Econ 713. Discussion. Week #4. 02-16-06
Danqing Hu & Michael Rapp
1 Subgame Perfect Equilibrium (SPE)
De…nition 1 (Subgame) A subgame 0of an extensive form game is a subset of which
1. Begins at a node x0
0;and contains this node, all of its successors, and no other nodes.
2. Is closed under information sets, i.e., if x20; x 2h; and y2h, then y20:
De…nition 2 (SPE) A Po…le is a SPE of if it induces a NE in every subgame of
2 Perfect Bayesian Equilibrium (PBE)
De…nition 3 (Belief) Beliefs are a map :D7! [0;1] for all decision nodes which satis…es P
x2h
(x) = 1
for all information sets h:
De…nition 4 (Sequentially Rational) A strategy pro…le is sequentially rational given beliefs if for
each player i and information set h2Hi;player i’s behavior conditional on h being reached maximizes his
expected utility given iand :
De…nition 5 (Bayesian beliefs) The beliefs are Bayesian given pro…le if
If x2h; then (x) = P(x)
P(h)whenever P(h)>0:
De…nition 6 (PBE) The strategy-beliefs pair (; )is a (weak) Perfect Bayesian Equilibrium if
1. is sequentially rational given :
2. is Bayesian given :
Exercise 1 Consider the fol lowing interaction between two entrepreneurs (players 1 and 2) who are working
on a joint project and a venture capitalist (player 3) who is a potential investor in the project. First,
the entrepreneurs simultaneously decide whether to devote high or low e¤ort to preliminary work on the
project. They then make a presentation to the venture capitalist. If both entrepreneurs chose high e¤ort, the
presentation goes well; otherwise it goes poorly. The venture capitalist only observes whether the presentation
goes well or poorly; he does not directly observe the entrepreneurs’e¤ ort levels. The payo¤s are as follows.
Each entrepreneur obtains $5 if the venture capitalist invests and $0 otherwise. In addition, choosing high
e¤ort costs an entrepreneur $1, while choosing low e¤ort is free. Investing costs the venture capitalist $2,
but if he invests he gains $3 for each entrepreneur who chose high e¤ort. If the venture capitalist does not
invest, his payo¤ is $0. All players are risk neutral.
1. Draw an extensive form representation of this game.
2. Find al l perfect Bayesian equilibria in which al l players choose pure strategies.
1
