Revenue Management with Forward-Looking Buyers Posted Prices and - - PowerPoint PPT Presentation

Introduction Model Single Unit Allocation Implementation Applications The End

Revenue Management with Forward-Looking Buyers

Posted Prices and Fire-sales

Simon Board Andy Skrzypacz

UCLA Stanford

June 4, 2013

Introduction Model Single Unit Allocation Implementation Applications The End

The Problem

Seller owns K units of a good

◮ Seller has T periods to sell the goods. ◮ Buyers enter over time. ◮ Privately known values.

Introduction Model Single Unit Allocation Implementation Applications The End

The Problem

Seller owns K units of a good

◮ Seller has T periods to sell the goods. ◮ Buyers enter over time. ◮ Privately known values.

Big literature on revenue management

◮ Typically assume buyers are myopic.

Forward looking buyers

◮ Agents delay if expect prices to fall. ◮ Prefer to buy sooner rather than later.

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Applications

RM is hugely successful branch of market design

◮ Historically: Airlines, Seasonal clothing, Hotels, Cars ◮ Online economy: Ad networks, Ticket distributors, e-Retailers

Buyers strategically time purchases

◮ Clothing (Soysal and Krishnamurthi, 2012) ◮ Airlines (Li, Granados and Netessine, 2012) ◮ Redzone contracts (e.g. YouTube) ◮ Price prediction sites (e.g. Bing Travel)

Questions

◮ What is the optimal mechanism? ◮ Is there a simple way to implement it?

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Price and Cutoffs with One Units

Prices and Sales for a Sample Product Sales and prices over time

–50 50 150 250 350 450 550 650 750 850 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Weeks Sales

–10 40 90 140 190 240

Prices

1st markdown 2nd markdown Sales Prices

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Results

Allocations determined by deterministic cutoffs.

◮ Only depend on (k, t), ◮ Not on # of agents, their values, when sold units.

When demand gets weaker over time

◮ Cutoffs satisfy one-period-look-ahead property.

Implement in continuous time via posted prices

◮ With auction at time T. ◮ Relies on cutoffs being deterministic.

Prices depend on when previous units were sold.

◮ Cutoffs are easy; prices are hard.

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Outline

• 1. Allocations

◮ General demand - Cutoffs are deterministic ◮ Decreasing demand - One-period-look-ahead property

• 2. Implementation

◮ General demand - Use posted prices ◮ Decreasing demand - Prices given by differential equation

• 3. Applications

◮ Retailing - Storage costs ◮ Display ads - Third degree price discrimination ◮ Airlines - Changing distribution of arrivals ◮ House selling - Disappearing buyers

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Literature

Gallien (2006)

◮ Infinite periods; Inter-arrival times have increasing failure rate. ◮ No delay in equilibrium.

Pai and Vohra (2013), Mierendorff (2009)

◮ Privately known value, entry time, exit time; No discounting. ◮ Show how to simplify problem, but do not fully characterize.

Aviv and Pazgal (2008), Elmaghraby et al (2008)

◮ Similar model to ours; only allow for two prices.

MacQueen and Miller (1960), McAfee and McMillan (1988)

◮ Optimal policy for single unit.

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Model

Introduction Model Single Unit Allocation Implementation Applications The End

Model

◮ Time discrete and finite t ∈ {1, . . . , T} ◮ Seller has K goods. ◮ Seller can commit to mechanism.

Entrants

◮ At start of period t, Nt buyers arrive ◮ Nt independently distributed, but distribution may vary ◮ Nt observed by seller but not other buyers

Preferences

◮ Buyer has value vi ∼ f(·) for one unit. ◮ Utility is (v − pt)δt

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Mechanisms

◮ Buyer makes report ˜

vi when enters market.

◮ Mechanism τi, TRi describes allocation and transfer. ◮ Feasible if award after entry, K goods, adapted to seller’s info

Buyer’s problem

◮ Buyer chooses ˜

vi to maximise ui(˜ vi, vi, ti) = E0

• viδτi(˜

vi,v−i,t) − TRi(˜

vi, v−i, t)

• vi, ti
• where Et is expectation at the start of period t.

Mechanism is (IC) and (IR) if

(INT) ui(vi, vi, ti) = E0[ vi

v δτi(z,v−i,t) dz|vi, ti]

(MON) E0[δτi(v,t)|vi, ti] is increasing in vi.

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Buyer’s expected rents

◮ Taking expectations over (vi, ti) and integrating by parts,

E0[ui(vi, vi, ti)] = E0

• δτi(v,t) 1 − F(vi)

f(vi)

• Seller’s problem

◮ Define marginal revenue, m(v) := v − (1 − F(v))/f(v). ◮ Seller chooses mechanism to solve

Profit = E0

i

TRi

• = E0

i

δτi(v,t)m(vi)

• ◮ Assume m(v) is increasing in v, so (MON) satisfied.

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Example: One Unit, IID Arrivals

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Single Unit

Proposition 0.

Suppose K = 1 and Nt is IID. The seller awards the good to the buyer with the highest valuation exceeding a cutoff xt, where m(xt) = δEt+1[max{m(v1

t+1), m(xt)}]

for t < T m(xT ) = 0 These cutoffs are constant in periods t < T, and drop at time T. (i) Cutoffs deterministic: depend on t; not on # entrants, values. (ii) Characterized by one-period-look-ahead rule. (iii) Constant for t < T: Seller indifferent between selling/waiting. If delay, face same tradeoff tomorrow and indifferent again. Hence assume buy tomorrow.

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Implementation in Continuous Time

◮ Buyers enter at Poisson rate λ. ◮ Optimal cutoffs are deterministic:

rm(x∗) = λE

• max{m(v) − m(x∗), 0}
• Implementation via Posted Prices

◮ At time T hold SPA with reserve m−1(0). ◮ The final posted price

pT = E0

• max{v2

≤T , m−1(0)}

• v1

≤T = x∗ ◮ Posted price for t < T,

˙ pt = −(x∗ − pt)

• λ(1 − F(x∗)) + r

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Price and Cutoffs with One Units

Assumptions: Buyers enter with λ = 5 and have values v ∼ U[0, 1]. Total time is T = 1 and the interest rate is r = 1/16.

0.2 0.4 0.6 0.8 1 0.5 0.6 0.7 0.8 0.9 1

Time, t Prices Cutoffs Auction

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Implementation via Contingent Contract

Contingent Contract

◮ Netflix wishes to buy ad slot on front page of YouTube ◮ Buy-it-now price pH ◮ Pay pL to lock-in later if no other buyer

Implementation

◮ Fix price path pt above, with final price pT ◮ When buyer enters, bids b ◮ If b ≥ pT , buyer locks-in contract at time min{t : pt = b} ◮ If b < pT , this is treated as bid in auction at T

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Many Units: Allocations

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Preliminaries

Seller has k units at start of period t

◮ Let y := {y1, y2, . . . , yk} be highest buyers at time t.

Lemma 1.

The optimal mechanism uses cutoffs xj

t(y−(k−j+1)), j ≤ k. ◮ Across buyers, seller allocates to high value buyers first ◮ For one buyer, allocations monotone in values ◮ Unit j awarded iff yk−ℓ+1 ≥ xℓ t(yk−ℓ+1) for ℓ ∈ {j, . . . , k}

Highest values (y1, . . . , yk) act as state

◮ Buyer’s ti doesn’t affect allocation, so seller need not know ◮ Optimal allocations independent of when past units sold

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◮ “Continuation profit” at time t with k units is

Πk

t (y) := max τi≥t Et i

δτi(y)−tm(vi)

• ˜

Πk

t (y) := max τi≥t Et+1 i

δτi(y)−tm(vi)

• Lemma 2.

Suppose xj

t(·) are decreasing in j. Then unit j is allocated iff

yk−j+1 ≥ xj

t(yk−j+1)

Idea

◮ If want to sell jth unit then want to sell units {j + 1, . . . , k}

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◮ ∆˜

Πk

t (y) := ˜

Πk

t (sell 1 today) − ˜

Πk

t (sell 0 today) ◮ Cutoff xj t(·) is deterministic if it is independent of y−(k−j+1)

Lemma 3.

Suppose {xj

s}s≥t+1 are deterministic and decreasing in j. Then:

(a) ∆˜ Πk

t (y) is independent of y−1

(b) ∆˜ Πk

t (y1) is continuous and strictly increasing in y1

(c) ∆˜ Πk

t (y1) is increasing in k.

Idea

(a) Allocation to yj determined by rank relative to no. of goods. Decision today does not affect when yj gets good. Hence value of yj does not affect difference ∆˜ Πk

t (y).

(b) A higher y1 is more valuable if sell earlier. (c) The option value of waiting declines with more goods.

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Deterministic Allocations

Theorem 1.

The optimal cutoffs xk

t are deterministic, decreasing in k and

uniquely determined by ∆˜ Πk

t (xk t ) = 0 ◮ At T, m(xk T ) = 0. By induction, suppose xk t (y−1) > xk−1 t

0 ≥ ∆˜ Πk

t (xk t (y−1)) > ∆˜

Πk

t (xk−1 t

) ≥ ∆˜ Πk−1

t

(xk−1

t

) = 0 Using (i) ˜ Πk

t (sell ≥ 1 today) ≥ ˜

Πk

t (sell 1 today)

(ii) monotonicity of ∆˜ Πk

t (y1) in y1

(iii) monotonicity of ∆˜ Πk

t (y1) in k

(iv) induction.

◮ As xk t (y−1) ≥ xk−1 t

, ∆˜ Πk

t (xk t (y−1)) = 0 and xk t deterministic ◮ Hence seller need not elicit y−1 to determine allocation.

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Decreasing Demand

◮ D ˜

Πk

t (y1) := ˜

Πk

t (sell 1 today) − ˜

Πk

t (sell ≥ 1 tomorrow) ◮ Note D ˜

Πk

t (y1) ≥ ∆˜

Πk

t (y1), with equality if xk t ≥ xk t+1

Theorem 2.

Suppose Nt is decreasing in FOSD. Then xk

t are decreasing in t,

and determined by a one-period-look-ahead policy, D ˜ Πk

t (xk t ) = 0. ◮ If {xk s}s≥t+1 are decreasing in s, then D ˜

Πk

t+1(y1) ≥ D ˜

Πk

t (y1).

Idea: Option value lower when fewer periods.

◮ By contradiction, if xk t < xk t+1 then

0 ≤ D ˜ Πk

t (xk t ) < D ˜

Πk

t (xk t+1) ≤ D ˜

Πk

t+1(xk t+1) = 0. ◮ Using (i) ˜

Πk

t (sell 0 today) ≥ ˜

Πk

t (sell ≥ 1 tomorrow)

(ii) monotonicity of D ˜ Πk

t (y1) in y1

(iii) monotonicity of D ˜ Πk

t (y1) in t

(iv) induction.

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Decreasing Demand: Indifference Equations

The optimal cutoffs xk

t are given by local indifference conditions ◮ At time T,

m(xk

T ) = 0 ◮ At time T − 1,

m(xk

T−1) = δET−1

• max{m(xk

T−1), m(vk T )}

• ◮ At time t < T − 1,

m(xk

t ) + δEt+1

• ˜

Πk−1

t+1 (vt+1)

• = δEt+1
• max{m(xk

t ), m(v1 t+1)}

• + δEt+1
• ˜

Πk−1

t+1 ({xk t , vt+1}2 k)

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Implementation with Posted Prices

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General Demand

◮ Assume Poisson arrivals λt, discount rate r, period length h ◮ Price mechanism: Single posted price in each period; if there

is excess demand, good is rationed randomly.

Theorem 3.

Suppose λt is Lipschitz continuous. Then lost profit from using posted prices and auction for final good in final period is O(h). (i) Cutoffs do not jump down by more than O(h) Idea: If t < T − h, follows from continuity of λt. For t = T − h, have m(xk

t ) ≈ 0 for k ≥ 2

(ii) Prices wrong because (1) don’t adjust cutoffs within a period; and (2) may ration incorrectly. But the prob. of 2 sales in one period is O(h2).

◮ Poisson arrivals important since imply common expectations

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Decreasing Demand: Allocations

Poisson rate λt decreasing in t.

◮ Optimal cutoffs given by infinitesimal-period-look-ahead rule:

rm(xk

t ) = λtEv

• max{m(v) − m(xk

t ), 0} + Πk−1 t

• min{v, xk

t }

• − Πk−1

t

(v)

• m(xk

T ) = 0

where v is drawn from F(·)

End game, t → T

◮ If k ≥ 2, then xk t → m−1(0). ◮ If k = 1, then xk t jumps down to m−1(0)

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Decreasing Demand: Prices

Period T

◮ For k = 1, hold SPA with reserve m−1(0) ◮ Final posted price

pT = E0

• max{y2, m−1(0)}
• y1 = lim

h→0 x1 T−h, {sT (x)}x≤y1

• where sT (x) is last time the cutoff went below x.

◮ For k ≥ 2, pt → m−1(0) as t → T.

For t < T, prices determined by

˙ pk

t =

• ˙

xk

t

t

st(xk

t )

λs ds

• f(xk

t )−λt(1−F(xk t ))

xk

t − pk t − U k−1 t

(xk

t )

• −r
• xk

t − pk t

• ◮ If other units purchased earlier, pk

t is higher. ◮ Price falls over time but jumps with every sale.

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Price and Cutoffs with Two Units

0.2 0.4 0.6 0.8 1 0.5 0.6 0.7 0.8 0.9 1

Penultimate Unit

Time, t Prices Cutoffs 0.2 0.4 0.6 0.8 1 0.5 0.6 0.7 0.8 0.9 1

Last Unit

Time, t Prices Cutoffs

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Probability of Sale

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Time, t Last Unit Penultimate Unit

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Forward-Looking vs. Myopic Buyers

Myopic Buyers

◮ Buyers buy when enter, or leave forever ◮ Cutoffs m(xk t ) = δ(V k t+1 − V k−1 t+1 ), where V k t is value in (k, t). ◮ Implement with prices equal to cutoff.

Under forward-looking buyers

◮ Profits higher ◮ Total sales higher ◮ Sales later in season

Retailing data suggest forward-looking buyers

◮ Price reductions lead to large numbers of sales ◮ Burst of sales quickly dies down ◮ Prices fall rapidly near the end of season

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Cutoffs, Prices and Sales with Myopic Buyers

0.2 0.4 0.6 0.8 1 0.5 0.6 0.7 0.8 0.9 1

Cutoffs/Prices

Time, t Last Unit Penultimate Unit 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Probability of Sale

Time, t Last Unit Penultimate Unit

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Applications

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Retail Markets - Inventory Costs

◮ Inventory cost ct if good held until time t. ◮ Assume marginal cost ∆ct = ct+1 − ct is increasing in t.

Cutoffs are deterministic and decreasing over time.

◮ For t = T, m(xk T ) = −∆cT . For t < T,

m(xk

t ) + Et+1

• ˜

Πk−1

t+1 (vt+1)

• = Et+1
• max{m(xk

t ), m(v1 t+1)}

• + Et+1
• ˜

Πk−1

t+1 ({xk t , vt+1}2 k)

• − ∆ct

◮ In continuous time,

˙ ct = λtE

• max{m(v) − m(xk

t ), 0} + Πk−1 t

• min{v, xk

t }

• − Πk−1

t

(v)

• ˙

pk

t =

• ˙

xk

t

t

st(xk

t )

λsds

• f(xk

t ) − λt(1 − F(xk t ))

xk

t − pk t − U k−1 t

(xk

t )

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Display Ads - Price Discrimination

◮ Rich media ad buyers have values v ∼ fR ◮ Static ad buyers have values v ∼ fS

Solving the problem

◮ Letting mi ∈ {mR, mS}, the seller maximizes

Profit = E0

i

δτimi(vi)

• ◮ State variable is now k highest marginal revenues

◮ Cutoffs are deterministic in marginal revenue space

Implementation

◮ Use two price schedules for two types of buyer ◮ If rich media buyers have higher values, their marginal

revenues are lower and prices are higher.

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Airlines - Changing Distributions

◮ Demand ft gets stronger over time ◮ Seller maximizes

E0

i

δτimti(vi)

• Optimal discriminations

◮ If ti observed, have cohort specific cutoffs/prices. ◮ Bias towards earlier cohorts. ◮ This is (IC) if ti not observed. ◮ e.g. If ft ∼ exp(µt), then issue coupon of µt for cohort t.

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Selling a House - Disappearing Buyers

◮ Buyers exit the game with probability ∈ (0, 1). ◮ Now need to carry around all past entrants as state

Cutoffs no longer deterministic

◮ If delay buyer y1 may disappear, so value of y2 matters ◮ Prices no longer optimal ◮ Explanation for indicative bidding in real estate

Also have problem if

◮ Buyers have different discount rates ◮ Mix of myopic and forward-looking buyers ◮ General problem: ranking of buyer’s values changes

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Conclusion

Optimal cutoffs

◮ Deterministic (only depend on k and t). ◮ Characterised by one-period-look-ahead rule.

Implemented by posted prices

◮ Sequence of prices with auction at time T. ◮ Prices depend on when sold previous units.

Extensions

◮ Nt correlated (e.g. learning) ◮ Different quality of ad slots ◮ Cost of paying attention to prices.