Regret (decision theory)
In decision theory, on making decisions under uncertainty—should information about the best course of action arrive after taking a fixed decision—the human emotional response of regret 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 with a powerful social and reputational component, and is central to how humans learn from experience and to the human psychology of risk aversion. Conscious anticipation of regret creates a feedback loop that transcends regret from the emotional realm—often modeled as mere human behavior—into the realm of the rational choice behavior that is modeled in decision theory.
Description
Regret theory is a model in
It incorporates a regret term in the
For independent lotteries and when regret is evaluated over the difference between utilities and then averaged over the all combinations of outcomes, the regret can still be transitive but for only specific form of regret functional. It is shown that only
Regret aversion is not only a theoretical economics model, but a cognitive bias occurring as a decision has been made to abstain from regretting an alternative decision. To better preface, regret aversion can be seen through fear by either commission or omission; the prospect of committing to a failure or omitting an opportunity that we seek to avoid.[7] Regret, feeling sadness or disappointment over something that has happened, can be rationalized for a certain decision, but can guide preferences and can lead people astray. This contributes to the spread of disinformation because things are not seen as one's personal responsibility.
Evidence
Several experiments over both incentivized and hypothetical choices attest to the magnitude of this effect.
Experiments in
In decisions over lotteries, experiments also provide supporting evidence of anticipated regret.[9][10][11] 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.[12][13] 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).[12]
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,[14] and the disposition effect,[15] among others.
Minimax regret
The
- Hypothesis testing
- Prediction
- Economics
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
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:
Return | Interest rates rise | Static rates | Interest rates fall | Worst return |
---|---|---|---|---|
Stocks | −4 | 4 | 12 | −4 |
Bonds | −2 | 3 | 8 | −2 |
Money market | 3 | 2 | 1 | 1 |
Best return | 3 | 4 | 12 |
The crude
The regret table for this example, constructed by subtracting actual returns from best returns, is as follows:
Regret | Interest rates rise | Static rates | Interest rates fall | Worst regret |
---|---|---|---|---|
Stocks | 7 | 0 | 0 | 7 |
Bonds | 5 | 1 | 4 | 5 |
Money market | 0 | 2 | 11 | 11 |
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 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
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
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
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 and in particular a numerical solution can be efficiently calculated.[17] Similar ideas can be used when is random with uncertainty in the covariance matrix.[18][19]
Regret in principal-agent problems
Camara, Hartline and Johnsen
Collina, Roth and Shao[21] improve their mechanism both in running-time and in the bounds for regret (as a function of the number of distinct states of nature).
See also
- Regret-free mechanism
- Competitive regret
- Decision theory
- Info-gap decision theory
- Loss function
- Minimax
- Wald's maximin model
References
- JSTOR 2232669.
- .
- ISBN 90-277-1420-7.
- ^ S2CID 36505167.
- hdl:10419/150148.
- .
- ^ "Why do we anticipate regret before we make a decision?". The Decision Lab.
- S2CID 51815774.
- .
- .
- S2CID 254978441.
- ^ S2CID 748553.
- S2CID 7524552.
- hdl:2142/28707.
- S2CID 153522835.
- .
- S2CID 16417895.
- S2CID 15596014.
- S2CID 16732469.
- S2CID 221640554.
- arXiv:2311.07754 [cs.GT].
External links
- "TUTORIAL G05: Decision theory". Archived from the original on 3 July 2015.