## Play Safe Strategies

In game theory, a player adopts a 'play safe' strategy if he looks for the worst possible outcome that could result from the choices he makes. He then picks the choice that results in the least worst option. We can determine the play safe strategy from a payoff matrix. The matrix below shows the outcomes for each player in a two player game. The outcomes are given as ordered pairs, with the first ordered pair for player A and the second for player B.

 Player B's Strategy Player A's Strategy 1 2 3 1 (2,1) (3,7) (4,9) 2 (3,0) (4,5) (5,9) 3 (7,5) (3,9) (2,2) 3 (1,4) (5,1) (6,1)

The worst outcomes for each player are shown below. For player A playing strategy 1, the worst outcome is 2, when player B plays strategy 1.

 Player B's Strategy Player A's Strategy 1 2 3 Worst Outcome for Player A 1 (2,1) (3,7) (4,9) 2 2 (3,0) (4,5) (5,9) 3 3 (7,5) (3,9) (2,2) 2 3 (1,4) (5,4) (6,1) 1 Worst Outcome for Player B 0 4 1

If player A chooses strategy 2, his worst outcome is a win of 3. A's play safe strategy is to choose option 2.

If player B chooses strategy 2, his worst outcome is a win of 4. B's play safe strategy is to choose option 2.

Notice that player A chose the strategy corresponding with the maximum of the minimum values in each row, while B chose the maximum of the minimum values in each column.