Answers to Problem Set 4: Correction Economics 703 Spring 2016

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.