Regret and the Weighting of Small Probabilities

The weighting function, and where it reaches the portfolio

Expected value weights each outcome by its probability. Prospect theory found that people do not: they transform the probabilities first, overweighting small ones and underweighting moderate and large ones. A 1% chance behaves like something closer to 3%, and a 90% chance behaves like something closer to 80%. The pattern explains a pair of behaviours that look contradictory and are in fact the same behaviour: people buy lottery tickets, accepting a negative expected value, and buy insurance, paying more than the expected loss. In both cases the small probability is doing too much work. The market version is documented rather than theoretical. Stocks whose recent returns include an extreme maximum — the lottery-like payoff — are systematically overpriced and underperform, and the investors who hold them disproportionately are the ones who also concentrate their portfolios in a few names and trade actively. On the options side the same preference shows up as demand for far out-of-the-money contracts, which is one contributor to the shape of the volatility skew: the tail that people want to own is the tail that is priced richly. A trader buying either is not making an arithmetic error in the usual sense — they are paying a premium for a payoff distribution they find appealing, and the premium is real and recurring. The consequence is a specific, compounding drag. Each long shot is individually small, so the cost is invisible in a month and large in a year, and the winners are memorable in a way the losers are not. That is the pattern of the beginner rung’s overconfidence seen through a different instrument: the frequency is the cost, and the visible tail is the reason the frequency persists. Two long shots, priced — Lottery ticket: $100 for 1 in 100 at $5,000: Expected value $50 — 50% of the price ← · The same ticket at a 0.03 decision weight: 0.03 × $5,000 = $150, which reads as a $50 edge ← · One-week call: $200 premium, 4% chance of $3,000: Expected value $120 — 60% of the premium · The same call at a 0.10 decision weight: 0.10 × $3,000 = $300, which reads as an edge Neither instrument is mispriced by accident. The weighting is what the seller is selling to, which is why the same preference can be harvested by writing the option rather than buying it — and why the harvest has its own tail risk, which the Options subject spends several lessons on.

Regret: a prediction error wearing a decision rule

Regret theory observes that people anticipate the pain of regret and let it shape the choice, which means they can violate expected utility without any error in probability. The strongest form of it is the act-versus-omission asymmetry: an action that produces a bad outcome is expected to feel worse than an equally costly inaction. That expectation is what keeps a loser unsold — selling is the act that admits the error, so holding postpones the regret — and it is what produces the mirror behaviour of buying a name that has run, because the anticipated regret of watching it go further is vivid and immediate. The empirical problem is that the anticipation is badly calibrated. Studies of affective forecasting find that people overestimate the intensity and the duration of future regret, a bias that survives the fact that the same person has been wrong about it before. The regret of the act is not only smaller than expected, it is mostly gone within a short period, while the cost of the decision stays on the account. So the trader who holds a loser to avoid an anticipated feeling has paid a real amount to avoid a smaller one, which is the same structure as the beginner rung’s lesson on loss aversion and arrives from the different direction of the choice rather than of the accounting. The correction is to demote regret from a rule to a consideration. It is information about the trader, and it belongs in the design of the process: written exits reduce the scope for regret to decide, pre-commitment removes the moment, and rules about position size mean the outcome that would be regretted is bounded in advance. What does not work is optimising against regret, because a portfolio chosen to minimise the worst feeling is not the same portfolio as one chosen to maximise expected value, and the difference is paid in return — a point that the mastery rung’s audit happens to resolve in numbers, since the trades taken to avoid a feeling are exactly the ones with a consistent negative expectancy. • Weight outcomes by their probabilities, not by how the probabilities feel. • Ask whether a bet would be attractive repeated a hundred times, which removes the single-tail appeal. • Separate the regret of an act from the regret of an omission; both are real, and the second is the one that compounds. • Put expected value in front of regret when they disagree, and record the case so the disagreement is visible later. • Design the process to reduce the moments where regret could decide — written exits, pre-commitment, bounded size. Regret aversion also protects against genuinely irreversible errors, and irreversibility is worth paying something for. The distinction is whether the cost is a premium for a real option, like preserving capital you cannot replace, or a premium for a feeling about an outcome you will have stopped thinking about by next month.

Action regrets fade; inaction regrets do not

Regret is not symmetric in time, and the asymmetry runs in a direction that changes what a good rule looks like. Research on the way people remember their own regrets finds a consistent pattern: in the short run, people regret **actions** more than inactions — the trade taken that lost, the purchase that went wrong, the thing done. Over longer horizons the ordering reverses, and people regret the things they did not do: the position never taken, the career path not followed, the offer declined. The short-run regret is loud and the long-run regret is quiet, which means the loud one gets to vote first. For a portfolio the consequence is specific. A stop taken that turns out unnecessary is an action regret, felt immediately and with a price attached that everyone can see. A position never entered because the moment felt risky is an inaction regret, felt years later and calibrated against nothing. A trader who is graded continuously on the first and never on the second will slowly drift toward excessive caution, because the costs of acting are visible daily and the costs of not acting are invisible by construction. That is the mechanism behind the research finding that the most heavily scrutinised decision-makers take less risk than the plan requires — it is not timidity, it is a regime in which only one kind of error is scored. The corrective is to make the un-taken decisions visible. A log of the setups that were declined, with the price and the condition that caused the decline, is the only way an inaction cost ever enters the review — and it converts a felt sense of caution into a measurable one. If the log shows that most of what was declined would have worked, the process is too tight. If it shows what was declined would mostly have failed, the discipline is earning its keep. Neither conclusion is available to a person who only scores the trades that happened, which is why the missing log is the bias rather than the regret. Regret minimisation is not a decision rule, and it is also not evenly distributed across time. The practical fix is to log the trades you did not take, so that the quiet regret has a number the loud one has to compete with.

Regret-proofing the decision before it is made

The asymmetry in this lesson — action regrets fade while inaction regrets do not — has a practical consequence that runs the other way from what it suggests: the feeling that a decision will be regretted is not information about the decision, it is information about which regret the mind is currently simulating. The imagination is asymmetric in the same direction as the memory: it produces vivid scenarios for the move you take and vaguer ones for the world in which doing nothing was wrong, because the action is concrete and the alternative is not. The remedy is not to suppress the feeling but to make the comparison explicit while there is still time to run it, which is what the exercise below does. Four questions, asked before the position and written down, convert the regret from a post-hoc verdict into an input. **What is the worst case if I act and the other one if I do not?** — stated in money, not adjectives, because a small loss and a missed doubling are different quantities and comparing them without units is what produces the wrong choice. **Would I regret the loss more or the miss more, and by how much?** — a comparison, not a ranking, since the whole distortion in this lesson is the over-weighting of one tail. **Is the regret symmetric if I take the action and it works?** — the point being that the anticipated regret for acting is almost never priced against the anticipated regret for the outcome where acting was right and the position was too small. And **would I make this decision the same way if it were someone else’s money?** — which strips the self-image out of the choice, since inaction regrets are largely about identity and the story a person tells about their nerve. The output is a rule rather than a feeling, and that is the difference between this exercise and rumination. **Regret minimisation is a tiebreaker, never the objective.** When two options have the same expected value, choosing the one whose worst case is easier to live with is rational and cheap; when they do not, the decision belongs to the expectancy calculation and the regret is a tax on acting on it. The concrete form this takes in a plan is a **regret allowance**: a stated small size at which a position may be taken purely because not taking it would be intolerable, permanently capped, written down, and excluded from the performance audit — because its purpose is not return, it is the prevention of a decision made badly later. That is a different thing from size chosen to avoid regret, which is the mistake this lesson exists to prevent, and the boundary between the two is precisely the cap: the position is allowed to satisfy the feeling as long as the arithmetic of the book is untouched by it. • State the regret on both sides in money before the position exists. • Compare the two rather than ranking them; the imagined one is always more vivid. • Ask the someone-else test to strip identity out of the choice. • Use regret minimisation only to break a tie in expected value. • Allow a capped regret allowance for the intolerable miss, and exclude it from the audit. The connection to the loss-aversion lesson is that the same mechanism — a reference point — is doing the work in both cases. There the reference is the price paid and here it is the price imagined, and in both the fix is to state the reference explicitly rather than let it be set by whichever comparison is easiest to picture.

Sources

• Prospect theory and the probability weighting functionKahneman & Tversky (1979); Tversky & Kahneman (1992), “Advances in Prospect Theory”
• Regret theory as an alternative to expected utilityLoomes & Sugden (1982), Economic Journal
• Who gambles in the stock market?Kumar (2009), Journal of Finance
• Immune neglect and the forecasting of regretGilbert et al. (1998), Journal of Personality and Social Psychology