Regret (decision theory)

Regret theory is a model in theoretical economics simultaneously developed in 1982 by Graham Loomes and Robert Sugden,[1] David E. Bell,[2] and Peter C. Fishburn.[3] Regret theory models choice under uncertainty taking into account the effect of anticipated regret. Subsequently, several other authors improved upon it.[4]

It incorporates a regret term in the utility function 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,[5] although most satisfy a weaker version.[4]

Several experiments over both incentivized and hypothetical choices attest to the magnitude of this effect.

Experiments in first price auctions show that by manipulating the feedback the participants expect to receive, significant differences in the average bids are observed.[6] 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 he or 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.[7][8][9] 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.

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,[12] and the disposition effect,[13] among others.

The minimax regret approach is to minimize the worst-case regret, originally presented by Leonard Savage in 1951.[14] 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
• 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 ordinal measurements (ranking), as in standard minimax.

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 ${\displaystyle x}$ from its noisy linear measurement with known noise covariance structure. The loss of reconstruction of ${\displaystyle x}$ is measured using the mean-squared error (MSE). The unknown parameter vector is known to lie in an ellipsoid ${\displaystyle E}$ centered at zero. The regret is defined to be the difference between the MSE of the linear estimator that doesn't know the parameter ${\displaystyle x}$, and the MSE of the linear estimator that knows ${\displaystyle x}$. 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 ${\displaystyle y}$ and the unknown deterministic parameter vector ${\displaystyle x}$ are tied by the linear model

${\displaystyle y=Hx+w}$

where ${\displaystyle H}$ is a known ${\displaystyle n\times m}$ matrix with full column rank ${\displaystyle m}$, and ${\displaystyle w}$ is a zero mean random vector with a known covariance matrix ${\displaystyle C_{w}}$.

Let

${\displaystyle {\hat {x}}=Gy}$

be a linear estimate of ${\displaystyle x}$ from ${\displaystyle y}$, where ${\displaystyle G}$ is some ${\displaystyle m\times n}$ matrix. The MSE of this estimator is given by

${\displaystyle MSE=E\left(||{\hat {x}}-x||^{2}\right)=Tr(GC_{w}G^{*})+x^{*}(I-GH)^{*}(I-GH)x.}$

Since the MSE depends explicitly on ${\displaystyle x}$ 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 ${\displaystyle x}$, i.e., the matrix ${\displaystyle G}$ can explicitly depend on ${\displaystyle x}$:

${\displaystyle {\hat {x}}^{o}=G(x)y.}$

The MSE of ${\displaystyle {\hat {x}}^{o}}$ is

${\displaystyle MSE^{o}=E\left(||{\hat {x}}^{o}-x||^{2}\right)=Tr(G(x)C_{w}G(x)^{*})+x^{*}(I-G(x)H)^{*}(I-G(x)H)x.}$

To find the optimal ${\displaystyle G(x)}$, ${\displaystyle MSE^{o}}$ is differentiated with respect to ${\displaystyle G}$ and the derivative is equated to 0 getting

${\displaystyle G(x)=xx^{*}H^{*}(C_{w}+Hxx^{*}H^{*})^{-1}.}$

Then, using the Matrix Inversion Lemma

${\displaystyle G(x)={\frac {1}{1+x^{*}H^{*}C_{w}^{-1}Hx}}xx^{*}H^{*}C_{w}^{-1}.}$

Substituting this ${\displaystyle G(x)}$ back into ${\displaystyle MSE^{o}}$, one gets

${\displaystyle MSE^{o}={\frac {x^{*}x}{1+x^{*}H^{*}C_{w}^{-1}Hx}}.}$

This is the smallest MSE achievable with a linear estimate that knows ${\displaystyle x}$. 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 ${\displaystyle G}$ is equal to

${\displaystyle R(x,G)=MSE-MSE^{o}=Tr(GC_{w}G^{*})+x^{*}(I-GH)^{*}(I-GH)x-{\frac {x^{*}x}{1+x^{*}H^{*}C_{w}^{-1}Hx}}.}$

The minimax regret approach here is to minimize the worst-case regret, i.e., ${\displaystyle \sup _{x\in E}R(x,G).}$ This will allow a performance as close as possible to the best achievable performance in the worst case of the parameter ${\displaystyle x}$. Although this problem appears difficult, it is an instance of convex optimization and in particular a numerical solution can be efficiently calculated.[15] Similar ideas can be used when ${\displaystyle x}$ is random with uncertainty in the covariance matrix.[16][17]

• Competitive regret
• Decision theory
• Info-gap decision theory
• Loss function
• Minimax
• Regret (emotion)
• Wald's maximin model

• "TUTORIAL G05: Decision theory". Archived from the original on 3 July 2015.