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?
A particular two-player game starts with a pile of diamonds and a pile of rubies. Onyour turn, you can take any number of diamonds, or any number of rubies, or an equalnumber of each. You must take at least one gem on each of your turns. Whoever takesthe last gem wins the game. For example, in a game that starts with 5 diamonds and10 rubies, a game could look like: you take 2 diamonds, then your opponent takes 7rubies, then you take 3 diamonds and 3 rubies to win the game.You get to choose the starting number of diamonds and rubies, and whether you gofirst or second. Find all starting configurations (including who goes first) with 8 gemswhere you are guaranteed to win. If you have to let your opponent go first, what arethe starting configurations of gems where you are guaranteed to win? If you can’t findall such configurations, describe the ones you do find and any patterns you see.
Recall the game of chess is played on an 8 by 8 square board and a king can move from a given square to any adjacent square vertically, horizontally or diagonally. What is the maximum number of kings that can be placed on a chessboard without any two of them attacking each other (i.e. being able to move to the others’ square)?
A pet store sells three different starter kits for 10-gallon aquariums.
The accompanying chart shows the contents of each kit. The store
has 72 filters, 144 pounds of gravel, and 54 packages of fish food
available. If the store makes as many of kit I as kits II and
Ill together, how many of each kit should be created to
maximize profit?
The pet store should create
(Type integers or decimals.)
of kit 1,
of kit II, and
of kit III.
Filters
Gravel (pounds)
Fish food (packages)
Profit
Kit I
1
3
1
8
Kit II Kit III
2
2
0
13
132
17
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.