Question

[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 $c_{ij}$ and the payoff to the column player is $-c_{ij}$. 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 $x_1$, $x_2$, and $x_3$ be probabilities with which the column player will select the various strategies over many plays of the game. Thus $x_1 + x_2 + x_3 = 1$, $x_1$, $x_2$, $x_3 ge 0$. 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 $2x_1 - x_2$. If the row player selects row 2, her payoff is $-3x_1 + 2x_2 + x_3$. Wishing to minimize the maximum expected payoff to the row player, the column player should solve the following linear program: Minimize $z$ subject to $x_1 + x_2 + x_3 = 1$ $2x_1 - x_2 le z$ $-3x_1 + 2x_2 + x_3 le z$ $x_1, quad x_2, quad x_3 ge 0$ unrestricted. Transposing the variable $z$ to the left-hand-side, we get the column player's problem: Maximize $z$ subject to $x_1 + x_2 + x_3 = 1$ $z - 2x_1 + x_2 ge 0$ $z + 3x_1 - 2x_2 - x_3 ge 0$ $x_1, quad x_2, quad x_3 ge 0$ 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.

          [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 $c_{ij}$ and the
payoff to the column player is $-c_{ij}$. 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 $x_1$, $x_2$, and $x_3$ be probabilities with which the column player will select the
various strategies over many plays of the game. Thus $x_1 + x_2 + x_3 = 1$, $x_1$, $x_2$,
$x_3 ge 0$. 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 $2x_1 - x_2$. If the row player
selects row 2, her payoff is $-3x_1 + 2x_2 + x_3$. Wishing to minimize the maximum
expected payoff to the row player, the column player should solve the following
linear program:
Minimize $z$
subject to
$x_1 + x_2 + x_3 = 1$
$2x_1 - x_2 le z$
$-3x_1 + 2x_2 + x_3 le z$
$x_1, quad x_2, quad x_3 ge 0$
unrestricted.
Transposing the variable $z$ to the left-hand-side, we get the column player's
problem:
Maximize $z$
subject to
$x_1 + x_2 + x_3 = 1$
$z - 2x_1 + x_2 ge 0$
$z + 3x_1 - 2x_2 - x_3 ge 0$
$x_1, quad x_2, quad x_3 ge 0$
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.

[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 ge 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 le z
-3x1 + 2x2 + x3 le z
x1, quad x2, quad x3 ge 0
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 ge 0
z + 3x1 - 2x2 - x3 ge 0
x1, quad x2, quad x3 ge 0
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.

Added by Linda R.

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