MMP.8:4.4 - Construct or refute a usable choice

For each circumstance w, let A(w) be the actions satisfying the requirement under the modeled relations. For a fixed robust choice, seek an action in the intersection of A(w) over the admitted circumstances. An empty intersection refutes that fixed-choice requirement.

For an observation-dependent choice, group circumstances by the observation they produce. For each obtainable observation o, intersect A(w) over the circumstances with h(w)=o. An action in this intersection works throughout that observationally indistinguishable group. In a finite problem, choosing one such action for each observation constructs a policy. One empty intersection proves that this observation cannot support the required policy.

For a noisy observation and an all-cases guarantee, construct the allowed pairs (w,o) from the observation mechanism’s admitted realizations. This relation, rather than a positive probability for each individual report, determines compatibility. For example, if O=w+E with E uniform on [-1,1] and all errors in that closed interval are admitted, the boundary error E=1 remains in an all-cases guarantee despite having probability zero. An almost-sure or specified-probability requirement is a different claim; state that choice.

For a received report o, first check that at least one circumstance is compatible with it. If none is compatible, return the conflict between the report and the observation model under C.16.IR:4.4. A universal statement over an empty set supplies no guarantee for the situation that produced the report. With a nonempty compatible set, intersect A(w) over its members. For a probability-of-success requirement, instead calculate the event under the conditional family or joint law chosen in :4.3; the set of allowed pairs alone supplies no probability weights.

For infinite spaces or long-running interaction, these relations still specify the question, but a usable policy requires the corresponding mathematical and computational construction. A pointwise existence argument does not automatically provide an effective rule. C.29.2 addresses that obtaining work; the present method retains the information restrictions in what it asks the computation to produce.

When several outcomes remain possible for one action and circumstance, apply the required quantifier to those outcomes as well. Derive or inspect a violating outcome when refuting a guarantee. If the missing factor is an unmodeled selection mechanism, obtain it or retain the conditional answer rather than allowing the solver to invent a favorable mechanism.