Portfolio mandates and minimax regret

What you will learn

Derive menu-relative minimax regret, reconstruct statewise comparators after mandate changes, and certify sharp opportunity bounds and all tied minimizing policies.

Prerequisites: Stress envelopes and reverse loss thresholds · Decision robustness · Target returns and the feasible frontier

Separate the mandate from the decision criterion

A portfolio committee can agree on every scenario payoff and still disagree about a decision. One member protects the worst absolute result; another wants to limit the opportunity missed relative to what could have been achieved in the same state. These are different objectives. This lesson makes the second objective precise using minimax regret. The word regret means a numerical opportunity gap, not an emotion, a probability or a realized cash payment. We compare complete policies chosen before the state is revealed. A hindsight comparator is a measuring device and is not an executable promise to know tomorrow's state.

Assume one period, 100 initial fictional chips, no flows, financing, fees or interim trades. Exactly one of two stipulated states occurs: expansion or contraction. No probabilities are supplied. The available indivisible policies are Reserve, Balanced and Growth. Their net chip gains, in that state order, are (2,2), (8,−2) and (14,−8). These are complete portfolio policies, not securities whose weights can be mixed. Randomizing between policies and splitting capital are excluded. A mandate is a hard floor on each policy's stipulated worst gain. It is tested before selecting a policy and applies to the hindsight comparator too. It does not guarantee a floor outside these two modeled states.

Build a statewise opportunity ledger

For an admissible menu A and state s, define M(s)=max over a in A of g(a,s). The regret of policy a in that state is r(a,s)=M(s)−g(a,s). The maximum regret R(a) is the larger of its two state regrets. Choose every admissible policy attaining the smallest R. The order of operations matters: first compare policies within each state to build the column maxima, then subtract each policy's gain, then take a maximum across states, and only then minimize across policies. Subtracting a policy's own best outcome compares different worlds and does not measure a missed opportunity in one world.

With floor −8, all three policies qualify. Expansion has comparator 14 and contraction has comparator 2. Reserve therefore has regrets (12,0), Balanced (6,4), and Growth (0,10). Their maximum regrets are 12, 6 and 10. Balanced uniquely minimizes the maximum at six chips. The calculation certifies a bound: Balanced never trails the admissible state winner by more than six chips within this menu and state set. Equality is attained in expansion, where Growth gains fourteen and Balanced gains eight. This witness makes the six-chip bound sharp; it does not say Balanced actually loses six chips.

Reconcile regret with absolute protection

Independently check the ledger by adding gain and regret within every cell. Balanced gives 8+6=14 in expansion and −2+4=2 in contraction. Growth gives 14+0=14 and −8+10=2. Each sum recovers its column maximum, so a sign or column mismatch would be exposed. Terminal wealth provides another check: Balanced ends at 108 or 98, while the statewise winners end at 114 or 102. Differences in wealth reproduce the same six and four chip regrets because the common initial wealth cancels.

Maximin instead compares each policy's minimum gain: Reserve two, Balanced minus two, Growth minus eight. It selects Reserve, even though Reserve can miss twelve chips in expansion. Neither rule dominates the other as a universal preference. Maximin answers which policy has the best stipulated absolute floor. Minimax regret answers which has the smallest worst missed opportunity. Neither computes expected utility without probabilities and utility assumptions. In particular, counting two scenarios does not make them equally likely. A committee receipt should name the adopted rule and explain why an absolute loss floor is imposed separately, rather than hiding the value judgment inside a score.

Recompute the comparator after a mandate change

Now tighten the floor to −3. Growth fails because its contraction gain is −8. Reserve and Balanced remain. Rebuild the column maxima using only this admissible menu: they are now (8,2). Reserve's regrets become (6,0) and Balanced's become (0,4), so Balanced again wins, but its certified maximum regret falls from six to four. No Balanced payoff improved. The opportunity set changed. Keeping fourteen as the expansion comparator would charge the committee for failing to choose a policy that its own mandate forbids. Such a comparison can be reported separately as an unrestricted benchmark, but it cannot be labeled regret against the admissible menu.

This menu dependence is a substantive modeling choice. Removing a policy can reduce the regret of every surviving policy without improving their wealth. A statement that a review reduced maximum regret therefore needs both the old and new admissible sets. If the floor rises to zero, only Reserve qualifies and its regret against itself is zero in both states. That zero certifies optimality within a singleton menu; it says nothing about competitiveness against excluded opportunities. In a general review the menu might even become empty, in which case no feasible recommendation exists. Do not manufacture a best policy by minimizing an empty list.

Stress the opportunity and retain all ties

Hold the loose floor at −8 and vary only Growth's expansion gain u. For 10≤u≤18, the comparator remains (u,2). Reserve has maximum regret u−2, Balanced has max(u−8,4), and Growth has ten. At u=10 these are eight, four and ten; Balanced wins at four. At u=14 they are twelve, six and ten. At u=18 they are sixteen, ten and ten: Balanced and Growth tie. Returning only the first smallest array entry would silently impose an unannounced tie breaker. Report both policies, with their different absolute floors, and ask for a declared additional preference if one action must be chosen.

Use the workshop to construct the two column maxima and three maximum regrets, writing “excluded” for a forbidden policy. Certify the baseline, save it, tighten the mandate and repair the ledger. Then restore the loose floor and set u=18 to obtain a tied optimum. Saved certificates let you compare the changing menu, opportunity bound and selected set during this visit. A common amount added to every policy within one state cancels from regret because both comparator and policy gain shift equally, provided admissibility is held fixed. A floor tied to absolute gains could change admissibility under that shift, so the qualification is essential. Missing scenarios, nonlinear costs and implementability restrictions still require independent review. The final conclusion is conditional on the stated menu, not a forecast of investment success.

Hands-on: Build a mandate-relative regret ledger

Change eligibility and opportunity, construct column benchmarks and row regret maxima, then retain contrasting certificates and tied decisions.

Common mistakes

• Using a forbidden policy as an admissible comparator.
• Subtracting gains across different states.
• Calling opportunity loss an actual cash loss or probability.
• Assuming zero regret in a singleton menu proves universal superiority.
• Dropping tied minimizers or silently allowing capital mixtures.

Keep these ideas

• Filter policies by the mandate before forming column maxima.
• Regret is the same-state comparator minus policy gain.
• Minimize the row maximum and retain all ties.
• Changing the menu can change regret without changing surviving payoffs.

Further reading

• MIT 16.422: Classical Decision Theory, slides 12–15 ↗

  Read PDF pages 12–15 on maximin, statewise opportunity loss and regret comparison. Original portfolio menu, mandate filtering, sharp certificates, sensitivity and diagnostics; no video review or copied exercise. Checked 2026-10-25.