Check for and eliminate dominated strategies and remove them.
Check for a stable solution (if max row min = min column max).
If no stable solution, find the optimal mixed strategy:
Let player 1’s option
This means player 1’s option
Create an expression for each of player 2’s options.
Find the point(s) where all the expressions are equal.
Of these points, find the one where the lowest lines at that value of
Solve these two lines simultaneously to find
Player 2 plays D Player 2 plays E Player 1 plays A 1 2 Player 1 plays C 3 -1
player 2 plays randomly between D and E.
Let player 1 play A with probability
They therefore play C with probability
If player 2 plays D, player 1 wins
If player 2 plays E, player 1 wins
Solve simultaneously:
Answer : player 1 should play A with probability
Remember that player 1 must play these strategies as randomly as possible.
That’s because they don’t want player 2 to be able to predict what they will
do.
Player 2 plays C Player 2 plays D Player 2 plays E Player 1 plays A 0 -1 2 Player 1 plays B 2 3 -2
Add the row minima and column maxima:
Player 2 plays C Player 2 plays D Player 2 plays E Row minima Player 1 plays A 0 -1 2 -1 Player 1 plays B 2 3 -2 -2 Column maxima 2 3 2
Let player 1 play A with probability
They therefore play B with probability
If player 2 plays C, player 1 wins
If player 2 plays D, player 1 wins
If player 2 plays E, player 1 wins
Drawing the graph, the point where all the lines are the highest is at the
intersection of
Solve simultaneously:
Answer : they should play A with probability