Compare minimax regret cost with a worst bill

Minimax regret cost can select a different action from the one that keeps the largest possible bill smallest. The difference becomes visible in a table with only four invented costs. Choosing the smallest worst bill gives action A; choosing the smallest worst missed opportunity gives action B. Both calculations are correct under their stated objectives. Deciding which objective describes your concern comes before using either result.

The example below is a self-contained exercise. Every number is stipulated in abstract cost units, with no connection to a product price or a forecast. There are two available actions and two possible states. You must select one action before learning the state. That timing stays fixed throughout the calculation, including when you calculate what would have looked preferable afterward.

Keep the choice and the unknown state separate

Call the actions A and B, and call the states S and T. If you select A, your cost is 10 in S and 10 in T. If you select B, your cost is 0 in S and 11 in T. Lower cost is preferable. Those four entries are the complete cost account for this exercise; there are no later charges, refunds or additional consequences to add.

Write the table with actions as rows and states as columns. The A row reads 10, 10. The B row reads 0, 11. Keeping the columns aligned matters because each column describes the alternatives under one identical state. Comparing B in S with A in T would compare different circumstances, so it cannot establish which action would have been preferable after one particular state occurred.

You receive no signal before acting and have no opportunity to revise the choice. The permitted decisions are exactly A and B. A rule that says to select B when S occurs and A when T occurs would require information unavailable at decision time. Random mixtures are outside this exercise as well. Adding them would enlarge the allowed decision set and require a separate calculation.

Find the action with the smaller worst bill

For A, the larger of its two costs is 10. For B, the larger is 11. A person minimizing the largest absolute cost therefore chooses A, since 10 is less than 11. This is a direct comparison of the maximum entry in each row. The calculation makes no claim about how often either state will occur.

A also supplies a simple absolute bound in this table: whichever state occurs, its cost is 10. B can cost less, but it can also cost 11. The worst-bill objective treats that higher possible total as decisive. It does not give B credit for its zero entry once each row has been reduced to its largest entry. That follows from the chosen objective, rather than from an arithmetic omission.

A is not cheaper in every state. In S, B costs 0 while A costs 10; in T, A costs 10 while B costs 11. Neither row is everywhere as cheap as the other. This crossing leaves room for different comparison rules to disagree. If one action had been at least as cheap in both columns, the present reason for choosing between competing advantages would disappear.

Calculate a benchmark inside each state

For the regret calculation, first find the lowest offered cost separately in each column. In S, the minimum of 10 and 0 is 0, supplied by B. In T, the minimum of 10 and 11 is 10, supplied by A. Record the benchmark row as 0, 10. Each benchmark belongs to its own state and must stay in that column during subtraction.

These benchmarks describe what the better offered action would cost if you knew the state. They do not provide a new action available beforehand. In particular, the benchmark row is not a third selectable plan costing 0 in S and 10 in T. It combines the better outcome from different actions for the limited purpose of measuring the gap left by your fixed choice.

Define regret for a cell as that cell's cost minus the benchmark in the same column. The word names this numerical shortfall only. It does not assert how disappointed a person will feel. Because each benchmark is the minimum of the offered entries in its column, no cell can have negative regret, and at least one entry in each column has regret zero.

Subtract before taking the worst regret

For A in S, subtract 0 from 10 to get regret 10. For A in T, subtract 10 from 10 to get regret 0. Its regret row is therefore 10, 0. For B in S, subtract 0 from 0 to get 0. For B in T, subtract 10 from 11 to get 1. Its regret row is 0, 1.

Now take the largest regret in each row. A has worst regret 10 and B has worst regret 1. Minimizing that maximum chooses B. Selecting B limits the gap from the better offered action in the realized state to at most one cost unit. It also allows an absolute cost of 11, which remains part of the original table.

The two results can be checked without any terminology. The first rule compares 10 with 11 and keeps A. The second compares 10 with 1 and keeps B. In state S, choosing A misses a ten-unit opportunity to do better. In T, choosing B misses a one-unit opportunity. The regret rule gives those state-specific missed opportunities priority over the larger absolute bill.

Explain why the ranking changes

Within a column, subtracting the same benchmark from both entries preserves their order and their difference. In S, B stays ten units cheaper than A. In T, A stays one unit cheaper than B. The regret transformation has not made a more expensive action cheaper than its competitor within either state. It has changed the baseline used when comparing outcomes across states.

Those column baselines differ: one is 0 and the other is 10. Taking a row maximum after different subtractions can therefore select a different row from taking the maximum before them. For B, the cost 11 in T becomes regret 1 after its column benchmark is removed. A's cost 10 in S remains regret 10 because that column benchmark is zero. That asymmetry produces the reversal.

Subtracting one common constant from every cell would behave differently. If every cost fell by the same amount, every row maximum would fall by that amount, leaving the ordering of row maxima unchanged. The present transformation uses a separate minimum for each state. Checking which quantities share a baseline prevents a valid within-column observation from being stretched into an invalid statement about maxima across columns.

Check what the benchmark leaves out

Suppose a further stipulated overhead of 100 units applies to both actions whenever T occurs, while S is unchanged. The new cost rows are A=(10,110) and B=(0,111). The state benchmarks become 0 and 110. Subtracting them gives the same regret rows as before. Shared state-specific overhead disappears from regret even though the actual total payable has increased substantially in T.

That example shows why a small regret bound cannot serve as a spending limit. Regret measures the gap from the offered same-state alternative, and a shared unavoidable charge adds no such gap. Someone who must keep the total below a specified ceiling needs to inspect absolute costs as well. The original table makes this concrete: B's worst regret is one, while its largest absolute cost is eleven.

The benchmark also depends on which actions are included. Its definition here considers exactly A and B. Adding another feasible action could reduce a column minimum and change the regret table. A comparison that silently uses a different action set in different cells would no longer follow the stated definition. Keep the offered alternatives fixed while constructing and checking all four entries.

Keep other cost questions in their own models

The separate average-demand pack budget example compares the mean of stepped bills with pricing a mean demand. Its declared equal-weight scenarios support an expectation calculation. This table supplies no probabilities and uses row maxima.

The unknown-duration first-buy threshold example studies a sequential rent-or-buy rule through a worst cost ratio. Here there is one fixed advance action, two states and an additive same-state gap. No switching threshold enters this calculation.

Assigning probabilities would answer another question. An expected-cost comparison would multiply each cost by its state probability and add within a row. Neither conservative rule used here performs that operation. Equal-looking column widths on a page do not supply equal probabilities, and the absence of probability estimates does not justify inserting one-half merely to obtain an average.

Keep listing observations outside the invented table

The Monogram Small Foot key ring listing describes a case of random character key rings. Its captured gallery shows sealed character packages. These are listing observations; neither a package count nor a character assortment supplies a cost entry or state probability for this exercise.

The Kelly Waters GP3793 key ring listing names a different key ring. Both gallery views show a ring attached to a rectangular tag; one adds a ruler. This article draws no price, material, size, warranty or performance conclusion from those views.

The actions A and B remain abstract labels. They do not stand for those two listings, and S and T do not describe possible orders or product quality. Making such a connection would require evidence about complete comparable costs and feasible decisions. The arithmetic works because its four inputs are expressly given, not because a catalog image establishes what a future buyer will pay.

Audit the small table and choose the objective

A finite arithmetic check can test the subtraction procedure on every two-action, two-state table whose entries are integers from zero through four. There are five choices for each of four entries, giving 625 tables. For each table, take the two column minima, subtract each from its column and check nonnegative results with at least one zero in each column. The saved checks satisfy those conditions.

Those 625 controls check the procedure on that finite domain. The general reason for nonnegative regret is already contained in the definition of a minimum: every offered cost is at least its column minimum. A cell attaining the minimum subtracts itself and gives zero. Neither the finite controls nor this argument implies that worst absolute cost and worst regret always agree; the 10,10 versus 0,11 example explicitly shows disagreement.

Before choosing, state the quantity you want to constrain. If the concern is the largest bill in this invented table, A gives the smaller bound of 10. If the concern is the largest extra cost relative to the better offered action in the realized state, B gives the smaller bound of 1. Keep the original costs beside the regret calculations so the selected objective and its limits remain visible after the state is known.

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