There are some deals where it is important for declarer to exploit uncertainty on the part of the defenders, but these are definitely the exception as opposed to the rule.

This suggests that Proposition 5.0.5 is doing a reasonable job of modeling the defenders' cardplay, but the combination function for the maximizer needs to be modified to reflect the imperfect-information nature of his task.

To understand this, let us return to our putative expert, who suggested at the beginning of this section that he might be playing for West to hold the spade queen. What he might say in a bit more detail is, "I could play for each opponent to hold exactly three hearts, or I could play for West to hold the spade queen. The latter was the better chance."
This suggests that the value assigned to the position by the maximizer is not a single set of situations (those in which he can make the contract), but a set S of sets of situations.

Each set S E S corresponds to one set of situations that the maximizer could play for, given his incomplete knowledge of the positions of the opposing cards.

Extending the notation used earlier in this section, we will denote the set of sets of situations by 2 25 . The maximizer's combination function on 2 25 is given by

## Max( F, G) = F U G

(8)
where each of F and G are sets of sets of situations. This says that if the maximizer is on play in a situation p, and he has one move that will allow him to select from a set F of things to "play for" and another move that will allow him to select from a set G, then his choice at p is to select from any element of F ∪ G .

The minimizer's function is a bit more subtle. Suppose that at a node p, the minimizer can move to a successor with value F = {Fi}, or to a successor with value G = {Gi}. What value should we assign to p?

Since the minimizer has perfect information, he will always guarantee that the maximizer achieves the minimum value for the actual situation. Whatever element of F i ∈ F or G i ∈ G
is eventually selected by the maximizer, the eventual value of p will be the minimum of F i and G i . In other words min({Fi}, {Gj}) = {min(Fi, Gj)}
(9)
where the individual minima are computed using the perfect information rule (7).

Definition 5.0.6  Let G be the set of positions in an imperfect information game, a set of pairs (p, Z) where p is a position from the point of view of the maximizing player and Z is the set of perfect information positions consistent with p. The imperfect information game for G is the game (G, V, pi, s, ev, f+, f-) where:
1. The value set V is 22 2. The initial position pi is (po, S), where po is the initial imperfect information position and S is the set of all perfect information positions consistent with it.

3. The successor function is described as follows:
(a) If the maximizer is on play in the given position, the successors are obtained by enumerating the maximizer's legal plays and leaving the elements of the set Z of sttuations unchanged.