AUCTIONS WITH DYNAMIC POPULATIONS: EFFICIENCY AND REVENUE MAXIMIZATION Maher Said November 2012
MOTIVATION Many real-world markets are asynchronous. This introduces dynamic trade-ofgs: Said (2012): Auctions with Dynamic Populations ▶ Not all buyers and sellers available at same time. ▶ Arrive at the market at difgerent times. ▶ Transactions occur at difgerent times. ▶ Competition in future can be higher or lower. ▶ Opportunities to trade in future may be greater or fewer. ▶ Transact now, or wait until future?
MOTIVATION Additional strategic element: competition across time. 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 ▶ May face same competitors repeatedly. ▶ Individuals may want to learn about others. ▶ May also be concerned about others’ use of information.
EXAMPLE: EBAY Consider a buyer searching for a product on eBay. Willingness to pay depends on expectations about future. Future supply random, as is future competition: Said (2012): Auctions with Dynamic Populations ▶ Buyer arrives on the market. ▶ Can choose to participate in an auction immediately. ▶ May choose to “wait and see” instead. ▶ When is the next auction starting? ▶ How high will demand be?
EXAMPLE: EBAY Buyer also observes competitor behavior: Conversely, buyer also concerned with how her bids afgects others. Said (2012): Auctions with Dynamic Populations ▶ Bid amounts, prices observable. ▶ Should incorporate this information into behavior. ▶ 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?
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: How should spot-market be organized? What auction format should be used? Said (2012): Auctions with Dynamic Populations ▶ Amazon recently introduced “Spot Instances.” ▶ Runs an auction every hour for excess capacity. ▶ Supply fmuctuates hour-to-hour (or even faster). ▶ So does demand….
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: Said (2012): Auctions with Dynamic Populations ▶ 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.
PREVIEW: EFFICIENCY We can achieve effjcient outcomes in this setting. Dynamic analogue of Vickrey-Clarke-Groves. Charge buyers prices corresponding to externalities. Said (2012): Auctions with Dynamic Populations Externality price accounts for current and future impact on market.
PREVIEW: AUCTIONS We can also use a sequence of auctions for effjciency. But interaction across time generates interdependent values. Using ascending price auctions does work Said (2012): Auctions with Dynamic Populations ▶ 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.
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: Revenue maximization via “effjcient” mechanisms with optimal reserve. Said (2012): Auctions with Dynamic Populations ▶ Dynamic version of incentive compatibility mirrors static. ▶ Revenue Equivalence Theorem continues to hold. ▶ Trade-ofgs between revenue and effjciency same as in static world.
otherwise. MODEL: BUYERS Preferences are quasilinear and time-separable. Values are persistent over time. Said (2012): Auctions with Dynamic Populations Countable set I of risk-neutral buyers. Each buyer i has single-unit demand. Value is v i ∈ V . Flow payofg when paying p i,t in period t : � v i − p i,t if i receives an object at time t , u i,t = − p i,t Common discount factor δ ∈ (0 , 1) .
MODEL: BUYER ARRIVALS Buyers arrive stochastically to the market. Buyers remain on market until allocated an object. otherwise. Said (2012): Auctions with Dynamic Populations In each period t , N t buyers arrive. N t is distributed according to λ t . Denote set of arriving buyers by I t . a t : I → { 0 , 1 } indicates presence at time t : � 1 if i is present, a t ( i ) = 0
MODEL: OBJECTS Objects are homogeneous and indivisible. Also arrive randomly to the market. Objects are non-storable: unallocated objects cannot be carried over to future periods. Said (2012): Auctions with Dynamic Populations k t ∈ K := { 0 , 1 , . . . , K } objects arrive in period t . µ t ( k ) is probability of exactly k ∈ K objects available
RELATED LITERATURE 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 Effjcient dynamic mechanism design: Athey-Segal (2007); Bergemann-Välimäki (2010);
EFFICIENT MECHANISMS Want an effjcient mechanism. Plan: 1. Characterize effjcient policy. 2. Effjcient direct mechanism. 3. Corresponding “real-world” indirect mechanism. Said (2012): Auctions with Dynamic Populations Effjciency ⇐ ⇒ maximize social welfare.
PLANNER'S PROBLEM max s.t. Said (2012): Auctions with Dynamic Populations Goal is to maximize allocative effjciency: Consider a social planner who commits to a feasible mechanism M = { x t , p t } t ∈ N 0 at time 0: ▶ x i,t is probability of allocating to i at time t . ▶ p i,t is payment made by i at time t . � � ∞ �� � � δ t x i,t v i E { x i,t } t =0 i ∈I � x i,t ≤ k t for all t, i ∈I � ∞ x i,t ≤ 1 for all i, t =0 x i,t = 0 if a t ( i ) = 0 .
EFFICIENT POLICY Objects perishable = Buyers’ values persistent = Said (2012): Auctions with Dynamic Populations ⇒ no benefjt to “withholding.” ⇒ delay cost increasing in v i . Effjcient policy x ∗ is an assortative matching. In each t , allocate to k t highest-valued buyers present.
VICKREY-CLARKE-GROVES VCG-like mechanisms: Said (2012): Auctions with Dynamic Populations ▶ 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.
MARGINAL CONTRIBUTION Said (2012): Auctions with Dynamic Populations Consider an arbitrary agent i ∈ I t . Social welfare when i arrives: � ∞ � � � δ s − t x ∗ W ( ω t , v t ) := E j,s ( ω s , v s ) v j . s = t j ∈I Social welfare when removing i from the market: ∞ � � W − i ( ω − i δ s − t x ∗ j,s ( ω − i . t , v t ) := E s , v s ) v j s = t j ∈I\{ i } i ’s marginal contribution to the social welfare: w i ( ω t , v t ) := W ( ω t , v t ) − W − i ( ω − i t , v t ) .
MARGINAL CONTRIBUTION Effjcient policy = Effjcient policy = Said (2012): Auctions with Dynamic Populations Suppose a single item is available in each period t . ⇒ allocate to highest-valued buyer ( i ). Thought experiment: remove buyer i from market. ⇒ allocate to 2nd-highest buyer ( j ).
MARGINAL CONTRIBUTION = = = Said (2012): Auctions with Dynamic Populations But x j,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.
MARGINAL CONTRIBUTION current period future periods Said (2012): Auctions with Dynamic Populations i ’s marginal contribution can be decomposed into two efgects. Presence of i leads to a gain today: v i − v j > 0 . Presence of i also leads to a gain in future: δ E [ w j ( ω t +1 , v t +1 ) |· ] > 0 . � �� � j ’s future contribution Marginal contribution of i is then w i ( ω t , v t ) = v i − v j + δ E [ w j ( ω t +1 , v t +1 ) |· ] . � �� � � �� �
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