Minimax criterion

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Minimax criterion
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Min-max criterion - is a decision-making criterion presented in 1954 by Leonard Savage. This criterion minimizes the expected loss associated with making worse than optimal decision, for a given state of nature. In the decision-making process such strategy should be selected, for which the relative loss is the smallest.


Fig.1. Minimax criterion procedure (example)

According to the criterion you must first find a matrix of relative loss (a matrix of grief). Loss is defined as the difference between the greatest win, for a particular state of nature, and winning that corresponds to the predicted results of current decision. The calculations are made in columns in each of them going to the largest value, and then typed in the array of grief, the difference between the highest value and the currently calculated. This process can be expressed in following equation\[y_{ij} = \underset{i}{max}\{a_{ij}\} - a_{ij}\]

The next step is, specifying maximum loss for each strategy, and at the end selection of the strategy for which the maximum loss, will be the largest. Model\[v = \underset{i}{min}\{\underset{j}{max}\{y_{ij}\}\}\]


Example: we have an array of payments from the four possible decisions and three possible states:

\( \underset{Strategies}{} \diagdown \overset{States\ of\ nature} {} \) I II III
T1 200 100 120
T2 0 300 −200
T3 0 300 500
T4 −100 200 0

On this basis cost matrix of relative loss is calculated. Each value in column is subtracted from max values in scenario I/II/III:

\( \underset{Strategies}{} \diagdown \overset{States\ of\ nature} {} \) I II III
T1 0 200 380
T2 200 0 700
T3 200 0 0
T4 300 100 500

We are looking for the maximum losses for each strategy, and then select the smallest:

Strategies \(\underset{j}{max}\{a_{ij}\}\)
T1 380
T2 700
T3 200
T4 500

Summing up: According to the Min-max manager should choose a strategy T3.