Answer to Question #150944 in Operations Research for AHMED

Question

a) A and B play a game in which each player has three coins a sh 5 coin, sh 10 coin and sh 20 coin. Each player selects a coin without the knowledge of the others choice. If the sum of the coins is an odd number, A wins B’s coin. If the sum is even, B wins A’s coin.
Required
i) Draw the payoff matrix for the game.
ii) Does this game have a saddle point?
iii) Find the value of the game

Answer 1

i)

(B)

5 10 20 Row Min.

5 -5 10 20 -5

(A) 10 5 -10 -10 -10

20 5 -20 -20 -20

Column max 5 10 20

(A) 2 4 Column max

8 9 (minimax)

ii)Here maximin of -5 is not equal to minimax of 5. Hence, no saddle point. 

iii) The last column is dominated by second columns - we delete it

and the last row is dominated by second row - we delete it

Now consider a matrix

$\def\arraystretch{1.5} \begin{array}{c:c:c:c} & 5 & 10 & oddmeters \\ \hline 5 & -5 & -10 & 15 \\ \hdashline 10 & 5 & -10 & 15\\ \hdashline oddmeters & 10 & 20 \end{array}$

p1 = p2 = 15/(15+15) = 1/2

q2 = 10/(10+20) = 1/3

q1 = 20/(10+20) = 2/3

Value of game = (a11 * a22 - a21 * a12)/[(a11+a22)-(a21+a12)] = ((-5)*(-10) - 5*10)/[(-5+(-10)) - (5+10)] = 0