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Answers to Problem Set 4: Correction Economics 703 Spring 2016 2. d) The original version of the answers says that (cf, eh), (af, ei), and (af, di) are all weak perfect Bayesian equilibria. This is incorrect: only (cf, eh) and (af, ei) are weak perfect Bayesian. To see why (af, di) is not, consider 2 at his first information set. Let µ be the probability his beliefs give the left–hand node. Then 2’s expected payoff to playing d is 1 − µ. His expected payoff to the strategy ei is µ, suggesting that 2 is sequentially rational if µ ≤ 1/2. However, at his first information set, 2 evaluates his full continuation strategy. In particular, 2 can change to the strategy eh. This strategy yields expected payoff µ + 2(1 − µ) = 2 − µ > 1 − µ. Hence 2’s strategy at this information set is not sequentially rational. To see why taking into account changes of the full strategy doesn’t affect our conclusion about (af, ei), again consider that first information set for player 2. Now suppose 2’s belief puts probability 1 on the node on the left. In this case, regardless of what 2 expects in the subgame, e is the best strategy at this information set. Put differently, given this belief by 2, his expected payoff to the strategy d (or, more completely, dh or di) is 0, while his expected payoff to eh or ei is 1. Hence both eh and ei are sequentially rational at this information set. 1