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Find all Nash equilibria using the table method. Use / for column dominance and \ for row dominance. Put both / and \ if both apply; i.e. check every cell for both row and column elimination. After using algebra to resolve any surviving cells, plug your strategies back into the game and compute payoffs to verify your solutions. Show all your work. NO CALCULATORS or decimal answers - use fractions!

Find all Nash equilibria using the table method. Use / for column dominance and \ for row dominance. Put both / and \ if both apply; i.e. check every cell for both row and column elimination. After using algebra to resolve any surviving cells, plug your strategies back into the game and compute payoffs to verify your solutions. Show all your work. NO CALCULATORS or decimal answers - use fractions!

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Find all Nash equilibria using the table method. Use / for column dominance and \ for row dominance. Put both / and \ if both apply; i.e. check every cell for both row and column elimination. After using algebra to resolve any surviving cells, plug your strategies back into the game and compute payoffs to verify your solutions. Show all your work. NO CALCULATORS or decimal answers - use fractions! 1,2 0,3 -1,-2 2,-2 1,1 1,0 2,1 -3,0 -1,-1 1 2 3 12 13 23 123 1 2 3 12 13 23 123
Transcribed Image Text:Find all Nash equilibria using the table method. Use / for column dominance and \ for row dominance. Put both / and \ if both apply; i.e. check every cell for both row and column elimination. After using algebra to resolve any surviving cells, plug your strategies back into the game and compute payoffs to verify your solutions. Show all your work. NO CALCULATORS or decimal answers - use fractions! 1,2 0,3 -1,-2 2,-2 1,1 1,0 2,1 -3,0 -1,-1 1 2 3 12 13 23 123 1 2 3 12 13 23 123
Find all Nash equilibria using the table method. Use / for column dominance and \ for row dominance. Put both / and \ if both apply; i.e. check every cell for both row and column elimination. After using algebra to analyze any surviving cells, plug your strategies back into the game and compute payoffs to verify your solutions. Show all your work. NO CALCULATORS or decimal answers - use fractions! Thank you! 0 1/3 1/2 2/7 1/2 1,2 1/2 3/7 0 0 2/7 1/2 1 1 A A A A 2 3 12 13 1 23 1 123 1 -1,-2 3,1 2 1 1 A 1 1 A 1 3/2 0/2 E: need p2(x,c1)=p2(x,c2)=p2(x,c3) and p1(r1,y) = p1(r2,y) = p1(r3,y). yB makes all 3 rows pay the same, and it is the unique mixture that does so, so there are no NE in cell E. 3 7 1 ^ 1 A 1 1 12 1 1 1 / B I C 5/6 2/3 1,3 2,-2 0,0 3/2 1/2 -x1+0x2+1(1-x1-x2) = 0 -2x1 - x2 + 1 = 0 2x1 + x2 = 1 13 1 A / A 1 1 1 1/6 0 2,1 1 7/6 -3,0 1 7/6 -1,-1 1-1/6 0 1/2 23 123 1 1 / 1 A D 1 A 1 A / E 1 A: XA=(0,0,1)= r3, yA=(1,0,0)= c1 B: xB=(1/2, 0, 1/2), yB=(1/3, 2/3, 0) C: yC=yB, xC must satisfy p2(x,c1)=p2(x,c2)>=p2(x,c3) 2x1-2x2+1(1-x2-x3) -[3x1-2x2+0( -[1x1+0x2-1( D: xD must satisfy p2(x,c2)=p2(x,c3)>=p2(x,c1) X~ 3x1-2(1-x2) -[ 1x1+0(1-x1) 2x1-2(1-x1)= 0 x1 = x2 = 1/2 yD must satisfy p1(r1,y)=p1(r2,y)>=p1(r3,y) 1y2+2(1-y2) -[2y2-3(1-y2) -y2+5(1-y2)= 0 5-6y2 = 0 The set of x's that satisfy both the equation and the inequalities. NE = {(x,y) | y = (1/3, 2/3, 0), 2x1-2x2+1(1-x2-x3) x1-2x2+2(1-x1-x2) >= 0 -x1-4x2 + 2 >= 0 x1 + 4x2 <= 2 y1 = 0, y2 = 5/6, y3 = 1/6 xB C: continued...p2(x,c2)>=p2(x,c3) 3x1-2x2+0( -[1x1+0x2-1( x = axB+ (1-a)x- 0 <= a <=1} 2x1-2x2+1(1-x1-x2) >= 0 x1 - 3x2 + 1 >= 0 -x1 + 3x2 <= 1 C: continued...p2(x,c1)>=p2(x,c3) 2x1 + x2 = 1 x1 + 4x2 = 2 x1 = 2/7, x2 = 3/7, x3 = 2/7 x~= (2/7,3/7/2/7)
Transcribed Image Text:Find all Nash equilibria using the table method. Use / for column dominance and \ for row dominance. Put both / and \ if both apply; i.e. check every cell for both row and column elimination. After using algebra to analyze any surviving cells, plug your strategies back into the game and compute payoffs to verify your solutions. Show all your work. NO CALCULATORS or decimal answers - use fractions! Thank you! 0 1/3 1/2 2/7 1/2 1,2 1/2 3/7 0 0 2/7 1/2 1 1 A A A A 2 3 12 13 1 23 1 123 1 -1,-2 3,1 2 1 1 A 1 1 A 1 3/2 0/2 E: need p2(x,c1)=p2(x,c2)=p2(x,c3) and p1(r1,y) = p1(r2,y) = p1(r3,y). yB makes all 3 rows pay the same, and it is the unique mixture that does so, so there are no NE in cell E. 3 7 1 ^ 1 A 1 1 12 1 1 1 / B I C 5/6 2/3 1,3 2,-2 0,0 3/2 1/2 -x1+0x2+1(1-x1-x2) = 0 -2x1 - x2 + 1 = 0 2x1 + x2 = 1 13 1 A / A 1 1 1 1/6 0 2,1 1 7/6 -3,0 1 7/6 -1,-1 1-1/6 0 1/2 23 123 1 1 / 1 A D 1 A 1 A / E 1 A: XA=(0,0,1)= r3, yA=(1,0,0)= c1 B: xB=(1/2, 0, 1/2), yB=(1/3, 2/3, 0) C: yC=yB, xC must satisfy p2(x,c1)=p2(x,c2)>=p2(x,c3) 2x1-2x2+1(1-x2-x3) -[3x1-2x2+0( -[1x1+0x2-1( D: xD must satisfy p2(x,c2)=p2(x,c3)>=p2(x,c1) X~ 3x1-2(1-x2) -[ 1x1+0(1-x1) 2x1-2(1-x1)= 0 x1 = x2 = 1/2 yD must satisfy p1(r1,y)=p1(r2,y)>=p1(r3,y) 1y2+2(1-y2) -[2y2-3(1-y2) -y2+5(1-y2)= 0 5-6y2 = 0 The set of x's that satisfy both the equation and the inequalities. NE = {(x,y) | y = (1/3, 2/3, 0), 2x1-2x2+1(1-x2-x3) x1-2x2+2(1-x1-x2) >= 0 -x1-4x2 + 2 >= 0 x1 + 4x2 <= 2 y1 = 0, y2 = 5/6, y3 = 1/6 xB C: continued...p2(x,c2)>=p2(x,c3) 3x1-2x2+0( -[1x1+0x2-1( x = axB+ (1-a)x- 0 <= a <=1} 2x1-2x2+1(1-x1-x2) >= 0 x1 - 3x2 + 1 >= 0 -x1 + 3x2 <= 1 C: continued...p2(x,c1)>=p2(x,c3) 2x1 + x2 = 1 x1 + 4x2 = 2 x1 = 2/7, x2 = 3/7, x3 = 2/7 x~= (2/7,3/7/2/7)
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