Maximax, maximin and minimax regret

Maximin, maximax and minimax regret are three approaches to decision making under uncertainty.

Illustration

Payoff tables show the payoff (profit or loss) for the range of possible outcomes based on two factors:

1. Different decision choices
2. Different possible real world scenarios

For example, suppose Geoffrey Ramsbottom is faced with the following pay-off table. He has to choose how many salads to make in advance each day before he knows the actual demand.

• His choice is between 40, 50, 60 and 70 salads.
• The actual demand can also vary between 40, 50, 60 and 70 with the probabilities as shown in the table - e.g. P(demand = 40) is 0.1.
• The table then shows the profit or loss - for example, if he chooses to make 70 but demand is only 50, then he will make a loss of $60.

The question is then which output level to choose.

Maximax

The maximax rule involves selecting the alternative that maximises the maximum payoff available.

This approach would be suitable for an optimist, or 'risk-seeking' investor, who seeks to achieve the best results if the best happens. The manager who employs the maximax criterion is assuming that whatever action is taken, the best will happen; he/she is a risk-taker. So, how many salads will Geoffrey decide to supply?

Looking at the payoff table, the highest maximum possible pay-off is $140. This happens if we make 70 salads and demand is also 70. Geoffrey should therefore decide to supply 70 salads every day.

Maximin

The maximin rule involves selecting the alternative that maximises the minimum pay-off achievable. The investor would look at the worst possible outcome at each supply level, then selects the highest one of these. The decision maker therefore chooses the outcome which is guaranteed to minimise his losses. In the process, he loses out on the opportunity of making big profits. 

This approach would be appropriate for a pessimist who seeks to achieve the best results if the worst happens.

So, how many salads will Geoffrey decide to supply? Looking at the payoff table,

• If we decide to supply 40 salads, the minimum pay-off is $80.
• If we decide to supply 50 salads, the minimum pay-off is $0.
• If we decide to supply 60 salads, the minimum pay-off is ($80).
• If we decide to supply 70 salads, the minimum pay-off is ($160).

The highest minimum payoff arises from supplying 40 salads. This ensures that the worst possible scenario still results in a gain of at least $80.

Minimax regret

The minimax regret strategy is the one that minimises the maximum regret. It is useful for a risk-neutral decision maker. Essentially, this is the technique for a 'sore loser' who does not wish to make the wrong decision.

'Regret' in this context is defined as the opportunity loss through having made the wrong decision.

To solve this a table showing the size of the regret needs to be constructed. This means we need to find the biggest pay-off for each demand row, then subtract all other numbers in this row from the largest number.

For example, if the demand is 40 salads, we will make a maximum profit of $80 if they all sell. If we had decided to supply 50 salads, we would achieve a nil profit. The difference or 'regret' between that nil profit and the maximum of $80 achievable for that row is $80. 

Regrets can be tabulated as follows : 

 The maximum regrets for each choice are thus as follows (reading down the columns):

• If we decide to supply 40 salads, the maximum regret is $60.
• If we decide to supply 50 salads, the maximum regret is $80.
• If we decide to supply 60 salads, the maximum regret is $160.
• If we decide to supply 70 salads, the maximum regret is $240.

A manager employing the minimax regret criterion would want to minimise that maximum regret, and therefore supply 40 salads only.

Note that the above techniques can be used even if we do not have probabilities. To calculate expected values, for example, we will need probabilities.

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