Introduction
Irrelevance
Implementation
Conclusions
Dynamic Contracting: An Irrelevance Result
Péter Eső and Balázs Szentes
Oxford and LSE
Introduction
Irrelevance
Implementation
Conclusions
Model
at t = 0, . . . , T the agent learns type qt 2 [q t , q t ];
then takes a hidden action, at 2 R;
this generates a contractible public signal, st 2 R;
then a contractible decision xt 2 Xt ⇢ Rn is taken;
agent’s payoff: u
e ( q T , aT , s T , x T )
p; [superscript for history]
a contract is offered by the principal after q0 is realized.
Assumption 0
cdf of qt is Gt (· | q t 1 , at 1 , x t 1 ), full support, bdd derivatives;
pdf of st is ht (· | ft (qt , at )); for all qt , qbt and at there exists b
at such
that ft (qt , at ) = ft (qbt , b
at ); [cannot detect action directly]
u
e and ft are C 2 with bounded derivatives. [Lipschitz]
Introduction
Irrelevance
Implementation
Conclusions
An application
Agent: private banker. Principal: investor (eg., Oxford college).
qt is AR(1), the banker’s ability to achieve a greater return.
at
0 is the banker’s effort to engage in ‘ethical investment’.
xt is the invested amount; kxt for agent, (1
Agent’s payoff:
Principal’s:
ÂT
t =0
ÂT
t =0
h
h
1 ca2
2 t
qt kxt
i
k )xt for principal.
+ wt .
(qt + at + x t )(1
k )xt
i
1 rx 2
2 t
wt .
st = qt + at + x t , the principal’s perceived return, is contractible;
x t : noise; c: agent’s effort cost; r : principal’s cost of financing;
wt is the agent’s period-t wage (in addition to return on kxt ).
Contract signed right after q0 is realized (payment p ⌘
Introduction
Irrelevance
Ât wt ).
Implementation
Conclusions
Orthogonalization
For all t define # t = Gt (qt | q t 1 , at 1 , x t 1 ).
Note: Each # t is uniform on [0, 1] conditional on (q t 1 , at 1 , x t 1 ).
Call # t the agent’s orthogonalized type at t. [new information]
The agent’s type in the original model can be recovered as
qt = yt (#t , at 1 , x t 1 ) = Gt 1 # t |yt 1 (#t 1 , at 2 , x t 2 ), at 1 , x t 1 .
Redefine payoff:
T
T
T
T
⇣
T
T
u (# , a , s , x ) = u
e y (# , a
T 1
,x
T 1
T
T
), a , s , x
T
⌘
.
The model with orthogonalized types is equivalent to the original.
Introduction
Irrelevance
Implementation
Conclusions
Revelation principle
A direct mechanism is (xT , aT , p), where
at : E t ⇥ S t 1 ! R, xt : E t ⇥ S t ! Xt , and p : E T ⇥ S T ! R.
at (#t , s t 1 ) is the action suggested to the agent by the principal
(needs to be made incentive compatible, i.e., obedience);
xt (#t , s t ) is the contracted decision;
p(#T , s T ) is the total payment made by the agent.
Without loss consider incentive compatible (IC) direct mechanisms.
(xT , aT ) implementable if there exists p such that (xT , aT , p) is IC.
Introduction
Irrelevance
Implementation
Conclusions
Preview of results
[1] We derive necessary conditions of implementing (xT , aT ).
[2] Compare with benchmark case, where # 1 , . . . , # T are observable:
The cost of implementing (xT , aT ) is the same!
The benchmark is a static problem, so the availability of dynamic
deviations in the original problem is irrelevant for rents.
[3] Towards a recipe for solving dynamic contracting problems:
1
Solve the benchmark case.
2
Check if solution is implementable in the original case.
For step 2, we derive sufficient conditions of implementability.
Introduction
Irrelevance
Implementation
Conclusions
Related literature
Static Bayesian mechanism design – Vickrey, Mirrlees, Myerson ...
Necessary and sufficient conditions of incentive compatibility.
Dynamic mechanism design: Baron & Besanko (1984) with iid types,
irrelevance result noted. Besanko (1985) with AR(1).
More general type processes in specific applications:
Courty & Li (2000), Eso & Szentes (2007a,b), several others.
Pavan, Segal & Toikka (2014) solve a general hidden info model.
Garrett & Pavan (2012) solve a specialized hidden info + action case.
What’s new: Most general setup (hidden action, public signal), and
unlike Pavan, Segal & Toikka (2014) we state the irrelevance result.
Introduction
Irrelevance
Implementation
Conclusions
Necessary conditions for implementation
Idea: Restrict the agent’s deviation strategies such that
the agent may report b
# 0 instead of # 0 , but then
she reports # 1 , . . . , # T truthfully, and
she masks her initial lie by setting at = b
at (#t , b
# 0 , s t 1 ) such that
⇣
⌘
⇣
⌘
t
t
1
t
t
1
b
ft qt , at (b
#0, # 0, s
) = ft q t , b
at (# , b
#0, s
) ,
⇣
with qbt = yt b
# 0 , #t
⇣
and qt = yt #t , b
at
t 1 (b
# 0 , #t 01 , s t 2 ), xt 1 (b
# 0 , #t 01 , s t 1 )
0, a
⌘
t
1
1 ( #t 1 , b
t
2
t
1
t
1
#0, s
), x (b# 0 , # 0 , s
) .
⌘
The principal cannot detect the agent’s time-0 misreport b
# 0 on # 0 from
the public signals even if she reports (or he observes) # 1 , . . . , # T .
Introduction
Irrelevance
Implementation
Conclusions
Necessary conditions for implementation
Suppose (xT , aT , p) is IC, and denote the agent’s payoff by
⇥
P0 (# 0 ) = E u #T , aT (#T , s T
⇤
p( #T , s T ) # 0 .
1 ) , s T , xT ( # T , s T )
Proposition. If (xT , aT , p) is IC in the original or benchmark, then
P0 ( # 0 ) = P0 (0) + E
+
Z #0 T
0
Â
t =0
∂
∂at u
⇣
Z
#0
0
∂
∂# 0 u
⇣
y, #
T
T
0, a
(y , #
T
0 ), x
y , #T 0 , aT (y , #T 0 ), xT (y , #T 0 )
⌘
T
(y , #
T
0)
⌘
dy
∂
at (y , #t 0 , y ) dy
∂# 0 b
#
,
with (y , #t 0 ) ⌘ (y , # 1 , . . . , # t ) and s t suppressed in the notation.
Introduction
Irrelevance
Implementation
Conclusions
Proof
A restricted deviation (the agent reports b
# 0 6= # 0 , but then truthfully
reveals # 1 , . . . , # T and takes lie-masking actions) cannot be profitable.
The deviation payoff is W (# 0 , b
# 0 ) = U (# 0 , b
#0)
P (b
# 0 ), where
h ⇣
⌘
i
T 1
T T b
T
T
b
U (# 0 , b
# 0 ) = E u #T , b
aT (#T
,
#
,
s
)
,
s
,
x
(
#
,
#
,
s
)
#
0
0 ,
0 0
0
h
P (b
# 0 ) = E p(b
#0
, #T
Envelope Theorem:
0
, sT )
d P0 ( # 0 )
d #0
i
.
=
k
∂W (# 0 ,b
#0 )
∂# 0
b
# 0 =# 0
=
k
∂U (# 0 ,b
#0 )
∂# 0
b
# 0 =# 0
.
P0 is Lipschitz continuous; integrate it up from its derivative.
The same result in the benchmark where # 1 , . . . , # T are observable. ⇤
Introduction
Irrelevance
Implementation
Conclusions
The irrelevance result
Theorem. Suppose (xT , aT ) is implementable. The principal’s
maximal revenue (or least cost) of implementing (xT , aT ) subject to
participation constraint P0 (# 0 ) 0 is the same as in the benchmark.
Interpretation: The agent earns no rents for post-contractual, truly
“new” private information (i.e., no rents for observing # 1 , . . . , # T ).
Generalization of Baron & Besanko’s (1984) insight.
The availability of dynamic deviation strategies is irrelevant for the
agent’s payoff (also for the principal’s profit, if it is quasilinear.)
Introduction
Irrelevance
Implementation
Conclusions
Proof of the Theorem
Suppose (xT , aT , p) is incentive compatible in the original model,
and let p be the payment rule implementing (xT , aT ) in the
benchmark for maximal expected transfer. By the Proposition,
E
h
p( #T , s T )
#0
, aT , xT
i
E
h
p( #T , s T )
Consider mechanism (xT , aT , p + c ):
#0
, aT , xT
i
= c.
IC in the original model and induces participation;
has the same expected transfer as p implementing (xT , aT ).
Since the max transfer in the benchmark is an upper bound on the
max transfer in the original problem, both are the same. ⇤
Introduction
Irrelevance
Implementation
Conclusions
Recipe for solving dynamic contracting problems
1. Orthogonalize the information structure.
2. Solve the benchmark (static) problem.
 BUT: The benchmark solution may not be implementable! The two
problems (the original one and benchmark case) are not equivalent.
“if (xT , aT , p) is IC then you can’t do it cheaper in the benchmark”
What is implementable in the original problem? To be discussed next.
3. In regular models check the appropriate monotonicity condition.
Introduction
Irrelevance
Implementation
Conclusions
Implementation: Pure adverse selection
Assumption 1: (i) qt has continuous cdf Gt (·|qt 1 ). [Markov]
(ii) Gt (· | qt 1 )  Gt (· | qbt 1 ) whenever qt 1 qbt 1 . [FOSD]
t
Assumption 2: (i) ũ (q T , aT , x T ) = ÂT
t =0 ũt ( qt , at , x ). [t-separable]
∂
(ii) ∂q∂t u
et > 0, ∂q∂t u
et (qt , at , x t )
et (qt , at , b
x t ) for x t b
x t . [SC]
∂qt u
Proposition. Suppose u
et doesn’t vary with at and st is uninformative.
A decision rule e
xT is implementable if e
xt (q t ) is increasing for all t.
Follows from Theorem 2 of Pavan, Segal and Toikka (2014).
Idea: In a regular, Markovian environment, if the agent lies at t then
she undoes it at t + 1. Continuation payoff pinned down: induction.
Introduction
Irrelevance
Implementation
Conclusions
Implementation: No public signal
Assumption 3: (i) ut is concave in at ;
(ii)
(iii)
∂
et (qt , at , x t )
∂qt u
∂
et (qt , at , x t )
∂ at u
∂
et (qt , b
at , x t ) for at b
at ;
∂qt u
∂
et (qt , at , b
x t ) whenever x t
∂qt u
b
xt.
Define at⇤ (qt , x t ) = arg maxat ũt (qt , at , x t ). [agent-optimal action]
Then let vt (qt , x t ) ⌘ u
et (qt , at⇤ (qt , x t ), x t ).
Proposition: Suppose st is not informative of ft (qt , at ). (e
xT , e
aT ) is
implementable if e
xt is increasing in q t and e
at (q t ) = at⇤ (qt , e
xt (q t )).
Proof: Verify Assumptions 1-2 for felicity function vt (qt , x t ).
Then apply the previous Proposition with {vt }t . ⇤
Introduction
Irrelevance
Implementation
Conclusions
Implementation: Generic public signal
Assume at affects ut and st is informative of ft .
Assumption 4: The distribution of st conditional on ft (qt , at ) = ȳ is
not the average of distributions of st conditional on other values of ft .
Generic, introduced by McAfee & Reny (1992).
Assumption 5: (i) u
etat (qt , at , x t ) < 0.
(ii) There exists a K 2 N such that ftat (qt , at ) , ft qt (qt , at ) > 1/K .
(iii) fta2 (qt , at ) ft qt (qt , at )  ftat (qt , at ) ftat qt (qt , at ) .
t
(iv) u
et qt xt (qt , at , x t ) ftat (qt , at )
u
etat xt (qt , at , x t ) ft qt (qt , at ) .
NB (ii)-(iii): distro of st FOSD increases in at and qt ; an increase in at ,
holding ft constant, weakly decreases the marginal impact of at on ft .
Introduction
Irrelevance
Implementation
Conclusions
Implementation: Generic public signal
Proposition: (e
xT , e
aT ) is approximately (= within e for agent’s payoff)
implementable if e
xt and e
at are both increasing in q t .
Proof: Define āt (qt , yt ) such that ft (qt , āt (qt , yt )) ⌘ yt .
Let wt (qt , yt , x t ) = u
et (qt , āt (qt , yt ), x t ) — as if y were contractible!
By A2-A5 felicity wt satisfies A2 (with yt labeled as at ).
e
yt (q t ) = ft (qt , e
at (q t )) is increasing, hence if y is contractible then
(e
xt , e
yt ) is implementable by some p(q T ). [Proposition for pure AS]
By A4, and Theorem 2 of McAfee & Reny (1992), 9Dt : St ⇥ Yt ! R,
Est [Dt |ft = yt ] ⌘ 0 so that p
e( q T , s T ) ⌘ p( q T ) + ÂT
yt (q t ))
t = 0 D t ( st , e
approximately implements (e
xT , e
aT ). [truth is e-optimal for agent] ⇤
Introduction
Irrelevance
Implementation
Conclusions
Solution in the application
Assume qt = lqt 1 + (1
Hence qt = (1
u
et = qt kxt
so ut # 0 = k (1
l)# t , with q 1 ⌘ 0 and # t iid uniform.
l) Âtk =0 lt k # k .
1 ca2
2 t
() ut = (1
l ) lt
l)lt xt and utat =
⇣
Âtk =0 l k # k
cat .
⌘
kxt
1 ca2 ,
2 t
Lie-masking action: qbt + at (b
# 0 , #t 0 , s t 1 ) ⌘ qt + b
at ( # t , b
# 0 , s t 1 ),
where qbt = (1 l)lt b
# 0 + (1 l) Âtk =1 lt k # k . Hence
b
at ( # t , b
# 0 , s t 1 ) = at (b
# 0 , #t 0 , s t 1 ) + (1
and so b
at # 0 ( # t , b
#0, s t 1) =
(1
l ) lt .
l)lt (b
#0
# 0 ),
Introduction
Irrelevance
Implementation
Conclusions
Solution in the application
P0 ( # 0 ) = P0 (0) + E
"Z
#0 T
E [P0 (# 0 )] = P0 (0) + E
max
T
ÂE
t =0
h
 (1
0
t =0
"
T
 (1
h
l)lt kxt (y , #t 0 , s t ) + cat (y , #t 0 , s t
l ) lt (1
t =0
(qt + at + x t ) (1
1 2
2 cat
k )xt
(1
h
# 0 ) kxt (#t , s t ) + cat (#t , s t
1 2
2 rxt
1
) dy # 0
)
i
#
+ qt kxt
l ) lt (1
# 0 ) (kxt + yat )
If x ⌘ 0 (no noise in st ) then pointwise FOC in at and xt yields
x̃t⇤ (q t ) =
1
i
i
P0 (0)
qt + lt q0 (1 l ) lt
, ãt⇤ positive affine in x̃t⇤ .
rc (1 k )2
Both increasing if rc > (1 k )2 , hence by last Proposition approximately
implementable in the original and benchmark problems.
Introduction
Irrelevance
Implementation
Conclusions
Conclusions
No rent for # 1 , . . . , # T (the orthogonalized, “new” information).
Recipe: reduce to a static (benchmark) problem, even with hidden
action and contractible public signal about a summary of qt and at .
Tricks — orthogonalization, lie-masking actions, undoing deviations
in regular Markovian dynamic problems — may be useful elsewhere.
#