In
decision theory
Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
, on making
decisions under
uncertainty
Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable or ...
—should information about the best course of action arrive ''after'' taking a fixed decision—the human emotional response of
regret
Regret is the emotion of wishing one had made a different decision in the past, because the consequences of the decision were unfavorable.
Regret is related to perceived opportunity. Its intensity varies over time after the decision, in regard ...
is often experienced, and can be measured as the value of difference between a made decision and the optimal decision.
The theory of regret aversion or anticipated regret proposes that when facing a decision, individuals might ''anticipate'' regret and thus incorporate in their choice their desire to eliminate or reduce this possibility. Regret is a negative
emotion
Emotions are mental states brought on by neurophysiological changes, variously associated with thoughts, feelings, behavioral responses, and a degree of pleasure or displeasure. There is currently no scientific consensus on a definition. ...
with a powerful social and
reputation
The reputation of a social entity (a person, a social group, an organization, or a place) is an opinion about that entity typically as a result of social evaluation on a set of criteria, such as behavior or performance.
Reputation is a ubiquitous ...
al component, and is central to how humans learn from experience and to the human psychology of
risk aversion
In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more ce ...
. Conscious anticipation of regret creates a
feedback loop
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled c ...
that transcends regret from the emotional realm—often modeled as mere
human behavior
Human behavior is the potential and expressed capacity ( mentally, physically, and socially) of human individuals or groups to respond to internal and external stimuli throughout their life. Kagan, Jerome, Marc H. Bornstein, and Richard M. L ...
—into the realm of the
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
choice behavior that is modeled in decision theory.
Description
Regret theory is a model in
theoretical economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
simultaneously developed in 1982 by
Graham Loomes
Graham Loomes, (born 5 August 1950) is a British economist and academic, specialising in behavioural economics. Since 2009, he has been Professor of Economics and Behavioural Science at the University of Warwick. He previously worked at the Un ...
and
Robert Sugden
Robert Sugden (also Sugden-Dingle) is a fictional character from the British ITV soap opera ''Emmerdale''. The character originally appeared on the show regularly between 22 April 1986 and 3 October 2005. During that time he was first played ...
, David E. Bell, and
Peter C. Fishburn. Regret theory models choice under uncertainty taking into account the effect of anticipated regret. Subsequently, several other authors improved upon it.
It incorporates a regret term in the
utility function
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...
which depends negatively on the realized outcome and positively on the best alternative outcome given the uncertainty resolution. This regret term is usually an increasing, continuous and non-negative function subtracted to the traditional utility index. These type of preferences always violate
transitivity in the traditional sense, although most satisfy a weaker version.
Evidence
Several experiments over both incentivized and hypothetical choices attest to the magnitude of this effect.
Experiments in
first price auction
A first-price sealed-bid auction (FPSBA) is a common type of auction. It is also known as blind auction. In this type of auction, all bidders simultaneously submit sealed bids so that no bidder knows the bid of any other participant. The highest bi ...
s show that by manipulating the feedback the participants expect to receive, significant differences in the average bids are observed. In particular, "Loser's regret" can be induced by revealing the winning bid to all participants in the auction, and thus revealing to the losers whether they would have been able to make a profit and how much could it have been (a participant that has a valuation of $50, bids $30 and finds out the winning bid was $35 will also learn that she could have earned as much as $15 by bidding anything over $35.) This in turn allows for the possibility of regret and if bidders correctly anticipate this, they would tend to bid higher than in the case where no feedback on the winning bid is provided in order to decrease the possibility of regret.
In decisions over lotteries, experiments also provide supporting evidence of anticipated regret. As in the case of first price auctions, differences in feedback over the resolution of the uncertainty can cause the possibility of regret and if this is anticipated, it may induce different preferences.
For example, when faced with a choice between $40 with certainty and a coin toss that pays $100 if the outcome is guessed correctly and $0 otherwise, not only does the certain payment alternative minimizes the risk but also the possibility of regret, since typically the coin will not be tossed (and thus the uncertainty not resolved) while if the coin toss is chosen, the outcome that pays $0 will induce regret. If the coin is tossed regardless of the chosen alternative, then the alternative payoff will always be known and then there is no choice that will eliminate the possibility of regret.
Anticipated regret versus experienced regret
Anticipated regret tends to be overestimated for both choices and actions over which people perceive themselves to be responsible.
People are particularly likely to overestimate the regret they will feel when missing a desired outcome by a narrow margin. In one study, commuters predicted they would experience greater regret if they missed a train by 1 minute more than missing a train by 5 minutes, for example, but commuters who actually missed their train by 1 or 5 minutes experienced (equal and) lower amounts of regret. Commuters appeared to overestimate the regret they would feel when missing the train by a narrow margin, because they tended to underestimate the extent to which they would attribute missing the train to external causes (e.g., missing their wallet or spending less time in the shower).
Applications
Besides the traditional setting of choices over lotteries, regret aversion has been proposed as an explanation for the typically observed overbidding in first price auctions, and the
disposition effect
The disposition effect is an anomaly discovered in behavioral finance. It relates to the tendency of investors to sell assets that have increased in value, while keeping assets that have dropped in value.
Hersh Shefrin and Meir Statman identified ...
, among others.
Minimax regret
The
minimax
Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When de ...
regret approach is to minimize the worst-case regret, originally presented by
Leonard Savage
Leonard Jimmie Savage (born Leonard Ogashevitz; 20 November 1917 – 1 November 1971) was an American mathematician and statistician. Economist Milton Friedman said Savage was "one of the few people I have met whom I would unhesitatingly call a g ...
in 1951.
The aim of this is to perform as closely as possible to the optimal course. Since the minimax criterion applied here is to the regret (difference or ratio of the payoffs) rather than to the payoff itself, it is not as pessimistic as the ordinary minimax approach. Similar approaches have been used in a variety of areas such as:
*
Hypothesis testing
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...
*
Prediction
A prediction (Latin ''præ-'', "before," and ''dicere'', "to say"), or forecast, is a statement about a future event or data. They are often, but not always, based upon experience or knowledge. There is no universal agreement about the exact ...
*
Economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...
One benefit of minimax (as opposed to expected regret) is that it is independent of the probabilities of the various outcomes: thus if regret can be accurately computed, one can reliably use minimax regret. However, probabilities of outcomes are hard to estimate.
This differs from the standard minimax approach in that it uses ''differences'' or ''ratios'' between outcomes, and thus requires interval or ratio measurements, as well as
ordinal measurement
Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to dependent and independent variables, variables. Psychologist Stanley Smith Stevens developed the best-known class ...
s (ranking), as in standard minimax.
Example
Suppose an investor has to choose between investing in stocks, bonds or the money market, and the total return depends on what happens to interest rates. The following table shows some possible returns:
The crude
maximin choice based on returns would be to invest in the money market, ensuring a return of at least 1. However, if interest rates fell then the regret associated with this choice would be large. This would be 11, which is the difference between the 12 which could have been received if the outcome had been known in advance and the 1 received. A mixed portfolio of about 11.1% in stocks and 88.9% in the money market would have ensured a return of at least 2.22; but, if interest rates fell, there would be a regret of about 9.78.
The regret table for this example, constructed by subtracting actual returns from best returns, is as follows:
Therefore, using a minimax choice based on regret, the best course would be to invest in bonds, ensuring a regret of no worse than 5. A mixed investment portfolio would do even better: 61.1% invested in stocks, and 38.9% in the money market would produce a regret no worse than about 4.28.
Example: Linear estimation setting
What follows is an illustration of how the concept of regret can be used to design a linear
estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the ...
.
In this example, the problem is to construct a linear estimator of a finite-dimensional parameter vector
from its noisy linear measurement with known noise covariance structure. The loss of reconstruction of
is measured using the
mean-squared error
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwe ...
(MSE). The unknown parameter vector is known to lie in an
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...
centered at zero. The regret is defined to be the difference between the MSE of the linear estimator that doesn't know the parameter
, and the MSE of the linear estimator that knows
. Also, since the estimator is restricted to be linear, the zero MSE cannot be achieved in the latter case. In this case, the solution of a convex optimization problem gives the optimal, minimax regret-minimizing linear estimator, which can be seen by the following argument.
According to the assumptions, the observed vector
and the unknown deterministic parameter vector
are tied by the linear model
:
where
is a known
matrix with
full column rank
In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. p. 48, § 1.16 This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the di ...
, and
is a zero mean random vector with a known covariance matrix
.
Let
:
be a linear estimate of
from
, where
is some
matrix. The MSE of this estimator is given by
:
Since the MSE depends explicitly on
it cannot be minimized directly. Instead, the concept of regret can be used in order to define a linear estimator with good MSE performance. To define the regret here, consider a linear estimator that knows the value of the parameter
, i.e., the matrix
can explicitly depend on
:
:
The MSE of
is
:
To find the optimal
,
is differentiated with respect to
and the derivative is equated to 0 getting
:
Then, using the
Matrix Inversion Lemma
In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury, says that the inverse of a rank-''k'' correction of some matrix (mathematics), matrix can be computed by doing a rank-''k'' correction to the ...
:
Substituting this
back into
, one gets
:
This is the smallest MSE achievable with a linear estimate that knows
. In practice this MSE cannot be achieved, but it serves as a bound on the optimal MSE. The regret of using the linear estimator specified by
is equal to
:
The minimax regret approach here is to minimize the worst-case regret, i.e.,
This will allow a performance as close as possible to the best achievable performance in the worst case of the parameter
. Although this problem appears difficult, it is an instance of
convex optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization probl ...
and in particular a numerical solution can be efficiently calculated. Similar ideas can be used when
is random with uncertainty in the
covariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
.
See also
*
Competitive regret
*
Decision theory
Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
*
Info-gap decision theory
Info-gap decision theory seeks to optimize robustness to failure under severe uncertainty,Yakov Ben-Haim, ''Information-Gap Theory: Decisions Under Severe Uncertainty,'' Academic Press, London, 2001.Yakov Ben-Haim, ''Info-Gap Theory: Decisions Unde ...
*
Loss function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...
*
Minimax
Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When de ...
*
Regret (emotion)
Regret is the emotion of wishing one had made a different decision in the past, because the consequences of the decision were unfavorable.
Regret is related to perceived opportunity. Its intensity varies over time after the decision, in regard ...
*
Wald's maximin model
In decision theory and game theory, Wald's maximin model is a non-probabilistic decision-making model according to which decisions are ranked on the basis of their worst-case outcomes – the optimal decision is one with the least bad worst outco ...
References
External links
* {{cite web, url=http://philosophy.hku.hk/think/strategy/decision.php, title=TUTORIAL G05: Decision theory, archive-url=https://web.archive.org/web/20150703104008/http://philosophy.hku.hk/think/strategy/decision.php, archive-date=3 July 2015
Choice modelling
Optimal decisions
Decision theory