## Minimax Regret Decision Rule

minimax regret criterion

Decision choice method that minimizes the maximum possible regret (opportunity loss) associated with a wrong decision after the fact opportunity loss

Difference between a given payoff and the highest possible payoff for the resulting state of nature

A second useful decision criterion focuses on the opportunity loss associated with a decision rather than on its worst possible outcome. This decision rule, known as the minimax regret criterion, states that the decision maker should minimize the maximum possible regret (opportunity loss) associated with a wrong decision after the fact. This criterion instructs one to minimize the difference between possible outcomes and the best outcome for each state of nature.

To illustrate this decision technique, the concept of opportunity loss, or regret, must be examined in greater detail. In game theory, opportunity loss is defined as the difference between a given payoff and the highest possible payoff for the resulting state of nature. Opportunity losses result because returns actually received under conditions of uncertainty are frequently lower than the maximum return that would have been possible had perfect knowledge been available beforehand.

Table 14.5 shows the opportunity loss or regret matrix associated with U-Pump's gasoline-pricing problem. It was constructed by finding the maximum payoff for a given state of nature and then subtracting from this amount the payoffs that would result from various decision alternatives. Opportunity loss is always a positive figure or zero, because each alternative payoff is subtracted from the largest payoff possible in a given state of nature. For example, if U-Pump's competitor reduced its price, the best possible decision for that state of nature would be for U-Pump to have also reduced prices. After the fact, U-Pump would have no regrets had it done so. Should U-Pump maintain its current price, the firm would experience a \$1,500 opportunity loss, or regret. To calculate this amount, subtract the \$1,000 payoff associated with U-Pump's maintaining its current price despite a competitor price reduction from the \$2,500 payoff that it would have received from matching the competitor's price reduction. Similarly, if U-Pump would reduce its price while its competitor maintains the current price, U-Pump would experience a \$2,000 opportunity loss or regret after the fact.

The minimax regret criterion would cause U-Pump to maintain the current retail price of gasoline because this decision alternative minimizes the maximum regret, or opportunity loss. The maximum regret in this case is limited to the \$1,500 loss that would result if the competitor reduced its current price. If U-Pump were to reduce its price while the competitor maintained its current price, U-Pump's opportunity loss would be \$2,000 per week, \$500 more than the maximum regret from U-Pump maintaining its current price.