Introduction
Model
Analysis
Continuous types
Robust Predictions in Dynamic Screening Daniel Garrett, Alessandro Pavan and Juuso Toikka TSE, Northwestern, MIT
March 2017
Conclusions
Introduction
Model
Analysis
Continuous types
Conclusions
Dynamic Mechanism Design Mechanism Design: procurement, regulation, employment/compensation, taxation, etc...
Standard model: one-time information, one-time decisions
Many settings - information arrives over time: information is serially correlated - sequence of decisions
Dynamic mechanism design (DMD)
Introduction
Model
Analysis
Continuous types
Dynamic Mechanism Design: Example
Manufacturer (principal) procures input from supplier (agent) Agent has private information on cost (of quality/quantity each period) that evolves stochastically with time
Conclusions
Introduction
Model
Analysis
Continuous types
Conclusions
Standard approach: "relaxed approach" How to solve for principal’s pro…t-maximizing mechanism? Usual approach solves a "relaxed program" Account only for certain necessary conditions for IC (derived from "local" IC constraints)
Verify ex-post that solution to relaxed program is in fact globally IC
Relaxed approach (when it works) often permits complete characterization of optimal mechanism In some cases, optimal mechanism derived in closed form Mimics tractability of Myerson’s (1981) optimal auctions work
Introduction
Model
Analysis
Continuous types
Conclusions
Existing predictions under stringent conditions All predictions in literature to date hinge on validity of the "relaxed approach" Relaxed approach works under restrictive conditions on evolution of agent information See Battaglini and Lamba (2015) Restrictive conditions derived by "reverse engineering": not re‡ective of economic considerations
Central prediction of existing literature: "vanishing distortions" Optimal (pro…t-maximizing) allocations converge to …rst-best with time E.g.: Besanko (1985), Battaglini (2005), Pavan-Segal-Toikka (2014), Battaglini and Lamba (2015), Bergemann and Strack (2015)
Introduction
Model
Analysis
Continuous types
Conclusions
Our focus: Do vanishing distortions hold more generally? Consider broad class of processes where agent types are not fully persistent In particular: Regular Markov processes (have a steady state)
"Vanishing distortions" a general feature of dynamic contracting? Key idea: Distortions in allocations to reduce information rents earned by agent due to private information at time of contracting If private information is less than fully persistent, then distortions later in the relationship have little e¤ect on information rents
Introduction
Model
Analysis
Continuous types
Conclusions
Contribution: Two positive answers (a di¤erent approach) Approach: Identify admissible perturbations to any IC and IR mechanism Such perturbations should not increase pro…ts ( ! necessary conditions for optimality)
Finding 1: Expected marginal bene…t approaches expected marginal cost as relationship progresses Holds for any discount factor
Finding 2: Allocations approach e¢ cient ones in probability as relationship progresses Holds for su¢ ciently large discount factors (or processes that are not too persistent)
Introduction
Model
Analysis
MODEL
Continuous types
Conclusions
Introduction
Model
Analysis
Continuous types
Conclusions
Dynamic Environment Players: Procurer (principal) and supplier (agent) t = 1; 2; : : : (some results also available for …nite horizon) qt supplied by supplier to procurer at each date t; total payment xt Common discount factor
2 (0; 1)
Introduction
Model
Analysis
Continuous types
Dynamic Environment Principal values quality according to B : (0; q) ! R. Strictly increasing and strictly concave, twice-continuously di¤erentiable function satisfying lim B (q) =
q&0
1.
Agent cost of quality ht qt + C (qt ) C ( ) : (0; q) ! R+ a strictly increasing and strictly convex, twice-continuously di¤erentiable function satisfying lim C (q) = +1
q%q
Agent date-t "type", ht 2 0<
0
<
<
N
= f 0; : : : ;
Ng
Conclusions
Introduction
Model
Analysis
Continuous types
Dynamic Environment Principal intertemporal payo¤ X UP =
t 1
(B (qt )
(xt
ht q t
xt )
t 1
Supplier intertemporal payo¤ X t UA =
1
C (qt ))
t 1
Agent outside option equal to zero
ht privately observed by agent at the beginning of period t Fairly canonical procurement set-up (inspired, e.g., by Baron-Myerson, 1982)
Conclusions
Introduction
Model
Analysis
Continuous types
Conclusions
Types process Evolution of types governed by regular Markov process (Markov chain is aperiodic and irreducible)
Time-invariant (exogenous) transition matrix A (element ij, prob. of reaching j from i) (Regularity: A has only positive elements for some integer
1) N +1
Existence of a unique limiting/stationary distribution aS 2 (0; 1)
N +1
Initial types according to a vector of probabilities a1 2 (0; 1) Not necessarily equal to aS
Introduction
Model
Analysis
Continuous types
Conclusions
Mechanisms
Direct mechanism: a collection of functions 1 M = hqt (ht ) ; pt (ht )it=1 , where ht = (h1 ; h2 ; : : : ; ht ) 2 represents types up to date t
t
For each ht , let xt = pt (ht ) + C (qt (ht )) Agent is always reimbursed C qt ht payment pt ht
and receives additional
Introduction
Model
Analysis
Continuous types
Conclusions
Incentive compatibility IC: Markov process plus discounting ! enough to check one-stage deviations from truth-telling Agent payo¤ from date-t onwards under truth-telling "1 X s t ~s ~ s qs h ~s V t ht = E ps h h s=t
jht
#
IC: requirement that, for all t, all ht 1 ; ht 2 t , all h0t 2 , 1 3 2 0 t 1 0 ~s 1 X p h ; h ; h s t >t s t A jht 5 @ V t ht 1 ; h t E4 ~s ~ s q s ht 1 ; h 0 ; h h t >t s=t
Introduction
Model
Analysis
Continuous types
Conclusions
E¢ cient mechanism
The e¢ cient allocation is q E ( ) s.t. B 0 q E (ht ) = ht + C 0 q E (ht ) for all ht 2 One possible e¢ cient (IC) mechanism is obtained by letting pE (ht ) + C q E (ht ) = B q E (ht ) at each t (Here, ignoring participation constraints; i.e., assuming agent willing to participate)
Introduction
Model
Analysis
Continuous types
Conclusions
Principal’s problem Date-1 participation constraint V1 (h1 )
0 for all h1 2
. 1
Principal seeks to maximize by choice of hqt (ht ) ; pt (ht )it=1 its expected payo¤ 2 3 X t 1 ~t ~t ~ t )5 E4 (B qt h C qt h pt h t 1
subject to incentive and (date-1) participation constraints. 1
Let M = hqt (ht ) ; pt (ht )it=1 denote a solution to principal’s problem
Introduction
Model
Analysis
ANALYSIS
Continuous types
Conclusions
Introduction
Model
Analysis
Continuous types
Convergence in expectation Proposition An optimal mechanism M exists, with the allocation rule qt (ht ) uniquely determined. Ith satis…es h i i 0 t ~ ~t + C0 q h ~t E B qt h !E h as t ! +1. t While e¢ ciency calls for marginal bene…t B 0 q E (ht ) equal to marginal cost ht + C 0 q E (ht ) , convergence here is only with respect to ex-ante expectations. Result leaves the possibility that distortions persist in the long run (Although, convergence to e¢ cient allocations in probability if distortions are always in one direction, say downwards)
Conclusions
Introduction
Model
Proof (sketch):
Analysis
Continuous types
Conclusions
An admissible perturbation 1
Suppose hqt (ht ) ; pt (ht )it=1 is an optimal (hence IC and IR) policy Note all values qt ht are interior to (0; q).
Considering increase qt ( ) uniformly at arbitrary date t, say by small amount > 0 Recall that the additional cost C qt ht + reimbursed at date t
C qt ht
Increase date-1 payments h i p1 ( ) uniformly by t 1 ~ maxh1 2 E ht jh1
will be
The new mechanism is incentive compatible and date-1 individually rational ^ 1 an additional expected rent equal to It leaves initial typehh i h i t 1 ~ t jh1 ~ t jh ^1 maxh1 2 E h E h
Introduction
Model
Analysis
Continuous types
Conclusions
Proof (sketch): Proof by contradiction If the result is not true, there exists a subsequence of dates (tn ) s.t. h i ~ tn ~t + C0 q ~ tn E B 0 qtn h h > tn h n
forh all dates tn in the sequence; or, s.t. ~ tn ~t + C0 q ~ tn h E B 0 qtn h tn h n appropriate > 0.
i
<
for an
First case (as an example). Increase qtn ( ) uniformly at arbitrary date tn , by arbitrarily small amount n > 0 ( n can be chosen small enough, depending on n, such that the new policy remains interior).
Adjust payments as described above, leaving new mechanism IC and (date-1) IR
Introduction
Model
Analysis
Continuous types
Conclusions
Proof (sketch) The new mechanism increases expected surplus by at least ( n small enough)
tn 1
n
The new mechanism leaves an additional expected rent to an agent ^ 1 equal to whose date-1 type is h tn 1
n
n h io ~ t jh1 max E h n h1
^1 2 , Since, for all h n h io ~ t jh1 max E h n h1
h i ~ t jh ^1 . E h n
h i ~ t jh ^ 1 ! 0 as tn ! 1 E h n
the increase in surplus dominates for tn large enough.
Introduction
Model
Analysis
Continuous types
Conclusions
Convergence to e¢ ciency in probability Proposition Fix the stochastic process and technology. any 2 ; 1 , the following are true: h ~ tq h ~t ~t 1 h limt!1 E B qt h t h ~t ~ t qE h = limt!1 E B q E h ~t qt h
There exists
such that, for
~t + C qt h ~ t + C qE h ~t h ~t qE h
i
i .
2
For any
1
Expected loss in surplus from distortions vanishes in the long run
2
Optimal allocation converges to e¢ cient one in probability
> 0, limt!1 Pr
>
= 0.
Introduction
Model
Analysis
Continuous types
Proof (sketch): E¢ cient mechanism Approach focuses on combining putative optimal mechanism M with e¢ cient dynamic mechanism E¢ cient dynamic mechanism: To …x ideas C q E (ht ) at each t, ht 2 pE (ht ) = B q E (ht ) This is a repetition of static mechanisms Continuation of e¢ cient mechanism thus una¤ected by current reports IC constraints equivalent to static ones
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Introduction
Model
Analysis
Continuous types
Conclusions
Proof (sketch): Going to e¢ ciency too fast 1
Idea 1 (FAILED!): Suppose M = hqt (ht ) ; pt (ht )it=1 is an optimal mechanism such that result does not hold. Replace the payment and allocation rules with those of the e¢ cient mechanism from date t onwards. Adjust payments, to ensure satisfaction of date-1 participation constraints. Such an adjustment can be made s.t. (expected) increase in surplus dominates any increases in information rents (when t large enough) Assuming new mechanism is IC
Problem: New mechanism need not be IC. IC from date t onwards, but not necessarily at earlier dates.
Introduction
Model
Analysis
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Conclusions
Proof (sketch): Approaching e¢ ciency gradually Idea 2 (WORKS!): Approach e¢ ciency gradually Observation A: E¢ cient mechanism has a …xed amount of slack at all histories Exists ! > 0 s.t., for any pE ( j ) pE ( k )
j;
E ( j) E q ( k) j
jq
k
with
j
6= !j
k,
j
! min f
kj j
k
:
j
6=
kg .
Observation B: For any 1, any linear convex combination of 1 1 hqt (ht ) ; pt (ht )it= and q E (ht ) ; pE (ht ) t= satis…es IC at all histories ht 1 for t . At least some …xed amount of slack in all IC constraints (according to ! and linear weights)
Introduction
Model
Analysis
Continuous types
Conclusions
Proof (sketch): Approaching e¢ ciency gradually Observations permit gradual growth in weight on e¢ ciency For example, put (q1new (h1 ) ; pnew (h1 )) 1
=
+ qtnew ht ; pnew ht t
=
2, where 0 <
1
The new mechanism is IC if 2
For …xed 1
If = decrease
2 1
2
2
1
(q1 (h1 ) ; p1 (h1 ))
E
q (h1 ) ; pE (h1 ) , and 2
1 +
for t
1
1
2
qt ht ; pt ht
q E (ht ) ; pE (ht )
1.
is not too much larger than
, mechanism is IC from date 2 onwards.
, then there is slack in IC at date 1. Hence, can 2 below by a small amount.
1
.
Introduction
Model
Analysis
Continuous types
Conclusions
Proof (sketch): Condition on discount factor
Why we need
close to 1?
Proposed "new" mechanism approaches e¢ ciency gradually. Positive weight on e¢ cient policy at early dates may increase information rents (by relatively large amount) When is small, gains in surplus at later dates need not exceed these increased rents.
Introduction
Model
Analysis
Continuous types
Conclusions
Corollary: any discount factor and small persistence Proposition Fix the technology, initial type distribution a1 , and discount factor . There exists m s.t., for all m m, if the transition matrix is Am : h i ~t ~t ~ tq h ~t 1 limt!1 E B qt h C qt h h t i h ~t . ~t ~t ~ t qE h C qtE h h = limt!1 E B q E h 2
For any
> 0, limt!1 Pr
~t qt h
~t qE h
I.e., convergence to e¢ ciency holds for any process is not very persistent
>
= 0.
2 (0; 1) provided
Related: Convergence always holds if principal and agent meet (su¢ ciently) infrequently (say every m periods)
Introduction
Model
Analysis
Continuous types
CONTINUOUS TYPES
Conclusions
Introduction
Model
Analysis
Continuous types
Conclusions
Continuous types: Stochastic process Suppose date-t type ht is now drawn according to time-invariant Markov chain F = (Ft ); = ; R+ F1 is (abs. continuous) cdf of initial distribution; density f1 Ft ( jht
1)
cdf of ht , given ht
1
2
(Ft ( jht
1)
full support on
Stochastic monotonicity: Ft jh0t 1 …rst-order stochastically dominates Ft ( jht 1 ) for h0t 1 > ht 1 Time-invariance: Ft ( j ) = Fs ( j ) all t; s > 1, all Stationarity assumption: 9! invariant distribution sup A2B( )
F t (A; )
2 s.t., for all
(A) ! 0 as t ! 1.
2
)
Introduction
Model
Analysis
Continuous types
Continuous types: Auxiliary shock representation Stochastic process can be represented via "auxiliary shocks" Shocks independent of initial private information Follow Eso, Szentes (2007), Pavan, Segal, Toikka (2014)
ht = z (ht
1 ; "t ),
where " = ("t ) are i.i.d. random variables
E.g., take "t to have uniform distribution, and determine z via "probability integral transform" result Assume additionally that bounded
@z(ht 1 ;"t ) @ht 1
exists and is continuous and
Conclusions
Introduction
Model
Analysis
Continuous types
Conclusions
Continuous types: Impulse responses Let (Z
;t )t
a collection of functions s.t. ht = Z
;t
(h ; ") for t
Impulse responses: I
ht =
!t
@Z
(h ; ") @h
;t
(where vector " derived from ht using function z) AR(1) example: ht
= ht 1 + "t = Z ;t (h ; ") = ! I !t ht = t
t
h + .
t
1
"
+1
+
+ "t
1
+ "t
Introduction
Model
Analysis
Continuous types
Conclusions
Characterization of IC Theorem (Pavan, Segal, Toikka, 2014) 1
Mechanism M = hqt (ht ) ; pt (ht )it=1 IC i¤, for all t 0, all ht 1 , Vt (ht ) is Lipschitz continuous in ht with 2 3 t X @Vt (h ) s t ~ s qs h ~ s jht 5 a.e., = E4 It!s h @ht s t
^ t, and, for all ht 1 , ht , h Z ht h Dt ht 1 ; x ; x
Dt
^t h
ht
1
^t ;x ;h
i
dx
0
where Dt ht ; y
2 X E4 s t
s t
3
~ s qs h ~ s ; y jht 5 . It!s h t
Introduction
Model
Analysis
Continuous types
Dynamic virtual surplus
Use result (and integration by parts) to write "dynamic virtual surplus" 2 0 13 ~t ~t C qt h B qt h X 6 C7 t 1B E4 ~1) @ A5 F1 (h t t ~ ~ ~ qt h ht + f h~ I1!t h t 1 1( 1) V1
(FOSD ensures IC binds at , so can put V1 optimum).
= 0 for an
Conclusions
Introduction
Model
Analysis
Continuous types
Conclusions
Relaxed approach "Relaxed approach"/"First-order approach": Pointwise maximization B 0 q t ht
FOSD (I1!t
= C 0 q t ht
+ ht +
F1 (h1 ) I1!t ht f1 (h1 )
0) implies distortions always downwards
Distortions driven by impulse responses Solution must be checked against "integral monotonicity" constraint
Introduction
Model
Analysis
Continuous types
Conclusions
Convergence in expectation Proposition 1
An optimal policy exists, with the optimal allocation rule (qt (ht ))t=1 uniquely determined. Suppose that, for each t, 0 < t qt (ht ) t
~t E B 0 qt h
~t C 0 qt h
~t h
i
= !
Provided F1 = , convergence is monotone.
2
E4
~1 F1 h f1
~1 h
3
~t 5 I1!t h
0 as t ! +1.
Introduction
Model
Analysis
Continuous types
Conclusions
Convergence in expectation: Comments With continuum of types, we are (to date) unable to guarantee that optimal policy qt (ht ) remains bounded away from the boundary (0 and q)
However, if this holds, we can quantify expected distortions (gap between MB and MC) for any t Interiority could be guaranteed (result applies) if restrict qt ( ) to be Lipschitz (with …xed but large Lipschitz constant)
Introduction
Model
Analysis
Continuous types
Conclusions
In progress...
Version of convergence to e¢ ciency for Allocations restricted to be Lipschitz (with …xed, but large constant) su¢ ciently large (for …xed stochastic process and Lipschitz constant)
Introduction
Model
Analysis
CONCLUSIONS
Continuous types
Conclusions
Introduction
Model
Analysis
Continuous types
Conclusions
Conclusions Earlier literature derived convergence to e¢ cient allocations applying relaxed/…rst-order approach Heavily restricted classes of processes
This paper: Su¢ cient conditions for convergence to e¢ ciency Discrete types, regular Markov process, su¢ cient degree of patience Also weaker notion of convergence (distortions vanish in expectation) Built on Garrett/Pavan (2015, JET): 2 periods with risk averse agent
Results seem to con…rm observations made with relaxed/…rst-order approach
