Transcript Static Games of Incomplete Information . Mechanism design • Typically a 3-step game of incomplete info Step 1: Principal designs mechanism/contract Step 2: Agents accept/reject.

Static Games of Incomplete
Information
.
Mechanism design
• Typically a 3-step game of incomplete info
Step 1: Principal designs mechanism/contract
Step 2: Agents accept/reject the mechanism
Step 3: Agents that have accepted, play the
game specified by mechanism
• Constant theme: Incomplete information and
binding individual rationality constraints
prevent efficient outcomes
Nonlinear pricing
• A monopolist produces good at marginal cost c
and sells quantity q
• Consumer transfers T to seller and has utility
u1(q, T, θ)= θV(q)-T, V(0)=0, V/>0, V//<0
• θ is private knowledge for buyer
• Seller knows that θ=  w.p. p and θ=  w.p. p
• The game:
1. Seller offers tariff T(q): specifies a price for qty q
2. Consumer accepts/rejects
• If seller knows θ, she will charge T= θV(q), her
profit, θV(q)-cq. This is maximized at some q given
by θV/(q)=c
Nonlinear pricing
• Let (q, T ) be bundle for type  and (q, T ) for type 
• Seller’s expected profit: Eu0  p(T  cq)  p(T  cq)
• Seller faces two constraints:
1. Individual Rationality (IR): Consumer should be
willing to purchase
2. Incentive Compatibility (IC): Consumer should
consume the bundle intended for his type
• IR1: V (q) T  0 ; and IR2: V (q) T  0
• IC1:V (q)  T  V (q)  T ; and IC2: V (q)  T  V (q)  T
• First step: To show that only IR1 and IC2 are binding
Nonlinear pricing
• First note: IR1 and IC2 imply IR2
• IR2 can’t be binding unless q =0
• However, IR1 must bind. Else seller can increase
T & T by same amount and increase revenue
• Also, IC2 must be binding, else seller can increase
T , satisfy all constraints and increase revenue
• The high-type’s indifference curve is always
steeper than the low type’s for any allocation
• This implies that high type consumes more than
low type: q  q
Nonlinear pricing
• Eliminating transfers, principal’s objective function is:
max{([ p  p(   )]V (q)  pcq)  p(V (q)  cq)}
q ,q
/
q
:

V
(q)  c /(1 
• FOC wrt
p(   )
)
p
• FOC wrt q : V / (q)  c
• Check that IC1 is satisfied
• Note: Quantity purchased by high-type is optimal
Quantity purchased by low-type is sub-optimal
• Seller sacrifices efficiency for rent-extraction!
Auctions
•
•
•
•
Seller has unit of good and there are two bidders
Each bidder can have types  ,  with  < 
Corresponding probabilities are p and p
Buyer’s expected probability of getting the good are
X & X and payments are T & T
• The constraints are:
IR1:  X T  0 ;
IR2:  X T  0
IC1:  X  T   X  T ; IC2:  X  T   X  T
• What is seller’s optimal contract?
Auctions
• Seller’s expected profit is: pT  pT
• Again, IR1 and IC2 are binding. The seller’s profit:
Eu0  (  p ) X  p X
• Also, ex-ante prob of a player getting good,
• Moreover, X  p  p
pX  pX 
1
2
2
p


p

X

0
X

p

• Case 1:
. The seller sets
and
2
Optimal mechanism: Not to sell if both announce low-type;
sell to high-type if they announce different types; sell wp ½ to
each if both announce high type
X  p
p
2
• Case 2:   p . The seller sets X  p / 2 and
Optimal mechanism: Sell to high-type if bidders announce
different types, and sell wp ½ to each if they both announce
high-type or low-type
Moral Hazard
•Consider a Principal and an agent who can
exert costly effort, e
•Let e  {0, 1}, with costs: ψ(0)=0, ψ(1)= ψ
•Agent receives transfer, t, and has utility;
U=u(t)- ψ(e), with u/>0, u//<0.
•Production is stochastic, and production level,
q~ {q, q}, q  q
• Stochastic influence of effort on production:
Pr{q~  q e  0}   0 ; Pr{q~  q e  1}  1 ,  1   0
Moral Hazard
• Principal can offer a contract, {t(q~ )}, that depends
on observed, random output q~
• With two possible outcomes, contract is: t if output
is q and t if output is q
• Let Principal’s profit with qty q be S(q)
• His profit when agent expends effort e=0 is:
V0   0 [S (q)  t ]  (1   0 )[S (q)  t ]
• His profit when agent expends effort e=1 is:
V1  1[S (q)  t ]  (1  1 )[S (q)  t ]
Incentive Feasible Contracts
• Induce positive effort and ensure participation
• Incentive constraint:
1u(t )  (1 1 )u(t )    0u(t )  (1  0 )u(t )
• Participation constraint:
1u(t )  (1  1 )u(t )   0
Complete Information Benchmark
• Complete info or First-Best: Principal observes effort
• Principal’s problem is:
max 1 ( S  t )  (1   1 )(S  t )
{( t ,t )}
•
•
•
•
subject to: 1u(t )  (1  1 )u(t )   0
Using Lagrangian, μ, and from FOCs we have,
1
1
 / * 
0
*
u (t ) u / (t )
*
*
*
From the above equations, we have that: t  t  t
Thus, Agent obtains full insurance!
The optimal transfer is: t*= u-1(ψ)=h(ψ), where h=u-1
First Best Case
• When there is complete information
• Principal’s profit from inducing effort e=1:
V1= 1 S  (1  1 )S  h( )
• If agent exerted 0 effort, principal would earn:
V0= 0 S  (1   0 )S
• Inducing effort is optimal for principal if:
S  h( ), where   1   0 ; S  S  S
• Principal’s First-Best cost of inducing effort is: h(ψ)
Second-Best: In terms of transfers
• Agent is risk-averse
• Principal’s problem, P, is:
• (P):
max1 (S  t )  (1  1 )(S  t )
{(t ,t )}
subject to: 1u(t )  (1  1 )u(t )   0 , and
1u(t)  (1 1)u(t)    0u(t)  (1  0 )u(t)
• First ensure concavity of (P): Let
u  u(t ); u  u(t )
Second-Best: In terms of utilities
• The Principal’s program can be rewritten in terms of
utilities
• (P/): max ( S  h(u ))  (1   )(S  h(u ))
{( u ,u )}
1
1
subjectto :  1 u  (1   1 )u    0 u  (1   0 )u
 1 u  (1   1 )u   0
• Principal’s objective function is concave in (u, u )
because h(.) is convex, and the constraints are linear
• The KKT conditions are necessary and sufficient
Both IR and IC are binding
• Let λ & μ be Lagrange multipliers for IC & IR
• The FOCs, upon rearranging terms, are:
1

1

  
; / SB    
SB
 1 u (t )
1  1
u / (t )
where, t
SB
,t
SB
are second-best optimal transfers
1  1
1
• From these,   / SB  SB  0 , so IR is binding
u (t ) u / (t )
• Also,   1 (1  1 ) ( 1SB  1SB )  0 , so IC is binding
 u / (t ) u / (t )
Second-Best Solution
• The variables ( t , t , λ, μ ) are solved
simultaneously from two FOCs, IC and IR
• The second-best optimal transfers are:
SB
t
SB
SB
SB


SB
 h(  (1   1 )
); t  h(   1
)


• t  t : contract does not provide full insurance
SB
SB
nd
SB

t

(
1


)
t
• 2 Best cost of inducing effort: C = 1
1
• Clearly, for the Principal, CSB> CFB. So Principal
induces high effort (e=1) less often than in first-best
• There is under-provision of effort in the second-best
SB
Mechanism design with a single agent
• Agent’s type  [ ,  ] with distribution/density P( ) / p( )
• Type-contingent allocation is fn.   y( )  ( x( ), t ( ))
• Defn: A decision function x :  
is X
implementable if
there exists a transfer t(.) such that allocation y(.) is
incentive-compatible, i.e.
u1 ( y( ), )  u1 ( y(ˆ), ),  ( ,ˆ) [ , ][ , ]
• Theorem: A piecewise C1 decision fn x(.) is
implementable only if
 u1 / xk dxk
(
)
0

u1 / t d
k 1 
n
whenever x  x( ), t  t ( ) and x is differentiable at θ
Mechanism design with a single agent
• Sketch of proof: Type θ announces ˆ to maximize
(ˆ, )  u1 ( x(ˆ),t (ˆ), )
( , )
 2( , )
 0,
0
The FOC and SOC are
ˆ
ˆ2


Totally differentiating the first equation,
 2 ( , )
0
ˆ
 
The (local) SOC becomes
  u1  dxk   u1  dt



0






x
d




t
d



k 1
 k
n
 2 ( , )  2( , )

0
2
ˆ
ˆ


or,
 u1  dxk  u1  dt



0



x
d


t
d



k 1 
k 
n
Rewrite the FOC we get,
Eliminating, dt/dθ,
  u1  u1   u1  u1 u1 dxk

{[ 

]/
}
0



  xk  t   t  xk t d
k 1
n
Mechanism design with a single agent
• The sorting/ single crossing/ constant sign (CS) condition is:
u1 / xk
u1 / t
  u1 / xk

  u1 / t

  0

• Note that
is agent’s marginal rate of substitution
between decision k and transfer t
u1
• Consider x to be output supplied by agent, i.e., x  0
• Then sorting condition means that the agent’s indifference
curve in (x, t) space, u1 / x , is decreasing in θ
u1 / t
• If θ2> θ1 , y(θ1)=(x(θ1), t(θ1)), y(θ2)=(x(θ2), t(θ2)), then y(θ2)>y(θ1)
• Theorem: If decision space is 1-dim and CS holds, then a
necessary condition for x(.) to be implementable is that it is
monotonic.
• What about sufficiency?
Optimal mechanisms for one agent
• The assumptions:
A1: Reservation utility u independent of type
A2: Quasi-linear utilities:
Principal: u0(x, t,θ)= V0(x, θ)-t; Agent: u1(x, t,θ)= V1(x, θ)+t
 2V1
0
A3: n=1, i.e., decision is 1-dim and CS holds.
x
A4:
V1
0

A5:
 2V0
0
x
A6:
 3V1
 3V1
 0, & 2
0
2
x
x 
Optimal mechanisms for one agent
• The problem: Principal maximizes his expected utility
max E u0 ( x( ), t ( ),  )
{ x ( ),t ( )}
subject to: (IR) u1(x(θ), t(θ), θ)≥ u =0, for all θ
(IC) u1(x(θ), t(θ), θ)≥ u1 ( x(ˆ), t (ˆ), )  ( , ˆ)
• From A1 & A4, if IR satisfied at    , it is satisfied everywhere
• IR binding at    . Thus, u1 ( x( ), t ( ), )  u  0
• Let U1 ( )  maxu1 ( x(ˆ), t (ˆ),  )  u1 ( x( ), t ( ),  )
ˆ

• From Envelope theorem,
• This implies that,
dU 1 u1 V1


d


~ ~
V1 ( x( ,  ) ~
U1 ( )  u  
d
~



Optimal mechanisms for one agent
• Further, u0= V0+ V1- U1≡ Social surplus-Agent’s utility
• Principal’s objective function:
~ ~
V1 ( x( ), ) ~
 [V0 ( x( ), )  V1 ( x( ), )   ~ d ] p( )d 



 [V0 ( x( ), )  V1 ( x( ), ) 

1  P( ) V1 ( x( ), )
] p( )d
p( )

• Since monotonicity is necessary and sufficient for
implementability, Principal’s optimization program becomes

1  P( ) V1 ( x,  )
max  [V0 ( x,  )  V1 ( x,  ) 
] p ( )d
{x(.)}
p( )


s.t. x(.) is monotonic
Optimal mechanisms
• We solve the principal’s program ignoring monotonicity
• The solution to the relaxed program is
V0 V1 1  P( )  2V1


x
x
p( ) x
• The principal faces a trade-off between maximizing total
surplus (V0+ V1) and appropriating the agent’s info rent (U1)
• When is it legit to focus on relaxed program?
When solution x*(θ) to above eq is monotonic. Differentiating,
 2V0  2V1 1  P( )  3V1 dx*
( 2  2 
)
2
x
x
p( ) x  d
 2V0 1  P( )  3V1
 2V1 d 1  P( )

[ (
)  1] 

x d p( )
x
p( ) x 2
When Hazard rate is monotone:
d  p( ) 

0
d  1  P( ) 