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Dynamic games, beliefs,
and sequential rationality
Consider an altered entry deterrence game played by an entrant and a monopolist.
Suppose there are two strategies that an entrant can use to enter: in1 and in2
Suppose that the monopolist is unable to tell which strategy the entrant uses to enter
– so the monopolist observes entry or not but doesn’t observe in1 or in2
Now there are 2 NE:
1) (E plays out, M cuts if E enters)
2) (E plays in1, M doesn’t cut if E enters)
Both NE are subgame perfect
- because the only subgame is the entire game.
To eliminate the unreasonable equilibrium, we will introduce beliefs.
Monopolist will not cut prices
SPNE is not an effective refinement.
- NE1 involves a non-credible threat
- both NE are SPNE here since the only sub-game is the entire game
- both nodes are in the same information set
- we need a different type of refinement to remove non-credible threats
The basic content of the new equilibrium refinement:
- require that players have beliefs about the likelihood of being at the various nodes in
information sets with multiple nodes and that players optimize given their beliefs
Require the monopolist’s strategy be optimal for some belief about the strategy E has used.
- Cutting prices is not optimal no matter which entry strategy E used
- so for any probability λ that E used in1
(1 -λ ) that E used in2
the monopolist would optimally not cut prices
System of Beliefs
Defn: A system of beliefs  in extensive form game  E is a specification of a
probability   x   0,1 for each decision node x in  E such that
  x   1
 information sets H .
x H
If an information set contains 2 nodes, the player must assign probabilities to each
node such that the probabilities sum to 1.
Let E U i H ,  ,  i ,   i  denote player i ' s expected utility starting at information set H
if his beliefs regarding the conditional probabilities of being at the various nodes in H
are given by  , if he follows strategy  i , and all other players use   i .
Sequentially rational given beliefs
Defn: A strategy profile    1 ,... n  in extensive form game  E is
sequentially rational at information set H given a system of beliefs 
if, denoting by i  H  the player who moves at information set H , we have

 
E U i H  H ,  ,  i H  ,  i H   E U i H  H ,  ,  i H  ,  i H 


for all  i  H    Si  H  .

If strategy profile  satisfiies this condition for all
allows for mixing
information sets H , then we say  is sequentially rational given belief system  .
Strategy profile
A strategy profile    1 ,... n  is sequentially rational if no player finds it worthwhile,
once one of his info sets has been reached, to revise his strategy given his beliefs about
what has already occurred    and his rival's strategies   i  .
Player i has to be optimizing given what everyone is doing.
This is a best response notion with beliefs that apply for information sets with multiple nodes
Now two factors affect the optimal choice:
1) everyone else’s strategy
2) the player’s beliefs
Weak Perfect Bayesian Equilibrium
Defn: A profile of strategies and system of beliefs  ,   is a
weak perfect Bayesian equilibrium in extensive form game  E
if it has the following properties:
1)
2)
The strategy profile  is sequentially rational given belief system 
The system of beliefs  is derived from strategy profile 
through Bayes' Rule whenever possible
So for any information set H such that Prob  H    0 (the prob. of reaching H
under  is positive) we must have:
 x  
Prob  x  
Prob  H  
for all x  H .
2nd Property
Just as an example, this is not an equilibrium strategy,
Suppose E puts probability :
5/12 on staying out
1/3 on in1
1/4 on in2
H
Then the information set H is reached with positive probability, since both in1 and in2 are
played with positive probability.
The second property requires that M form beliefs using Bayes' Rule
Prob  H    1 3  1 4  7 12 (probability of reaching H given E 's strategy)
Prob  E uses in1    1 3 ;
Prob  E uses in2    1 4
E’s strategies
If E enters,
M must assign the following probabilities to the nodes following in1 and in2.
1
1
3
  node following in1 
;
  node following in2   4
7
7
12
12
Consider another strategy,
Suppose E stays out with probability 1, then H is never reached.
In this case, WPBE puts no restrictions on M’s beliefs about the probability of
being at the node following in1 and in2
It still requires that M have some beliefs and that M’s strategy be optimal
(sequentially rational) given those beliefs.
There is no system of beliefs for which cutting price is sequentially rational for M
So the equilibrium where E stays out can be ruled out using WPBE
Unique WPBE
WPBE  NE
- NE only require sequential rationality on the equilibrium path
- if an information set is never reached in equilibrium then play
does not have to be optimal there
The NE where (E plays in1, M does not cut P if E enters) is a WPBE.
-in the WPBE, M assigns prob 1 to being at the node following in1 if entry occurs
-given this belief, the strategies are sequentially rational
E plays in1, M updates beliefs that he is at node in1 w/ prob.1 
Unique WPBE : 

punish
not
does
M


More Generally,
Suppose E uses strategy : out with prob. 1  p1  p2  , in1 with p1 , and in2 with p2
Prob  H    p1  p2 ; Prob  E uses in1    p1 ;
  in1 
p1
;
p1  p2
  in2  
Prob  E uses in2    p2
p2
p1  p2
M needs to optimize given E's strategies and M's beliefs.
- M's optimal (sequentially rational) strategy is not punish
Suppose M does not punish, what is E's optimal strategy?
For E to use a mixed strategy, E must be indifferent between all of its pure strategies that
are played with positive probability. But this indifference cannot hold given M's strategy
E plays in2  2; in1  3; out  0.
We get a unique WPBE : in1 is clearly optimal so p1  1, p2  0,
1  p1  p2   0