s155-zerosum-example.pdf - Example(KP Exercise 2.3 Find the...

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Example (KP Exercise 2.3).Find the value of the following zero-sum game and determine someoptimal strategies for each of the players:A¨˝834147160385˛.Solution.1.Check for pure Nash equilibria.The row minima are 1,1,0; the column maxima are 8,7,8,6. Thusmaximinjaijmaxt1,1,0u “1 which is strictly larger than minjmaxiaijmint8,7,8,6u “6. Thisimplies that there is no pure Nash equilibrium.2.Check for dominated rows.In the payoff matrixA, canrow1be dominated by any convex combinationof the other two rows? That is, is there anyx1P r0,1ssuch thatx1row2` p1´x1qrow3ěrow1? Theanswer is no, by looking at the first entry of each row:rx1row2` p1´x1qrow3s1x1¨4` p1´x1q ¨0ď4ă8“ prow1q1.Similarly,row2cannot be dominated by any convex combination of the other two rows, by lookingeither at the second entryprow2q27, or at the fourth entryprow2q46. Likewise,row3cannot bedominated by any convex combination of the other two rows, by looking at the third entryprow3q38.We conclude that none of the rows can be dominated.3.Check for dominated columns.Similarly, none ofcol1,col3,col4can be dominated. Forcol2it is lessclear — do there exist non-negative numbersy1, y3, y4, withy1`y3`y41, such thaty1col1`y3col3`y4col4ďcol2? Note that the desired inequality always holds in the second entry:ry1col1`y3col3`y4col4s2y1¨4`y3¨1`y4¨6ď6ă7“ pcol2q2.

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