Regret¶
Core Idea¶
Regret is the value gap an agent registers between the outcome it actually obtained and the better outcome that a forgone alternative would have produced — a comparison made after the fact against a counterfactual reference rather than against prior expectations, a structure Savage (1951) first made formal in proposing the minimax-regret criterion. [1] The defining structure is retrospective and relative: the same realized outcome can be coded as a triumph or a regret depending solely on which unchosen path it is compared against. A bet that returns ten percent is a success measured against the bank but a regret measured against the stock that doubled; nothing about the realized payoff has changed, only the benchmark. Anticipated regret then feeds backward into choice, so the pattern couples a backward-looking evaluation to a forward-looking decision rule — the very anticipation of the future gap reshapes which act is taken now, a coupling that the regret-theory of Loomes and Sugden (1982) treats as a primitive of choice under uncertainty. [2] Regret thus names a particular closed loop: an outcome is realized, compared against the best forgone alternative, scored as a shortfall, and that score is fed forward to bias the next decision. It is neither pure emotion nor pure calculation but the structural object both share — the chosen-versus-best-unchosen difference that drives learning and avoidance alike.
How would you explain it like I'm…
Wishing You'd Picked the Other One
The Gap from What Could Have Been
Regret
Structural Signature¶
Regret encodes a structural pattern: choice over alternatives → realized outcome → comparison against the best forgone branch → scalar shortfall → feedback into future choice. It separates two evaluations that everyday language fuses — shortfall against expectation (disappointment) and shortfall against the best available alternative (regret) — and names only the second, a separation that the online-learning literature operationalizes as cumulative regret against the best fixed action in hindsight. [3]
Recurring features:
- Value gap between obtained outcome and best forgone alternative
- Retrospective comparison against a counterfactual benchmark, not against expectation
- Scalar shortfall summarizing an entire counterfactual tree
- Anticipated regret fed backward into present choice
- Cumulative gap to the best fixed action in hindsight
- Minimax shortfall under unknown states of the world
- Closed loop from outcome through comparison to learning
The structural insight is robust across substrates within the agentic cluster: a gambler weighing the path not taken, a bandit algorithm bounding its lifetime reward gap, a clinician second-guessing a treatment window, and a bidder shading against the winner's curse all instantiate the same comparison — not outcome-versus-expectation but outcome-versus-best-alternative. [4] What unifies them is that the benchmark is itself counterfactual and only knowable after the fact, so regret is always an evaluation against a reference the agent could not have certified in advance.
What It Is Not¶
Regret is not disappointment. Disappointment measures realized outcome against what the agent expected or hoped — a vertical drop from an internal forecast. Regret measures realized outcome against the best alternative the agent could have chosen — a horizontal comparison across the branches of a decision. The two come apart cleanly: an investor who expected ten percent and earned ten percent feels no disappointment, yet may feel sharp regret if a passed-over option earned thirty. [5] Naming regret is precisely the act of holding these two evaluations distinct.
Nor is regret merely a negative emotion or a synonym for sadness. The prime names a comparative structure, not an affective tone. A person can register the regret-gap dispassionately — "the other clinic would have caught it sooner" — without an emotional storm, and an algorithm registers regret with no affect at all. The feeling that often accompanies the gap in humans is a downstream consequence, not the defining content.
Regret also does not claim that a good outcome cannot be regretted. This is a frequent misconception. Because the benchmark is the best forgone branch rather than a zero or a baseline, a winning bet placed too small carries regret, a successful surgery performed too late carries regret, a profitable sale that left surplus on the table carries regret. The sign of the realized outcome and the sign of the regret are independent.
Finally, regret does not assert that minimizing regret is the same as maximizing expected value, nor that it is always the right objective. Minimax-regret and expected-value maximization are genuinely different decision rules that can recommend different acts; the prime simply names the quantity that regret-based rules optimize, leaving open whether doing so is wise in any particular setting.
Broad Use¶
Decision theory: Regret-minimization criteria — Savage's minimax regret — select acts that bound the worst-case shortfall against the best act in hindsight, providing a decision rule that needs no prior over states of the world, an alternative to expected-utility maximization analyzed at length by Bell (1982) as a descriptively richer account of choice. [5]
Behavioral economics: Anticipated regret explains a family of otherwise puzzling behaviors — buying insurance against low-probability losses, holding losing stocks to avoid realizing the regret of a bad purchase, refusing to switch from defaults, and the asymmetry between regret of action and regret of inaction documented in the omission-bias literature. [6]
Reinforcement learning and online learning: "Regret" is the formal performance objective — the cumulative gap between an algorithm's accumulated reward and that of the best fixed action chosen in hindsight; bandit and online-learning algorithms are rated by their regret bounds, with no-regret guarantees meaning the per-round gap vanishes as the horizon grows. [3]
Medicine: Clinicians weigh decision regret — acting versus waiting, screening versus not — especially when a missed window produces an irreversible outcome, and validated instruments such as the Decision Regret Scale measure patients' retrospective regret over medical choices. [7]
Auction and market design: Bidders shade bids to avoid the regret of the winner's curse (paying more than the object's value) or the regret of leaving surplus on the table (bidding too little and losing a profitable object); mechanism designers engineer formats — sealed-bid versus ascending, for instance — partly to manage the regret exposure participants anticipate. [8]
Clarity¶
Naming regret separates two evaluations that ordinary "disappointment" conflates: shortfall against what you expected versus shortfall against what you could have chosen. [5] This is the prime's central clarifying contribution. Once the distinction is in hand, practitioners can say precisely that a good outcome can still be regretted (a winning bet placed too small), that a bad outcome need not be regretted (the best available choice that simply turned out poorly), and that minimizing regret is a distinct objective from maximizing expected value. Each of these statements is nearly impossible to express crisply without the prime, because the everyday vocabulary of "let-down" and "wishing things were different" runs the expectation-benchmark and the alternative-benchmark together.
The clarity also redirects attention from the agent's mood to the decision's geometry. Instead of asking "how bad does this feel," regret-language asks "against which forgone branch is this outcome being scored, and is that the right benchmark?" — turning a diffuse complaint into a specifiable comparison whose reference point can be examined and, sometimes, contested.
Manages Complexity¶
Regret compresses a whole counterfactual tree into a single scalar — the distance from the best forgone branch — turning the open-ended question "what else might I have done" into a comparable quantity. [2] A decision may have dozens of unchosen branches, each with its own realized-in-the-alternate-world payoff, but regret bounds the evaluation to the one comparison that matters: chosen outcome versus the maximum over alternatives. This is a drastic and deliberate reduction. It throws away the full distribution of counterfactuals and keeps only the gap to the best one, which is exactly the quantity relevant to evaluation and to learning what to do next time.
This compression is what makes regret tractable as both a cognitive heuristic and a formal objective. A person cannot hold an entire decision tree in mind, but can ask "did I leave the best option unchosen, and by how much?" An algorithm cannot enumerate every policy, but can accumulate a single running regret figure and optimize against it. In both cases the scalar replaces a combinatorial object with a one-dimensional signal that can be tracked, bounded, and minimized.
Abstract Reasoning¶
Once regret is recognized, minimax-regret and no-regret reasoning become available: an agent can commit to decision rules whose hindsight shortfall is provably bounded even under adversarial conditions, without needing a prior over states. [3] This unlocks a distinctive style of robust reasoning. Rather than estimating probabilities of states and maximizing expected payoff — which requires a prior that may be unavailable or untrustworthy — the agent reasons directly about the worst-case gap to the best act it could have taken, and selects to keep that gap small whatever the world does. No-regret learning extends this into repeated play: an agent that guarantees vanishing average regret can match the best fixed strategy in hindsight without ever knowing the environment in advance, a result with deep ties to the existence of correlated equilibria in game theory.
Regret-recognition also licenses inferences about behavior that expected-utility models miss. Default stickiness, omission bias, the reluctance to sell at a loss, and the extra premium people pay to avoid foreseeable second-guessing all become predictable once anticipated regret is admitted as an argument in the agent's objective. The prime thus does double duty: it supports a normative robust-decision toolkit and a descriptive theory of systematic departures from expected-utility behavior.
Knowledge Transfer¶
The online-learning notion of bounding cumulative regret transfers directly to robust policy design: minimize the worst-case gap to the best benchmark you could not have known in advance. A policymaker uncertain over future states, like a bandit algorithm uncertain over arm rewards, can adopt rules that guarantee bounded shortfall against the best fixed policy in hindsight, importing the no-regret machinery wholesale. [9] Conversely, the psychological finding that anticipated regret distorts choice transfers into mechanism and market design, where formats are deliberately engineered to reduce bidders' or consumers' regret exposure — choosing auction rules, default settings, or disclosure regimes that lower the regret a participant will anticipate and therefore the distortion it induces. [4] The same scalar gap that an algorithm minimizes mathematically is the object a designer reshapes psychologically, and a practitioner fluent in one reading often recognizes the structure in the other.
Examples¶
Formal/abstract¶
Multi-armed bandit (online learning): An algorithm repeatedly chooses among K slot-machine arms with unknown reward distributions. After T rounds, its regret is defined as the difference between the cumulative reward of the single best arm in hindsight and the cumulative reward the algorithm actually earned. A "no-regret" algorithm guarantees that this gap grows sublinearly in T, so the per-round regret tends to zero: in the long run the algorithm performs as well as if it had known the best arm from the start. The benchmark — the best fixed arm — is counterfactual and only identifiable after all the draws are seen; the algorithm must do well against a comparison it cannot compute in advance. Mapped back: This is the prime in its purest form. The realized outcome (the algorithm's reward) is scored against the best forgone alternative (the best fixed arm), the gap is compressed to a single scalar (cumulative regret), and minimizing that scalar is the entire design objective. No affect, no expectation-benchmark, only the chosen-versus-best-unchosen difference fed forward into the next round's choice.
Minimax regret in decision theory: A decision-maker must choose an act without a probability distribution over the states of the world. For each act and each possible state, she computes the regret — how much worse that act's payoff is than the best act she could have chosen had she known the state. She then selects the act whose maximum regret across states is smallest. The criterion needs no prior; it asks only "across everything that could happen, which choice keeps my largest possible second-guess the smallest?" Mapped back: Again the structure is exact. The reference is the best act in hindsight, computed state by state; the evaluation is the gap to that reference; and the chosen rule is the one that bounds the worst such gap. The example also shows the prime's independence from expected value: a minimax-regret act can differ sharply from the act a probability-weighted expected-payoff calculation would recommend, because the two optimize different quantities.
Applied/industry¶
Investing and the disposition effect: A retail investor holds a stock that has fallen below its purchase price. Selling would realize a loss and, with it, the regret of having made a bad purchase; holding keeps the loss unrealized and the regret deferred. Empirically, investors disproportionately sell winners and hold losers — the disposition effect — in large part to avoid the sharp regret of crystallizing a mistake. The benchmark generating the regret is the counterfactual of having never bought, or having sold earlier; the realized loss is scored against that forgone path. Mapped back: The realized outcome (a paper loss) is being compared not against expectation but against the best forgone alternative (not buying, or selling sooner), and the anticipation of that regret-gap feeds backward into the present choice to hold. The investor is, in effect, minimizing anticipated regret rather than maximizing expected wealth — a textbook case of the prime's backward-to-forward coupling distorting choice.
Patient treatment decisions in medicine: A patient facing a screening or treatment decision — a prophylactic surgery, an early intervention with side effects, a watchful-waiting protocol — weighs not only expected health outcomes but the regret they will feel under each branch. If a missed window leads to an irreversible bad outcome, the anticipated regret of inaction can dominate; instruments like the Decision Regret Scale later measure how much retrospective regret the realized choice produced. Clinicians increasingly elicit anticipated regret explicitly, because it predicts decisions that pure expected-utility models do not. Mapped back: The structure holds across the agentic substrate: the patient scores the realized (or anticipated) outcome against the best forgone branch — "the other choice would have spared me" — and that counterfactual gap, anticipated in advance, reshapes the decision made now. As in the bandit case, the benchmark is knowable only after the fact, yet it governs choice before the fact.
Structural Tensions¶
T1: Regret is a clean scalar in formal settings but a contested benchmark in human ones. In online learning the comparison class — the set of fixed actions against which regret is measured — is specified in advance, so the gap is unambiguous. In human decisions the relevant counterfactual is rarely fixed: an agent may compare against the best option actually available, the best option imaginable, or the best option a reasonable person would have foreseen. Because regret is defined relative to a benchmark, who gets to set the benchmark determines how much regret there is, and that choice is frequently disputed after the fact.
T2: Anticipating regret can improve choice or paralyze it. A modest dose of anticipated regret functions as useful foresight, steering an agent away from options it would bitterly second-guess. But because every action forecloses alternatives, sufficiently vivid anticipation of regret can attach to every branch, producing decision avoidance, default stickiness, and the very inaction whose regret the agent was trying to dodge. The same backward-to-forward coupling that sharpens decisions can freeze them.
T3: Minimizing regret and maximizing expected value pull apart. Because regret scores against the best hindsight benchmark rather than against an expectation, regret-minimizing acts can be systematically more conservative — or in adversarial settings more robust — than expected-value-maximizing acts. An agent who optimizes one quantity is not optimizing the other, and a practitioner who silently switches benchmarks (treating a regret-minimizing rule as if it maximized expected payoff) will mispredict behavior and mis-evaluate outcomes.
T4: The hindsight benchmark is unknowable in advance, yet governs choice in advance. Regret is always measured against the best forgone alternative, which can only be identified once outcomes are realized. This is structurally awkward: the quantity that anticipated regret asks an agent to minimize is defined by information the agent does not yet have. Formal settings resolve this by fixing a comparison class and proving worst-case bounds; human reasoning resolves it by guessing at the eventual benchmark, and the guess is often wrong, producing either unwarranted regret or culpable complacency.
T5: Compressing the counterfactual tree to a scalar discards information that may matter. Regret keeps only the gap to the single best forgone branch and throws away the rest of the distribution of alternatives. This is what makes it tractable, but it can mislead: two decisions with identical regret may differ enormously in how many alternatives beat the chosen one, how close the runner-up was, or how variable the forgone payoffs were. An agent who tracks only the scalar may over-weight a single spectacular missed opportunity while ignoring that the chosen path dominated almost everything else.
T6: Regret can drive learning or entrench loss-chasing. Fed forward as a learning signal, regret is corrective — it tells an agent which forgone option to favor next time, and no-regret dynamics converge toward good benchmarks. But the same feedback can be pathological: an agent who keeps comparing against a tantalizing forgone path may chase losses, escalate commitment, or refuse to accept a sunk outcome, letting the counterfactual benchmark distort rather than improve subsequent choice. Whether regret-feedback is adaptive depends on whether the benchmark it enshrines is one the agent could actually have achieved.
Structural–Framed Character¶
Regret sits toward the structural side of the structural–framed spectrum, with some framing: at its core it is a value gap an agent registers between the outcome actually obtained and the better outcome a forgone alternative would have produced — a retrospective comparison against a counterfactual reference. The same realized outcome can be coded as triumph or regret depending solely on which unchosen branch it is measured against.
The counterfactual comparison itself is recognizable structure, formalized neutrally in online learning as a cumulative-gap objective and definable without invoking the institutions of any field. What pulls it toward framing is a psychological-evaluative gloss: the vocabulary of regret and anticipated regret carries affect, the comparison presupposes an agent who values outcomes, and naming a gap "regret" rather than "difference" marks it as bad. The structural skeleton and the affective overlay sit together, leaving it mixed but structural-leaning.
Substrate Independence¶
Regret is a moderately substrate-independent prime — composite 3 / 5 on the substrate-independence scale. Its defining structure — a retrospective value gap measured against a counterfactual benchmark and fed back into future choice — is cleanly substrate-agnostic and has a strong formal incarnation in online-learning regret bounds and minimax regret, alongside cognitive and economic uses. That gives it genuine leverage across formal, cognitive, and social substrates. What limits it is that the structure presupposes a valuing agent making counterfactual comparisons, so it does not reach physical or biological substrates without going metaphorical — real multi-substrate reach, but confined to the agentic cluster, lands it at 3.
- Composite substrate independence — 3 / 5
- Domain breadth — 3 / 5
- Structural abstraction — 4 / 5
- Transfer evidence — 3 / 5
Relationships to Other Primes¶
Parents (3) — more general patterns this builds on
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Regret presupposes Comparison
Regret is the difference an agent registers between the outcome obtained and the better outcome a forgone alternative would have produced. Computing that gap requires placing the actual and counterfactual outcomes under a shared evaluative frame and reading off a greater-lesser relation. Without Comparison as the underlying operation, no value gap can be formed, so regret presupposes the alignment-and-difference machinery that comparison supplies, applied retrospectively to chosen versus unchosen paths.
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Regret presupposes Decision
Regret is meaningful only after an agent has selected one alternative from a set and foreclosed the others; it then evaluates the realized path against unchosen counterfactuals. Without a prior Decision — the act of collapsing deliberation into commitment — there is no chosen-versus-foregone pair to compare and no commitment whose costs can be registered. Regret therefore presupposes the decision event as the structural precondition of its own evaluative geometry.
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Regret presupposes Modal Reasoning
Regret presupposes modal reasoning because its defining structure is retrospective comparison of the realized outcome against the better outcome a forgone alternative would have produced. That comparison requires quantifying over a space of possible-but-unrealized alternatives and evaluating a counterfactual conditional, which is exactly what modal reasoning supplies via its operators over accessible alternatives. Without the prior availability of structured reasoning over what could-have-been, regret reduces to a brute affective response with no benchmark against which to register the gap between actual and alternative.
Path to root: Regret → Comparison
Neighborhood in Abstraction Space¶
Regret sits among the more crowded primes in the catalog (23rd percentile for distinctiveness): several abstractions describe nearly the same structure, so a description that fits it will tend to fit its neighbors too — transporting it usually means disambiguating within this family rather than landing on it exactly.
Family — Language, Symbol & Cultural Form (32 primes)
Nearest neighbors
- Temporal Inconsistency and Preference Reversals — 0.83
- Optimism Bias — 0.83
- Decision — 0.82
- Paradigmatic vs. Syntagmatic Relations — 0.82
- Self Control — 0.80
Computed from structural-signature embeddings · 2026-05-29
Not to Be Confused With¶
Regret must be distinguished from Risk Aversion, with which it is most often confused because both shape choice under uncertainty and both can make an agent favor the safer option. The decisive difference is temporal and referential. Risk aversion is a prospective preference defined over the uncertain magnitudes of outcomes themselves: a risk-averse agent, looking forward, dislikes the spread of a gamble and will trade expected value for reduced variance, preferring a sure forty to a fifty-fifty shot at a hundred. The reference is the distribution of the act's own possible payoffs, assessed before anything is realized. Regret, by contrast, is a retrospective comparison against a forgone alternative: it is defined only once an outcome is realized (or anticipated as if realized) and scored against the best branch the agent did not take. A risk-averse agent and a regret-averse agent can be driven to opposite choices by the same gamble — regret aversion can make an agent take a risky option precisely to avoid the regret of having played it safe and watched the bold path pay off, whereas risk aversion pushes uniformly toward the low-variance option. The two also differ in what information they require: risk aversion needs the agent's utility curvature over outcomes; regret needs a counterfactual benchmark of forgone alternatives. They can coexist in one agent, but they are distinct arguments in the objective, and conflating them mispredicts exactly the cases — bold choices made out of regret aversion — where they diverge.
Regret is also not Loss Aversion, though both involve a reference point and an asymmetry around it. Loss aversion, in the prospect-theory tradition, codes outcomes against a status-quo (or expectation-based) reference point and weights losses below that point more heavily than equivalent gains above it; the reference is where the agent currently stands or expects to stand, and the asymmetry is between gains and losses relative to that anchor. Regret codes outcomes against a different reference entirely — the best unchosen option — and its asymmetry is between the chosen path and the best forgone path, not between gains and losses. The two references can point in opposite directions. An outcome can be a gain relative to the status quo (no loss aversion triggered) yet a sharp regret relative to a forgone alternative that gained even more; conversely, an outcome can be a loss relative to the status quo yet carry little regret if no available alternative would have done better. Loss aversion explains why people overweight downside relative to a baseline; regret explains why people second-guess relative to the road not taken. Treating them as one collapses two distinct reference structures — baseline-relative and alternative-relative — that systematically diverge in mixed cases.
Finally, regret is not Counterfactual Reasoning, which is in effect its parent. Counterfactual reasoning is the general cognitive and logical operation of considering what would have happened under alternatives to what actually occurred — "if I had left earlier, I would have caught the train" — and it spans causal attribution, learning, blame, explanation, and emotion with no particular valuation attached. Regret is the specific valuation that picks out one counterfactual structure: the chosen-versus-best-forgone value gap, scored as a shortfall and fed back into choice and learning. Every regret rests on a counterfactual, but the vast majority of counterfactual reasoning is not regret — counterfactuals can support relief ("good thing I didn't take that flight"), causal inference, or pure curiosity with no regret component at all. The parent operation is value-neutral and open-ended over all alternatives; the prime is value-laden and bounded to the single best alternative whose shortfall drives evaluation. Keeping them distinct matters because remedies differ: a problem of generating the right counterfactuals (the parent) calls for different intervention than a problem of over-weighting the best-forgone-branch comparison in choice (regret).
Solution Archetypes¶
No catalogued solution archetypes reference this prime yet.
Notes¶
Regret operates at multiple scales and across the agentic substrate, but it does not reach physical or biological substrates non-metaphorically: it presupposes a valuing agent capable of representing and comparing alternatives. A molecule does not regret; a thermostat does not regret. This is why the prime's substrate independence sits at the middle of the scale rather than at the top — it travels cleanly across formal, cognitive, and economic instances but stops at the boundary of agency.
The formal and psychological readings of regret are tightly linked but not identical. The online-learning sense is a precise mathematical quantity (cumulative gap to the best fixed action) with provable bounds; the psychological sense is an affect-laden anticipation that distorts choice. The shared structural object — the chosen-versus-best-forgone gap — is what lets insight transfer between them, but practitioners should not assume that results about one reading carry over to the other without argument. A no-regret bound is a theorem; regret aversion is an empirical regularity.
A recurring subtlety is the benchmark problem: regret is only as well-defined as the comparison class it is measured against. In formal settings this class is specified in advance and the regret figure is unambiguous. In human and policy settings the benchmark is contested — best available option, best foreseeable option, best imaginable option — and much of the practical difficulty of reasoning about regret is really difficulty about which counterfactual deserves to be the reference.
Finally, regret's compression to a scalar is both its power and its hazard. The same reduction that makes it tractable discards the shape of the counterfactual distribution, so an agent that optimizes regret alone can be blindsided by considerations (variance of forgone payoffs, number of dominating alternatives, achievability of the benchmark) that the scalar silently omits. Regret is best used as one argument in an objective, not the whole of it.
References¶
[1] Savage, L. J. (1951). The theory of statistical decision. Journal of the American Statistical Association, 46(253), 55–67. Introduces the minimax-regret criterion, the formal origin of regret as a prior-free decision objective scored against the best act in hindsight. ↩
[2] Loomes, G., & Sugden, R. (1982). Regret theory: An alternative theory of rational choice under uncertainty. The Economic Journal, 92(368), 805–824. Treats anticipated regret as a primitive of choice under uncertainty and models regret as a comparison against the forgone alternative; supports the backward-to-forward coupling and the counterfactual-tree compression. ↩
[3] Cesa-Bianchi, N., & Lugosi, G. (2006). Prediction, Learning, and Games. Cambridge University Press. Canonical treatment of online learning: defines regret as the cumulative gap to the best fixed action in hindsight and establishes no-regret bounds for bandit/sequential-prediction algorithms under adversarial conditions. ↩
[4] Zeelenberg, M., & Pieters, R. (2007). A theory of regret regulation 1.0. Journal of Consumer Psychology, 17(1), 3–18. Comprehensive theory distinguishing regret from related emotions across psychology, economics, and marketing; grounds the cross-substrate outcome-versus-best-alternative comparison and the transfer of anticipated-regret findings into market/consumer design. ↩
[5] Bell, D. E. (1982). Regret in decision making under uncertainty. Operations Research, 30(5), 961–981. Formalizes regret as the shortfall against the best forgone alternative—distinct from disappointment against expectation—and analyzes regret-incorporating choice as a descriptively richer alternative to expected-utility maximization. ↩
[6] Ritov, I., & Baron, J. (1990). Reluctance to vaccinate: Omission bias and ambiguity. Journal of Behavioral Decision Making, 3(4), 263–277. Documents omission bias—the asymmetry between regret of action and regret of inaction—exemplifying how anticipated regret distorts choice toward defaults and inaction. ↩
[7] Brehaut, J. C., O'Connor, A. M., Wood, T. J., Hack, T. F., Siminoff, L., Gordon, E., & Feldman-Stewart, D. (2003). Validation of a decision regret scale. Medical Decision Making, 23(4), 281–292. Develops and validates the Decision Regret Scale measuring patients' retrospective regret over medical choices. ↩
[8] Kagel, J. H., & Levin, D. (1986). The winner's curse and public information in common value auctions. American Economic Review, 76(5), 894–920. Experimental demonstration that adding bidders worsens the winner's curse — large fields (6–7 bidders) bid more aggressively and earn negative profits — confirming that more competitors make winning worse news and require a larger downward correction, a diagnostic marker of the curse. ↩
[9] Hart, S., & Mas-Colell, A. (2000). A simple adaptive procedure leading to correlated equilibrium. Econometrica, 68(5), 1127–1150. Establishes that regret-matching (no-regret) dynamics converge to correlated equilibrium, the result tying cumulative-regret bounding to robust repeated-play policy design. ↩