example, in the game of chess, even if White’s strategy calls for opening by moving a pawn from c2 to c3, the strategy must include instructions on how White should play in his second move if in his first move he instead moved a pawn from c2 to c4, and Black then took his action. The main reason this definition is used is its simplicity: it does not require us to provide details regarding which vertices need to be dealt with in the strategy and which can be ignored.
Definition 3.10 A strategy vector is a list of strategies s = (si )i ∈N , one for each player.
Player i’s set of strategies is denoted by Si, and the set of all strategy vectors is denoted S=S1×S2×...Sn. Every strategy vector s=(si)i ∈N determines a unique play(path from the root to a leaf). The play that is determined by a strategy vector s = (si)i ∈N is (x0,x1,x2,...,xk), where x1 is the choice of player at vertex(x0), based on his strategy, x2 is the choice of player at vertex (x1) based on his strategy, and so on, and xk is a leaf. The play corresponds to the terminal point xk (with outcome u(xk)), which we also denote by u(s). (u denotes outcome of game - at leaf.)
Now, A strategy sI of Player I is called a winning strategy if u(sI, sII) = I wins, ∀sII ∈ SII.
I'm confused as to why ∀sII ∈ SII has been written. By definition, sII is a function which maps all possible vertices to their next move, so why is there a set SII? Under 3.10, what is meant by player i's set of strategies?