The mini-max regret choice

An alternative is to calculate the ``regret'' associated with making the ``wrong'' choice under each possible outcome. For example, if we knew that the actual outcome would be low damage, the best choice would be to enact moderate pollution controls, because it's the least expensive. We'd lose 5 units if we chose either lax or stringent regulation. The ``regret'' matrix is shown in Table 2. Each entry is calculated by identifying the smallest number in a column and subtracting it from all of the numbers in that column.

Table 2: ``Regrets'' associated with different combinations of regulation and forest damage (based on Table 1).

	No damage	Low damage	High damage
Stringent control	$9-0=9$	$10-5=5$	$11-11=0$
Moderate control	$1-0=1$	$5-5=0$	$14-11=3$
Lax control	$0-0=0$	$10-5=5$	$30-11=19$

Now we might pick the alternative that makes our maximum ``regret'' as small as possible, i.e., the one the minimizes our maximum ``regret.'' Looking at Table 2, it's easy to see that (a) the maximum regret for the stringent control choice is 9, for the moderate control choice is 3, and for the lax control choice is 19 and that (b) the choice that minimizes the maximum ``regret'' is moderate control.

It's vital to note that neither mini-max on cost nor mini-max on ``regret'' can be said to be the ``right'' choice. They balance tradeoffs in different ways related to how risk averse you might be. It's also important to notice that even if you can estimate probabilities and calculate the expected cost associated with each action, you might still prefer a mini-max choice based on either cost or ``regret''.