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

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

Player 2 plays DPlayer 2 plays E
Player 1 plays A12
Player 1 plays C3-1

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)

Player 2 plays CPlayer 2 plays DPlayer 2 plays E
Player 1 plays A0-12
Player 1 plays B23-2

Add the row minima and column maxima:

Player 2 plays CPlayer 2 plays DPlayer 2 plays ERow minima
Player 1 plays A0-12-1
Player 1 plays B23-2-2
Column maxima232

flashcards

QuestionAnswer
Player 1 plays A with probability pPlayer 1 therefore plays B (or C) with probability 1-p
What is the first step before finding an optimal mixed strategy?Check for and eliminate dominated strategies, then check for a stable solution (max row min = min column max).
When should you find the optimal mixed strategy?When there is no stable solution (max row min ≠ min column max).
How do you create expressions for each of player 2’s options?For each of player 2’s options, multiply the payoff when player 1 plays A by p plus the payoff when player 1 plays B (or C) by 1-p, then simplify.
How do you find the value of p for player 1’s mixed strategy?Set the expressions for player 2’s options equal to each other at the point where the lowest lines at that p are the highest possible, then solve the two selected equations simultaneously.
In a 2 \times 2 game with payoffs: (A,D)=1, (A,E)=2, (C,D)=3, (C,E)=-1, how do you find player 1’s optimal p?Set 1p+3(1-p) = 2p+(-1)(1-p) → 3-2p=3p-1 → 5p=4 → p=4/5 (play A with 0.8, C with 0.2).
Why must player 1 play their mixed strategy as randomly as possible?So player 2 cannot predict what they will do.
In a 2 \times 3 game with payoffs: (A,C)=0, (A,D)=-1, (A,E)=2, (B,C)=2, (B,D)=3, (B,E)=-2, what are the row minima and column maxima?Row minima: -1 (row A), -2 (row B). Column maxima: 2 (col C), 3 (col D), 2 (col E). No stable solution because max row min (-1) ≠ min column max (2).
For the 2 \times 3 game above, write the expressions for player 2’s options in terms of p (player 1 plays A with p).Player 2 plays C: 2-2p. D: 3-4p. E: 4p-2.
For the 2 \times 3 game above, which two lines intersect at the optimal p?2-2p and 4p-2 (the point where the lowest lines are the highest).
Solve 2-2p=4p-2 to find p for the 2 \times 3 game.2-2p=4p-2 → 4=6p → p=2/3 (play A with 2/3, B with 1/3).