The maximin criterion is optimistic while the maximax criterion is pessimistic

The Hurwicz Criterion, presented in in 1951, is probably the earliest novel contribution to the field of economics for which Leo has been recognized. It provides a formula for balancing pessimism and optimism in decision-making under uncertainty – that is, when future conditions are to some extent unknown. A defining feature of the Hurwicz Criterion is that it allows the decision maker to simultaneously take into account both the best and the worst possible outcomes. To do this, the decision maker chooses a “coefficient of pessimism”, called alpha (α), which is a decimal number between 0 and 1. This number determines the emphasis on the worst possible outcome. Then the number (1-α) determines the emphasis to be placed on the best outcome. So, if the coefficient of pessimism is .4, then the emphasis on the best outcome will be .6.

If alpha determines the emphasis to be placed on the best outcome, it may be called a “coefficient of optimism.” Either a “coefficient of pessimism” or a “coefficient of optimism” may also be called a “coefficient of realism.”

This contrasts with other approaches, such as:

• Maximin (pessimistic), which looks only at the worst possible result in each scenario, and chooses the “best of the worst”.
• Maximax (optimistic), which looks only at the best possible result in each scenario, and chooses the “best of the best”

The Hurwicz Criterion is sometimes confused with Minimax Regret, which compares what I actually did with what I would have done if I could have predicted the future. Another way of putting this is that Minimax Regret looks at the maximum possible regret: the maximum difference, for each scenario, between what I actually did and what I “coulda-shoulda-woulda” done. It then takes the path that minimizes potential regret.

The table below shows assumed “pay-offs” (economic results expressed in monetary units such as dollars) for building either no reservoir, a small reservoir or a large reservoir under three scenarios: low, medium and high climate-change impacts. (I got the idea for this example from Green and Weatherhead, “Coping with climate change uncertainty for adaptation planning“. However, my example is hypothetical, is mine alone, and does not make use of their data or the novel strategy that they propose.) For my hypothetical example, we’re assuming that the pay-offs are known. What isn’t known is the degree to which climate change will impact the activities, such as agriculture, that would be supported by the reservoir. The numbers in red represent the result that should be chosen under each approach.

“Optimistic” in this context means “maximizing return on investment.” It does not mean a belief that “all will be well.” For instance, a large investment in insurance may give the best return in case of a catastrophe. This is, in fact, my thinking in the example below.

Taking the different approaches one at a time, from right to left in the table:

• Using Maximin, the option that has the largest minimum is a small reservoir, with a minimum pay-off of 600 (which will occur under low impacts).
• Under the Hurwicz Criterion, with a coefficient of pessimism of .4 (slightly optimistic), the large reservoir is built, based on a minimum pay-off of 400, a maximum pay-off of 800, and a Hurwicz Criterion pay-off of 640 (which “beats” -20 and 630).
• Under the Hurwicz Criterion, with a coefficient of pessimism of .5 (neutral), the small reservoir is built, based on a minimum pay-off of 600, a maximum pay-off of 650, and a Hurwicz Criterion pay-off of 625 (which “beats” -25 and 600).
• Under the Hurwicz Criterion, with a coefficient of pessimism of .6 (slightly pessimistic), the small reservoir is built, again based on a minimum pay-off of 600 and a maximum pay-off of 650, but this time with a Hurwicz Criterion pay-off of 620 (which “beats” -30 and 560).
• Under Maximax, the large reservoir is built, because it has the highest possible pay-off of any option in the table, namely 800. This pay-off occurs under high climate change impacts, so this choice reflects “optimism” only in the sense of maximizing potential return on investment in the limited context of building (or not building) a reservoir.
• Under Minimax Regret, the small reservoir is built, because the largest disparity between actual and “coulda-shoulda-woulda” for the small reservoir is 150 (occurring under high impacts, where the small reservoir pay-off is 650 and the large reservoir pay-off would have been 800). Potential regrets for other scenarios are bigger. For example, if we don’t build a reservoir, then under the high impacts scenario, the potential regret is a whopping 850 (the difference between -50 and 800). The large reservoir has less potential for regret, but still more than 150: The largest “spread” there is 200, occurring in the low impacts scenario.

Table of pay-offs for building a reservoir under differing climate scenarios with Hurwicz Criterion alpha = coefficient of pessimism

The paper in which the Hurwicz Criterion was originally stated is: “The Generalised Bayes Minimax Principle: A Criterion for Decision Making Under Uncertainty,” Cowles Commission Discussion Paper 355, February 8, 1951. 7p.

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What is the maximin criterion?