Decision Making Under Regret 9

1. Core Concept Overview & Intuition

Decision making under regret is the branch of decision analysis in which the decision maker does not merely ask "which action yields the largest payoff?" but instead asks "how much would I regret choosing this action if a particular state of nature turned out to be true?" Regret is the shortfall between the payoff actually obtained from a chosen action and the best payoff that could have been obtained in that same state of nature had the correct action been known in advance. The central intuition is that a good decision is one that keeps the worst possible regret as small as possible, rather than one that chases the largest possible gain.

This lesson sits within the broader decision-making environment. A decision problem consists of selecting the best of the available alternatives or actions, the possible events (states or outcomes of a random process), the probabilities of these events, and the criteria (decision rule) according to which the best alternative is selected. The common elements of every such problem are alternative courses of action and uncertainty. Uncertainty arises when the outcome of some action is not known in advance, and when there are many possible outcomes of events, also called states of nature, one cannot predict with certainty what will happen. Decision analysis falls under two categories: decision making under uncertainty and decision making under risk. Regret analysis belongs to the uncertainty setting, where probabilities of the states of nature are not used to weight the outcomes; the criterion operates directly on the payoff structure.

The pedagogical purpose of the lesson is to develop the ability to construct a regret table from a payoff table and to apply a conservative decision rule to it. The lesson learning outcomes are therefore stated as follows.

> Lesson Learning Outcomes. At the end of this lesson the learner will be able to: (i) explain the notion of regret in a decision problem; (ii) construct a regret (opportunity-loss) table from a given payoff table; (iii) apply the minimax regret criterion to select an optimal action; and (iv) interpret the resulting decision in the language of the decision-making environment.

The assessment component of the lesson tests exactly these skills: given a payoff table with alternatives as rows and states of nature as columns, the learner must compute the column maxima, form the regret entries, identify the maximum regret in each row, and then select the action whose maximum regret is smallest. The assessment is deliberately computational and interpretive, requiring both the arithmetic of the regret table and a written justification of the chosen action.

A useful way to build intuition is to contrast the regret criterion with the more familiar optimistic and pessimistic rules. The optimistic (maximax) rule selects the action with the largest best payoff; the pessimistic (maximin) rule selects the action with the largest worst payoff. The minimax regret rule is a third, distinct posture: it selects the action whose worst shortfall from the ideal is smallest. Regret therefore converts a payoff problem into a loss problem, and the decision maker then behaves conservatively with respect to loss. This is why the regret table is sometimes called the opportunity-loss table: each entry measures the opportunity forgone by not having chosen the best action for that state of nature.

The credit-management scenario from the module illustrates the decision environment in which such tables arise. A credit manager must decide whether to extend credit to a potential new customer. The decision alternatives are to extend credit and not to extend credit; the states of nature are the actual credit record of the customer, classified as poor risk, average risk, or good risk. The payoff table records the monetary consequence of each alternative under each state of nature, with the prior probabilities of the states recorded beneath the table. In the uncertainty setting these probabilities are not used to weight the payoffs; the regret criterion works from the payoff entries alone. The same structure, with the same three states of nature, is the natural vehicle for a regret analysis, and it is the structure used in the worked exemplar of Section 4.

2. Mathematical Formalization & Core Definitions

The formal apparatus of regret analysis is built from a finite decision matrix. The following definitions fix the notation used throughout the lesson.

> Definition (Decision Matrix). Let $A={a_1,a_2,\dots,a_m}$ be the set of $m$ alternative courses of action and let $S={s_1,s_2,\dots,s_n}$ be the set of $n$ states of nature. A payoff table is a matrix $P=(p_{ij})$ of order $m\times n$, where $p_{ij}$ denotes the payoff received when action $a_i$ is chosen and state $s_j$ occurs.

> Definition (Column Maximum). For each state of nature $s_j$, the column maximum is the largest payoff attainable in that state over all actions:

$$
M_j=\max_{1\le i\le m}p_{ij}.
$$

> Definition (Regret). The regret associated with action $a_i$ under state $s_j$ is the shortfall between the column maximum and the payoff actually received:

$$
r_{ij}=M_j-p_{ij}.
$$

> The regret matrix is $R=(r_{ij})$ of order $m\times n$. Each entry satisfies $r_{ij}\ge 0$, and in every column $j$ at least one entry equals zero, attained by any action that achieves the column maximum $M_j$.

> Definition (Maximum Regret of an Action). The maximum regret of action $a_i$ is the largest regret that action can produce across all states of nature:

$$
R_i=\max_{1\le j\le n}r_{ij}.
$$

> Definition (Minimax Regret Criterion). The minimax regret criterion selects the action whose maximum regret is smallest:

$$
a^{*}=\arg\min_{1\le i\le m}R_i.
$$

> The value $R^{*}=\min_{1\le i\le m}R_i$ is the minimax regret of the decision problem.

The regret matrix has a simple structural property that is worth stating explicitly because it drives every calculation. Since $M_j$ is the maximum of column $j$, the regret entries are non-negative, and the action that attains $M_j$ has zero regret in that column. Consequently the regret table can be constructed column by column: subtract every entry of a column from that column's maximum. No row or column totals are required, and no probabilities enter the computation.

The relationship between the payoff table and the regret table can be summarized as follows. The payoff table records gains; the regret table records forgone gains. The minimax regret criterion is applied to the regret table exactly as the maximin criterion is applied to the payoff table, with the direction of preference reversed because regret is a loss. This duality is the reason the criterion is sometimes described as a conservative rule: it protects the decision maker against the worst possible shortfall rather than maximizing the best possible gain.

The lesson's assessment is framed around this computational chain. The learner is given a payoff table, must compute the column maxima, must form the regret entries by subtraction, must identify the row maxima of the regret table, and must then select the action with the smallest row maximum. Each step is a direct application of one of the definitions above, and the final answer is a single action together with its minimax regret value.