Finite maths

SCHAUM’S OUTLlNE OF THEORY AND PROBLEMS OF FINITE MATHE RIATIC S BY SEYMOUR LIPSCHUTZ, Ph.D Associate Professor of Mathematics Temple University SC€IAUMfS OUTLINE SERIES McGRAW-HILL BOOK COMPANY New York, St Louis San Francisco, Toronto, Sydne) Copyright @ 1966 b j McGraw-Hill, Inc All Rights Reserved Printed in the United States of America No p a r t of this publication may be reproduced stored in a rerrieval sy%tem, or transmitted, in any form or by any means electronic, mechanical, photocopying, recording or otherwise, without the prior written permissioii of the imblisher 579hT SHSH Finite mathematics has in recent years become an integral part of the mathematical background necessary for such diverse fields as biology, chemistry, economics, psychology, sociology, education, political science, business and engineering This book, in presenting the more essential material, is designed for use as a supplement to all current standard texts or as a textbook for a formal course in finite mathematics The material has been divided into twenty-five chapters, since the logical arrangement is thereby not disturbed while the usefulness as a text and reference book on any of several levels is greatly increased The basic areas covered are: logic; set theory; vectors and matrices: counting - permutations, combinations and partitions; probability and Markov chains; linear programming and game theory The area on vectors and matrices includes a chapter on systems of linear equations; it is in this context that the important concept of linear dependence and independence is introduced The area on linear programming and game theory includes a chapter on inequalities and one on points, lines and hyperplanes; this is done to make this section self-contained Furthermore, the simplex method is given for solving linear programming problems with more than two unknowns and fo r solving relatively large games In using the book it is possible to change the order of many later chapters or even t o omit certain chapters without difficulty and without loss of continuity Each chapter begins with a clear statement of pertinent definitions, principles and theorems together with illustrative and other descriptive material This is followed by graded sets of solved and supplementary problems The solved problems serve to illustrate and amplify the theory, bring into sharp focus those fine points without which the student continually feels himself on unsafe ground, and provide the repetition of basic principles so vital t o effective learning Proofs of theorems and derivations of basic results are included among the solved problems The supplementary problems serve as a complete review of the material in each chapter More material has been included here than can be covered in most first courses This has been done t o make the book more flexible, to provide a more useful book of reference and to stimulate further interest in the topics I wish to thank many of my friends and colleagues, especially P Hagis, J Landman, B Lide and T Slook, for invaluable suggestions and critical review of the manuscript I also wish to express my gratitude t o the staff of the Schaum Publishing Company, particularly to N Monti, for their unfailing cooperation S LIPSCHUTZ Temple University June, 1966 CONTENTS Page Chapter PROPOSITIONS AND TRUTH T A B L E S Statements Compound statements Conjunction, p Negation, -p Propositions and truth tables Chapter A q Disjunction, p v q ALGEBRA OF PROPOSITIONS 10 Tautologies and contradictions Logical equivalence Algebra of propositions Chapter CONDITIONAL STATEMENTS Conditional, p Chapter + q 18 Biconditional, p ++ q Conditional statements and variations ARGUMENTS, LOGICAL IMPLICATION 26 Arguments Arguments and statements Logical implication Chapter S E T THEORY 35 Sets and elements Finite and infinite sets Subsets Universal and null sets Set operations Arguments and Venn diagrams Chapter PRODUCT SETS 50 Ordered pairs Product sets Product sets in general Truth sets of propositions Chapter RELATIONS 58 Relations Relations a s sets of ordered pairs Inverse relation Equivalence relations Partitions Equivalence relations and partitions Chapter FUNCTIONS 66 Definition of a function Graph of a function Composition function One-one and onto functions Inverse and identity functions Chapter VECTORS 81 Column vectors Vector addition Scalar multiplication Row vectors Multiplication of a row vector and a column vector Chapter 10 MATRICES Matrices Matrix addition Scalar multiplication Matrix multiplication Square matrices Algebra of square matrices Transpose 90 CONTENTS Page Chapter II LINEAR EQUATIONS 104 Linear equation in two unknowns Two linear equations in two unknowns General linear equation General system of linear equations Homogeneous systems of linear equations Chapter 12 DETERMINANTS OF ORDER TWO AND THREE 127 Introduction Determinants of order one Determinants of order two Linear equations in two unknowns and determinants Determinants of order three Linear equations in three unknowns and determinants Invertible matrices Invertible matrices and determinants Chapter 13 THE BINOMIAL COEFFICIENTS AND THEOREM 141 Factorial notation Binomial coefficients angle Multinomial coefficients Chapter 14 Binomial theorem Pascal's tri- PERMUTATIONS, ORDERED SAMPLES 152 Fundamental principle of counting Permutations Permutations with repetitions Ordered samples Chapter 15 COMBINATIONS, ORDERED PARTITIONS 161 Combinations Chapter 16 Partitions and cross-partitions Ordered partitions TREE DIAGRAMS 176 Tree diagrams Chapter 17 PROBABILITY 184 Introduction Sample space and events Finite probability spaces Equiprobable spaces Theorems on finite probability spaces Classical birthday problem Chapter 18 CONDITIONAL PROBABILITY INDEPENDENCE 199 Conditional probability Multiplication theorem for conditional probability Finite stochastic processes and tree diagrams Independence Chapter 19 INDEPENDENT TRIALS, RANDOM VARIABLES 216 Independent or repeated trials Repeated trials with two outcomes Random variables Probability space and distribution of a random variable Expected value Gambling games Chapter 20 MARKOV CHAINS 233 Probability vectors Stochastic and regular stochastic matrices Fixed points of square matrices Fixed points and regular stochastic matrices Markov chains Higher transition probabilities Stationary distribution of regular Markov chains Absorbing states Chapter 21 INEQUALITIES The real line Positive numbers Order (Finite) Intervals Infinite intervals Linear inequalities in one unknown Absolute value 257 CONTENTS Page Chapter 22 POINTS, LINES AND HYPERPLANES 266 Cartesian plane Distance between points Inclination and slope of a line Lines and linear equations Point-slope form Slope-intercept form Parallel lines Distance between a point and a line Euclidean nz-space Bounded sets Hyperplanes Parallel hyperplanes Distance between a point and a hyperplane Chapter 23 CONVEX SETS AND LINEAR INEQUALITIES 281 Line segments and convex sets Linear inequalities Polyhedral convex sets and extreme points Polygonal convex sets and convex polygons Linear functions on polyhedral convex sets Chapter 24 LINEAR PROGRAMJIIKG 300 Linear programming problems Dual problems Matrix notation Introduction t o the simplex method Initial simplex tableau Pivot entry of a simplex tableau Calculating the new simplex tableau Interpreting the terminal tableau Algorithm of the simplex method Chapter 25 THEORY OF GAMES 321 Introduction to matrix games Strategies Optimum strategies and the value of a game Strictly determined games X matrix games Recessive rows and columns Solution of a matrix game by the simplex method X m and i i i x matrix games Summary INDEX 337 Chapter I Propositions and Truth Tables STATEMENTS A statement (or verbal assertion) is any collection of symbols (or sounds) which is either true or false, but not both Statements will usually be denoted by the letters P, (1, r, The truth or falsity of a statement is called its truth value Example 1.1: Consider t h e following expressions: (i) P ar i s is in England (ii) +2 = (iii) Where a r e you going? (iv) P u t the honiework on the blackboard 6) are statements; the first is false and the second is true The expressions (i) and The expressions (iii) and [iv) a r e not statements since neither is either true o r false COMPOUND STATEMENTS Some statements are composite, that is, composed of substatements and various logical connectives which we discuss subsequently Such composite statements are called compound statements Example 2.1: “Roses a r e red and violets a r e blue” is a compound statement with substatements “Roses a r e red” and “Violets a r e blue” Example 2.2: “He is intelligent o r studies every night” is, implicitly, a compound statement with substatements “He is intelligent” and “He studies every night” The fundamental property of a compound statement is that its truth value is completely determined by the truth values of its substatements together with the way in which they are connected to form the compound statement We begin with a study of some of these connectives CONJUNCTION, p A Q Any two statements can be combined by the word “and” to form a compound statement called the conjunction of the original statements Symbolically, P A denotes the conjunction of the statements p and q, read “ p and q” Example 3.1: Let p be “It is raining” and let q be “The sun is shining” Then p A q denotes the statement “It is raining and the sun is shining” The truth value of the compound statement p ~ satisfies q the following property: [TI] If p is true and Q is true, then p A q is true; otherwise, 11 A q is false In other words, the conjunction of two statements is true only in the case when each substatement is true PROPOSITIONS AND TRUTH T A B L E S Example 3.2: [CHAP Consider the following four statements: (i) +2 +2 +2 Paris is in France and (ii) P a r i s is in France and (iii) P a r i s is in England and (iv) Paris is in England and = = = - = By property [Ti], only the first statement is true Each of the other statements is false since at least one of its substatements is false A convenient way to state property [Ti] is by means of a table as follows: Here, the first line is a short way of saying that if p is true and q is true then p A is true The other lines have analogous meaning We regard this table as defining precisely the truth value of the compound statement p A q as a function of the truth values of p and of q DISJUNCTION, 11 q Any two statements can be combined by the word “or” (in the sense of “and/or”) to form a new statement which is called the disjunction of the original two statements Symbolically, P”4 denotes the disjunction of the statements p and q and is read ‘ip or q” Example 4.1: Let p be “Marc studied French at the university”, and let q be “Marc lived in France” Then p v q is the statement “Marc studied French at t h e university o r (Marc) lived in France” The truth value of the compound statement p v q satisfies the following property: IT,] If p is true or q is true o r both p and q a r e true, then q is true; otherwise p v q 13 is false Accordingly, the disjunction of two statements is false only when both substatements a r e false The property LT,] can also be written in the form of the table below, which we regard as defining 11 v q: Example 4.2: Consider the following four statements: +2 +2 +2 2+2 P a r i s is in France o r = (ii) P a r i s is in France o r = ii) iiii) P a r i s is in England or iiv) P a r i s is in England o r = = By property [T2], only (iv) is false Each of the other statements is t r u e since a t least one of its substatements is true 324 THEORY O F GAMES [CHAP 25 If we circle the minimum entry in each row and put a square about the maximum e ntry in each column, we obtain A = The point is both circled and “boxed” and so is a saddle point of A Thus w = is the value of the game, and optimum strategies po for R and qo for C are as follows: pa = (O,l,Oj and qo = ( O , O , , O j T h a t is, R consistently plays row 2, and C consistently plays column We now show t h at po, q and w satisfy the required properties to be optimum strategies and the value of the game: -4 -3 (i) pOA = (O,l,O)(f )!2 = (2,3,1,2j (1,1,1,1) = (w,v,v,wj -1 We remark that the proof of Theorem 25.1 is similar to the proof above f o r the special case x MATRIX GAMES +ll Consider the matrix game A = If A is strictly determined, then the solution is indicated in the preceding section Thus we need only consider the case in which A is non-strictly determined We first state a useful criterion to determine whether or not a X matrix game is strictly determined Lemma 25.2: The above matrix game A is non-strictly determined if and only if each of the entries on one of the diagonals is greater than each of the entries on the other diagonal: (i) a,d > b and a,d > c ; or (ii) b, c > a and b,c > d The following theorem applies Theorem 25.3: Suppose the above matrix game A is non-strictly determined Then p o = (xl,xz) is an optimum strategy for player R, qo = (y,,y2) is a n optimum strategy for player C, and v is the value of the game where d-c x, = a + d - b - c ’ II‘, = d-b y1= a+d-b-c’ and 2,= zJ2 a-b a+d-b-c a-c = a+d-b-c ad - bc ~ + d - b - ~ 325 THEORY O F GAMES CHAP 251 Note that the five quotients above have the same denominator, and that the numerator of the formula f o r v is the determinant of the matrix A Example 3.1 Consider the following matrix game (see Example 1.2): Using the above formula, we can obtain the value v 2: of the game: - - (-3).(-3) = -1 12 Thus the game is not f a i r and is in favor of the column player C Optimum strategies p o f o r R and qo for C a r e as follows: po = (L ' 1A) and qo = f - f (&,A) RECESSIVE ROWS AND COLUMNS Let A be a matrix game Suppose A contains a row r asuch that r l L r , for some other row r, Recall that r l L r l means that every entry of r z is less than or equal t o the corresponding entry of T, Then r z is called a yecessive row and r, is said to dominate it Clearly, player R would always rather play row r, than row rl since he is guaranteed to win the same o r a greater amount in every possible play of the game Accordingly, the recessive row r? can be omitted from the game On the other hand, suppose A contains a column Ca such that c , h c for some other column cj Then cl is called a recessive coLumri and c, is said to dominate it For analogous reasons player C would always rather play column c, than column el Hence the recessive column c, can be omitted from the game We emphasize that a recessive row contains numbers smaller than those of another row, whereas a recessive column contains numbers larger than those of another column Example 4.1 Consider the game A = Note that ( - , - , l ) (2,-1,2), i.e every entry -2 in the first row is f the corresponding entry in the second row Thus the first row is recessive and can be omitted from the game and the game may be reduced to - I Now observe t h a t the third column is recessive since each entry is second column Thus the game may be reduced to the X game A* = Fl the corresponding entry in the The solution to A* can be found by using the formulas in Theorem 25.3 and is Thus the solution to the original game A is 326 THEORY O F GAMES JCHAP 25 SOLUTIOS OF A MATRIX GAME BY THE SIMPLEX METHOD Suppose we are given a matrix game A We assume that A is non-strictly determined and does not contain any recessive rows or columns We obtain the solution to A as follows (1) Add a sufficiently large number h to every entry of A to form, say, the following matrix game which has only positive entries: I A' 1 x ult'! nil a22 a2,!( /all \ ah1 a12 \ * ah!,,1 ak2 (The purpose of this step is to guarantee that the value of the new matrix game is positive.) (2) Form the following initial tableau: PI Pr * P,,, ' 0 a21 a.2 U2r,1 0 - a k l ak2 ' ahnl a11 a12 al,,, 1 ' * * * ' ' -1 -1 -1 I 0 ~~ * ' 0 ~ ~ This tableau corresponds to the linear programming problem: Maximize f = subject t o and cc'i - xz + - + T , ( ~ + - + ui,,,Tl,l + + + + + + ali?-l ~~12x2 * f ar13'1 t a22xz * * ahi.1'1 nk2x2 * ' s.1'0, X S S O , Cllr,!5m ' ., ahnis)'itr X,,,'O (3) Solve the above linear programming problem by the simplex method Let P be the optimum solution to the maximum problem, Q the optimum solution to the dual minimum problem, and 2' = f ( P )= g ( Q ) the entry in the lower right hand corner of the terminal tableau Set 1 (i) p o = -Q (ii) qo = +P -k (iii) 2: = v* V* ' Then p o is an optimum strategy f o r player R in game A , qo is an optimum strategy f o r player C, and c is the value of the game We remark that the games ,4 and A have the same optimum strategies for their respective players and that their values differ by the added constant k ; that is, p o is an optimum strategy f o r player R in game A * , 4" is an optimum strategy for player C in game A*, and 2' 4h = 1/v* is the value of the matrix game A Example 5.1 Consider the mati I X game THEORY O F GAMES CHAP 25! 327 Add to each entry of A t o form the matrix game The initial simplex tableau and successive tableaux a r e as follows: (iii) Thus P = C O , & , i ) , Q = (i,d) and z- = 4; hence l/v* = 81'3 Then for the original game A , p o = (8/3)& = (2/3, 1/3) is a n optimum strategy f o r player R, qo = ( / ) P = (0, 1/3, 2/3) is a n optimum strategy f o r player C, v = 8/3 - = -1/3 is the value of the game Observe t h a t the game is favorable t o the column player C ExampIe 5.2 Consider the matrix game A shown below: A = ; -1 -1 Adding t o each entry of A , we obtain the matrix game A' on the right The initial tableau and t h e successive tableaux a r e as follows: P, P1 P, P, 1 0 1 1 0 -1 -1 -1 ~~ (iii) P, P, 328 THEORY O F GAMES [CHAP Thus P = (1/22,3/22,9/22), Q = (3/22,1/22,9/22) and v* = 13/22; hence l/w* = 22/13 f o r the original game A , po = (22/13)Q = (3/13, 1/13, 9/13) is a n optimum strategy for R , qo = (22/13)P = (1/13, 3/13, 9/13) is a n optimum strategy f o r C, w = 22/13 - = -4/13 Accordingly, is the value of the game x m AND m x MATRIX GAMES Consider the x m matrix game Suppose we apply the simplex method t o the above game Note that the optimum solution of the corresponding maximum linear programming problem will contain at most two non-zero components; hence so will an optimum strategy f o r the column player C In other words, the x m matrix game A can be reduced to a x game A* I n the next example we show a way of solving such a x nz game A and, in particular, how to reduce the game to a x game A X A similar method holds f o r any m X matrix game Example 6.1 Consider the following matrix game (see Example 5.1): -1 A Let p0 = (2,1-2) = (-2 -:) denote an optimum strategy for R and let v be the value of the game Then or 32 - ( - ) w -x+(l-x) v f v O-(l-x) L 5x-2 V or v f V L Let us plot the three linear inequalities in w and x on a coordinate plane where x 1, as shown in the diagram The shaded region depotes all possible values of v; however, R chooses his optimum strategy so t h a t v is a maximum This occurs at the point W which is the intersection of v = -2x+ and u = x - 1; t h a t is, where x = Q and v = -& Thus t h e value of the game is -$ and the optimum strategy f o r R is ($,Q) f +1 -22 2-1 f To find a n optimum strategy f o r C, we can set qo = (y, z, - y - x ) and use the condition qoAt (w,v, v) = -Q, -Q) However, a simpler way is as follows Observe that t h e two lines intersecting at W a r e determined by only the second and third columns of A Omit the other column and reduce the game t o t h e X niatrix game (-4, A* = El -1 -2 -3 -4 By Theorem 25.3, the solution t o A* i s V' = -9, po' = (3, Q) and q0' = (I ) 3' CHAP 251 329 THEO.RY O F G A M E S An optimum strategy f o r the column player C in the matrix game A is then v = v’ and po = PO‘, as expected.) q0 = (O,&,$) (Note tha t Remark: In the diagram above, the three hounding lines intersect the line E = at -2, and -1 respectively, and the line r = l at 3, -1 and 0, respectively Observe tha t these numbers correspond t o the components of the r o w of the original matrix A SUMMARY I n finding a solution to a matrix game, the reader should follow the following steps: (1)Test to see if the game is strictly determined (2) Eliminate all recessive rows and columns (3) I n the case of a x game, use the formulas in Theorem 25.3 (4) In the case of a Example 6.1 X m game or an ni X game, reduce the game to a X game as in ( ) Use the simplex method in all other cases Solved Problems MATRIX GAMES 25.1 Which of the following games are strictly determined? For the strictly determined games, find the value v of the game and find an optimum strategy po for the row player R and an optimum strategy qo for the column player C I -8 -1 -3 (i) - (iii) (ii) In each case, circle the minimum entries in each row and put a box around the maximum entries in each column as follows: @ (i) (i) mi01 I -3 (iii) (ii) The -1 in row and column is a saddle point; hence z1 = -1, p0 = (O,l,O), q0 = (0,1,0) (ii) There is no saddle point; hence the game is non-strictly determined Note t h a t both 3’s in the first column a r e “boxed”, since both a r e maximum entries in the column (iii) The in row and column is a saddle point; hence v = 2, p o = (0,1, O), QO = (0,0,1,0) 330 25.2 THEORY O F GAMES [CHAP 25 Find the solution to each of the following x games (ii) po (iv) If the game is non-strictly determined, use the following formulas of Theorem 25.3, where ( y l , x2) and 40 = (ul, ur) a-b d - b d -c XI = ‘ d x2 = a + c l - b - c ’ a+d-b-c’ b -c’ Llq (ii (iii) a-e = a f d - b - ~ ’ u= The game is non-strictly determined F i r s t compute ad - bc a + d - - b - ~ (L td -b -c = + - + = Then That is, (g, +) is an optimum strategy f o r R and [g,Q I is a n optimum strategy for C The 2.2-0.(-1) - L value of the game is ti = - Observe t h a t the game I S not Pair and is in favor of t h e TOW player R lii) This game is strictly determined with saddle point -1 Thus w = -1; and p @ = ( , O ) qo = ( , l), i.e R should always play row and C should always play column and (iii) The game is non-strictly determined, F i r s t compute a - d - b - c = 1- 43 ’1 = 10 Then yl = 8/10 = 8, x2 = , y1 = 6, yz = Tha t is, (A,.2) is a n optimum strategy for R and 1.5- (-3)*(-1) - i.6, 4) is an optimum strategy f o r C The value of the game is t = 10 The game is not f r and, since v is positive, the game is in favor of the row player R + iivj The game is non-strictly determined First compute a d - b - c = -3 - - - = -13 Then x1 = 5/13, y = 8/13, y = 8/13, y, = 5/13 Thus (&,&) is a n optimum strategy f o r R (-3).(-3)-2.5 - and is a n optimum strategy f o r C The value of the game is L’ = - 13 13 * The game is not f r and is in favor of the row player R (A,&) 25.3 Find a solution to the matrix game First circle each row minimum and box each column maximum: Observe t h at there is no saddle point and so the game is non-strictly determined We now te st f o r recessive rows and columns Note t h a t the third column is recessive since each entry is larger than the corresponding entry in the second column; hence the third column can be omitted to obtain the game -5 CHAP 251 331 THEORY OF G A M E S Note t h a t the second row is recessive since each entry is smaller than the corresponding entry in the first row Thus the second row can be omitted to obtain the game I 1 - Using the formulas in Theorem 25.3, the solution to the above v' = -13/11, = (2 11' 1 ) (Io' = ,o' t game is - f i (11' -) 11 Thus a solution to the original game is v = -13/11, 25.4 pa = Find a solution t o the matrix game - (fi.O,,), (I" -1 A = Set p o = (:r, - x ) Then p o A (-;-;-;-;) (x,1-2:) or -2% +1 = (5,h ) -3 -2 -2 is equivalent to iil, I ' , c , i ' j (V,V,C,") v ~ - 32 3x-2 % v -7x+5 v Plot t h e above four linear inequalities a s shown in the diagram Note t h a t t h e maximum L' satisfying the inequalities occurs a t t h e intersection of the lines -2x+ = 2; and ~ - = v, t h a t is, a t the point x= and L' = -L Thus v = -8 is the value of the game, and p o = (2, 8) is a n optimum strategy f o r player R Since the two lines determining L* come from the first and third columns of t h e matrix game A , omit the other columns t o reduce A t o the X matrix game A :1 - F l 1 A solution of A " , by Theorem -2 (g, , b, ) Thus q = in game A 25.5 I is a n optimum strategy f o r player C Note t h a t v = v' and po 1P O ' , as expected.) Find a solution t o the matrix game or A = 47-1 -3.r -1 23 El v f 21 f v 332 THEORY O F GAMES [CHAP 25 Plot the above three inequalities (where O‘x‘l) as shown in the diagram W e seek the miximum v satisfying the given inequalities; note t h at i t occurs a t the intersection of the lines -3% = v and 2x = v, i.e at the and v = % Thus v = is the value of point r = the game and qo = (&,Q) i s an optimum strategy f o r player C Since the two lines determining v come from the second and third rows of A , omit the other row to obtain + the X + Fl matrix game A:: = A solution of A’?,by Theorem 23.5, is 3’ = , = (2 ) qo’ 5’5 ’ = (I, +) mi Thus po = (0,$, 8) is a n optimum strategy for player R (Note t h at u = v’ and qo = q o ’ , as expected.) 25.6 Find a solution to the matrix game -1 The reader should first verify t h a t the game is non-strictly determined and tha t there a re no recessive rows or columns We find a solution by the simplex method F i r s t add to each entry t o obtain the game The initial and subsequent tableaux follow: Pl p2 Pl p3 pz p3 (i) (ii) (iii) (Renzayk: Since the l as t row in (iii) has no negative indicators, the tableau is a terminal one and so all of its entries need not be computed.) Thus P = (L li’ 17’ O ) , Q = (0’ ’ ”) 17 and ~ : hence 1/~:,: !?, Accordingly, po = ( $ ) Q = ( ,i ,~ 40 = ( ) y ) P, = (i ) and 2: = 17 - = -5 5’5’ 5’ =&; 25.7 Two players R and C simultaneously show or fingers If the sum of the fingers shown is even, then R wins the sum from C; if the sum is odd, then R loses the sum to C Find optimum strategies f o r the players and to whom the game is favorable THEORY OF GAMES CHAP 251 The matrix of the game is 333 The game is non-strictly determined; hence the formulas in Theorem 25.3, Page 234, apply First compute a d - b - c = 5 = 20 Then + + + + (g, &), is an optimum strategy f o r R and (&,%) is a n optimum strategy f o r C The value T h a t is, = - - ( - )20' ( - ) - -L of the game is , and, since i t is negative, the game is favorable t o C 25.8 Two players R and C match pennies If the coins match, then R wins; if the coins not match then C wins Determine optimum strategies for the players and the value of the game The matrix of the game is H T The game is non-strictly determined Using the formulas in Theorem 25.3, Page 234, we obtain Thus (g, &) is an optimum strategy for both R and C The value of x1 = x, = y1 = y2 = = 1.1 - ( - l ) ( - i ) = 0: hence the game is fair the game is 4 THEOREMS 25.9 Let p o be a given strategy for player R in a matrix game A Show that the following conditions are equivalent: v, for every strategy q for C; (ii) p o A (v,v, , v) (i) p o A qt Assume that iii) holds Then, for any strategy q f o r C, pOA qt (v,v, , v) qt = v On the other hand, assume t h a t (i) holds and t h a t pOA = ( a l , a , , a k ) strategy q = (1,0, ,O), we have Choosing the pure pOA qt = (al,aL, , ak)qt = a , s v Similarly, a2 w, , ak v I n other words, @ A (v, v, , w) 25.10 Let p = ( a l , a r , ., ak) and p* = ( b l ,bz, , bk) be probability vectors Show that each point on the line segment joining p and p * , i.e t p -t (1- t)pa where g t 1, is also a probability vector Observe that + (1- t ) b z , , t a L + (1- t ) b k ) Since t , - f, the a , and the b, are all non-negative, ta, + (1- t ) b , is also non-negative every i Furthermore, f a , + (1- t ) b , + ta2 + (1- t ) b , + + tak + (1- t)bh + (1- t ) b k = ta, + ta, + + tah + (1 t ) b , + (1 t ) b z + (1- t)p" = (tal + (1 - t ) b l , ta2 * ' ' * - - * = t ( a , + a + ~ ~ ~ + a ~ ) + ( - t ) ( b l + b , + ~ ~=~ + t +b (~ l)- t ) = for 334 T H E O R Y O F GAMES 'CHAP 25.11 Let p o and p be optimum strategies for player R in a matrix game A with value c Show that p' = t p o + (1- t ) p , where t 1, is also an optimum strategy for R p A By the pieceding problem, p is a probability vector, hence we need only show t h a t (L,I , ,v) We a r e given t h a t pOA Thus (L, I!, v) p A and p'A == (tpo - i l - t ) p ) A tpOA + (1 t ) A~ - t ( v , v , , l = (v,w, ,v) ( L , L, (1 - t ) ( C , w, , w ) ) ,L) 25.12 Show that if a and b are saddle points of a matrix A , then a = b If n and b a r e in the sanie row, then each is a minimum of the row and so a = ti Similarly if u and b a r e in the same column, then n and b a r e maxima of the column and so a = b Now assume t h a t a and b appear in different columns and different rows as illustrated below: *: :,: ::i ::: \:,: \* * :L * :: ., .' Q r;;l ::: f :,: pJ :.: * * Q k.; Q * * @ : : p J :.k * *\ :I :,: :.: :: :: :i i' *I i:: Since a is a saddle point, a G d and c f a ; and since b is a saddle point, b s c and d E b Thus a ' d s b ' c a Accordingly, the equality relation holds above: 25.13 Show that the game A = a = d = b = c is strictly determined Observe t h a t each a is a minimum of the first row If a and if u c then the second a is a saddle point b then the first a is a saddle point, Thus n e are left with the case t h a t b > a and c > a, i.e where b and c a r e maxima of their respective columns Now if b f then b is a saddle point, and if c b then c is a saddle point (3 Accordingly, in all cases A has a saddle point and so A is strictly determined 25.14 Prove Lemma 25.2: The matrix game A = is non-strictly determined if and only if each of the entries on one of the diagonals is greater than each of the entries on the other diagonal: (i) a,d > b and a,d > c; or (ii) b,c > a and b,c > d If ( i ) or l i i ) holds, then there is no saddle point for A and so A is non-strictly determined Now suppose A is non-strictly determined By t h e preceding problem, a # b We consider two cases Case (1): a < b If a c then a is a saddle point; hence IL < c We need only show t h a t and d < c Suppose d b: then d > c o r else d is a saddle point But a < c and d > c implies c is a saddle point Accordingly, d % b leads to a contradiction, and so d < b Similarly, d < c d b The proof is similar t o t h e proof in Case (1) CHAP 251 THEORY O F GAMES 335 +l+l 25.15 Prove Theorem 25.3: Suppose the matrix game A = is not strictly determined Then a-c a-b a-b a-c ?Jo = ( a + d - b - c ' a + d - b - c ad - bc u = a+d-b-c are respectively an optimum strategy f o r R, a n optimum strategy for C and the value of the game Set p0 = (x,1-x) and q0 = (y,1-y), +c ( b - d ) ~+ d or (a-c)z Then pOA v v and (w,w) and qoAf (w,w) is equivalent to (a- b)y +b (c-d)y 4- d f t? f v Solving the above inequalities as equations f o r the unknowns , y and v, we obtain the required result We remark t h a t a b - c - d # by the preceding problem, i.e by Lemma 25.2 We also note t h a t in the case of a X non-strictly determined matrix game A , we have the following stronger result: poA = (w,v) and qoAt = ( t i , w), + 25.16 Let A be a matrix game with optimum strategy p o f o r R, optimum strategy qo f o r C, and value v ; and let k be a positive number Show that f o r the game kA, p o is an optimum strategy for R, qo is an optimum strategy for C, and kv is the value of the game We a r e given t h at pOA (w,w, , c) and qOA (w,W , , c) Then pO(kA) = k(p0A) k(w,v, , w) = (kv,kv, , kv) qO(kA)t = qOkAt = k(q0At) f k ( ~ W,, , W ) = (lit],kv, , kw) , f which was to be shown Supplementary Problems MATRIX GAMES 25.17 Find a solution t o each game, i.e an optimum strategy po for R , an optimum strategy qo f o r C, and t h e value of the game (9 25.18 Find a solution to each game (4 25.19 -4 - -3 -4 Find a solution t o each game (ii) (iii) -1 -2 336 25.20 THEORY O F GAMES [CHAP 25 Find a solution to each game (i) -1 Find a solution to the game if iii a < 1, (ii) < a < 3, (iii) a > 25.21 Consider the game 25.22 Each of two players R and C h a s a dime and a quarter They each show a coin simultaneously If the coins ar e the same, R wins C's coin; if the coins a r e different, C wins R's coin Represent the game as a matrix game and find a solution 25.23 Each of two players R and C has a penny, nickel and dime They each show a coin simultaneously If the total amount of money shown is even, R wins C's coin; if i t is odd, C wins R's coin Represent the game as a matrix and find a solution THEOREMS 25.24 Suppose every en t r y in a matrix game A is increased by a n amount k Show tha t the value of the game also increases by k , but t h a t the optimum strategies remain the same 25.25 Show t h at if every entry in a matrix game is positive, then the value of the game is positive Answers to Supplementary Problems (i) p o = (0, l), q0 = (0,l), w = 1; (ii) p o = (l,O), q0 = (0,I), v = 2; = ( l , O ) , v = (iii) 25.18 (i) po = (,7,.3), qo = (.4,.6), w = 2; (ii) p o = (3/4,1/4), (iiij p o = (7/8,1/8), qo = (5/8,3/8), v = 3/8 = (5/12,7/12), 25.19 (i) p o = (5/8,3/8), qo = (3/8,0,5/8), w = 1/8; (ii) PO = (5/7,2/7), q0 = (5/7,2/7,0), v = 3/7; (iii) P O = (2/7,0,5/7), q = (4/7,3/7), w = 13/7 25.21 (i) 25.17 p0 can be any strategy, yo 25.22 p o = (l,O), q = (O,l), w = 1; (ii) p o = (0,1), q0 = (0,1), w = a ; =I; p o = (5/7, / ) , q' = 25.23 qo = 10 10 (i,g), w ; po = ( l O / l l , 0, l / i l ) , yo = (1/2, 0, 1/2), -10 j -10 I 10 w = v = 1/4; INDEX De Morgan’s laws, 11 Degenerate equations, 107 Dependent vectors, 114 Detachment, law of, 26 Determinants, 127 Disjoint sets, 37 Disjunction, exclusive, 3, Distance, 266 Distribution, 218 binomial, 217 Domain of a function, 66 Dot product, 84 Dual problem, 301 Duality theorem, 303 Absolute value, 259 Absorbing states, 240 Algebra of propositions, 11 Algebra of sets, 38 Algorithm of the simplex method, 308 Angle of inclination, 266 Argument, 26 Bayes’ theorem, 207 Biconditional statement, 19 Binomial coefficients, 141 theorem, 142 Binomial distribution, 217 Birthday problem, 188 Bounded sets, 269 Bounding hyperplane, 282 Echelon form, 109 Element, 35 Elementary event, 184 Empty set, 37 Equations, linear, 104 system of, 107 Equiprobable space, 186 Equivalence class, 60 Equivalence relation, 59 Equivalent, logically, 10 Euclidean space, 268 Event, 184 Exclusive disjunction, 3, Expected value, 220 of a game, 221 Extreme point, 282 C player, 321 Cartesian product, 50 plane, 266 Cauchy-Schwarz inequality, 278 Cell, 60 Class, 37 equivalence, 60 Closed half space, 282 Closed interval, 258 Co-domain of a function, 66 Coefficients, binomial, 141 of equations, 106 Collection, 37 Column vector, 81 Combinations, 161 Complement of a set, 37 Components of vectors, 81, 83 Composition function, 68 Compound statements, Conclusion of an argument, 26 Conditional probability, 199 Conditional statement, 18 Conjunction, Consistent equations, 109 Constant, 106 Contradiction, 10 Contrapositive, 19 Converse statement, 19 Convex sets, 281 polygonal, 283 polyhedral, 282 Corner point, 282 Counting, fundamental principle of, 152 Cross-partitions, 162 Factorial, 141 F a i r game, 221, 323 Fallacy, 26 False statement, Family, 37 of lines, 268 Feasible solution, 300 Finite set, 36 Fixed point, 234 Frequency, relative, 184 Function, 66 composition of, 68 identity, 70 inverse, 70 linear, 67, 281 objective, 300 polynomial, 68 337 INDEX Games, gambling, 221 f a i r , 221 Games, matrix, 321 f a i r , 323 strictly determined, 323 value of, 323 Graph of a function, 67 Half space, 282 plane, 282 Homogeneous equations, 111 Hyperplane, 269 Identity function, 70 Image, 66 Implication, 28 Inclination of a line, 266 Inconsistent equations, 109 inequalities, 285 Independent events, 201 trials, 216 Independent vectors, 114 Inequalities, 258 linear, 259 Infinite set, 36 Initial simplex tableau, 304 Inner product, 84 Intersection of sets, 37 Intervals, 258 Inverse, function, 70 matrix, 132 relation, 59 statement, 19 Invertible matrix, 131 Joint denial, 16 Line, in the plane, 267 real, 257 segment, 281 Linear, equation, function, 67, 281 inequality, 259 programming, 300 space, 83 Logical implication, 28 Logically equivalent, 10 Mapping, 66 Markov chain, 236 Matrix, 90, 112 addition of, 90 game, 321 multiplication of, 92 non-singular, 133 regular, 234 scalar multiplication of, 91 singular, 133 square, 93 stochastic, 233 Matrix (cont.) transition, 236 transpose, 94 Member, 35 Minkowski’s inequality, 278 Mixed strategy, 322 Multinoniial coefficients, 144 Mutually exclusive events, 185 Negation, Negative numbers, 257 New simplex tableau, 306 Non-singular matrix, 133 Norm of a vector, 269 Null set, 36 Numbers, real, 256 Objective function, 300 Odds, 188 One-to-one function, 69 Onto function, 69 Open half space, 282 Optimum solution, 300 Optimum strategy, 322 Order, 257 Ordered pair, 50 Ordered partition, 163 Origin, 257 Pair, ordered, 50 Parallel lines, 266 Parameter, 268 Partitions, 60, 162 ordered, 163 Pascal’s triangle, 143 Permutation, 152 Perpendicular lines, 266 Pivot, 305 Pivotal column, 305 row, 305 Plane, Cartesian, 266 Plane, half, 282 Play of a game, 321 Players, R and C, 321 Point, 266 corner, 282 extreme, 282 fixed, 234 saddle, 323 Polygonal convex set, 283 Polyhedral convex set, 282 Polynomial, 68 Premises, 26 Primal problem, 301 Probability, 184 conditional, 199 distribution, 238 vector, 233 Product set, 50 Proper subset, 36 Proposition, P u r e strategy, 322 Quotient set, 60