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GT_Week_6

# GT_Week_6 - Game Theory Solutions to Exercises The...

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Game Theory Solutions to Exercises The Principle of Indiﬀerence Jan-Jaap Oosterwijk Fall 2007

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3 The Principle of Indiﬀerence 3.1 Consider the game with matrix - 2 2 - 1 1 1 1 3 0 1 . (a) Note that this game has a saddle point. (b) Show that the inverse of the matrix exists. (c) Show that II has an optimal strategy giving positive weight to each of his columns. (d) Why then, don’t equations (16) give an optimal strategy for II? Solution: (a) Denote the above matrix by A := ( a ij ) 1 i,j 3 . Then a 23 = 1 is the minimum of the elements in its row and the maximum of the elements in its column, hence a saddle point. (b) We ﬁnd that the determinant of A , expanding on its middle column, is det( A ) = ± ± ± ± ± ± - 2 2 - 1 1 1 1 3 0 1 ± ± ± ± ± ± = - 2 · ± ± ± ± 1 1 3 1 ± ± ± ± + ± ± ± ± - 2 - 1 3 1 ± ± ± ± = - 2(1 · 1 - 1 · 3) + ( - 2 · 1 - ( - 1) · 3) = 5 , so det( A ) 6 = 0 and hence its inverse exists. (c) Since a 23 = 1 is a saddle point, we know the value of the game to be 1. Hence, any optimal strategy q for Player II must guarantee this value. i.e. - 2 q 1 + 2 q 2 - q 3 1 q 1 + q 2 + q 3 1 3 q 1 + q 3 1 Since q 1 + q 2 + q 3 = 1, we have equality in the second equation. Hence q 3 = 1 - q 1 - q 2 , which we can substitute in the other two equations to eliminate one variable. This gives us - q 1 + 3 q 2 - 1 1 2 q 1 - q 2 + 1 1 , i.e. q 2 1 3 q 1 + 2 3 , and q 2 2 q 1 . Finally, we require that 0 q 1 1, 0 q 2 1 and also 0 q 3 1 which is equivalent to 0 q 1 + q 2 1. You can make a diagram of this and see that the set of optimal strategies of Player II is exactly ² q R 3 : q 2 0 ,q 2 1 3 q 1 + 2 3 ,q 2 1 - q 1 ,q 2 2 q 1 , and q 3 = 1 - q 1 - q 2 ³ . The optimal strategies giving positive weight to each of his columns are those for which q 1 > 0, q 2 > 0, and q 3 > 0, i.e. (see diagram) ² q R 3 : q 2 > 0 ,q 2 < 1 3 q 1 + 2 3 ,q 2 < 1 - q 1 ,q 2 > 2 q 1 , and q 3 = 1 - q 1 - q 2 ³ . 2
(d) The only condition of Theorem 3.2 that we haven’t checked yet is the one at the very beginning of the section: Player I must not have an optimal strategy giving positive weight to each of the rows. 3.2 Consider the diagonal matrix game with matrix (18). (a) Suppose one of the diagonal terms is zero. What is the value of the game? (b) Suppose one of the diagonal terms is positive and another is negative. What is the value of the game? (c) Suppose all diagonal terms are negative. What is the value of the game? Solution: Let m N and, for each 1 i m , d i R . Deﬁne A := ( a ij ) 1 i,j m := d 1 0 ··· 0 0 d 2 ··· 0 . . . . .

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GT_Week_6 - Game Theory Solutions to Exercises The...

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