Econ 713. Discussion. Week #5. 02-23-06
Danqing Hu & Michael Rapp
1 Sequential Equilibrium
De…nition 1 (Sequential Equilibrium) A strategy-beliefs pair ( ; )is a sequential eqiulibrium if
1. is sequentially rational given :
2. is consistent given :
9a sequence of strategy-beliefs pairs k; ksuch that
(a) each kis completely mixed
(b) each kis Bayesian given k, and k; k! (; ):
Theorem 1 Sequential equilibria exist and are subgame perfect.
Exercise 1 Last week, we looked for the set of PBE in the game depicted below:We found (L; l; (I; d)) and
H; h; (I ; d) ; (z)1
3. Indicate whether those equilibria are sequential or not.
Entrepreneurs’game
2 Perfect Equilibrium
De…nition 2 (Perturbed Game) For any game Gand 2RjSij
++ , the perturb ed game Gis the game G
with the restriction that each strategy simust be played with probability at least si.
De…nition 3 (Perfect Equilibrium) Pro…le is a perfect equilibrium of Gif there exists a sequence of
k!
0and a sequence of pro…les ksuch that
1. Each kis a Nash equilibrium of Gk;
2. k!
De…nition 4 ("perfect pro…le) For "2R;pro…le is "perfect if it is completely mixed and if si=2
Bi(i))i(si)"
Proposition 2 is a perfect equilibrium if and only if it is the limit of a sequence of "perfect pro…les
with "!0.
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