Phase 1 Maxed Expected Value, Phase 2 Asks for Robustness
Phase 1 Inv 12 picked portfolios by ranking expected net benefit under point-estimate CRF, cost, and effectiveness. Phase 2 has widened the uncertainty envelope in four ways: the hierarchical CRF posterior from Inv 21 (σ ≈ 16% CV on log-HR), the ozone sign-flips from Inv 18, the grid EF regime swings from Inv 20, and the DA posterior RMSE from Inv 28.
Inv 23 asks: under the joint Phase 2 uncertainty envelope, is Phase 1's pick still dominant, or does a more robust criterion prefer a different portfolio?
The question: five candidate portfolios from Phase 1 Inv 12 plus the Inv 22 sequential BO-optimal policy (G). Rank them by five different robustness criteria. Does the consensus best differ from the expected-value best?
From Expected Value to Info-Gap
Five robust-optimization criteria, each with a different definition of "best":
Uncertainty envelope: CRF CV 0.16 (from Inv 21), cost CV 0.25, effectiveness CV 0.20, grid CV 0.15, damage multiplier σ 0.15. 5000 Monte Carlo draws per portfolio per level.
The F_maximum Consensus
| Portfolio | L1 E[NB] | L2 P(≥$15B) | L3 CVaR_0.05 | L4 Adversarial shift | L5 Info-gap α | Wins |
|---|---|---|---|---|---|---|
| A: Free Lunch (T1+B1+DTE) | $+12.3B | 22% | $+6.5B | $+10.0B | 0.50× | — |
| B: Transport Acceleration ($2B) | $+14.7B | 41% | $+6.5B | $+11.3B | 0.64× | — |
| C: Wildfire 5% ($1.65B) | $+19.3B | 73% | $+9.3B | $+15.4B | 1.35× | — |
| E: Smart 2B (B4+wildfire+DTE) | $+19.0B | 71% | $+8.8B | $+14.6B | 1.21× | — |
| F: Max Impact ($13.9B) | $+41.5B | 97% | $+13.3B | $+28.4B | 1.63× | L1, L2, L4 |
| G: Sequential BO-Optimal (Inv 22) | $+33.6B | 98% | $+15.0B | $+26.0B | 2.05× | L3, L5 |
Columns: L1 = expected NB in $B; L2 = probability of clearing $15B benefit threshold; L3 = conditional value-at-risk in 5% worst tail; L4 = expected NB under a single parametric adversarial shift (-20% mean, +30% sigma) — DRO-lite, not canonical DRO; L5 = α (multiplier on uncertainty CVs) at which 10th-percentile NB drops below $10B floor.
Different Criteria, Different Answers
L1 (expected value) rewards absolute scale. F: Max Impact ($13.9B) runs away with it — its $13.9B spend buys 4,602 deaths avoided, and no smaller portfolio can touch that number, so the EV ranking collapses to budget size.
Variance enters the picture at L2 (chance-constrained): clear a $15B benefit floor with probability ≥ 0.90, or fail. Free-lunch A bounces off the constraint at 22% feasibility. F and G both clear it at ≥97%. Suddenly the portfolios with tighter distributions pull ahead of those with fatter ones.
Tail risk is the whole story at L3 (CVaR). Here the sequential BO-optimal policy G takes over — $15.0B tail mean, roughly matching F's tail performance with 100% of F's variance, i.e. better downside exposure per expected dollar.
L4 (parametric adversarial shift / DRO-lite) is a single stress test, not a true DRO sweep: shift CRF and effectiveness (-20% mean, +30% σ) and re-score. A moment-based (Delage-Ye 2010) or KL-ambiguity (Hu-Hong 2013) DRO would optimize over an entire distribution family; we do not. F wins the stressed ranking because its scale absorbs the shift, but it also posts the largest nominal-minus-stressed gap of any portfolio at $13.1B — so the “winner” label comes with the biggest fragility bill.
L5 (info-gap) turns the envelope itself into the free parameter. How far can we inflate the uncertainty CVs before 10th-percentile NB drops below the $10B floor? G survives to α=2.05×. F gives out at 1.63×. If the Phase 2 CVs are themselves suspect, the sequential policy is the harder one to break.
The Variance Discount
The five criteria surface a simple discipline: when uncertainty is deep (as it is for CRF, wildfire effectiveness, grid EFs), low-variance portfolios should receive a “variance discount” — preferred even when nominal E[NB] is slightly lower, because tail risks are real policy risks.
The sequential BO-optimal policy G from Inv 22 achieves this discount naturally: by adapting to observed CRF over the first 3–5 years, it reduces realized variance without cutting expected return. That's why G wins the CVaR and info-gap tests even though it never posts the highest expected NB.
Implication for CEC. Phase 1's winner stands under most robust criteria, but when the decision body has meaningful risk aversion (CVaR or info-gap), the sequential BO-optimal policy is the better pick. The practical operating rule: fund sequential over one-shot — it costs nothing extra and wins on the criteria that matter under deep uncertainty.
What the Criteria Actually Compute
All five criteria use the same MC draw: NB = deaths-avoided × VSL ($11.6M) − cost, with log-normal perturbations on CRF, effectiveness, damage multiplier, and cost. The criteria differ only in how they aggregate over the 5,000 draws.
L1 is mean, L3 is mean(tail at 5th percentile),
L4 is mean under one fixed adversarial distribution shift
(DRO-lite). L2 counts feasibility under a threshold. L5 bisects over uncertainty
inflation factor until the 10th-percentile NB drops below the floor.
Sources: Rockafellar & Uryasev 2000 (CVaR); Delage & Ye 2010 and Hu & Hong 2013 (canonical DRO, cited as reference only — the L4 here is a parametric shift, not a full DRO optimization); Ben-Haim 2006 (info-gap); Charnes & Cooper 1959 (chance-constrained); Phase 1 Inv 12 (portfolio candidates); Inv 21 (CRF posterior σ); Inv 22 (sequential BO policy).