AUCTIONS WITH DYNAMIC POPULATIONS: EFFICIENCY AND REVENUE - - PowerPoint PPT Presentation

AUCTIONS WITH DYNAMIC POPULATIONS: EFFICIENCY AND REVENUE MAXIMIZATION Maher Said November 2012

MOTIVATION

Many real-world markets are asynchronous. ▶ Not all buyers and sellers available at same time. ▶ Arrive at the market at difgerent times. ▶ Transactions occur at difgerent times. This introduces dynamic trade-ofgs: ▶ Competition in future can be higher or lower. ▶ Opportunities to trade in future may be greater or fewer. ▶ Transact now, or wait until future?

Said (2012): Auctions with Dynamic Populations

MOTIVATION

Additional strategic element: competition across time. ▶ May face same competitors repeatedly. ▶ Individuals may want to learn about others. ▶ May also be concerned about others’ use of information. How do population dynamics afgect competition, price determination, effjciency, revenue? How should markets and institutions be designed to account for dynamics and its efgects?

Said (2012): Auctions with Dynamic Populations

EXAMPLE: EBAY

Consider a buyer searching for a product on eBay. ▶ Buyer arrives on the market. ▶ Can choose to participate in an auction immediately. ▶ May choose to “wait and see” instead. Willingness to pay depends on expectations about future. Future supply random, as is future competition: ▶ When is the next auction starting? ▶ How high will demand be?

Said (2012): Auctions with Dynamic Populations

EXAMPLE: EBAY

Buyer also observes competitor behavior: ▶ Bid amounts, prices observable. ▶ Should incorporate this information into behavior. Conversely, buyer also concerned with how her bids afgects others. ▶ May try to strategically alter others’ expectations. ▶ Can submit a high bid to try to signal high future competition. ▶ How much information should be revealed to current and future opponents?

Said (2012): Auctions with Dynamic Populations

EXAMPLE: AMAZON.COM

Amazon operates a “cloud computing” business. A large portion of demand is pre-reserved. But there is also a lot of excess capacity: ▶ Amazon recently introduced “Spot Instances.” ▶ Runs an auction every hour for excess capacity. ▶ Supply fmuctuates hour-to-hour (or even faster). ▶ So does demand…. How should spot-market be organized? What auction format should be used?

Said (2012): Auctions with Dynamic Populations

QUESTIONS

Two main questions:

• 1. What outcomes are attainable in markets with dynamic populations of privately

informed buyers?

• 2. Can we achieve these outcomes using natural/simple “real-world” institutions?

Approach: ▶ Develop a model of a general dynamic environment. ▶ Privately informed buyers arrive at random times. ▶ Buyers persist on the market, waiting to transact. ▶ Uncertain supply: future object availability is stochastic.

Said (2012): Auctions with Dynamic Populations

PREVIEW: EFFICIENCY

We can achieve effjcient outcomes in this setting. Dynamic analogue of Vickrey-Clarke-Groves. Charge buyers prices corresponding to externalities. Externality price accounts for current and future impact on market.

Said (2012): Auctions with Dynamic Populations

PREVIEW: AUCTIONS

We can also use a sequence of auctions for effjciency. But interaction across time generates interdependent values. ▶ Need information revelation to achieve effjcient outcomes. ▶ In contrast to static settings, second-price auctions are not ideal. ▶ Second-price auction does not reveal enough information. Using ascending price auctions does work

Said (2012): Auctions with Dynamic Populations

PREVIEW: INFORMATION RENEWAL

New entrants to the market are asymmetrically informed. Incentives for information revelation difger across groups of buyers. “Memoryless” behavior provides correct incentives to all. We can restore symmetry by throwing away information. Allows information “renewal”: full revelation of private information in every period.

Said (2012): Auctions with Dynamic Populations

PREVIEW: REVENUE MAXIMIZATION

Revenue maximization is also possible. Static intuitions carry through to dynamic setting: ▶ Dynamic version of incentive compatibility mirrors static. ▶ Revenue Equivalence Theorem continues to hold. ▶ Trade-ofgs between revenue and effjciency same as in static world. Revenue maximization via “effjcient” mechanisms with optimal reserve.

Said (2012): Auctions with Dynamic Populations

MODEL: BUYERS

Countable set I of risk-neutral buyers. Each buyer i has single-unit demand. Value is vi ∈ V. Flow payofg when paying pi,t in period t: ui,t =

• vi − pi,t

if i receives an object at time t, −pi,t

• therwise.

Preferences are quasilinear and time-separable. Common discount factor δ ∈ (0, 1). Values are persistent over time.

Said (2012): Auctions with Dynamic Populations

MODEL: BUYER ARRIVALS

Buyers arrive stochastically to the market. In each period t, Nt buyers arrive. Nt is distributed according to λt. Denote set of arriving buyers by It. Buyers remain on market until allocated an object. at : I → {0, 1} indicates presence at time t: at(i) =

• 1

if i is present,

• therwise.

Said (2012): Auctions with Dynamic Populations

MODEL: OBJECTS

Objects are homogeneous and indivisible. Also arrive randomly to the market. kt ∈ K := {0, 1, . . . , K} objects arrive in period t. µt(k) is probability of exactly k ∈ K objects available Objects are non-storable: unallocated objects cannot be carried over to future periods.

Said (2012): Auctions with Dynamic Populations

RELATED LITERATURE

Effjcient dynamic mechanism design: Athey-Segal (2007); Bergemann-Välimäki (2010); Cavallo-Parkes-Singh (2009); Dolan (1978); Bloch-Houy (2010); Ünver (2010). Optimal dynamic mechanism design: Baron-Besanko (1984); Battaglini (2005); Board-Skrzypacz (2010); Deb (2009); Pavan-Segal-Toikka (2009). Dynamic auctions and revenue management: Gershkov-Moldovanu (2009, 2010); Pai-Vohra (2009); Vulcano-van Ryzin-Maglaras (2002). Sequential auctions: Jeitschko (1998, 1999); Kittsteiner-Nikutta-Winter (2004); Lavi-Nisan (2005), Lavi-Segev (2009); Milgrom-Weber (2000); Said (2011).

Said (2012): Auctions with Dynamic Populations

EFFICIENT MECHANISMS

Want an effjcient mechanism. Effjciency ⇐ ⇒ maximize social welfare. Plan:

• 1. Characterize effjcient policy.
• 2. Effjcient direct mechanism.
• 3. Corresponding “real-world” indirect mechanism.

Said (2012): Auctions with Dynamic Populations

PLANNER'S PROBLEM

Consider a social planner who commits to a feasible mechanism M = {xt, pt}t∈N0 at time 0: ▶ xi,t is probability of allocating to i at time t. ▶ pi,t is payment made by i at time t. Goal is to maximize allocative effjciency: max

{xi,t}

• E

∞

• t=0
• i∈I

δtxi,tvi

• s.t.
• i∈I

xi,t ≤ kt for all t,

∞

• t=0

xi,t ≤ 1 for all i, xi,t = 0 if at(i) = 0.

Said (2012): Auctions with Dynamic Populations

EFFICIENT POLICY

Objects perishable = ⇒ no benefjt to “withholding.” Buyers’ values persistent = ⇒ delay cost increasing in vi. Effjcient policy x∗ is an assortative matching. In each t, allocate to kt highest-valued buyers present.

Said (2012): Auctions with Dynamic Populations

VICKREY-CLARKE-GROVES

VCG-like mechanisms: ▶ Buyers report value on arrival. ▶ Mechanism allocates objects effjciently according to x∗. ▶ Charge each buyer a price equal to the externality imposed on the market. ▶ Leaves each buyer with net payofg equal to marginal contribution to the social welfare.

Said (2012): Auctions with Dynamic Populations

MARGINAL CONTRIBUTION

Consider an arbitrary agent i ∈ It. Social welfare when i arrives: W(ωt, vt) := E ∞

• s=t
• j∈I

δs−tx∗

j,s(ωs, vs)vj

• .

Social welfare when removing i from the market: W−i(ω−i

t , vt) := E

 

∞

• s=t
• j∈I\{i}

δs−tx∗

j,s(ω−i s , vs)vj

  . i’s marginal contribution to the social welfare: wi(ωt, vt) := W(ωt, vt) − W−i(ω−i

t , vt).

Said (2012): Auctions with Dynamic Populations

MARGINAL CONTRIBUTION

Suppose a single item is available in each period t. Effjcient policy = ⇒ allocate to highest-valued buyer (i). Thought experiment: remove buyer i from market. Effjcient policy = ⇒ allocate to 2nd-highest buyer (j).

Said (2012): Auctions with Dynamic Populations

MARGINAL CONTRIBUTION

But xj,t = 1 = ⇒ cannot allocate to j in the future: = ⇒ Period t + 1: allocate to 3rd-highest buyer instead of 2nd-highest. = ⇒ Period t + 2: allocate to 4th-highest buyer instead of 3rd-highest. = ⇒ Period t + 3: …. Allocating to j today = ⇒ lose j’s future marginal contribution.

Said (2012): Auctions with Dynamic Populations

MARGINAL CONTRIBUTION

i’s marginal contribution can be decomposed into two efgects. Presence of i leads to a gain today: vi − vj > 0. Presence of i also leads to a gain in future: δE [wj(ωt+1, vt+1)|·]

• j’s future contribution

> 0. Marginal contribution of i is then wi(ωt, vt) = vi − vj

current period

+ δE [wj(ωt+1, vt+1)|·]

• future periods

.

Said (2012): Auctions with Dynamic Populations

DYNAMIC PIVOT MECHANISM

Incentives aligned by taking into account anticipated future re-orderings of allocations: vi − pi,t = wi(ωt, vt). The dynamic pivot mechanism M∗ := {x∗

t, p∗ t}t∈N0 is the direct mechanism where

• 1. x∗ is the effjcient allocation rule, and
• 2. p∗ is defjned by p∗

i,t(ωt, vt) := x∗ i,t(ωt, vt) (vi − wi(ωt, vt)) .

If i is not among the top kt, then x∗

i,t = 0 =

⇒ p∗

i,t = 0.

M∗ gives each buyer her fmow marginal contribution in each period: ui,s = wi(ωs, vs) − δE [wi(ωs+1, vs+1)] .

Said (2012): Auctions with Dynamic Populations

DYNAMIC PIVOT MECHANISM

Theorem (Bergemann and Välimäki 2010) The dynamic pivot mechanism M∗ := {x∗

t, p∗ t}t∈N0 is periodic ex post incentive compatible and

individually rational. Truthful reporting is not dominant strategy. ▶ Reports are periodic ex post optimal, given expectations about future arrivals/behavior. ▶ For any realized values, truthful reporting is a Nash equilibrium of the “complete information” subgame. ▶ Buyers have no regret, regardless of competitors’ valuations. Expected payofg of i ∈ It: E [wi(ωt, vt)]. Since wi ≥ 0, participation is optimal.

Said (2012): Auctions with Dynamic Populations

INDIRECT IMPLEMENTATION

The dynamic pivot mechanism is a direct mechanism. It requires a single report upon buyers’ arrival. Despite truthful revelation, may not be practical or useful in “real world.” Transparency of the mechanism is important.

Said (2012): Auctions with Dynamic Populations

INDIRECT IMPLEMENTATION

Banks, Ledyard, and Porter (1989): Thetransparencyofamechanism—theeasewithwhichanagentisabletoanticipatethe results of any particular strategy—is important in achieving more efficient allocations. Nalebufg and Bulow (1993): Even Ph.D. students have trouble understanding the [multi-unit Vickrey auction]…. The problem is that if people do not understand the payment rules of the auction then we do not have confidence that the end result will be efficient. Ausubel (2004): Many [economists] believe it is too complicated for practitioners to understand. Is there a natural indirect mechanism for dynamic VCG?

Said (2012): Auctions with Dynamic Populations

SEQUENTIAL AUCTIONS

Yes: sequential auctions. But what auction format? “Standard” analogue of VCG in static settings is the second-price sealed-bid auction. Sequential second-price sealed-bid auction does not correspond to dynamic VCG.

Said (2012): Auctions with Dynamic Populations

OPTIONS

Buyers in sequential auctions have an option: lose today and participate in future auctions. Value of future participation: δV . Rational bidding: shade bids by option value: max

bi

• Pr (win) E
• vi − max

j̸=i {bj}

• + Pr (lose) δV
• ⇐

⇒ max

bi

• Pr (win) E
• (vi − δV ) − max

j̸=i {bj}

• + δV

= ⇒ b∗

i = vi − δV .

Said (2012): Auctions with Dynamic Populations

INTERDEPENDENCE

But value of option depends on expected prices. Future prices determined by competitors’ values = ⇒ V = V (vi, v−i). Despite independent private values, market dynamics generate interdependence. Buyers must learn competitors’ values in order to correctly price options.

Said (2012): Auctions with Dynamic Populations

SECOND-PRICE AUCTION

Second-price sealed-bid auction does not reveal enough information = ⇒ b∗

i = vi − δE [V (vi, v−i)] .

Bidders arriving at difgerent times have difgerent beliefs: = ⇒ Asymmetry in expectations. = ⇒ Asymmetry in bids. = ⇒ Ineffjcient outcomes.

Said (2012): Auctions with Dynamic Populations

REVEALING BIDS

Second-price auction with revealed bids? Any period with new entrants = ⇒ uninformed buyers. Revealing bids after submission is too late. Can’t make use of information until next period. Information revelation too “slow” to be useful.

Said (2012): Auctions with Dynamic Populations

ASCENDING AUCTION

Need an open auction format = ⇒ ascending auction. Price clock rises continuously. Buyers observe competitors’ exits from auction. Can infer competitors’ valuations. Can incorporate information into current-period bidding: = ⇒ b∗

i = vi − δV (vi, v−i).

Said (2012): Auctions with Dynamic Populations

ASCENDING AUCTION

Use a multi-unit, uniform-price version of the Milgrom and Weber (1982) “button” auction. Sell all kt units at same price to highest kt bidders. In each period t: ▶ Price clock starts at zero, rises continuously. ▶ Number of active participants public. ▶ Exits publicly observable and irreversible. ▶ Auction ends when at most kt buyers remain active. ▶ Price paid is price of fjnal exit. But this generates an additional asymmetry…

Said (2012): Auctions with Dynamic Populations

MORE ASYMMETRY

Consider losing buyers in period t. Observe each others’ exit points = ⇒ infer values. At beginning of period t + 1: perfectly informed. But new entrants also arrive in period t + 1. Buyers are now asymmetrically informed. Want new entrants to reveal private information. How do we provide incentives to new buyers?

Said (2012): Auctions with Dynamic Populations

RESTORING SYMMETRY

Solution: information renewal. Full revelation of private information in every period. Achieved by using “memoryless” strategies. Incumbents disregard past observations, behave as though uninformed: ▶ Allows all buyers to behave symmetrically. ▶ Provides appropriate incentives for entrants to reveal private information.

Said (2012): Auctions with Dynamic Populations

RESTORING SYMMETRY

Note: equilibrium behavior, not a restriction on strategies. Why is it rational to throw away information? Buyers engage in information revelation in every period. Buyers expect information to be revealed again. No need to condition on past history if information will be disclosed anew.

Said (2012): Auctions with Dynamic Populations

BIDDING STRATEGIES

Let nt be number of buyers present. Let ym denote m-th highest value. Buyers stay in auction until indifgerent between winning today and participating tomorrow. Buyers expect to pay prices equal to the externality they impose. ▶ Equivalently, option value equals expected marginal contribution to social welfare: V (vi, v−i) = E [wi(ωt+1, vt+1)|vt = (vi, v−i)] .

Said (2012): Auctions with Dynamic Populations

BIDDING STRATEGIES

Bid up to price at which indifgerent between winning now and receiving marginal contribution in future: vi − p

win now

= δE [wi(ωt+1, vt+1)|vt = (vi, . . . , vi)]

• future marginal contribution

. Marginal contribution is increasing, but slower than value. = ⇒ Buyer with lowest value will be fjrst to drop out. Remaining buyers observe exit price, infer value ynt of lowest buyer.

Said (2012): Auctions with Dynamic Populations

BIDDING STRATEGIES

Buyers update beliefs about future. Bid up to price at which indifgerent between winning now and receiving marginal contribution in future: vi − p

win now

= δE [wi(ωt+1, vt+1)|vt = (vi, . . . , vi, ynt)]

• updated future marginal contribution

. Again, marginal contribution is increasing, but slower than value. = ⇒ Buyer with second-lowest value will drop out next. Remaining buyers observe exit, infer value ynt−1 of 2nd-lowest buyer.

Said (2012): Auctions with Dynamic Populations

BIDDING STRATEGIES

Proceeding inductively, buyers continue updating their beliefs as bidders drop out. Bid up to price at which indifgerent between winning now and receiving marginal contribution in future: βt

m,nt(ωt, vi, ym+1, . . . , ynt)

:= vi − δE [wi(ωt+1, vt+1)|vt = (vi, . . . , vi, ym+1, . . . , ynt)] . Buyer with m-th highest value drops out next; remaining buyers infer ym.

Said (2012): Auctions with Dynamic Populations

BIDDING STRATEGIES

Auction ends when only top kt buyers remain. Each pays price determined by buyer with (kt + 1)-th highest value: βt

kt+1,nt(ωt, ykt+1, . . . , ynt).

In following periods: ▶ Process repeats from beginning. ▶ Starting from βt+1

nt+1,nt+1(ωt+1, vi).

Said (2012): Auctions with Dynamic Populations

EQUILIBRIUM

Theorem Bidding according to the strategies βt

m,nt in each period t is a perfect Bayesian equilibrium of the

sequential ascending auction mechanism. Suppose buyer i with value vi > ykt+1 wins. Pays βt

kt+1,nt(ωt, ykt+1, . . . , ynt) =

ykt+1 − δE [wkt+1(ωt+1, vt+1)|vt = (ykt+1, . . . , ykt+1, . . . , ynt)] . This is exactly the externality imposed by i’s presence on the market. ▶ If i is removed, x∗ allocates to (kt + 1)-th highest buyer. ▶ The planner realizes a gain equal to that buyer’s value. ▶ But the planner forgoes that buyer’s future contributions.

Said (2012): Auctions with Dynamic Populations

OUTCOME EQUIVALENCE

i’s payofg is then vi − ykt+1

• current period

+ δE [wkt+1(ωt+1, vt+1)|·]

• future periods

= wi(ωt, vt)

• total contribution

. Theorem The equilibrium of the sequential ascending auction mechanism is outcome equivalent to the truth-telling equilibrium of the dynamic pivot mechanism. Sequence of allocations is identical = ⇒ effjcient. Sequence of payments is identical: ▶ Winning buyers pay externality. ▶ Losing buyers pay nothing.

Said (2012): Auctions with Dynamic Populations

RECAP

So far: ▶ Effjcient policy and dynamic VCG. ▶ Indirect implementation via sequential ascending auctions. What about revenue? ▶ What is the revenue-maximizing policy? ▶ Can it be implemented via a direct mechanism? ▶ Can it be implemented via an indirect mechanism?

Said (2012): Auctions with Dynamic Populations

MONOPOLIST'S PROBLEM

Consider a monopolist who commits to a feasible mechanism M = {xt, pt}t∈N0 at time 0: ▶ xi,t is probability of allocating to i at time t. ▶ pi,t is payment made by i at time t. Goal is to maximize revenue: max

• E

∞

• t=0
• i∈I

δtpi,t

• .

Seller needs to induce buyers to reveal private information. How can she provide the correct incentives?

Said (2012): Auctions with Dynamic Populations

BUYER PAYOFFS

Revelation principle holds = ⇒ direct revelation is wlog. Consider i ∈ It with value vi. Suppose all other buyers are reporting truthfully. Then i’s payofg from reporting v′

i is

Ui(v′

i, vi, ωt) := E

∞

• s=t

δs−t xi,s(v′

i, v−i)vi − pi,s(v′ i, v−i)

• .

Said (2012): Auctions with Dynamic Populations

IC AND IR

Incentive compatibility: truthful reporting is a best response. Individual rationality: participation is optimal. The mechanism M = {xt, pt}t∈N0 is incentive compatible if, for all t, all i ∈ It, and all ωt, Ui(vi, vi, ωt) ≥ Ui(v′

i, vi, ωt) for all vi, v′ i ∈ V.

M is individually rational if, for all t, all i ∈ It, and all ωt, Ui(vi, vi, ωt) ≥ 0 for all vi ∈ V.

Said (2012): Auctions with Dynamic Populations

REDUCING DIMENSION

Incentive problem simplifjed by reduction to a single-dimensional allocation problem Expected discounted probability of allocation: qi(v′

i, ωt) := E

∞

• s=t

δs−txi,s(v′

i, v−i)

• .

Expected discounted payment: mi(v′

i, ωt) := E

∞

• s=t

δs−tpi,s(v′

i, v−i)

• .

Said (2012): Auctions with Dynamic Populations

IMPLEMENTABLE MECHANISMS

Since buyers are risk-neutral, payofgs are quasilinear, payofg from truthful reporting is

• Ui(vi, ωt) := qi(vi, ωt)vi − mi(vi, ωt).

Immediately yields characterization of IC and IR à la Myerson (1981). Lemma M = {xt, pt}t∈N0 is incentive compatible and individually rational if, and only if, for all t, all i ∈ It, and all ωt:

• 1. qi(vi, ωt) is nondecreasing in vi;

2. Ui(vi, ωt) = Ui(0, ωt) + vi

0 qi(v′ i, ωt) dv′ i; and

3. Ui(0, ωt) ≥ 0.

Said (2012): Auctions with Dynamic Populations

IMPLEMENTABLE ALLOCATIONS

Static settings: ▶ Incentive compatibility: higher vi = ⇒ higher probability of receiving object. Here: ▶ Incentive compatibility: higher vi = ⇒ higher probability of receiving object sooner. ▶ Multiple opportunities to receive an object: qi(v′

i, ωt) = E

∞

• s=t

δs−txi,s(v′

i, v−i)

• .

Said (2012): Auctions with Dynamic Populations

REVENUE EQUIVALENCE

Revenue Equivalence Theorem holds in this setting. Payments pinned down by allocation rule alone. Corollary If M = {xt, pt}t∈N0 is incentive compatible, then for all t ∈ N0, all i ∈ It, and all ωt, the expected payment of type vi ∈ V of buyer i, conditional on entry, is mi(vi, ωt) = mi(0, ωt) + qi(vi, ωt)vi − vi qi(v′

i, ωt) dv′ i.

If M is also individually rational, then mi(0, ωt) ≤ 0.

Said (2012): Auctions with Dynamic Populations

MONOPOLIST'S PROBLEM

Recall monopolist’s optimization problem: max

• E

∞

• t=0
• i∈I

δtpi,t(ωt, v)

• .

Equivalently: max

• E

∞

• t=0
• i∈It

δtat(i)mi(vi, ωt)

• .

Said (2012): Auctions with Dynamic Populations

SOLUTION

Revenue Equivalence = ⇒ monopolist’s problem is: max

• E

∞

• t=0
• i∈It

δtat(i)mi(0i, ωt) +

∞

• t=0
• i∈It

δtat(i)qi(vi, ωt)ϕ(vi)

• ,

where virtual value ϕ is ϕ(vi) := vi − 1 − F(vi) f(vi) . Assume wlog ϕ increasing (otherwise iron).

Said (2012): Auctions with Dynamic Populations

SOLUTION

Maximize the two pieces separately. First term: individual rationality ⇐ ⇒ mi(0, ωt) ≤ 0. ▶ So set mi(0, ωt) = 0. Second term: E ∞

• t=0
• i∈It

δtat(i)qi(vi, ωt)ϕ(vi)

• = E

∞

• t=0
• i∈It

δtxi,t(ωt, v)ϕ(vi)

• .

Same as planner’s problem—but with virtual values. ▶ So allocate objects to available buyers with highest (non-negative) virtual values.

Said (2012): Auctions with Dynamic Populations

OPTIMAL POLICY

Optimal policy x is incentive compatible. Higher value = ⇒ higher virtual value = ⇒ greater likelihood of being in top kt. ▶ Implies ˜ xi,t nondecreasing in vi for all t. ▶ And also ˜ qi(ωt, vi) nondecreasing in vi. What payment rule supports this policy?

Said (2012): Auctions with Dynamic Populations

MYERSON (1981)

Myerson’s optimal auction can be reinterpreted in terms of VCG. Instead of maximizing surplus, maximize virtual surplus. ▶ Agents report values vi. ▶ Mechanism computes virtual values ϕ(vi). ▶ Applies VCG to virtual values. Optimal mechanism yields allocation and “virtual price.” ▶ Virtual price is winner’s marginal contribution to the virtual surplus. ▶ Virtual price is inverted into a “standard” price ⇐ ⇒ lowest winning value.

Said (2012): Auctions with Dynamic Populations

VIRTUAL SURPLUS

Let ˜ r := ϕ−1(0). Defjne virtual surplus: Π(ωt, vt) := E ∞

• s=t
• j∈I

δs−t˜ xj,s(ωs, vs) (vj − ˜ r)

• .

Measures contributions beyond the reservation value ˜ r.

• x is effjcient for a planner with this payofg function.

▶ So run dynamic VCG on virtual surplus.

Said (2012): Auctions with Dynamic Populations

VIRTUAL VCG

Defjne Π−i(ω−i

t , vt) := E

 

∞

• s=t
• j∈I\{i}

δs−t˜ xj,s(ω−i

s , vs) (vj − ˜

r)   . i’s marginal contribution to the virtual surplus:

• wi(ωt, vt) := Π(ωt, vt) − Π−i(ω−i

t , vt).

• wi is i’s “replacement value” over a buyer with value ˜

r.

Said (2012): Auctions with Dynamic Populations

DYNAMIC VIRTUAL PIVOT MECHANISM

Align incentives by giving each buyer her marginal contribution to the virtual surplus. Want vi − pi,t = wi(ωt, vt). The dynamic virtual pivot mechanism M := { xt, pt}t∈N0 is the dynamic direct mechanism where 1. x is the optimal allocation rule, and 2. p is defjned by ˜ pi,t(ωt, vt) := ˜ xi(ωt, vt) (vi − wi(ωt, vt)) .

Said (2012): Auctions with Dynamic Populations

DYNAMIC VIRTUAL PIVOT MECHANISM

In each period, give each buyer her fmow marginal contribution: ui,t = wi(ωt, vt) − δE [ wi(ωt+1, vt+1)] . i’s expected payofg is then Ui(ωt, vi) = E [ wi(ωt, vt)]. Theorem The dynamic virtual pivot mechanism M := { xt, pt}t∈N0 is periodic ex post incentive compatible and individually rational.

• M implements revenue-maximizing policy

x.

• M is very closely related to dynamic VCG.

▶ Is there an equivalent “natural” indirect mechanism?

Said (2012): Auctions with Dynamic Populations

INDIRECT IMPLEMENTATION

“Standard” static private-values setting: ▶ Effjciency via VCG ⇐ ⇒ Second-price auction. ▶ Revenue-maximization via Myerson (1981) ⇐ ⇒ VCG with a reserve ⇐ ⇒ Second-price auction with a reserve. By analogy, in this dynamic setting: ▶ Effjciency via dynamic VCG ⇐ ⇒ Sequential ascending auction. ▶ Revenue-maximization via M ⇐ ⇒ Dynamic VCG with a reserve ⇐ ⇒ Sequential ascending auction with a reserve.

Said (2012): Auctions with Dynamic Populations

SUMMARY

Develop an intuitive indirect mechanism: sequential ascending auctions. ▶ Simple institution that yields effjciency. ▶ Equilibrium is outcome equivalent to dynamic VCG. Extend results to revenue-maximization. ▶ Optimal mechanism is a pivot mechanism with a reserve. ▶ Optimal indirect mechanism is the sequential ascending auction with a reserve.

Said (2012): Auctions with Dynamic Populations

LESSONS: DYNAMIC ENVIRONMENTS

Many of our intuitions from the static world carry through. We can use pivot mechanisms to achieve effjciency. Auctions continue to be efgective institutions. Same trade-ofgs between effjciency and revenue.

Said (2012): Auctions with Dynamic Populations

LESSONS: INSTITUTIONAL DESIGN

But…information is more important in dynamic markets. Even with independent private values, interaction across time creates interdependence. Need information revelation to attain desirable outcomes. Must consider dynamic auction formats to generate enough information: second-price auction not optimal.

Said (2012): Auctions with Dynamic Populations

LESSONS: DYNAMIC POPULATIONS

Buyer arrivals introduce a new tension. Information revelation vs. incentive provision. Revelation creates asymmetry, but asymmetry creates incentive problems across groups of buyers. Information revelation alone is not suffjcient = ⇒ need information renewal.

Said (2012): Auctions with Dynamic Populations

LESSONS: SIMPLICITY

Complex setting: ▶ Dynamic population of buyers. ▶ Stochastic supply. ▶ Asymmetric incentive constraints. Simple solution: ▶ Ascending auctions are a natural, intuitive institution. ▶ Bidding strategies are memoryless, no history dependence. ▶ Able to achieve desirable outcomes.

Said (2012): Auctions with Dynamic Populations

FUTURE WORK

Interdependent values: ▶ Additional complication: history dependence. ▶ New buyers interested in values of previous winners. ▶ Information transmission becomes more diffjcult. ▶ How to communicate multi-dimensonal information with one-dimensional strategy? More general arrival processes: ▶ Relax assumption of exogenously-driven arrivals. ▶ Allow buyers to condition entry on market conditions, seller to adjust supply. ▶ Balance motive to induce entry with rent extraction. ▶ Building block for competing markets and platforms.

Said (2012): Auctions with Dynamic Populations