PROGRESSIVE SCREENING: LONG-TERM CONTRACTING WITH A PRIVATELY KNOWN STOCHASTIC PROCESS Maher Said with Ralph Boleslavsky (University of Miami) January 2013
LONG-TERM CONTRACTS Long-term contracts are employed widely across many settings: Boleslavsky and Said (2013): Progressive Screening ▶ Trade; ▶ Employment; ▶ Supply chain management; ▶ Financial contracts; ▶ Insurance; ▶ ….
LONG-TERM CONTRACTS But in all these settings, important features of the economic environment change over time: Boleslavsky and Said (2013): Progressive Screening ▶ Trade: Consumers’ values for goods; ▶ Employment: Workers’ productivity; ▶ Supply chain management: Supplier production costs; ▶ Financial contracts: Borrower’s wealth; ▶ Insurance: Chance of claim; ▶ ….
LONG-TERM CONTRACTS Changes are not necessarily unanticipated: all parties may be aware the environment is dynamic. Optimal long-term contracts need to incorporate new info over time. And to do so, need to incentivize information revelation! Boleslavsky and Said (2013): Progressive Screening But these changes need not be observed by all parties.
THIS PAPER These are the sorts of settings we’re interested in. But we also want to look at the efgect of an additional source of private information: Boleslavsky and Said (2013): Progressive Screening ▶ Agents have private information about the state of the economic environment. ▶ And also about the process governing its evolution.
EXAMPLE A manufacturer is introducing a new product to a market. But sale occurs via local retailers. The manufacturer has a good sense of broad national trends. The retailer may have better projections of how demand will evolve in its local market. Boleslavsky and Said (2013): Progressive Screening
QUESTION AND PREVIEW So in settings like this, what should the optimal contract look like? In particular, how do we incentivize revelation of both process and states? We characterize a class of environments in which the optimal contract takes an especially simple form: static contracts. Boleslavsky and Said (2013): Progressive Screening ▶ The optimal dynamic contract can be implemented by a deterministic sequence of ▶ The seller screens stochastic processes with a menu of such sequences. ▶ Each sequence “progressively” screens buyer’s values. ▶ Rents reduced by increasingly restricting supply.
Boleslavsky and Said (2013): Progressive Screening BASIC MODEL Buyer has single-unit demand in each of T ≤ ∞ periods. Flow utility in period t : v t q t − p t . Marginal utility v t evolves over time: v t = α t v t − 1 , where v 0 := 1 . In each period t = 1 , . . . , T , buyer privately observes α t ∈ { u, d } , where ∆ := u − d > 0 . α t is iid in each period: Pr ( α t = u ) = λ and Pr ( α t = d ) = 1 − λ. At time 0, buyer privately observes λ ∈ [0 , 1] . λ is distributed according to cdf F (with density f ).
BASIC MODEL Boleslavsky and Said (2013): Progressive Screening u 3 λ u 2 1- λ λ u 2 d u 1- λ λ λ ud λ 1- λ 1- λ λ ud 2 d 1- λ λ d 2 1- λ d 3
BASIC MODEL In period 0, the seller ofgers a long-term contract. One-sided commitment: Boleslavsky and Said (2013): Progressive Screening In each t ≥ 1 , the seller can produce 1 unit at 0 cost. Both seller and buyer discount future by δ ∈ (0 , 1] . ▶ Additional restriction: if T = ∞ , then δu < 1 . ▶ If buyer accepts: sales/consumption occur in t ≥ 1 . ▶ If buyer rejects: both parties receive 0 payofg. ▶ Seller has full commitment power. ▶ Buyer has outside option with 0 utility in every period.
RELATED LITERATURE Part of literature on optimal dynamic mechanism design: Deb (2009, 2011); Esö and Szentes (2007); Kakade, Lobel, and Nazerzadeh (2011); Pavan, Segal, and Toikka (2011); and others…. Most closely related paper: Courty and Li (2000). Also: Pavan, Segal, and Toikka (2011). Boleslavsky and Said (2013): Progressive Screening ▶ Baron and Besanko (1984); Battaglini (2005); Besanko (1985); Courty and Li (2000); ▶ Essentially our setting but with T = 1 . ▶ With T > 1 , incentive compatibility is tricky. ▶ Also, T > 1 lets us consider long-term characteristics of optimal contract. ▶ Their paper is more ambitious/general. ▶ In some settings, they have similar results. ▶ Our paper: suffjcient conditions on primitives of the environment.
RELATED LITERATURE Information acquisition in dynamic mechanisms: (2011); Shi (2011). Multidimensional screening: of (new) private info in each period. And our results are reminiscent of priority pricing/option contracts as in Harris and Raviv (1981) or Wilson (1993). Boleslavsky and Said (2013): Progressive Screening ▶ Bergemann and Välimäki (2002); Gershkov and Szentes (2009); Krähmer and Strausz ▶ Information arrival is exogenous in our world. ▶ Matthews and Moore (1987); Rochet and Stole (2003); others…. ▶ Key distinction: IC is easier here because of timing —with dynamics, single dimension
CONTRACTS Some notation: Revelation principle holds, so (wlog) consider direct mechanisms. A contract is then: Boleslavsky and Said (2013): Progressive Screening ▶ History of shocks: α t := ( α t , α t − 1 , . . . , α 1 ) . ▶ History of shocks after s : α t − s := ( α t , α t − 1 , . . . , α s +1 ) . − s ) := � t ▶ Value given sequence of shocks: v ( α t τ = s +1 α τ . ▶ A sequence of payments p = { p t ( α t , λ ) } T t =0 . ▶ A sequence of allocation probabilities q = { q t ( α t , λ ) } T t =1 .
SELLER'S PROBLEM The seller wants to maximizes profjts: s.t. IC and IR constraints. Boleslavsky and Said (2013): Progressive Screening max � T 1 � � �� � � � δ t p t ( α t , λ ) p 0 ( λ ) + E � λ dF ( λ ) � � t =1 0
INCENTIVE COMPATIBILITY On path, buyer must have no incentive to misreport new private information. But optimal continuation reporting may involve additional misreports! So unlike a static mechanism design problem, we have to worry about compound contingent deviations. (unless we impose further assumptions). Boleslavsky and Said (2013): Progressive Screening Truth has to be better than lying and optimal continuation reporting. ▶ Large and intractable set of IC constraints that must be satisfjed. ▶ The “brute force” approach of imposing these constraints directly is generally useless
"STANDARD" APPROACH Some dynamic mechanism design work (e.g., Baron and Besanko 1984; Courty and Li 2000; Krähmer and Strausz 2011) avoids this by looking at two-period models. Boleslavsky and Said (2013): Progressive Screening ▶ Relatively easy to solve second period for any history. ▶ Then backward induction is straightforward.
"STANDARD" APPROACH Most others (e.g., Esö and Szentes 2007; Pavan, Segal, and Toikka 2011; Kakade, Lobel, and Nazerzadeh 2011) use “independent shocks.” shock. Also typically impose a Markov structure on “true” types. Combine with a “strong monotonicity” condition. Jointly: optimal strategy after a misreport is a “correction.” This permits a one-shot deviation approach to IC. Boleslavsky and Said (2013): Progressive Screening ▶ Period- t value is a continuous function of period- ( t − 1) value and an independent ▶ This representation always exists via probability integral transform. ▶ Our context: if I misreported α t − 1 as α ′ t − 1 , will then report α t as ( α t − 1 α t )/ α ′ t − 1 .
"STANDARD" APPROACH = This doesn’t work in our setting. Boleslavsky and Said (2013): Progressive Screening And full independent shock approach is still non-Markov in values: Period- t value is not a suffjcient statistic for t + 1 value. ▶ α t +1 is correlated with λ . ▶ Defjne { ξ t } T t =1 to be a sequence of iid U [0 , 1] RVs. � v t − 1 ( ξ t − 1 , λ ) d ˜ if ξ t < 1 − λ, v t ( ξ t , λ ) = ⇒ ˜ v t − 1 ( ξ t − 1 , λ ) u ˜ if ξ t ≥ 1 − λ. ▶ λ still matters!
OUR APPROACH We take a slightly difgerent approach. Solve a relaxed problem; impose only a subset of IC constraints. deviation. We can pair this solution with a static pricing rule. The resulting mechanism decouples incentives from one period to the next, yielding “full” IC. Boleslavsky and Said (2013): Progressive Screening ▶ Consider “one-time” deviations from truth-telling: buyer must report truthfully after a The solution to this relaxed problem is “path independent”: it depends only on α t only through v ( α t ) .
IC CONSTRAINTS This is essentially a static IC constraint. Note: Boleslavsky and Said (2013): Progressive Screening At time t ≥ 1 , for all histories ( α t − 1 , λ ) and all α t , α ′ t : T � δ s − t E − t , α t , α t − 1 , λ ) v ( α s − t , α t , α t − 1 , λ ) � q s ( α s − t , α t ) − p s ( α s � � α t , λ � s = t T � δ s − t E q s ( α s − t , α ′ t , α t − 1 , λ ) v ( α s − t , α t ) − p s ( α s − t , α ′ t , α t − 1 , λ ) � � α t , λ � � ≥ . s = t ▶ Payofgs are quasilinear. ▶ v ( α s − t , α t ) = α t v ( α s − t , α t − 1 ) = ⇒ single-crossing.
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