MMP.8:4 - Solution

Separate choices from circumstances, express the relations that must hold, and state which information each choice may use. Formulate the requested result over those objects, then test the formulation against a small contrasting pair of circumstances. Preserve any useful conditional answer when a stronger requirement is unavailable.

MMP.8:4.1 - Name choices, circumstances and consequences

Start from the question in the work. Identify what a participant can change and what is supplied by the subject or environment. Let a denote the choice and w the circumstance. Give both their domains and meanings. A choice can be an arrangement, an input, a rule for later actions or another object the work can construct.

Use the subject relations to connect them to a result. For a deterministic model, a consequence may be written y=f(a,w). An implicit relation R(a,w,y) can retain several possible consequences. Additional unknowns can describe forces, flows, internal states or other quantities jointly constrained by the model. Acausal equations can constrain these quantities without making them free controls.

State what the question requires of the result. If it asks whether every permitted behavior meets a condition, a solver’s ability to find one favorable y does not settle it. If the work can select among the permitted consequences, represent the mechanism that gives it that choice. A.3.3.TR supplies the relevant state-change or interaction rule.

MMP.8:4.2 - Recover the order and availability of information

Describe what is observed before each decision and what arrives afterward. Express the available observation as h(w) when it is a deterministic description of the circumstance. It may reveal only part of w. For a random or noisy observation, specify its probability law for each admitted w. An unknown fixed w can index a family P_w of observation laws without having a probability distribution itself. A joint law is needed when the question also treats w as random and averages over it. MMP.7 constructs the recording law.

A policy pi selects an action from the information available: a=pi(h(w)). Two circumstances with the same h(w) must therefore receive the same action. This uses MATH.2’s condition that an answer remains constant on cases identified by a description. Here the observation determines those groups. The information restriction is often called nonanticipativity: the policy does not use distinctions the decision-maker has yet to observe.

For repeated interaction, use the observation history available at each decision. A policy may remember earlier observations or actions. Omitting that memory is a substantive model choice. Global termination is unnecessary when the question concerns a continuing response; specify the response or progress condition that matters under the admitted inputs.

If an earlier action changes what can be observed, make h depend on that action and describe the cost and timing of observation. If action also changes the circumstance distribution or evolution, include that relation. C.28 supplies the causal-use question and the policy’s permitted pre-action information; C.28.MR constructs the mechanism replacement. A changed distribution cannot be inferred from a favorable selection of historical cases alone.

MMP.8:4.3 - State the required quantifiers and performance criterion

For a deterministic success condition G(a,w), these are different questions:

Choose the statement from the work’s requirement. The first two can expose possibilities or limits even when the latter two fail. A stronger claim can require a different action or information source.

When the receiving use permits failures with a stated probability, name the event and the randomness over which that probability is calculated. For unknown fixed w and random observation O, the question may require P_w(failure of pi(O)) <= epsilon for every admitted w. If w itself is modeled as random and the question concerns performance averaged over circumstances, use the joint law of w and O. An expected cost needs the same choice of what is averaged. Neither a probability nor an objective follows just from listing possible circumstances. Feasibility, expected performance, tail risk and worst-case performance are alternative questions with different consequences. Use the common choice and characterization methods to decide which matters to the work.

Preserve dependence within the circumstance set. Independently combining several ranges can create impossible circumstances and overstate a requirement. Conversely, excluding an inconvenient circumstance changes the range of the conclusion and needs a subject reason or an agreed narrower use.

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.

MMP.8:4.5 - Interpret the result and change the working method

Follow the resulting action or policy through one modeled case and a consequential change. Return the consequence to the original requirement. If a planned response depends on a distinction absent from the available observation, revise the observation, defer the decision, choose a more tolerant action or change the stated goal with the responsible party.

Compare these options by what they change and cost. A more informative observation can enlarge the feasible policy set, but it may arrive too late or cost more than a sufficient fixed choice. C.11.DUA supplies that comparison. An infeasible guarantee can still leave a useful bounded, conditional or risk-qualified proposal.

For a working-method change, explain who or what supplies the observation, what result it provides, when it becomes available and what the receiving action does with it. ME supplies the composition and change of those methods. The mathematical policy then describes an obtainable contribution, rather than relying on an unstated observer or decision-maker.