Deciding under uncertainty: utility, Kelly, regret, and robust choices

This book spends a lot of pages answering questions like "which strategy maximizes the withdrawal rate" or "which allocation minimizes failure". This page steps back and asks the question that comes before them: what makes a decision good when the future is unknown? A retiree has a problem that averages ignore. You live one path. A simulator draws ten thousand, life draws one, and there is no second game. Maximizing an average computed over ten thousand worlds is not automatically the right goal for someone who will only ever live in one of them.

Decision theory is a century of work across economics, mathematics and psychology, and it answers that question directly. Expected gain turns out to be a poor criterion, and utility is what fixes it. Two risky plans can be compared in a single honest number, the certainty equivalent. The Kelly criterion, seductive as it is, does not transfer to a retirement as written. Regret and robustness, which sound like excuses, are grown-up criteria. All of it condenses into a protocol of five rules. This is the most conceptual page in the book and probably the most profitable one: picking the wrong criterion costs more than getting the arithmetic wrong.

Utility: why the average is a bad judge§

The classic starting point is a puzzle posed in 1713, the St. Petersburg paradox. It is a coin-flip game with an infinite expected payoff that nobody would pay more than a few dozen dollars to play. Daniel Bernoulli's answer sets up everything that follows. What counts is not money but the utility of money, and utility grows more and more slowly. Going from $500,000 to $1M changes a life. Going from $10M to $10.5M changes nothing. That concavity has a direct consequence: at equal expected value, less spread means more utility. Risk aversion is not an emotion, it is a theorem.

For a retiree the concavity is extreme and lopsided. The bottom of the distribution destroys the standard of living, the dignity, the options: those are the paths that graze or hit ruin (Failure probability: reading it, choosing it, and not letting it run you). The top, where you die on a pile of gold, adds almost nothing beyond a bigger bequest (Real spending in retirement (the retirement smile, Die With Zero)). A criterion that weighs those two zones equally, which is exactly what an average does, is structurally wrong for the job. Hence a habit that runs through this book: judge plans on the 5th percentile and the median, never on the mean (Reading a fan chart without fooling yourself).

The tool that makes all of this usable is the certainty equivalent, the guaranteed amount you would take in exchange for a risky distribution. Take two plans with the same average. One pays $42,500 a year for certain, the other pays $20,000 or $65,000 depending on the markets. To a reasonably cautious person, the second is "worth" maybe $36,000 guaranteed. The certainty equivalent turns any lottery into one comparable number, and it drops faster the worse the plan's bad scenarios are. You never have to compute it formally. Asking "what guaranteed income would this risky plan be worth to me?" is already the right mental move, and it deflates plans with a pretty average and disastrous bad years on the spot.

DECIDING UNDER UNCERTAINTYThe certainty equivalent: what a risky plan is really worthutility (well-being)annual income ($k)utility grows moreand more slowlybad world ($20k)good world ($65k)the 50/50 lotteryE = 42.5CE ≈ 36the price of the risk
On a concave utility curve, a 50/50 lottery between $20k and $65k is worth less than its expected value of $42.5k. Its certainty equivalent falls to about $36k. The gap between the two is the price of the risk.

Tolerance and capacity: two dials, not one§

Everyday language runs together two things a decision has to keep apart. Risk tolerance is psychological: the drop you can sit through without panicking or selling (The psychology of spending: why spending it is the hard part). Risk capacity is objective: the drop your plan absorbs without breaking, however calm you happen to be. A phlegmatic ex-trader running a tight 4.5% withdrawal has high tolerance and low capacity. An anxious person sitting on 50 times their spending has the opposite. The rule for combining them is simple and has no exceptions: the smaller of the two governs. Capacity can be computed, and that is exactly what a simulator does. Tolerance, unfortunately, is discovered mostly in real declines. That is what makes the fit tests each strategy comes with worth running (Choosing your strategy: criteria, comparison, worked case), along with questions asked while you are calm, such as "can your withdrawal drop 20% in a single year?".

Regret, good enough, and robustness§

Utility does not capture everything that matters. Three more ideas round out the toolbox, and all three are more respectable than their reputation.

Regret. Harry Markowitz, who invented portfolio optimization, said he put his own savings in a 50/50 split "to minimize my future regret" rather than at the optimum his own theory pointed to. The minimax regret criterion picks the option whose worst backward-looking regret is the smallest. It is perfectly rational when you only live one path. It explains why a cautious plan still holds stocks, out of regret at missing thirty years of gains, and why an aggressive plan still holds bonds, out of regret at the crash in year one. Those choices beat the corner solutions. Many of the lukewarm-looking answers in this book are exactly that: the size of the gold sleeve, the withdrawal corridor, the broadened 60/40. They are deliberate regret minimizations, and that is a compliment.

Good enough. Herbert Simon called it satisficing: look for a solution that is sufficient rather than the best one. A plan with a 90% success rate and an income that covers your life has nothing to gain from further optimization, and plenty to lose in complexity, fees and fragility. Simon's question ("good enough for what?") serves a retiree better than the optimizer's ("maximal under which model?"). The honest answer to that second one is always "under a model that is wrong" (Monte Carlo: strengths, weaknesses, and how to use it well).

Robustness. A robust decision stays good even when the model that produced it is wrong. Portfolio optima are famously fragile ridges: move the expected return by one point and the "optimal" allocation moves by twenty. Nature grants us one mercy, though, because the surface around the optimum is flat. Between 50 and 80% stocks the SWR barely moves (The stock and bond allocation in retirement), and the same holds between 5 and 15% in gold or trend. The practice follows from that terrain. Find the plateau, settle in the middle of it rather than at the edge where one modeling error tips everything over, and walk away from any recommendation that only exists on a ridge. That is robust optimization in the version you can run without mathematics: optimize against the worst plausible assumption, not for the best estimate.

The protocol: five rules for deciding§

All of it condenses into a protocol you can apply to every decision in the plan, from the allocation to the exit date (The one-more-year syndrome).

One: judge decisions on the process, never on the outcome of one path. A good decision can turn out badly and a stupid bet can pay off; over a single life, confusing the two is the most expensive mistake there is. Two: compare plans on the 5th percentile and the median together, your homemade certainty equivalent, not on the average and not on the best case. Three: prefer the middle of a plateau to the top of a ridge, and distrust any option whose advantage vanishes when you move one assumption by a point. Four: when the choice is close, minimize the regret you can see coming, in both directions, the crash and the rally you missed. Five: write the decision down, with the conditions for revisiting it, while you are calm (Building your plan step by step), because the best decision theory in the world does not survive a call made at 11 at night in March 2020.

The essentials§

Going further§