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ECO326H1 (3)
Lecture

# Topic 2 - Iterated Elimination and Rationalizability.pdf

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Department
Economics
Course
ECO326H1
Professor
Marcin Peski
Semester
Winter

Description
Topic 2 – Iterated Elimination and Rationalizability Sept 17, 2012 (Chapter 12.1-12.4, 2.8, 3.3)  Example 1 2 1 L R U 4,0 2,2 M 1,2 5,3 D 0,5 1,4 -- For Player 1, D is strictly dominated. So if Player 1 is rational, D won’t be played. 2 1 L R U 4,0 2,2 M 1,2 5,3 D 0,5 1,4 -- If Player 2 believes that Player 1 is rational and won’t play D, Player 2 will play R. (Eliminating D, R is strictly dominant for Player 2.), therefore L can be eliminated. 2 1 L R U 4,0 2,2 M 1,2 5,3 D 0,5 1,4 -- Now if Player 1 believes Player 2 is rational and will play R, then U is strictly dominated. 2 1 L R U 4,0 2,2 M 1,2 5,3 D 0,5 1,4 -- (M,R) will be the final prediction. -- We are able to reduce the game smaller and smaller.  Iterated Elimination for 𝐺 = 𝐼,𝐴 ,𝑈𝑖 𝑖 Step A: Find all strictly dominated strategies in the game 𝐷𝑖⊂ 𝐴 (𝑖ot ⊆ since not all actions are strictly dominated) 𝑁𝐸𝑊 𝑁𝐸𝑊 Step B: Replace game 𝐺 by 𝐺 in which 𝐺 = (𝐼,𝐴 \𝑖 ,𝑖 )𝑖 (I:same outcome; U :sime utility function; Ai\D ismaller action set,eliminating strictly dominated strategies); Question: any strictly dominated strategies in 𝐺 𝑁𝐸𝑊 ? 𝑁𝐸𝑊 If yes: 𝐺 → 𝐺, go back to Step A; If no: Stop.  Example 2: Model of Politics - Players: two opportunistic politicians - Assume voter population (10) are distributed evenly on the line: Left 1 2 3 4 5 6 7 8 9 10 Right - Actions: two politicians take position 1…10. (𝑁 =𝑖1,…,10) - Payoffs of politicians = number of votes he gets; (i.e. Their payoff is not necessarily associated with whether they win or not) P2 P1 1 2 3 4 5 6 7 8 9 10 1 5, 5 1, 9 1.5, 8.5 2, 8 2.5, 7.5 3, 7 3.5, 6.5 4, 6 4.5, 5.5 5, 5 2 9, 1 5, 5 2, 8 2.5, 7.5 3, 7 3.5, 6.5 4, 6 4.5, 5.5 5, 5 5.5, 4.5 3 8.5, 1.5 8, 2 5, 5 3, 7 3.5, 7.5 4, 6 4.5, 5.5 5, 5 5.5, 4.5 6, 4 4 8, 2 7.5, 2.5 7, 3 5, 5 4, 6 4.5, 5.5 5, 5 5.5, 4.5 6, 4 6.5, 3.5 5 7.5, 2.5 7, 3 6.5, 3.5 6, 4 5, 5 5, 5 5.5, 4.5 6, 4 6.5, 3.5 7, 3 6 7, 3 6.5, 3.5 6, 4 5.5, 4.5 5, 5 5, 5 6, 4 6.5, 3.5 7, 3 7.5, 2.5 7 6.5, 3.5 6, 4 5.5, 4.5 5, 5 4.5, 5.5 4, 6 5, 5 7, 3 7.5, 2.5 8, 2 8 6, 4 5.5, 4.5 5, 5 4.5, 5.5 4, 6 3.5, 7.5 3, 7 5, 5 8, 2 8.5, 1.5 9 5.5, 4.5 5, 5 4.5, 5.5 4, 6 3.5, 6.5 3, 7 2.5, 7.5 2, 8 5, 5 9, 1 10 5, 5 4.5, 5.5 4, 6 3.5, 6.5 3, 7 2.5, 7.5 2, 8 1.5, 8.5 1, 9 5, 5 Assume common knowledge of rationality. Round 1: 1 and 10 are strictly dominated by 2 for both Politician 1 and Politician 2. P2 P1 2 3 4 5 6 7 8 9 2 5, 5 2, 8 2.5, 7.5 3, 7 3.5, 6.5 4, 6 4.5, 5.5 5, 5 3 8, 2 5, 5 3, 7 3.5, 7.5 4, 6 4.5, 5.5 5, 5 5.5, 4.5 4 7.5, 2.5 7, 3 5, 5 4, 6 4.5, 5.5 5, 5 5.5, 4.5 6, 4 5 7, 3 6.5, 3.5 6, 4 5, 5 5, 5 5.5, 4.5 6, 4 6.5, 3.5 6 6.5, 3.5 6, 4 5.5, 4.5 5, 5 5, 5 6, 4 6.5, 3.5 7, 3 7 6, 4 5.5, 4.5 5, 5 4.5, 5.5 4, 6 5, 5 7, 3 7.5, 2.5 8 5.5, 4.5 5, 5 4.5, 5.5 4, 6 3.5, 7.5 3, 7 5, 5 8, 2 9 5, 5 4.5, 5.5 4, 6 3.5, 6.5 3, 7 2.5, 7.5 2, 8 5, 5 Round 2: 2 and 9 are strictly dominated by 3. P2 P1 3 4 5 6 7 8 3 5, 5 3, 7 3.5, 7.5 4, 6 4.5, 5.5 5, 5 4 7, 3 5, 5 4, 6 4.5, 5.5 5, 5 5.5, 4.5 5 6.5, 3.5 6, 4 5, 5 5, 5 5.5, 4.5 6, 4
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