Game Theory Dynamic Bayesian Games

1
Game Theory Dynamic Bayesian Games

• Univ. Prof.dr. M.C.W. Janssen
• University of Vienna
• Winter semester 2008-9
• Week 2 (January 7)

2
Difference Dynamics and Statics

• The only thing to learn in static game with
  asymmetric information is when types are
  correlated and then information about own type
  reveals info about types of other players
• Usually, independent types are assumed
• In dynamic games with asymmetric information
  players may learn about types of other players
  through actions that are chosen before they
  themselves have to make decisions

3
Important class of signaling games

• In signaling games there are two players, Sender
  and Receiver
• Type of Sender is private information, sender
  takes an action
• Strategy is action depending on type
• Receiver takes an action after observing action
  taken by the sender
• Type of sender may be inferred (revealed) on the
  basis of the action that is actually taken

4
A simple example
3,2
accept
reject
U
-1,0
apply
S
0,0
N.A
1/2
good
-1,-3
N
accept
apply
1/2
reject
U
-1,0
bad
S
N.A
0,0
5
What is an equilibrium in this example?

• Strategies for both players such that both
  players strategies are optimal given strategy of
  the other and the possibly updated beliefs
• Pooling and separating equilibria possible (as
  well as in more complicated games- semi-pooling
  or semi-separating equilibria)
• Pooling equilibria are equilibria where all types
  of the sender choose the same action
• (Fully) Separating equilibria are equilibria
  where all types choose different actions from one
  another (space of actions and types should allow
  this)

6
A Separating equilibrium

• Proposal University accepts, student applies if,
  and only if she is good
• Check Can a player benefit by deviating?
• Good student gets positive pay-off in
  equilibrium, if she deviates she gets 0
• Bad student does not want to apply as this would
  give a negative pay-off of -2 instead of the
  equilibrium pay-off of 0
• University can get a pay-off of 2 or -3 when
  accepting. How to weigh these pay-offs? Bayes
  Rule says that P(good student/application) 1.
  Thus, equilibrium pay-off is 2 and deviating
  gives lower pay-off (of 0)
• Thus, this is a separating equilibrium
• Are there other separating equilibria?

7
A Pooling equilibria

• Proposal University rejects, student never
  applies
• Check Can one of the players benefit by
  deviating
• University always gets pay-off of 0 if student
  does not apply. Deviating does not improve his
  situation.
• Student does not want to apply (whatever her
  type) knowing she will be rejected pay-off of -1
  instead of the equilibrium pay-off of 0)
• Thus, this is a pooling equilibrium
• Are there other pooling equilibria?

8
The example changed
3,2
accept
reject
U
-1,0
apply
S
0,0
N.A
1/2
good
-1,1
N
accept
apply
1/2
reject
U
-1,0
bad
S
N.A
0,0
9
Is pooling equilibrium reasonable?

• First, it is important to realize that pooling on
  not applying is still part of an equilibrium
• But, in this modified game, it does not seem
  reasonable. Why?
• Rejecting students always gives a pay-off of 0,
  whereas accepting gives a positive pay-off
• Thus, rejecting is an incredible threat
• How to get rid of incredible threats? (Usually,
  impose subgame perfection. But how many subgames
  are there?)

10
Perfect Bayes-Nash equilibrium

• Refinement of Bayes-Nash equilibrium
• Often, in games with private information some
  information sets are off-the-equilibrium path
• When an information set is off-the-equilibrium
  path, Bayes Rule cannot be applied (gives 0/0)
• Bayes-Nash equilibrium does not impose any
  restrictions on strategy after such info set
• Perfect Bayes-Nash equilibrium says that (i) some
  out-of-equilibrium beliefs have to be specified
  and (ii) given these beliefs, actions have to be
  optimal

11
Definition PBNE

• A Perfect Bayes-Nash equilibrium is a set of
  strategies s, one for each player, and
  out-of-equilibrium beliefs µ(./a) such that
• i. each player chooses an optimal strategy given
  strategies of other players evaluated at updated
  beliefs
• iia. µ(./a) is formed using Bayes Rule whenever
  possible, i.e., if ?? p(?)s(a/?) gt 0
• iib. µ(./a) is any (arbitrary) probability
  distribution over type space T if ?? p(?)s(a/?)
  0

12
Ruling out pooling equilibrium in modified example

• PBNE requires P(good student/apply) to be
  specified. Lets say it is µ, where 0 µ 1
• Thus, P(bad student/apply) 1 µ
• The expected pay-off for the University of
  accepting a student is then 2µ (1-µ), which is
  positive for any permissible value of µ.
• Therefore, University cannot reject a student if
  it receives an application, as this is not
  optimal given any out-of-equilibrium belief µ.
• With separating equilibrium, one does not need to
  specify out-of-equilibrium beliefs, as no
  information set is out-of-equilibrium.

13
Sequential Rationality

• Sequential rationality extends and refine PBNE
• Rationality condition of PBNE is extended to hold
  for any info set
• Beliefs condition is refined to include info sets
  that do not arise from asymmetric information

14
Example where sequential rationality is needed

• Two NE (T,U) and (B,D)
• SPE and PBNE do not have any bite
• Still (T,U) seems unreasonable as it is always
  best for player 2 to choose D
• But SPE and PBNE do not require to specify
  beliefs in this situation

2,2
T
0,0
U
M
D
0,1
B
1,0
U
D
3,1
15
Definition Sequential equilibrium

• An assessment (s,µ) is a sequential equilibrium
  if for any info set h and alternative strategy
  si(h)
• ui(h)(s/h,µ(h)) ui(h)((si(h) ,s-i(h))/ h,µ(h))
  
• (s,µ) is consistent in the sense that (s,µ)
  should be able to be considered as the limit of a
  sequence (sn,µn) where sn is a completely mixed
  strategy satisfying the first condition and such
  that in any info set any action is chosen with
  strictly positive probability and µn is obtained
  from sn using Bayes Rule (which is possible to
  apply for any sn
• Consider again example to see that (T,U) is not
  sequentially rational

16
Reconsider simple example
3,2
accept
reject
U
-1,0
apply
S
0,0
N.A
1/2
good
-1,-3
N
accept
apply
1/2
reject
U
-1,0
bad
S
N.A
0,0
16
17
Is pooling reasonable?

• It satisfies PBNE
• Specify out-of-eq beliefs P(good/apply) lt 3/5
• It satisfies sequential equilibrium
• Sequence sn where both types of students choose
  to apply with probability 1/n. Updating beliefs
  gives P(good/apply) ½ so that university has to
  reject along the sequence sn
• But still it seems that the out-of-equilibrium
  belief should be P(good/apply) 1. Why? To apply
  is a dominates strategy for bad student. (he has
  never incentive to apply whereas good student
  may have incentive if she is rejected)
• How to formalize this?