PartIIISec2_5

# PartIIISec2_5 - Game Theory Solutions to Exercises...

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Unformatted text preview: Game Theory Solutions to Exercises: Noncooperative Games Jan-Jaap Oosterwijk Fall 2007 2 Noncooperative Games 2.5.1 Strategic Equilibria Are Individually Rational. A payoff vector is said to be individually rational if each player receives at least his safety level. Show that if (˜ p , ˜ q ) is a strategic equilibrium for the game with matrices A and B , then ˜ p T A ˜ q ≥ v I and ˜ p T B ˜ q ≥ v II . Thus, the payoff vector for a strategic equilibrium is individually rational. Solution: This exercise concerns a two-person general-sum noncooperative game with an m × n payoff matrix ( a ij ,b ij ). Let (˜ p , ˜ q ) be a strategic equilibrium. Then m X i =1 n X j =1 ˜ p i ˜ q j a ij ≥ m X i =1 n X j =1 p i ˜ q j a ij (i.e. ˜ p T A ˜ q ≥ p T A ˜ q ) for all p ∈ X * and m X i =1 n X j =1 ˜ p i ˜ q j b ij ≥ m X i =1 n X j =1 ˜ p i q j b ij (i.e. ˜ p T B ˜ q ≥ ˜ p T Bq ) for all q ∈ Y * . (If Player I would change his strategy from ˜ p to some other strategy p and Player II would keep on using ˜ q , then the average payoff to Player I decreases or at best stays the same. Similarly, if Player II would change her strategy from ˜ q to some other strategy q and Player I would keep on using ˜ p , then the average payoff to Player II decreases or at best stays the same as well. This explains in which sense, when the players are using strategies ˜ p and ˜ q respectively, the game can be seen to be in an equilibrium from which neither one will want to escape on his own.) The safety level of each player is defined as, v I := max p ∈ X * min 1 ≤ j ≤ n m X i =1 p i a ij and v II := max q ∈ y * min 1 ≤ i ≤ m n X j =1 b ij q j . Due to the reasoning on p. II-33,34 resulting in formula (7) on p. II-35, this is equal to v I := max p ∈ X * min q ∈ Y * p T Aq and v II := max q ∈ Y * min p ∈ X * p T Bq Now, since ˜ p T A ˜ q ≥ p T A ˜ q for all p ∈ X * , we have that, in particular ˜ p T A ˜ q ≥ max p ∈ X * p T A ˜ q . Furthermore, for each p ∈ X * , p T A ˜ q ≥ min q ∈ Y * p T A ˜ q . Hence ˜ p T A ˜ q ≥ max p ∈ X * min q ∈ Y * p T A ˜ q = v I . The proof for the fact that ˜ p T B ˜ q ≥ v II is completely analogous. 2.5.2 Find the safety levels, the MM-strategies, and find all SE’s and associated vector payoffs of the following games in strategic form. (a) (0 , 0) (2 , 4) (2 , 4) (3 , 3) 2 Solution: The winnings of Player I and Player II are respectively given 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 its MM-strategy is p = (0 , 1). Matrix B T does not contain a saddle point, so the safety level of Player II is v II = Val( B T ) = · 3- 4 · 4- 4 + 3- 4 =- 16- 5 = 3 1 5 and its MM-strategy is q = ( q, 1- q ) with q = 3- 4- 5 = 1 5 . Using the labelling method to find pure SE’s as described in section 2.3, we get (0 , 0) (2 , 4 * ) (2 * , 4 * ) (3 * , 3) and see that there is one pure SE at...
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PartIIISec2_5 - Game Theory Solutions to Exercises...

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