Dynamic Games with Incomplete Information
In these lectures, we analyze the issues arise in a dynamics context in the presence of
incomplete information, such as how agents should interpret the actions the other parties
take. We define perfect Bayesian Nash equilibrium, and apply it in a sequential bargaining
model with incomplete information. As in the games with complete information,
now we will use a stronger notion of rationality – sequential rationality.
2 Perfect Bayesian Nash Equilibrium
Recall that in games with complete information some Nash equilibria may be based on
the assumption that some players will act sequentially irrationally at certain information
sets off the path of equilibrium. In those games we ignored these equilibria by focusing
on subgame perfect equilibria; in the latter equilibria each agent’s action is sequentially
rational at each information set. Now, we extend this notion to the games with incomplete
information. In these games, once again, some Bayesian Nash equilibria are based
on sequentially irrational moves off the path of equilibrium.
Consider the game in Figure 1. In this game, a firm is to decide whether to hire a
worker, who can be hard-working (High) or lazy (Low). Under the current contract, if
the worker is hard-working, then working is better for the worker, and the firm makes
profit of 1 if the worker works. If the worker’s lazy, then shirking is better for him, and
the firm will lose 1 if the worker shirks. If the worker is sequentially rational, then he
will work if he’s hard-working and shirk if he’s lazy. Since the firm finds the worker
1
Nature
High .7
Low .3
Firm Hire
W
Work
Shirk
Do not
hire
(1, 2)
(0, 1)
(0, 0)
Do not
hire
Hire
W Work
Shirk
(1, 1)
(-1, 2)
(0, 0)
Figure 1:
more likely to be hard-working, the firm will hire the worker. But there is another
Bayesian Nash equilibrium: the worker always shirks (independent of his type), and
therefore the firm does not hire the worker. This equilibrium is indicated in the figure
by the bold lines. It is based on the assumption that the worker will shirk when he
is hard-working, which is sequentially irrational. Since this happens off the path of
equilibrium, such irrationality is ignored in the Bayesian Nash equilibrium–as in the
ordinary Nash equilibrium.
We’ll now require sequential rationality at each information set. Such equilibria
will be called perfect Bayesian Nash equilibrium. The official definition requires more