PS30MaterialReview

PS30MaterialReview - PS 172 Week 2 Notes; 04/09/2008 PS30...

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PS 172 Week 2 Notes; 04/09/2008 PS30 Material Review—Part 2 MSNE with more than 2 Strategies Player 1 Player 2 X Y Z A 0, 5 10^, 5 5, 10* B 0, 10* 15^, 0 0, 5 C 5, 10 0, 15* 10^, 0 No PSNE found. Let’s try to find the MSNE. Remember the cardinal rule: your payoff with the other player’s probability . This is because what you get depends on the fact that the other player is going to play that strategy. Now, we attach probability values with each strategy. Player 1 Player 2 X (q1) Y (q2) Z (1-q1-q2) A (p1) 0, 5 10^, 5 5, 10* B (p2) 0, 10* 15^, 0 0, 5 C (1-p1-p2) 5, 10 0, 15* 10^, 0 Now we calculate the expected utility of each player’s strategy: EU(1A) = 0(q1) + 10(q2) + 5(1-q1-q2) = 10q2 + 5 – 5q1 – 5q2 = 5 – 5q1 + 5q2 EU(1B) = 0(q1) + 15(q2) + 0(1-q1-q2) = 15q2 EU(1C) = 5(q1) + 0(q2) + 10(1-q1-q2) = 10 – 5q1 – 10q2 Now we have 3 equations with 2 unknowns. We can solve for the probability values by first equating 2 expected utilities together: EU(1A) = EU(1C) 5 – 5q1 + 5q2 = 10 – 5q1 – 10q2 15q2 = 5 q2 = 1/3 The reason why I equate 1A and 1C is so that I can cancel as many terms as possible (in this case, a variable). Note that variables don’t always cancel out this nicely though! Now we can solve for q1: EU(1A) = EU(1B) 5 – 5q1 + 5(1/3) = 15(1/3) 5q1 = 5/3 q1 = 1/3 We can now solve for 1- q1 – q2. Do the same for p1, p2, and 1-p1-p2 for exercise. 1
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PS 172 Week 2 Notes; 04/09/2008 Pareto Efficiency in MSNE Multiple PSNE: I II A (q) B (1-q) A (p) 5*, 5* 0, 3 B (1-p) 3, 0 1*, 1* MSNE: For player 1: EU (A) = 5q + 0(1-q) EU (B) = 3q + 1(1-q) 5q = 3q + 1(1-q) = 3q + 1 – q = 2q + 1 3q = 1 q = 1/3 For player 2: since the game has symmetric payoffs, p = 1/3 We want to see if using the MSNE strategy will yield a Pareto efficient outcome. So, we need
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PS30MaterialReview - PS 172 Week 2 Notes; 04/09/2008 PS30...

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