ordering been ⟨ t, s⟩ instead, an alternative maximal aievable set of T would have been returned instead.

In any event, we have:
Proposition 7.1.6  Given a game G and a sequence S of situations, let A  be the achievable set induced by S for G. Then no proper superset of A in S is achievable. Proof. This is straightforward. For any element s ∈ S − A, we know that U ∪ { s } is not achievable for some U C A. Thus A U{s} is not achievable as well.

Algorithm 7.1.4 allows us to construct maximal achievable sets relative to our Monte Carlo sample; recall that we are taking our sequence S of situations to be any ordering of the sample itself. In practice, however, it is important not to focus too sharply on the sample itself, lest the eventual achievable set constructed overfit irrelevant probabilistic characteristics of that sample. This can be accomplished by replacing the simple union in step 2 of the algorithm with some more complicated operation that captures the idea of
"situations that are either like si or like those already in A."  In bridge, for example, A might be all situations where West has two or three hearts, and si might be some new situation where West has four hearts. The generalized union would be situations where West has two, three or four hearts. If this more general set is not achievable, another attempt could be made with the simple union. If we denote the "general union" by ⊕, Algorithm 7.1.4 becomes:
Algorithm 7.1.7 To construct an achievable set A from a sequence (so,..,sn) of situatio ns:
1. Set A = 0.

2. For i = 0, . . , n:
(a) If A ⊕ { s; } is achievable, set A = A ⊕ { s; }.

(b) Otherwise, if A U {s;} is achievable, set A = A U {s;}.

This algorithm can be used in practice to find achievable sets that are either maximal or effectively so over the set of all possible instances, not just those appearing in the Monte Carlo sample.

## 7.2 Maximizing The Payoff

It remains to find not just maximal achievable sets, but ones that approximate the solution to the game in question given a particular payoff function.

To understand how we do this, let me draw an analogy between the problem we are trying to solve and resource-constrained project seduling ( RCPS ). In RCPS, one has a list of tasks to be performed, together with ordering constraints saying that certain tasks need to be performed before others. In addition, each task uses a certain quantity of various resources; there are limitations on the availability of any particular resource at any particular time. As an example, building an aircraft wing may involve fabricating the top and bottom flight surfaces, building the aileron, and attaching the two. It should be clear that the aileron