solutions_hw6

# solutions_hw6 - Solutions to homework 6 2.5#1 Strategic...

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Unformatted text preview: Solutions to homework 6 2.5#1 Strategic Equilibria Are Individually Rational Consider a two-person noncooperative general-sum game with corresponding m × n payoff matrices A and B . Let ( ˜p , ˜q ) be a strategic equilibrium. This implies that ˜p > A˜q ≥ p > A˜q for all p ∈ X * . Furthermore, it implies that ˜p > A˜q ≥ max p ∈ X * p > A˜q . Trivially, for every p ∈ X * we have that p > A˜q ≥ min q ∈ Y * p > Aq . Hence, ˜p > A˜q ≥ max p ∈ X * min q ∈ Y * p > Aq = max p ∈ X * min 1 ≤ j ≤ n m X i =1 p i a ij = v I . The first equality follows from equation (7) on page II – 36, the second is just the definition of v I . The proof for the fact that ˜p > B˜q ≥ v II is almost identical. 2.5#2 Find the safety levels, the MM-strategies, and all SE’s and associated vector payoffs of the following games in strategic form (a) The bimatrix (0 , 0) (2 , 4) (2 , 4) (3 , 3) can be represented by A = 0 2 2 3 and B = 0 4 4 3 . Since A has a saddle point at a 21 = 2 the safety level of Player I is v I = Val( A ) = 2 and the MM-strategy is p = (0 , 1) > . B > does not have a saddle point, so the safety level of Player II is v II = Val( B > ) = · 3- 4 · 4- 4 + 3- 4 = 16 5 , and its MM-strategy is q = (1 / 5 , 4 / 5). Using the labeling method described in page III – 11 we get (0 , 0) (2 , 4 * ) (2 * , 4 * ) (3 * , 3) and hence there is one PSE at (2 , 1) with payoff (2 , 4). This is the only SE in fact, since if we try to solve for the equalizing strategy we get q 1 = 3...
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solutions_hw6 - Solutions to homework 6 2.5#1 Strategic...

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