Game Theory Dynamic Bayesian Games III

1
Game Theory Dynamic Bayesian Games III

• Univ. Prof.dr. M.C.W. Janssen
• University of Vienna
• Winter semester 2008-9
• Week 4 (January 19)

2
Further refinements

• There are many refinements proposed in the
  literature that go further than IC
• Some or most of these are however disputed for
  the fact that they are too restrictive,
  unrealistic, etc.
• To give you an idea of what these refinements do,
  I will discuss one further refinement D1

3
D1 refinement, basis idea

• So far we have looked at refinements where some
  types do not have an incentive to deviate where
  others do
• Incentive in terms of some notion of domination
• D1 talks about the strengths of the incentive to
  deviate in terms of the size of the set of
  strategies of the receiver
• If for one type the set of strategies of the
  receiver that makes deviation profitable is
  larger than for another type, then according to
  D1 it should be infinitely more likely that the
  deviation comes from this former type.

4
D1 example
1,3
U
M
R
-1,0
in
D
S1
2,2
0,0
out
1/2
up
-1,0
N
U
in
1/2
M
-1,3
down
D
S2
2,2
out
0,0
4
5
Consider IC in the context of the example

• Consider the pooling equilibrium where both types
  of the Sender choose out and the Receiver would
  choose M if the receiver would choose in
• This is consistent with IC.
• Equilibrium pay-off is 0
• Maximum possible deviation pay-off is 2 for both
  types (if e.g. player 2 chooses D and D is a
  possible best reponse if player 2 believes that
  1/3 lt µ(?/a1) lt 2/3 )
• In fact, any response by receiver is a possible
  best response

6
This pooling equilibrium is ruled out by D1

• What does D1 do?
• For which mixed reaction of Receiver would a
  certain type want to deviate?
• Write strategy of Receiver as (a,ß,1- a-ß)
• Pay-off of type S1 choosing in is then 2 - a
  -3ß. Thus if a 3ß lt 2 then this type would like
  to deviate
• Pay-off of type S2 choosing in is 2 - 3a -3ß.
  Thus if 3a 3ß lt 2 then this type would like to
  deviate
• Thus, if type S2 has an incentive to deviate,
  then type S1 also has an incentive
• D1 then says that after observing in, player 2
  should infer that it is type S1 that has deviated
  and thus should choose U after such a deviation.
  But then, S1 would like to deviate and choose in

7
More formal definition of D1

• Fix an equilibrium. Let u(?) the equilibrium
  pay-off of type ?
• Define D(?,a) the set of (possibly)
  mixed-strategy best responses to action a that
  make type ? to strictly prefer a to her
  equilibrium strategy
• A type ? is deleted for strategy a under
  criterion D1 if there is a type ? s.t.
• D(?,a) D(?,a)

8
More extensive example applying refinements
duopoly pricing signals quality?

• Two firms, either one produces either high or low
  quality
• Production cost cH gt cL
• Valuation VH gt VL and VH - cH gt VL cL
• Quality is private information to firms
• Other firm and consumers do now know it
• Ex ante probability of high quality is a
• Firms set prices given quality, consumers observe
  both prices, infer quality if possible, and then
  buy at the best deal.

9
Pooling equilibrium?

• p(H) p(L) p
• Constraints
• Consumers only buy if aVH (1-a)VL p
• High cost firms only set p if p cH
• Necessary condition aVH (1-a)VL cH
• Other conditions?
• All other prices are out-of-equilibrium.
• For PBE we have to specify out-of-equilibrium
  beliefs. Making consumer beliefs pessimistic
  creates favourable conditions for equilibrium
  existence as incentives for firms deviating are
  minimal µ(high quality / p) 0
• Consumers will buy at other prices p and these
  out-of-equilibrium beliefs if aVH (1-a)VL p
  VL p
• Will (low) quality firm want to deviate to pp-
  a(VH -VL)? Equilibrium pay-off (p-cL)/2.
  Deviation pay-off p- a(VH -VL)- cL.. This gives
  2a(VH -VL) cL cH

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Are out-of-equilibrium beliefs reasonable?

• Domination-based beliefs µ(high quality / p) 0
  for all p lt cH . Satisfied
• Intuitive Criterion?
• Equilibrium pay-offs (p-ci)/2 for i L,H
• Consumers rational responses to
  out-of-equilibrium prices p can be to buy
  (provided these prices are not too high) as they
  may believe p set by high quality firm. Both
  types may find it optimal to deviate for p gt cH
  . No further restrictions
• D1?
• Consider s(p) probability consumers buy at
  out-of-eq price p
• Both types finds it optimal to deviate if
  (p-ci)/2 lt s(p)(p-ci) or s(p) gt
  (p-ci)/2(p-ci) s(p). As ?s(p)/?c gt 0 iff p
  gt p, consumers should believe that if they
  observe a p gt p it comes from a high quality
  firm. They will buy at such a price if it is not
  too high and therefore firms have an incentive to
  deviate
• No pooling equilibrium satisfies D1!

11
Revealing equilibria?

• There cant be a revealing equilibrium in pure
  strategies
• If there was one, consumers would update their
  beliefs after observing p(L) and a firm knowing
  that it sells low quality believes there is a
  probability a that the other firm also sells low
  quality and then can gain by slightly
  undercutting
• On other hand, out-of-equilibrium beielfs may
  prevent undercutting argument for high quality
  prices

12
Characterizing revealing equilibria

• Suppose high quality sets some p(H)
• Then low quality does not sell (in a revealing
  equilibrium) if it sets p(L) gt p(H) - (VH -VL)
• Maximum price low quality p(H) - (VH -VL)
• Low quality mixes with F(p) with this upper bound
• Profit low quality firm when charging p is then
• (1-a)(1-F(p)) ap ap(H) - (VH -VL)
• Can be solved for mixed strategy distribution
  F(p) with lower bound ap(H) - (VH -VL)
• Profits high quality firm are a(p(H) - cH)/2

13
On/off equilibrium path

• p(H) and ap(H) - (VH -VL), p(H) - (VH -VL)
  are all on equilibrium path
• Now we also have prices off-the-equilibrium path
  in a fully revealing equilibrium
• Do firms have an incentive to deviate?
• Mimic other type should not optimal
• ap(H) - (VH -VL) - cL a(p(H) - cL)/2
• a(p(H) - cH)/2 ap(H) - (VH -VL) - cH
• Out-equilibrium belief restrictions