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Play Safe Strategy

When playing safe each player looks for the worst outcome that could happen and then picks the choice that results in the least worst option. The option one chooses is known as a play.

 B plays 1 B plays 2 B plays 3 Worst outcome for A A plays 1 (8,2) (0,9) (7,3) 0 A plays 2 (3,6) (9,0) (2,7) 2 A plays 3 (1,7) (6,4) (8,1) 1 A plays 4 (4,2) (4,6) (5,1) 4 Worst outcome for B 2 0 1

Looking at the worst outcomes for A we can see that the highest number is 4. Therefore A should play 4 and regardless of what B decides to play A would be guaranteed a value of 4 units. Looking at B’s worst outcome we can see that the highest number is 2 so B should play 1.

Therefore if A plays 4 and B plays 1 the play safe strategy gained A 4 units and B 2 units.

In the above example both A and B would benefit if they collaborated, if they had conferred between them, they may have decided for A to play 3 and B to play 2 therefore the result is (6,4) and both players increase their winnings. This is because the above example is a non-zero sum game.

In real life however most games are zero-sum. If I win you loose and vice versa. This can be shown if in each cell the values equal zero. In a zero-sum game you only need to write A’s winnings as this also represents the negative of B’s winnings.

A pay-off matrix is always written from the row player’s (usually A) point of view unless stated.

The play safe strategies (in a zero-sum game) are for player A (row) the row maximin. This is the maximum of the minimum values. For player B (column) it is the column minimax. This is the minimum of the maximum values (as shown below this is because the values are from A’s point of view and so should be negated).

 3 -4 2 -4 -1 4 -2 -2 -3 1 4 -3 1 -1 1 -1 3 4 4

Above is a pay-off matrix (from A’s perspective) if A were to play 1 and B were to play 2 then B would win 4. If A were to play 1 and B were to play 1 A would win 3.

The values to the right of the table show the row minimum (the worst that can happen) and the values beneath the table show the column maximum (the worst that can happen for B, these are the highest positive numbers as they have to be negated to get B’s result).

The row maximin (maximum of the minimum) is row 4; with a pay-off of -1. Therefore A would play 4. The column minimax (the minimum of the maximum) is column 1 = 3; hence B would play 1.

Usually we are given the pay-off matrix in terms of player A if you are told to use this pay-off matrix to determine the pay-off matrix for B then you would transpose the matrix and then multiply each term by -1 of Player A’s matrix.

Below is a pay-off matrix from A’s perspective; find B’s pay-off matrix:

 4 -2 -5 3

B’s pay off-matrix therefore is:

 -4 -5 -2 -3

Another example (note the transposition; the values in the column switch to become the values in the rows. Therefore one can only find another perspective of a matrix if the matrix contains the same number of options, e.g. A can play up to 5 and so can B).

 B plays 1 B plays 2 B plays 3 B plays 4 A plays 1 7 2 -3 5 A plays 2 4 -1 1 3 A plays 3 -2 5 2 -1 A plays 4 3 -3 -4 2

From B’s perspective the matrix would look like:

 A plays 1 A plays 2 A plays 3 A plays 4 B plays 1 -7 -4 2 -3 B plays 2 -2 1 -5 -3 B plays 3 3 -1 -2 4 B plays 4 -5 -3 1 -2

Note; you also have to multiply the cells by -1.