Collusion under Imperfect Price Information

14.126 Game Theory Muha met Yildiz

Model

 Infinitely repeated game with n firms

 Each firm ma xi mize s disco unted sum of its profit (  )

 Stage Game: each si multaneousl y produce q i ;

 Price p = P (  , Q ), where

  is i.i.d s hock

 Q = q 1 +… + q n total supply

 Fir m s observe p but not Q .

 Perfect Monitoring:  = E [  ]

 Imperfect Monitoring: each p is possi ble for each Q .

 Assum p ti on: Unique static NE: ( q N , q N ,…, q N )

 u i ( q ) = qE [ P (  , nq )] – q N E [ P (  , nq N )]

 u i ( q i ,q -i ) = q i E [ P (  , q i + ( n- 1) q -i )] – q N E [ P (  , nq N )]

SPE in Trigger strategies

 There are two modes: Collusion & War

 Collusion:

 Each produce q *;

 Switch to War if p < p *

 War: Each produce q N for T * periods, follo w ed by Co llu s io n

  ( Q ) = Pr( p  p *| Q )

Optimal Trigger strategy under perfect monitoring

 p * = P (  , nq *)

 Payoff:

v * = u i ( q *)

 Incentive constraint: for each q i ,

v *  (1-  ) u i ( q i ,q* ) +  T *+1 v *

 Optimal SPE in trig ger strategies:

 T * = 

 q* = arg m ax u i ( q *) s.t. u i ( q *)  (1-  ) u i ( q i ,q* )  q i

 Main Lesson: Punish as hard as possible!

SPE conditions

 SPE payoffs :

v = (1 -  ) u ( q *) +  ( nq *) v +  (1 -  ( nq *) )  T * v

v 

1  

1   ( q *)   T *  1 ( 1   ( q *))

u ( q *)

 SPE condition (I C): for all q i ,

v  (1 -  ) u ( q i , q *) +  ( q i +( n -1) q *) v +  (1-  ( q i +( n -1) q *))  T * v

 ( 1   T * )[  ( nq *)   ( q  ( n  1 ) q *)]

1   ( nq *)   T *  1 ( 1   ( nq *))

 Optimal SPE: maximiz e v subject to (IC).

  ( nq * ) < 1. [P rice wa rs ob served with p r obability 1]

 T* may b e <  [You may not want to punis h as ha rd a s po ss ib le .]

u i ( q i , q *)  u i ( q *)  i u ( q *)