Withholding a decision at a cost

Add a third action, construct the resulting policy regions and distinguish withholding from information acquisition.

15 minutes · advanced · Original Poker4Quant lesson

What you will learn

Compare three conditional expected losses, derive withholding boundaries and preserve all tied minimizing actions.

Prerequisites: Posterior beliefs and error costs

The action set can matter as much as the probability

A binary report is not the only possible decision model. Here a third action withholds the report and incurs a fixed immediate loss cW in either true state. It supplies no new evidence, no promised later answer and no additional consequence in this exercise. It is a terminal action with a stipulated loss, not a perfect-information purchase.

Positive and negative report losses remain(1−p)cFP and p cFN. Withholding has expected loss p cW+(1−p)cW=cW. Compare all three numbers. A third action cannot increase the attainable minimum expected loss because the old actions remain available, although it need not improve it for every p.

Construct the middle region rather than assuming it

With positive error costs, withholding weakly beats the negative report when p≥cW/cFN. It weakly beats the positive report when p≤1−cW/cFP. It is a minimizing action when BOTH inequalities hold. A nonempty region requires that the lower bound not exceed the upper bound. Strict interior points favor withholding over both reports; boundary equalities require ties.

For unit error costs and cW=1/4, the withholding region runs from p=1/4to3/4. At1/4, negative and withhold tie; at3/4, positive and withhold tie. Outside the region, the corresponding cheaper report wins. The direct positive/negative indifference point1/2is inside this region, where both reports lose to withholding.

Abstention is an objective-dependent action

Changing error costs can move the withholding region even when the supplied posteriors stay the same. With cFP=3,cFN=1,cW=3/8, its bounds become3/8and7/8. At p=3/8, withholding and reporting negative each cost3/8while reporting positive costs15/8.

If withholding were made too expensive, its region could disappear. If it triggered future investigation, delay costs or a later report, those consequences would need a new sequential model. Do not borrow the current flat-cost formula for an unstated follow-up process. The workshop explicitly inspects both possible true states under every action without pretending those counterfactuals happen together.

Worked example

With unit error costs and withholding cost1/4, compare all three actions at p=3/8. Then find what happens at p=3/4.

At p=3/8, positive-report loss is5/8, negative-report loss3/8and withholding loss1/4.
Withholding is uniquely best because1/4is less than both report losses.
At p=3/4, positive-report loss is1/4and negative-report loss3/4. Withholding still costs1/4.
At this boundary, positive and withhold are both minimizers. Selecting only one would omit a tied action.

The best action need not report the more likely state. An available terminal action can change the optimum without providing any new evidence.

Hands-on: Add a terminal withholding action

Construct three-action policy regions, retain tied minimizers and distinguish an immediate withholding cost from buying information.

Reading loss-policy workshop…

Common mistakes

Treating withholding as free truth revelation or a guaranteed correct later report.
Comparing withholding with only one report rather than both.
Using the direct report threshold as the entire three-action policy.
Forgetting boundary ties or assuming a withholding region always exists.

Keep these ideas

Compare all available actions under the same posterior and objective.
A fixed withholding cost is not the same mechanism as buying information.
The withholding region satisfies cW/cFN≤p≤1−cW/cFP when nonempty.
Tie boundaries and omitted future consequences belong in the model description.