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 nonzero sum
game.
In real life however most games are zerosum. If I win you
loose and vice versa. This can be shown if in each cell the values equal zero.
In a zerosum game you only need to write A’s winnings as this also represents
the negative of B’s winnings.
A payoff matrix is always written from the row player’s
(usually A) point of view unless stated.
The play safe strategies (in a zerosum 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 payoff 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
payoff 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 payoff matrix in terms of player A
if you are told to use this payoff matrix to determine the payoff matrix for
B then you would transpose the matrix and then multiply each term by 1 of
Player A’s matrix.
Below is a payoff matrix from A’s perspective; find B’s
payoff matrix:
B’s pay offmatrix therefore is:
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.
