Unit 13- Lesson 47- Mutual Grim Trigger as Nash Equilibrium PDF

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Spring 2016
Prof. Kathleen Brown...

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Lesson 47: Mutual Grim Trigger as Nash Equilibrium Need to show not just that GT is better than Unconditional Defect, but better than any other strategy. Cannot consider each possible alternative strategy individually - - they are infinite. Need to be to think about other able to group them. What about other ways of deviating from GT? Other Way #1. Cooperate after someone has defected. Given that other player is playing GT, this will result in sucker pay-off. For example: Alt 1: Defect, then cooperate, then return to GT 1

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Value (Dev2) = 5-2∂ Value (GT) = 3 + 3∂ + 3∂^2 + ... This way of deviating gives a lower pay-off than UD. V(A)(Dev 2) ≤ V(A)(Dev 1) because 5 - 2∂ < 5 whenever ∂ > 0. So whenever UD is not better than Grim Trigger (Whenever 5 ≤ [3 / (1-∂)], Alt 1 will also not be better (5 2∂ will also be less than [3 / (1-∂)]) Alt 2: Defect after cooperating for awhile? 1

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Value (Dev2) = 3 + 3∂ + 5∂^2 Value (GT) = 3 + 3∂ + 3∂^2 + ... Alt 2 is not an improvement if 3 + 3∂ + 5∂^2 ≤ 3 + 3∂ + 3∂^2 + 3∂^3 + ... The first two terms on each side cancel and we can factor out ∂^2 to get 5 ≤ 3 + 3∂ + 3∂^2 + 3∂^3 + ... which is a condition as UD Idea: If it’s worthwhile to defect at any point, it’s worth defecting right away Never need to compare one deviation to another Summary: Showing that strategies from an equilibrium is a...