Appendix: illustrative information studies

The worked studies isolate distinct mechanisms that can change an information purchase. Their inputs are authored illustrations and their outputs come from the same canonical Decision lab engine as the public app. They are not sampled projects, observed rates or estimates fitted to client records. The studies retain complete fixture, output and binding files in the replication desk. The tables below publish the decision-relevant input and result account so that a reader can understand the comparison before inspecting the full structured object.

The original action matrix is shared deliberately across several studies. This lets a change in the prior, signal likelihoods or acquisition price be examined while holding the response consequences fixed. The restricted-action study removes Commit; the reduced-upside study supplies a different matrix. Those are distinct decision-perimeter changes. The appendix shows the shared matrix once, identifies its applicable cases and preserves the alternative matrices separately. This avoids publishing repeated identical tables while keeping every fixture fully reproducible.

The useful-signal study provides the baseline mechanism. Stage is the best current action. A positive result selects Commit and a negative result selects Hold. The instrument therefore changes the conditional policy and creates a gross expected monetary gain. Its supplied acquisition cost leaves a positive net result. This finding is conditional on the prior, likelihoods, payoff perimeter and feasibility assumptions. It is not an empirical statement that a particular commercial study has that value or that a committee should make an actual commitment.

The no-policy-change study raises the prior while retaining the original matrix and instrument. Commit is already best and remains best in both branches. The signal updates belief but does not improve the implemented fixed-menu policy. Its gross gain is exactly 0.00 USDk in the current rational evaluation; the complete file preserves that equality and the numeric display. The supplied positive price therefore produces a negative net result. This case falsifies an unqualified claim that an informative or accurate signal always justifies a new research purchase. Its scope remains the supplied menu, state model and implementation conditions.

The expensive-signal study changes the price rather than the information mechanism. Its conditional actions and gross gain match the useful baseline because the prior, likelihoods and matrix are unchanged. The larger supplied price makes the net result negative. This case distinguishes the methodological usefulness of an instrument from the economic case for buying it. A committee could negotiate, redesign the scope or decline the commission, but the fixture does not model those procurement responses or infer the supplier's actual cost.

The false-positive-burden study combines a low prior with a weak negative-state filter. The positive signal raises the success probability but not enough to make a paid exposure preferable under the original matrix. Both branches select Hold. This route to redundancy differs from the already-favourable no-change case. It illustrates why a signal's likelihoods and the prior must be read together against the action boundary. A favourable direction of update does not by itself establish a sufficient condition for commitment.

The inverse-signal study deliberately contradicts the usual intuitive meaning of positive. Under its supplied likelihoods, a positive result lowers the success probability and selects Hold, while a negative result selects Commit. The signal still has positive net information value in the fixture because the conditional policy uses its actual association with the state. The case tests whether the reader follows likelihoods rather than labels. It does not validate a real negatively associated instrument or recommend using one without a relevant performance record.

The perfect-price-boundary study makes the signal perfectly informative within the binary illustration and supplies a price equal to its gross gain. Its net result is at the model boundary. The positive branch selects the success-state best action and the negative branch selects the failure-state best action. The case demonstrates the ceiling relationship exactly within the supplied arithmetic. It should not be read as evidence that perfect commercial observation is available at that price, or that a purchase with no margin for unresolved cost is prudent.

The restricted-action study removes Commit from the menu and retains the original likelihoods, prior and remaining payoff cells. The same instrument can now support Stage after a positive result and Hold after a negative result. The reduction in response opportunity lowers gross information value enough that the supplied price exceeds it. This case falsifies a purchase rationale that assumes an unavailable action. It does not calculate how to obtain authority, the chance of receiving approval or the value of an actual governance change.

The reduced-upside study keeps the instrument and prior but supplies smaller success payoffs for the active actions. The new matrix produces a different current value and information gain. It illustrates how an opportunity revision can require a new commission appraisal. No dated delay, discount rate or causal relationship is inferred. A real case linking study time to changed payoffs would need its own schedule and consequence evidence. The original and revised matrices are preserved so the reader can see exactly which assumption changed.

The branch account publishes mass, posterior and selected action for every study. Conditional expected payoff and weighted contribution are available in the full outputs; selected tables in the main argument show that weighting step. The appendix retains the complete branch interpretation without copying every display-oriented table from the app. The replication desk remains the full reproduction object. A reader can verify any scalar against its binding and rerun the named fixture through the paired canonical engine.

The policy sweep depends on the action matrix rather than the fixture's initial prior or signal performance. The original-matrix studies therefore share the same sweep. The restricted and reduced-upside matrices have different policy accounts. The graphical comparison publishes those distinct sweeps, with points selected from their actual canonical outputs. It is a picture of best current expected payoff across sampled priors. It does not plot EVSI, assign plausibility to the priors or imply that an empirical probability distribution has been estimated.

Each study should be challenged through its decisive assumption. For the useful baseline, ask whether the branch responses are feasible and the likelihoods apply. For no change, ask whether another consequential action has been omitted. For high price, ask whether the cost boundary is complete. For false positives, ask whether the prior and filter are supported. For the inverse case, ask whether the signal meaning is understood. For the boundary, ask about unresolved cost. For restriction and reduced upside, ask whether the decision perimeter has actually changed in the manner supplied.

The study slate can therefore select the next enquiry. A no-change result asks whether an active action boundary is reachable; an expensive result asks whether the purchase burden can be reduced or its purpose changed; a restricted-menu result asks whether the response belongs in the feasible set. An inverse signal asks whether names match likelihood meaning, while reduced upside asks whether the consequence account still represents the opportunity. These routes cannot be cured by the same additional report. The cases remain fictional controlled mechanisms, and their value is to locate the missing evidence or intervention without borrowing their numerical assumptions as an empirical baseline.

Economic break-even should describe an equality under the stated assumptions. The three price-precision fixtures retain the default payoff matrix, prior and likelihoods and change only acquisition price. The exact-price case sets cost 94.99 USDk and yields net information value 0. The nearby prices are 94.989999999 and 94.990000001 USDk. Their computed net values are 1e-09 and -1e-09. The first is positive and the second negative; neither is an exact break-even.

Joint signal-state masses and action payoffs are now evaluated as rational quantities derived from canonical decimal inputs. Branch actions are ranked before any division or numeric display, so posterior rounding cannot change a preferred response. The exported evaluation records exact EVSI, EVPI and net sign. This guarantee concerns arithmetic alone. A surplus of a billionth of USDk is not credible economic precision when probabilities, payoffs, permission or delay remain uncertain. A decision owner should challenge those assumptions and agree a substantive decision margin rather than let a software tolerance silently redefine equality.