Prerequisites to find the optimal mixed strategy

Prerequisites to find the optimal mixed strategy

1. Check for and eliminate dominated strategies and remove them.
2. Check for a stable solution (if max row min = min column max).
3. If no stable solution, find the optimal mixed strategy:

Finding the optimal mixed strategy

• Let player 1’s option A be played with probability p.
• This means player 1’s option B will be played with probability 1-p
• 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 P are the highest they can be.
• Solve these two lines simultaneously to find p.

Find the probabilities for a mixed strategy fro player 1, assuming player 2 plays randomly

• player 2 plays randomly between D and E.
• Let player 1 play A with probability p.
• They therefore play C with probability 1-p.
• If player 2 plays D, player 1 wins 1p+3(1-p)=3-2p.
• If player 2 plays E, player 1 wins 2p+(-1)(1-p)=3p-1.
• Solve simultaneously:
  • 3-2p=3p-1
  • 5p=4
  • p=4/5.
• Answer: player 1 should play A with probability 4/5 and C with probability 1/5. p=0.8.

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.

Find the probabilities for a mixed strategy for player 1, assuming player 1 plays randomly (2xn)

Add the row minima and column maxima:

• Let player 1 play A with probability p.
• They therefore play B with probability 1-p.
• If player 2 plays C, player 1 wins 0p+2(1-p)=2-2p.
• If player 2 plays D, player 1 wins -1p+3(1-p)=3-4p.
• If player 2 plays E, player 1 wins 2p+(-2)(1-p)=4p-2.
• Drawing the graph, the point where all the lines are the highest is at the intersection of 2-2p and 4p-2.
• Solve simultaneously:
  • 2-2p=4p-2
  • 4=6p
  • p=2/3.
• Answer: they should play A with probability 2/3 and B with probability 1/3. p=0.6667.