Dynamic Revenue Maximization:A Continuous Time Approach

Dirk Bergemann and Philipp Strack

February 2016Yale University

Introduction

• mechanism design with private values: pricing, auctions, etc

• from static view: one-off information, one-off decision ....

• ...to a dynamic view: recurrent information, recurrent decision• socially effi cient intertemporal allocationvia dynamic pivot mechanism (Bergemann & Välimäki (2010))

• today: revenue maximizing mechanism

Revenue Maximizing Mechanism: Applications

• leading examples:• recurrent license or rental contract• intertemporal provision of services (club membership, frequentbuyer contracts, calling plans)

• recurrent procurement, supply chain: manufacturer andretailer

• private information can be• current demand• trend of future demand• variance of future demand

• how does intertemporal pattern of prices, quantities,distortions depend on the nature of the private information

Revenue Maximizing Mechanism: Focus

• general issue in dynamic mechanism design:how to control for multi-period contingent deviations ?

• diffi cult to guarantee monotonicity of allocation• diffi cult to identify relevant incentive constraints• approach today:

1 separability of allocations over time

2 separability of information process and allocation process

Aside/Advertising

• recent and related work by Kruse and Strack (2012)"Optimal Stopping with Private Information"

• analysis of stopping problems rather than flow problems• in particular stopping today preempts stopping tomorrow

• single crossing condition of type and time together with...• ..."appropriate" derivative with respect to time• establishes relevant dynamic implementability conditions

Results

• necessary and suffi cient condition for revenue maximizingmechanism in terms of flow utilities

• closed form solution of revenue maximizing allocation andtransfers for a variety of stochastic processes, like

1 arithmetic brownian motion

2 geometric brownian motion

3 Ornstein—Uhlenbeck

4 bayesian learning

• characterization of the inter-temporal nature of the distortion• implementation of the revenue maximizing mechanism bysuitable modification of the second price auction.

Literature

• Battaglini (2005)• repeated selling by quantity/quality discriminating monopolist• two state Markov chain

• Eso and Szentes (2007)• single unit, single auction• orthogonal information/signal

• Pavan, Segal and Toikka (2011)• general allocation environment (many goods, many periods)• focus on Markov processes but gap in necessary and suffi cientcond.

• Boleslavsky and Said (2012)• repeated sale of single object to single agent• uncertainty about parameter of process, not state of Markovchain

• Kakade, Lobel and Nazerzadeh (2012)• repeated sale of single object to single agent• separable payoff in initial and subsequent signal

Flow Payoffs

• time is continuous t ∈ [0,T ], T <∞ or T =∞• n risk neutral agents, indexed by i ∈ {1, . . . , n} = N

• flow allocation x it ∈ X i ⊂ R+• cost of providing allocation xt =

(x1t , ..., x

nt

)is c (xt)

• flow transfer dπit ∈ R• flow net utility of agent i

v it · ui(t, x it

)− dπit

• willingness to pay, valuation: v it

Valuation and Private Information

• willingness to pay, valuation: v it

v it , φi (t, θi ,W it )

depends on private type θi and private signal W it

• at t = 0, agent i privately learns type θi ∈ Θ ⊂ R• type θi with distribution F i , strictly positive density f i , anddecreasing inverse hazard rate:

1− F i(θi)

f i(θi)

• at t > 0, agent i privately observes his valuation v it :

v it = φi (t, θi ,W it )

• Wt =(W 1t , ...,W

nt

)is a n dimensional Brownian Motion,

independent across agents

Discounted Payoffs

• common discount rate r > 0• the objective of the seller, or principal, is to maximizeexpected payments minus costs

E

[∫ T

0e−rt

(∑i∈N

dπit − c(xt)dt

)]

• the objective of agent i of type θi is to maximize expectedutility minus payments

E[∫ T

0e−rt

(v it · u(t, x it )dt − dπit

)| θi]

Continuity of Flow Payoffs

• allocation space X i ⊂ R and type space Θi ⊂ R are compact,convex subsets of R

• utility ui : R+ × X i → R+, continuous with u(t, 0) = 0 andnondecreasing in x i

• cost c : X → R, continuous and nondecreasing in everycomponent

• valuation φi : R+ ×Θi ×R→ R differentiable in w i and twicedifferentiable in θi .

Finite Discounted Payoffs

• the expected valuation of each agent is finite:

E[∫ T

0e−rt1{v it≥0}v

it · u

(t, x it

)dt | θi

]<∞

• influence of type on valuation grows at most linearly, i.e. thereexists two constants C ∈ R+, q ∈ (0, r) such that for everya < b

E[v it | θi = b

]− E

[v it | θi = a

]≤ Ceqt(b − a)

Monotonicity

• valuation v it :v it , φi (t, θi ,W i

t )

• higher types have higher valuation:

∂φ(t, θ,w)

∂θ, φθ(t, θ,w) ≥ 0 for all t, θ,w .

• higher signals lead to higher valuation:

φw (t, θ,w) > 0 for all t, θ,w .

• decreasing sensitivity of type θ, i.e.

φθ(t, θ,w)

φ(t, θ,w)is decreasing in w for all t, θ,w .

• decreasing influence of type θ, i.e.

φθ(t, θ,w)

φw (t, θ,w)is decreasing in θ for all t, θ,w .

Contracts

• without loss of generality we restrict attention to directrevelation mechanisms where all agents report their types θi

and valuations v i truthfully

• the principal can commit to a contract

(xt , πt)t∈[0,T ].

• At every point in time t the contract prescribes:

1 an allocation xt ∈ X ⊂ Rn+,2 a vector of payments πt ∈ Rn.3 xt and πt depend on the vector of types θ and all priorvaluations (vs )s≤t .

Implementability

• the objective of agent i of type θi is to maximize expectedutility minus payments

V ix ,π(θi ) , E[∫ T

0e−rt

(v it · ui (t, x it )dt − dπit

)| θi]

Definition (Implementability)

A contract (x , π) is implementable if it is rational for every agent iand type θi to accept the contract

V ix ,π(θi ) ≥ 0 ,

and optimal to report his type θi and the valuation v it truthfully atevery point in time t.

Social Welfare

• we define the social surplus s : R+ × Rn × X → R generatedat time t

s(t, vt , xt) ,∑i∈N

v it · ui (t, x it )− c(xt) .

• overall expected social surplus equals

E[∫ T

0e−rts(t, vt , xt)dt

].

Welfare Maximization

LemmaIf type profile θ and valuation profile v are publicly observable,then an allocation process x? is welfare maximizing if and only if

x?t ∈ argmaxx∈X

{s(t, vt , x)} for all t

Theorem (Welfare Maxizing Mechanism)

A welfare maximizing allocation x? : R+ × Rn → Rn can beimplemented by the dynamic pivot mechanism

dπi (t, vt) = maxx∈X

∑j 6=i

(uj (t, x jt )− u(t, x?(t, vt))

)v jt−(c(x)− c(x?(t, vt)))

• continuous time version of Bergemann and Välimäki (2010)

Welfare Maximization

note that ..

1 there always exists a welfare maximizing allocation that isstatic in the sense that xt conditions only on (t, vt).

2 the welfare maximizing allocation can be implemented bystatic payments (pivot mechanism, second price auction).

3 the dynamic welfare maximizing contract is equivalent to asequence of static welfare maximizing contracts

Revenue Equivalence

• objective of agent i of type θi is to maximize expected utilityminus payments

V ix ,π(θi ) = E[∫ T

0e−rt

(v it · ui (t, x it )dt − dπit

)| θi]

• we show that the value of the agent if he reports truthfully isLipschitz continuous

• necessary conditions for optimal mechanism

Consistent Deviation I

• valuation φ is strictly increasing in w• we implicitly define the function ω : R+ ×Θ× R→ R by

v = φ(t, θ, ω(t, θ, v)) for all (t, θ) ∈ R+ ×Θ

Definition (Consistent Deviation)

A deviation by agent i is a consistent deviation if type a ∈ Θ whoreports b ∈ Θ at t = 0 also reports:

v̂ it = φ(t, b, ω(t, a, v it )),

instead of his true valuation v it at all future dates t ∈ R+.

• we derive a necessary condition for incentive compatibilitythat is based only on the robustness of the mechanism toconsistent deviations

Consistent Deviation II

• note by construction:

W it = ω(t, a, v it )

• thus agent i’s report:

v̂ it = φ(t, b,W it )

equals the valuation he would have had if his type would havebeen b instead of a

• consequently the allocation agent i gets by consistentlyreporting b is the same allocation x it (b) an agent of type bgets if he reports truthfully

Indirect Utility Function

Proposition (Envelope Formula)

The indirect utility function V i of every agent i ∈ N in anyincentive compatible mechanism is absolutely continuous and hasthe weak derivative

V iθ(θ) = E[∫ T

0e−rtu(t, x it (θ))φθ(t, θi ,W i

t )dt]a.e. .

• the virtual valuation function J : R+ ×Θ× R→ R as:

J(t, θit , vit ) = v it −

1− F (θi )

f (θi )· φθ(t, θi ,w(t, θi , v i ))

• “impulse response”:

φθ(t, θi ,w(t, θi , v i ))

Revenue Equivalence

Theorem (Revenue Equivalence)

For any incentive compatible direct mechanism the expected payoffof the principal depends only on the allocation process (xt)t∈R+ :

E

[∫ T

0e−rt

(∑i∈N

πit − c(xt)

)dt

]=

E

[∫ T

0e−rt

(∑i∈N

J(t, θit , vit )u(t, x it )− c(xt)

)dt

]−∑i∈N

V i (θ) .

• as usual, virtual valuation J(t, θi , v it ) replaces valuation vit

• looking ahead: revenue optimization can be solved pointwisefor every (t, θ, vt)

• almost, but not quite, a static problem

Revenue Maximizing Allocations

Theorem (Revenue Maximizing Mechanism)

1 There exists an allocation process x? : R+ ×Θ× Rn → Rnthat only depends on (t, θ, vt) that maximizes revenue and isof the form

x?t ∈ argmaxx∈X

{s(t, J(t, θ, vt), x)} ,

and (x?)i is increasing in θi and v it .

2 There exists a payment process π? that implements therevenue maximizing allocation process x?. In addition, thepayment at time t depends only on t, θ, v.

Idea of Proof I: Allocation

• we saw that the optimal allocation

x?t (t, θ, vt)

can be determined pointwise• the monotonicity of x?t (t, θ, vt) depends on

J(t, θit , vit ) = v it −

1− F (θi )

f (θi )· φθ(t, θi ,w(t, θi , v i ))

or with v i = φ(t, θi ,w(t, θi , v i ) :

J(t, θit , vit ) = v it

(1− 1− F (θi )

f (θi )· φθ(t, θi ,w(t, θi , v i ))

φ(t, θi ,w(t, θi , v i ))

)• monotonicity conditions:

1 φθ/φ is decreasing in w ,2 φθ/φw is decreasing in θ

implies that J is increasing in θi , v i

Idea of Proof II: Incentives and Transfers

• monotonicity of J implies that the revenue-maximizingallocation (x∗ (t, θ, vt))i of agent i is increasing in θi andv i at every point in time t

• time separability implies that (x∗ (t, θ, vt))i does not dependon past reports (vs )s<t

• due to the monotonicity of x∗ in v it and the time-separabilityof allocation we can find payments at time t that ensuretruthful reporting of v it independent of truthfulness ofreported type θi and past reports (vs )s<t

• due to the monotonicity of x∗ in θi we can find time-zeropayments that ensure truthful reporting of the type θ,guaranteeing t = 0 truthtelling

Idea of Proof I

• monotonicity of φθ/φ and φθ/φw implies that J is increasingin θi , v i

• monotonicity of J implies that the revenue-maximizingallocation (x∗ (t, θ, vt))i of agent i is increasing in θi and v iatevery point in time t

• due to the monotonicity of x∗ in v it and the time-separabilitywe can find payments at time t that ensure truthful reportingof v it independent of the reported type θ

i and past reports(vs )s<t

• due to the monotonicity of x∗ in θi we can find time-zeropayments that ensure truthful reporting of the type θ

Revenue Maximization

Note that ..

1 the revenue maximizing allocation is dynamic and conditionson t, θ, vt .

2 the revenue maximizing allocation can be implemented byflow payments that depend only on t, θ, vt

3 in contrast, the effi cient mechanism, both allocation andtransfer, did depend only on t, vt

Advantages of Current Construction

• due to the construction of the process we can work with onlyone probability measure for all types

• impulse response functions are easily given in our setting as

φθ(t; θ;Wt)

• our conditions ensure that the impulse response functionsdepend only on θ, vt

Sequential Auctions

• at every t, the owner of a single unit of a, possibly divisible,object wishes to allocate it among the competing bidders,i ∈ {1, ..., n} = N.

• allocation space x it ∈ [0, 1] and∑i∈N x

it ≤ 1

• marginal cost of providing the object is zero• flow utility of each agent i is described by v it · x it − dπit .• v it immediately represents the willingness to pay of each bidder

Arithmetic Brownian Motion

• arithmetic Brownian motion v it is completely described by itsinitial value v i0, the drift µ and the variance σ of the diffusionprocess Wt

• willingness to pay of agent i therefore evolves according to

dv it = µdt + σdW it

• valuation of agent i can be represented as

v it = v0i + µt + σW it

Private Information: Initial Value

• given the above representation:

v it = θi + µt + σW it = φ(t, θi ,W i

t )

• we recall the general form of virtual utility

J(t, θit , vit ) = v it −

1− F (θi )

f (θi )· φθ(t, θi ,w(t, θi , v i )),

and so the partial derivative of φ with respect to θ is given byφθ = 1 :

J i (t, θi , v it ) = v it −1− F (θi )

f (θi )

• in every period t, the object is allocated to the agent i∗t withthe highest virtual utility (provided that the valuation ispositive)

i∗t ∈ argmaxi∈{1,...,n}

{v it −

1− F (θi )

f (θi ), 0}

Handicap Auction

• the associated flow transfers follow directly from the logic ofthe second price auction, and we have

dπit =

{maxj 6=i

{v jt −

1−F (θj )f (θj )

}+ 1−F (θi )

f (θi )if i = i∗t

0 if i 6= i∗t• flow net utility of the bidder is positive whenever he isassigned the object

v i∗t ≥ v

jt −

1− F (θj )

f (θj )+1− F (θi

∗)

f (θi∗)

,

• flow allocation proceeds as a “handicap” second price auction,where the price of the winner is determined by the currentvalue of the second highest bidder

• the “handicap” is computed as the difference between theconstant handicap of the current winner and the currentsecond highest bidder

• as in Eso and Szentes (2007) and Board (2007)

Private Information: Drift of the Process

• private information is with respect to the drift of theBrownian motion

• valuation v it then evolves according to:

vt = v i0 + θi t + σW it = φ(t, θi ,W i

t ),

• now the derivative of φ with respect to θ is given by φθ = t.• the virtual valuation is given by:

J(t, θi , v it ) = v it −1− F (θi )

f (θi )t.

• the handicap is increasing linearly in time, the distortion isgrowing deterministically over time

• note v it might be growing as well, so the increasing indistortion however does not allow us to conclude that theassignment of the object ends over time

Geometric Brownian Motion

• valuation v it of agent i by a geometric Brownian motion:

dv it = v it(µdt + σdW i

t

)• valuation of agent i can be represented as

v it = v i0 exp(

(µ− σ2

2)t + σW i

t

)= φ(t, θi ,W i

t )

• geometric Brownian motion v it is completely described by itsinitial value v i0, the drift µ and the variance σ of the diffusionprocess Wt

Private Information: Initial Value

• if the type is the initial valuation θi = v i0 the virtual valuationequals

J i (t, θi , v it ) = v it

(1− 1− F (v i0)

f (v i0)vi0

)• in every period t, the object is allocated to the agent i∗t withthe highest virtual utility:

i∗t ∈ argmaxi∈{1,...,n}

{v it

(1− 1− F (v i0)

f (v i0)vi0

)}• the associated flow payments dπit/dt are similar to those of asecond price auction:

dπitdt

=

max(0,maxj 6=i v

jt

1− 1−F (θj )f (θj )θj

1− 1−F (θi )f (θi )θi

)if i∗t = i

0 if i∗t 6= i

.

Private Information: Drift of the Process

• valuation v it then evolves according to:

v it = v i0 exp(

(θ − σ2

2)t + σW i

t

)= φ(t, θi ,W i

t )

• derivative of φ with respect to θ:

φθ = φt

• virtual valuation is given by:

J(t, θi , v it ) = v it

(1− 1− F (θi )

f (θi )t)

• distortion is still formed on the basis of a multiplicativehandicap, but is increasing linearly in time. It follows that thecontrast to the above case of an unknown starting value, thedistortion is growing over time.

Distortion over Time

• virtual valuation is given by:

J(t, θi , v it ) = v it

(1− 1− F (θi )

f (θi )t)

• virtual valuation is positive for small t

t ≤ f (θi )

1− F (θi )

and negative afterwards.

• the allocation process of agent i ends at the time

τ i =f (θi )

1− F (θi )

Allocation and Payments over Time

.

• the object is allocated to the agent i∗t with the highest virtualutility, provided that is positive:

i∗t ∈ argmaxi∈{0,1,...,n}

{v it (1−

1− F (θi )

f (θi )t), 0

}.

• the associated flow payments dπit are similar to those of asecond price auction:

dπit =

max(0,maxj 6=i v

jt

1− 1−F (θj )f (θj )

t

1− 1−F (θi )f (θi )

t

)if i∗t = i

0 if i∗t 6= i

.

Single Agent: Club Membership and Flow Services

• let n = 1 and the private information is the initial value of theprocess

Arithmetic Brownian Motion

• with arithmetic Brownian motion: the optimal allocation is togive the object to the agent if

v it ≥1− F (θi )

f (θi )

• allocation can be implemented by offering the good for sale atevery t at price:

1− F (θi )

f (θi )

• menu over usage prices and the initial member ship fee is:

E

[∫ T

0e−rt1

{v it≥1−F (θi )f (θi )

}

(v it −

1− F (θi )

f (θi )

)dt

]

Geometric Brownian Motion: Initial Value

• with geometric Brownian motion: the optimal allocation is togive the object to the agent if

v it

(1− 1− F (v i0)

f (v i0)vi0

)≥ 0,

• the revenue maximizing allocation gives the good forever toagents with high initial valuation

xt =

{1 if v i0 ≥ v?00 else

,

where

v?0 −1− F (v?0 )

f (v?0 )= 0

• the agent only makes a payment at time zero that equals

π0 =

{v?0r−µ if v i0 ≥ v?00 else

.

Geometric Brownian Motion: Drift

• if the private information, the type is the trend θi = µi , thenthe virtual valuation equals

v it

(1− 1− F (θi )

f (θi )t)

• the revenue maximizing allocation gives the good to the agentfor a fixed time

xt =

{1 if t ∈ [0, τ(µi )]

0 else

where

τ(µi ) =f (µi )

1− F (µi )

• the agent only makes a payment at time zero that equals

π0 =1− e−(r−µi )τ(µi )

r − µi v?0 .

Long-Run Behavior of the Distortion

• compare the expected social welfare generated by the revenuemaximizing allocation compared by the expected welfaregenerated by the socially optimal allocation

DefinitionThe distortion vanishes over time if the social welfare generated bythe revenue maximizing allocation converges to the social welfaregenerated by the welfare maximizing allocation,

limt→∞

E

s(t, vt , x(t, vt)︸ ︷︷ ︸Social

)− s(t, vt , x(t, θ, J(t, θ, vt))︸ ︷︷ ︸

Revenue

) = 0 .

Suffi cient Condition for Vanishing Distortion

Theorem (Vanishing Distortion)

The distortion vanishes in the long run if the expected valuation ofany type converges to the expected valuation of the lowest type

limt→∞

E[v it | θi = x

]− E [vt | θ]→ 0 .

• see Battaglini (2005) for two state ergodic Markov chain ...• ... and hence more generally for all ergodic Markov chains

Necessary Condition for Vanishing Distortion

• consider the case of n = 1

Theorem (Vanishing Distortion)

Let u be strictly concave and c strictly convex in x. The distortionvanishes in the long run if and only if the expected valuation ofany type converges to the expected valuation of the lowest type:

limt→∞

E [vt | θ]− E [vt | θ]→ 0 .

Discussion: Jump Processes

• we described results for continuous processes, the method canbe applied to discontinuous processes, say Poisson processes

• let (Zt)it∈R+ be a Poisson process with arrival rate λ ∈ R+• say valuation v it evolves deterministically, b (t), jumps up atrandom times, private information θi of agent i is sensitivityto jumps:

v it = φ(t, θi ,Z it ) = b(t) + θiZ it ,

• the partial derivative is

φθ = z =v − b(t)

θ.

• virtual valuation is given by

J(t, θi , v it ) = v it

(1− 1− F (θi )

f (θi )θi

)+1− F (θi )

f (θi )θib(t) .

• asE(φθ(t, θi ,Z it )) = E(Z it ) = λt,

it follows that the distortion is growing linearly over time

Conclusion

for the repeated sale of a non durable good

1 the static welfare maximizing contract is also the dynamicallymaximizing welfare.

2 the static revenue maximizing contract differs from thedynamic revenue maximizing contract.

3 the allocation is distorted in favor of the types with positivetime zero information.

4 the distortion vanishes in the long run if and only if the longrun valuation is independent of time zero valuation

Open Issues

• beyond time separable allocations• beyond separability of allocation and stochastic process• beyond state contingent to path contingent utility• open problem:

• how do stationary contract differ from current commitmentcontract

• agent may enter commitment tomorrow rather than today