Question

Why are the Maximin, Maximax, and the Minimax regret models? Not appropriate? When some conditions and probabilities have a moderate level of certainty or predictability?

Answer 1

Maximax (Optimist)

The maximax looks at the best that could happen under each action and then chooses the action with the largest value. They assume that they will get the most possible and then they take the action with the best case scenario. The maximum of the maximums or the "best of the best". This is the lotto player; they see large payoffs and ignore the probabilities.

Maximin (Pessimist)

The maximin person looks at the worst that could happen under each action and then choose the action with the largest payoff. They assume that the worst that can happen will, and then they take the action with the best worst-case scenario. The maximum of the minimums or the "best of the worst". This is the person who puts their money into a savings account because they could lose money at the stock market.

Minimax (Opportunist)

Minimax decision making is based on opportunistic loss. They are the kind that looks back after the state of nature has occurred and say "Now that I know what happened, if I had only picked this other action instead of the one I actually did, I could have done better". So, to make their decision (before the event occurs), they create an opportunistic loss (or regret) table. Then they take the minimum of the maximum. That sounds backward, but remember, this is a loss table. This similar to the maximin principle in theory; they want the best of the worst losses.

We can compare differences with the help of the following examples:-

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 maximizes 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 maximizes 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 minimize 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 minimizes the maximum regret. It is useful for a risk-neutral decision maker. Essentially, this is the technique for a 'sore loser' who do not wish to make the wrong decision.

'Regret' in this context is defined as the opportunity loss by 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 minimize that maximum regret and therefore supply 40 salads only.