A rectangular array of mn numbers arranged in n rows, each consisting of m columns, is said to contain a saddlepoint if there is a number that is both the minimum of its row and the maximum of its column. For instance, in the array 1 3 2 0 − 2 6 .5 12 3 the number 1 in the first row, first column is a saddlepoint. The existence of a saddlepoint is of significance in the theory of games. Consider a rectangular array of numbers as described previously and suppose that there are two individuals— A and B—who are playing the following game: 4 is to choose one of the numbers 1, 2,. .., n and B one of the numbers 1, 2,. . ., m. These choices are announced simultaneously, and if A chose i and B chose j. then A wins from B the amount specified by the number in the i th row, j th column of the array. Now suppose that the array contains a saddle point—say the number in the row r and column k call this number x r k . Now if player A chooses row r, then that player can guarantee herself a win of at least x r k (since x r k is the minimum number in the row r). On the other hand, if player B chooses column k, then he can guarantee that he will lose no more than x r k (since x r k is the maximum number in the column k). Hence, as A has a way of playing that guarantees her a win of x r k and as B has a way of playing that guarantees he will lose no more than x r k it seems reasonable to take these two strategies as being optimal and declare that the value of the game to player A is x r k . If the nm numbers in the rectangular array described are independently chosen from an arbitrary continuous distribution, what is the probability that the resulting array will contain a saddlepoint?
A rectangular array of mn numbers arranged in n rows, each consisting of m columns, is said to contain a saddlepoint if there is a number that is both the minimum of its row and the maximum of its column. For instance, in the array 1 3 2 0 − 2 6 .5 12 3 the number 1 in the first row, first column is a saddlepoint. The existence of a saddlepoint is of significance in the theory of games. Consider a rectangular array of numbers as described previously and suppose that there are two individuals— A and B—who are playing the following game: 4 is to choose one of the numbers 1, 2,. .., n and B one of the numbers 1, 2,. . ., m. These choices are announced simultaneously, and if A chose i and B chose j. then A wins from B the amount specified by the number in the i th row, j th column of the array. Now suppose that the array contains a saddle point—say the number in the row r and column k call this number x r k . Now if player A chooses row r, then that player can guarantee herself a win of at least x r k (since x r k is the minimum number in the row r). On the other hand, if player B chooses column k, then he can guarantee that he will lose no more than x r k (since x r k is the maximum number in the column k). Hence, as A has a way of playing that guarantees her a win of x r k and as B has a way of playing that guarantees he will lose no more than x r k it seems reasonable to take these two strategies as being optimal and declare that the value of the game to player A is x r k . If the nm numbers in the rectangular array described are independently chosen from an arbitrary continuous distribution, what is the probability that the resulting array will contain a saddlepoint?
Solution Summary: The author explains how the probability of a saddle point in an array of size n is calculated by using the following expression:
A rectangular array of mn numbers arranged in n rows, each consisting of m columns, is said to contain a saddlepoint if there is a number that is both the minimum of its row and the maximum of its column. For instance, in the array
1
3
2
0
−
2
6
.5
12
3
the number 1 in the first row, first column is a saddlepoint. The existence of a saddlepoint is of significance in the theory of games. Consider a rectangular array of numbers as described previously and suppose that there are two individuals— A and B—who are playing the following game: 4 is to choose one of the numbers 1, 2,. .., n and B one of the numbers 1, 2,. . ., m. These choices are announced simultaneously, and if A chose i and B chose j. then A wins from B the amount specified by the number in the
ith row, jth column of the array. Now suppose that the array contains a saddle point—say the number in the row r and column k call this number
x
r
k
. Now if player A chooses row r, then that player can guarantee herself a win of at least
x
r
k
(since
x
r
k
is the minimum number in the row r). On the other hand, if player B chooses column k, then he can guarantee that he will lose no more than
x
r
k
(since
x
r
k
is the maximum number in the column k). Hence, as A has a way of playing that guarantees her a win of
x
r
k
and as B has a way of playing that guarantees he will lose no more than
x
r
k
it seems reasonable to take these two strategies as being optimal and declare that the value of the game to player A is
x
r
k
. If the nm numbers in the rectangular array described are independently chosen from an arbitrary continuous distribution, what is the probability that the resulting array will contain a saddlepoint?
In a small office, there are m = 5 typists who need to use a single typewriter to complete their reports. Assume the time
each typist takes to prepare a report follows an exponential distribution with an average of 20 minutes per preparation
(A = 3 reports/hour), and the service time for the typewriter to type out a report also follows an exponential distribution,
averaging 30 minutes to complete a report (μ 2 reports/hour). Given that the number of typists is finite and all typists
=
share one typewriter, they will form a waiting queue.
(1). Describe this queuing system and explain how it fits the characteristics of the M/M/1/∞0/m model.
(2). Calculate the probability that any typist is using the typewriter at steady-state.
(3). Calculate the average number of typists waiting in the queue at steady-state.
(4). Considering the need to reduce waiting time, if an additional typewriter is introduced (turning into a two-server
system, or M/M/2/∞0/m model), analyze the expected impact,…
Can you tell the answer
Theorem 2.4 (The Hölder inequality)
Let p+q=1. If E|X|P < ∞ and E|Y| < ∞, then
.
EXY SEXY ≤ Xp Yq.
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