7. For the two-person game with matrix $\begin{pmatrix} 3 & -2 \\ 4 & -5 \\ -3 & 6 \end{pmatrix}$, give the LP formulation for finding the optimal mixed strategy for player I. Clearly indicate the meanings of any variables introduced.
Added by Ashley H.
Close
Step 1
The objective is to maximize the minimum expected payoff for player I. Let $v$ be the minimum expected payoff. Then the LP formulation is: Maximize $v$ subject to: $3x_1 + 4x_2 - 3x_3 \ge v$ $-2x_1 - 5x_2 + 6x_3 \ge v$ $x_1 + x_2 + x_3 = 1$ $x_1, x_2, x_3 \ge 0$ Show more…
Show all steps
Your feedback will help us improve your experience
Akash M and 82 other Algebra educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
6.27 Two players are involved in a competitive game. One player, called the row player, has two strategies available; the other player, called the column player, has three strategies available. If the row player selects strategy i and the column player selects strategy j, the payoff to the row player is cij and the payoff to the column player is -cij. Thus, the column player loses what the row player wins and vice versa; this is a two-person zero-sum game. The following matrix gives the payoffs to the row player: Let x1, x2, and x3 be probabilities with which the column player will select the various strategies over many plays of the game. Thus, x1 + x2 + x3 = 1, x1, x2, x3 >= 0. If the column player applies these probabilities to the selection of her strategy for any play of the game, consider the row player's options. If the row player selects row 1, then her expected payoff is 2x1 - x2. If the row player selects row 2, her payoff is -3x1 + 2x2 + x3. Wishing to minimize the maximum expected payoff to the row player, the column player should solve the following linear program: Minimize z subject to x1 + x2 + x3 = 1 2x1 - x2 <= z -3x1 + 2x2 + x3 <= z x1, x2, x3 >= 0 z unrestricted. Transposing the variable z to the left-hand-side, we get the column player's problem: Maximize z subject to x1 + x2 + x3 = 1 z - 2x1 + x2 >= 0 z + 3x1 - 2x2 - x3 >= 0 x1, x2, x3 >= 0 z unrestricted. a. Give the dual of this linear program. b. Interpret the dual problem in Part (a). (Hint: Consider the row player's problem.) c. Solve the dual problem of Part (a). (Hint: This problem may be solved graphically.) d. Use the optimal dual solution of Part (c) to compute the column player's probabilities. e. Interpret the complementary slackness conditions for this two-person zero-sum game.
Akash M.
Player I secretly chooses one of the numbers 1, 2, and 3, and player II tries to guess which. If player II guesses correctly, he wins 2; otherwise, he loses the absolute value of the difference between player I's choice and his guess. A) Set up the payoff matrix for player II as a column player. B) Set up the LP problem for the column player from the payoff matrix. C) Solve the LP problem.
Sri K.
Find: (a) the optimal mixed row strategy; (b) the optimal mixed column strategy, and (c) the expected value of the game in the event that each player uses his or her optimal mixed strategy. $$ P=\left[\begin{array}{ll} -2 & -1 \\ -1 & -3 \end{array}\right] $$
Matrix Algebra and Applications
Game Theory
Recommended Textbooks
Elementary and Intermediate Algebra
Algebra and Trigonometry
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD