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ps5(2010)_soln

# ps5(2010)_soln - MS&E 246 Game Theory with Engineering...

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Unformatted text preview: MS&E 246: Game Theory with Engineering Applications Feryal Erhun Winter, 2010 Problem Set # 5 Solutions 1. (25pts total) (a) (5pts) Π csc = P R Q xf ( x ) dx + R ∞ Q Qf ( x ) dx- cQ . Using the first-order condition (FOC): P (1- F ( Q ))- c = 0 ⇒ F ( Q ) = P- c P . (b) (5pts)Π R = P R Q xf ( x ) dx + R ∞ Q Qf ( x ) dx- wQ . Using the first-order condition (FOC): P (1- F ( Q ))- w = 0 ⇒ F ( Q ) = P- w P . (c) (5pts)Unless w = c , the wholesale price contract does not coordinate the supply chain. (d) (5pts)Π R = ( P- w e ) R Q xf ( x ) dx + R ∞ Q Qf ( x ) dx- w o Q . Using the first-order con- dition (FOC): ( P- w e )(1- F ( Q ))- w o = 0 ⇒ F ( Q ) = P- w e- w o P- w e . (e) (5pts) Let w o = λc and w e = (1- λ ) P , λ ∈ [0 , 1]. Then the DSC is coordinated using the options contracts. 2. (20pts total) (a) (5pts) Each player has two types. There are four pure strategies contingent on the type of a player: – AA: Always Attack. – AN: Attack if strong and Not attack if weak. – AA: Not attack if strong and Attack if weak. – NN: Never Attack. (b) (15 pts) Player 2 AA AN NA NN AA M 4- s + w 2 , M 4- s + w 2 M 2- s + w 4 , M 4- s 2 3 M 4- s + w 4 ,- w 2 M, AN M 4- s 2 , M 2- s + w 4 M- s 4 , M- s 4 M 2- s 4 , M- w 4 M 2 , Player 1 NA- w 2 , 3 M 4- s + w 4 M- w 4 , M 2- s 4 M- w 4 , M- w 4 M 2 , NN ,M , M 2 , M 2 , 1 Since s < w , if it is a best response to attack when weak, then it is also a best response to attach when strong. Thus, NA will never be played in a BNE. For each other strategy we compute the best response. Note that the payoff of not attacking is always 0....
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ps5(2010)_soln - MS&E 246 Game Theory with Engineering...

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