WO2009154724A1 - Mécanismes de vente aux enchères utilisant la connaissance extérieure - Google Patents

Mécanismes de vente aux enchères utilisant la connaissance extérieure Download PDF

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WO2009154724A1
WO2009154724A1 PCT/US2009/003584 US2009003584W WO2009154724A1 WO 2009154724 A1 WO2009154724 A1 WO 2009154724A1 US 2009003584 W US2009003584 W US 2009003584W WO 2009154724 A1 WO2009154724 A1 WO 2009154724A1
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player
players
information
goods
auction
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PCT/US2009/003584
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English (en)
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Silvio Micali
Jing Chen
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Silvio Micali
Jing Chen
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Publication of WO2009154724A1 publication Critical patent/WO2009154724A1/fr

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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
    • G06Q30/00—Commerce
    • G06Q30/06—Buying, selling or leasing transactions
    • G06Q30/08—Auctions

Definitions

  • This application is related to the field of auctions and auction mechanisms.
  • BACKGROUND OF THE INVENTION Auctions include a context and a mechanism.
  • the context comprises: a) A set of goods for sale: a single good, or a multiplicity of goods ⁇ in particular, multiple copies of the same good. (For instance, one may want to sell via an auction a given number of licenses under the same patent.) b) A set of players (e.g., individuals, firms, or combination thereof) who are desirous of buying (at least some of) the good(s). Players are typically denoted by 1, 2,..., n. c) The players' valuations.
  • the valuation of a player i essentially specifies how much i values the goods. For instance, i's valuation may specify a value v for each subset of the goods.
  • An outcome specifies how the goods are sold, that is which player wins which goods, and the (total) price each player pays.
  • An auction mechanism specifies the auctions "rules"; in essence what the players can do in the auction, and how the outcome is obtained from the actions of the players. What the players can do are also referred to as strategies, or sometimes bids (particularly when strategies consists of providing/announcing a string).
  • a rational player wishes to maximize his utility: typically, this consists of his value for the goods he receives minus the price he pays.
  • a mechanism should be carefully selected so that, when the players try to maximize their own utilities, the strategy they choose result in a desirable outcome.
  • a typically desired outcome is one generating reasonable revenue, that is the sum of the prices paid by all players. Because the players' valuations are typically not known or not well known to the seller or the designer of the auction mechanism, choosing good mechanisms is hard.
  • a player's strategy typically consisted of providing either publicly or privately (e.g., to an auctioneer) his own valuation, or information about his own valuation.
  • the famous second-price mechanism asks each player i to bid a valuation Vi, and then chooses as the winner of the good the player with the highest bid, the second highest bid as the winner's price, and O as the price of each of the other players.
  • Vi By bidding Vi each player i is ready to pay up to Vi if necessary, so that Vi essentially is valuation information relative to i himself.
  • Vi is ready to pay up to Vi if necessary, so that Vi essentially is valuation information relative to i himself.
  • each player i provides valuation information relative to i himself.
  • VCG mechanism which can be viewed as a generalization of the second price-mechanism to auctions in which multiple goods are for sale.
  • a player i's valuation information relative to himself as internal knowledge (or internal information). Auctions based on internal knowledge alone, however, typically fail to achieve several desiderata.
  • the famous VCG mechanism is vulnerable to collusion, provides no guarantee about revenue (even in the presence of fierce competition for the goods), and may be hampered by the fact that the players may be reluctant (e.g., for privacy reasons) to provide to anyone their own valuations.
  • an auction mechanism is provided that exploits a different type of player knowledge: namely, a player's external knowledge (or external information), in essence, a player's information about the valuations of the other players.
  • techniques of the system described herein may focus on just the players' external knowledge, so as to highlight to the maximum possible extent the difference between ours and the prior auction mechanisms.
  • an embodiment of the system described herein can be described in terms of at least two player stages, that is stages where at least one player takes action. (This is without loss of generality: more stages or a single stage could also be used.)
  • each player i provides information that includes information about the valuations of at least one other player.
  • information we mean at least one of: information that the players knows to be true, information that the player believes to be true, information claimed by the player, and other information.
  • This information may not be all the information in possession of the player, and is preferably information in a form that simplifies choosing a suitable auction outcome.
  • player i may "bid" (A) a partition the good(s) among the other players;
  • (C) a subset of the goods.
  • Item C essentially consists of an indication of how player i wishes to be rewarded.
  • item C may be considered as part of i's internal knowledge, since it might be in i's best interest to suggest the subset of the goods he values the most for his reward.
  • the price of any player who does not receive any goods is preferably 0. (This is without any limitation intended: in particular the mechanism can always charge any player with a fixed initial price as an "entrance fee.") Accordingly, item B of player i specifies a positive price only for those players who receive some goods according to item A of player i.
  • the information provided above by the player is publicly provided, so as to enable anyone to see how the outcome is selected without having to trust anyone.
  • the auction mechanism calls for identifying the star player, that is the player for which the sum of the prices in item B is maximum. Without loss of generality, the star player is unique. Else the mechanism may be called to randomly break ties among multiple players if needed.
  • each player j that, according to the partition of the goods in item A of the star player, is indicated to receive at least one good provides a "response information.”
  • this response information indicates whether player j is willing to receive the good(s) indicated for him in item A of the star player for the price p indicated for j in item B of the star player.
  • the auction mechanism determines an outcome according to the information provided by the star player in the first player stage, and the response information provided by the other players in the second player stage.
  • the auction mechanism may also determines the reward of the star player and possibly of all players. In particular, this can be done as follows.
  • the auction mechanism calls for the flip of a coin. If Heads comes up, the final outcome consists of assigning to the star player the subset of goods he provided in item C for a price of 0. No other goods are assigned to any other player, and the price of all other players is 0. If Tails comes up, the following happens. Let j be a player who is indicated, in items A and B of the star player, to receive a subset of goods Aj for a price Pj.
  • All players not indicated to receive goods in item A of the star player are assigned no goods and pay nothing. All goods not specified for any player in item A of the star player are not assigned to any player.
  • the mechanism may further provide additional rewards to all players as well, for their providing their "external knowledge". In particular, it may provide some cash. Preferably, such cash reward for a player is higher the higher is the revenue (i.e., the sum of the prices) provided by the player in item B.
  • the coin flip of the mechanism may occur before the second player stage.
  • this embodiment enables the seller to get revenue that, within a factor of 2, is at least equal to the revenue that the best informed player could generate by "selling the good(s) to the other players.” This is remarkable, because the seller himself (or the designer of the mechanism) may not have any information about the valuations of the players!
  • a SECOND EMBODIMENT in a second embodiment, another auction mechanism may be provided for the sale of a single good.
  • This auction mechanism too can be described in terms of at least two player stages. (This is again without loss of generality, since more stages or a single stage could also be used.)
  • each player i provides information that includes valuation information relative to all players. Since the current embodiment deals with the sale of a single good, the valuation information provided — publicly or privately — may consist of a single number, representing the value that a player may have for the single good for sale. In this case, to provide valuation information about all the n players, a player may provide a sequence ofn values, vl,...,vn, where vj is the value relative to player j, and could be interpreted as i providing the following information: "j's valuation for the good is at least vj.”
  • the (potential) winner of the good for sale is chosen to be the player, i, providing the highest valuation relative to himself.
  • i is the player providing the highest "self valuation”, therefore we may refer to the valuation provided by i relative to himself as i's self-valuation. (Again without loss of generality, this player is unique: in case of ties the mechanisms can randomly break them.)
  • R' the second-highest valuation of a player relative to himself — in a sense the second highest "self-valuation.”
  • the envisaged mechanism also computes an "external-knowledge" price p': in particular, the highest valuation relative to i provided by the other players.
  • the player who provides this valuation information relative to i is called the "best informed player", and referred in the next few line as player j for simplicity.
  • IfR' is greater than or equal to p' then, the good is assigned to player i for price R'.
  • the mechanism may reward player j, as in a sense he has been very helpful in generating more revenue.
  • j's reward may consists of cash, or other form of reward.
  • the reward can be a fixed amount of money, or may depend on the values of p', R', the valuation provided by i relative to himself, or a combination thereof.
  • j's reward may be proportional to p'-R'.
  • i's response information indicates that i is not willing to receive the good for a price p', then the good is not allocated (or not allocated to i), i may pay nothing, and player j may be punished/receive a negative reward (e.g., imposed a fine).
  • the mechanism may also reward other players or all players, in particular for their "external knowledge" as this enables more revenue to be generated.
  • all players may provide response information, rather the (potential) winner.
  • response information may be provided in all cases; in particular whether or not p' is greater or equal to R' .
  • the response information may be provided before hand, in particular before a player is aware of the valuation information relative to him provided by the other players. For instance, a player may "guess" what such valuation information might be.
  • the context specifies the players' types, all possible outcomes, and the players' utilities in these outcomes.
  • the mechanism specifies the players' strategy spaces and how strategies determine outcomes. Let us explain this for combinatorial auctions. Unrestricted Combinatorial Auctions
  • a valuation for a finite set of goods G is a function mapping each subset of G to a non-negative real, and that a profile is a vector indexed by the players.
  • TVi specifies the value that player i truly attributes to any of the 2 m subsets of the goods for sale, and is such 0.
  • P the price profile
  • Pj represents the amount player i pays. If P 1 is negative, then -P x represents the amount i receives.
  • COMBINATORIAL-AUCTION MECHANISMS COMBINATORIAL-AUCTION MECHANISMS.
  • a player's strategy ⁇ also called a bid
  • the mechanism also specifies a possibly probabilistic function from strategy profiles to outcomes, the outcome function, denoted too by M for simplicity.
  • underbidding may be the best thing to do.
  • the context may be such that (1) player i is the best informed player; (2) the second best informed player is j; and yet (3) player j is badly informed about player i: that is, RK 3 allocates to i a subset of goods .S 1 that i highly values for a reasonably low price. In this case, i would be better off if j announced RK 3 and were the star player.
  • underbidding may be far from being a dominated strategy.
  • our mechanism modifies the above basic procedure as follows Together with RK 1 , each player i also announces his (suppositively) favorite subset of the goods, S 1 . If i is declared the star player, then a coin toss of the mechanism determines whether the above basic procedure takes place or i receives S 1 for free, and all other goods remain unallocated. Second, the mechanism gives back to the players some amount of revenue: in particular, a fixed small amount, 0 ⁇ e ⁇ 1. Each player i actually gets a fraction of e proportional to the revenue of his announced ⁇ (Formally, this enables us to avoid any ambiguity about relying on "weakly dominated strategies" m our solution concept.)
  • TV continues to be the true-valuation profile, again an original and fundamental object
  • C, / describes the collusion structure, that is the sets of collusive players and the set of independent players
  • u describes the generalized utilities, that is the utility of each agent, that is collusive set or independent player, for each possible outcome
  • RKj describes the relevant knowledge, that is the part of the independent players' knowledge exploited by the mechanism
  • GKj describes the general knowledge, that is all the information available to and exploitable by the independent players.
  • a collusion structure consists of a pair (C 1 I), where C is a partition of the players, and I is the set of all players i such that ⁇ t ⁇ £ C.
  • Wc refer to a player in I as independent, to a player not in / as collusive, to any C 6 C of cardinality > 1 as a collusive set.
  • agent we use the term agent to denote either an independent player or a collusive set. Since each player i, collusive or not, belongs to a single set in C, for uniformity of presentation we may denote by Ci the set to which i belongs.
  • Ci the set to which i belongs.
  • u is a generalized utility function, for a set of players with true-valuation profile TV and collusion structure (C, I), if u is a vector, indexed by the subsets in C 1 of functions from outcomes to real numbers satisfying the following two properties
  • Wc refer to uc as Cs collective utility function. If i 6 I, we more simply write U 1 rather than u ⁇ y .
  • P j is a positive integer whenever A 1 ⁇ 0.
  • RK i The relevant knowledge of t, denoted by RK i: is defined to be the outcome in EKi having maximum revenue; that is,
  • RKi argmax RBv( ⁇ ).
  • any outcome ⁇ in EKi is a way for i to "offer the goods to the other players.”
  • property 1 guarantees that ⁇ offers non-empty subsets of the goods only to players other than i.
  • property 3 guarantees T Q to i that, so long as the generalized utility function is minimally monotone, each offer in ⁇ will be (rationally) accepted.
  • ⁇ essentially is a guaranteed way for i to generate revenue REV( ⁇ ) by selling the goods to the other players, no matter what the minimally monotone generalized utility function u may be.
  • Property 2 is a technical requirement with two purposes. The fact that "P j is positive" is used in our mechanism.
  • RK x is well defined. (If EKi had infinite cardinality, the outcome with the maximum revenue may not exist.) Of course, if breaking ties proves necessary, RK 1 is chosen to be the lexicographically first outcome with maximum revenue.
  • GK 1 consists of a subset of V_, (the set of all possible valuation subprofiles foi the players in — ⁇ ) such that TV_ t € GK 1 That is, GK 1 is the set of all possible candidates, in «'s opinion, for the other players' true valuations Such GK 1 is genuine in the sense that one of its candidates is the "right one " - 1 In this example,
  • Gif t consists of a "partial" probability distribution over V_, 2
  • one first computes EK ⁇ , as in example 1, for each valuation subprofile V whose submterval does not coincide with [0, 0], and then computes EK 1 and RK 1 accordingly
  • GK 1 could more simply specify the true probabilistic distribution from which TVl 1 is drawn but considering it as a partial probability distribution is more general our solution concept implementation in ⁇ V ⁇ / Strategies. But whether a strategy is dominated depends on additional factors (e.g., collusive sets actually present and their collective utility functions) about which arr agent may not have any information beyond what it knows about itself. In particular, some independent players may not know that there axe collusive players, while other players (independent or not) may have different knowledge about the actual collusive sets and their collective utility functions. We thus insist that each eliminated strategy for an agent A be dominated no matter what knowledge A may have, that is, in any generalized context compatible with ,4's knowledge. This of course cannot but increase the number of surviving strategy vectors Yet, it will not be a problem as long as we can guarantee our benchmark for all such vectors.
  • ⁇ is distinguishably T- A ), 4 and over ⁇ '. dominated over ⁇ '
  • ⁇ ⁇ is the same for any *£ compatible with A
  • a set of players C may be an agent in a collusive context ff, but not in another context ff 1 ' .
  • C may consist only of independent players in 'if'. In this case, let 10 be the number of players in C. Then, in "if, ⁇ may consist of a single collective strategy. While in ⁇ f', each independent player may have 2 uneliminated strategies, so that ⁇ . consists of 1024 strategy subprofiles )
  • M( ⁇ A U ⁇ _ ⁇ ) and M( ⁇ ' A U ⁇ _ ⁇ ) are distributions over outcomes, and the inequality means that the two distributions are different •
  • a strategy vector ⁇ is a S 1 ZS / play of a generalized auction ( ⁇ 1 M) if a e H ⁇ x ⁇ ⁇ c t €i v ⁇ ec*,
  • %' general knowledge GK may include some Bayesian information according to which the probability that TV j (S) > lOOp j is extraordinarily high, although not equal to 1
  • i may be bettei off risking a possible answer NO from j and announce an outcome ⁇ ⁇ allocating S to j for a price lOOp j Additional Properties of M
  • loss of privacy may alter the way games are played, and mechanisms should be designed so as to preserve as much privacy as possible.
  • tax-free is a crucial property of an auction mechanism.
  • Each player i can always use an efficient algorithm to approximate RKi without altering any incentives.
  • Lot M be a DST auction mechanism whose outcome function /, as for the VCG, is very hard to compute. Then, the
  • M could provide further privacy if it first asked each player i to announce just the revenue of ⁇ * in Stage 1, and then asked only the star player to reveal both ⁇ * and S* .
  • this alternative way of proceeding would enable the star player to announce Q* depending on the revenues announced by the other players, and thus an independent player i may have incentives to underbid.
  • Tails 5 ends up Tails, with the difference between R* and the “second-highest revenue.” (Of course these alternatives reduce our revenue benchmark. However the "sum of social welfare and revenue” will not be affected.)
  • the context specifies the players' types, all possible outcomes, and the players' utilities in these outcomes.
  • the mechanism specifies the players' strategy spaces and how strategies determine outcomes. Let us explain these terms for 1 x m auctions, that is auctions of m non-transferable copies of a single good, where each player maybe desirous of getting multiple copies.
  • a valuation ⁇ for a good g with supply m is a function mapping each integer between 1 and m to a non-negative integer such that v(l) > ⁇ • ⁇ > vim), and a profile is a vector indexed by the players.
  • 1 x m CONTEXTS The context for 1 x m auctions with players 1, 2, . . . , n is defined as follows.
  • the true valuation profile, TV TVi, ⁇ ⁇ ⁇ , TV n , is a profile of valuations such that for each player i and copy k, TVi(k) specifies the maximum marginal value that player i would like to pay for having a A;-th copy of the good,
  • the collusion structure, (C, I), is a pair where C is a partition of the players into collusive sets, and / is the set of players i such that ⁇ z ⁇ € C.
  • ⁇ (A, P), where a.
  • A the allocation, is a profile of non-negative integers such that A ⁇ H + A n ⁇ m, where A 1 is the number of copies allocated to player i and m — (Ai + ⁇ ⁇ ⁇ + A n ) is the number of unallocated copies.
  • P the price profile, is a profile of real numbers. (P, represents the amount player i pays. If P 1 is negative, ⁇ 0 then -P 1 represents the amount i receives.)
  • RK RK 1 , ⁇ ⁇ ⁇ , RK n , is such that a.
  • Each RK 1 is a valuation profile; b.
  • RKl TV 1 for each i; and c.
  • RK*(k) is the maximum value known to i guaranteed to be ⁇ TV- (A)-
  • RKl is referred as his internal knowledge
  • RK ⁇ _ % is referred as his external knowledge
  • GK GK 1 , . . . , GK n
  • GK ⁇ includes all of i's knowledge about the auction, e.g., about TV and (C, /).
  • RK 1 is deducible from GK 1 , and we define RK 1 explicitly for simplicity.
  • B is an upperbound on the players 1 true valuations, that is, TV 1 (Zc) ⁇ B for any player i and copy k (notice that such an upperbound can be very loose, and is quite realistic, for example, the sum of the GDP of the countries where the players' come from) .
  • Every player i announces three things publicly and simultaneously with each other: (1) a subset of players C x which includes i himself; (2) a valuation, that is a number, V 1 ; and (3) a valuation subprofile K]_ Ct for players not in C 1 .
  • C 1 is the collusive set to which i belongs
  • V 1 is the true valuation of i
  • K_ % Ct is the relevant knowledge of z about players in -C 1 .
  • C 1 is the collusive set to which i belongs
  • V 1 is the true valuation of i
  • K_ % Ct is the relevant knowledge of z about players in -C 1 .
  • C 1 is the collusive set to which i belongs
  • V 1 is the true valuation of i
  • K_ % Ct is the relevant knowledge of z about players in -C 1 .
  • CM doesn't abort
  • CP 1 be the CM price for i, as set by CM
  • KPy max, ⁇ c ⁇ K)/ be the known price
  • bipy argmax t ⁇ C / / Ky be the best informed player about j'.
  • Every player i announces three things publicly and simultaneously with each other: (1) a subset of players Ci which includes i himself; (2) a valuation V 1 ; and (3) a valuation subprofile K % __ Ci for players not in C » .
  • C » is the collusive set to which i belongs
  • V is the true valuation of i
  • K_ ⁇ Ci is the relevant knowledge of i about players in -C 1 .
  • C as i's declared collusive set, to V 1 as player i's self-declared valuation, and to .FCLc 4 5 ⁇ *' s declared knowledge.
  • CM • Run CM with inputs ((V 1 , Ci), . . . , (V n ,C n )). If CM aborts, then ABORT, with the same fine as CM charges the players. o • Else (CM doesn't abort), identify a partition C of the players as CM does (while with a different name PC), and for- each player i, let C 1 be the subset of players in C such that i 6 C[. Let ki be the number of copies allocated to player i, and CPj(A;) be the CM price for i's k-th. copy, as set by CM, and let W be the set of ordered winning pairs (i, k) with V l [k) decreasing.
  • W UcW ⁇ ;, and for each player j let k j ' be the number of pairs in W' which includes j.
  • Every player i announces two things publicly and simultaneously with each other: (1) a valuation Vi ⁇ and (2) a collusive set Ci which includes i himself. 0
  • this step is replaced by receiving inputs ((Vi 1 Ci), . .. , (V n , C n )) from M.
  • V is a number instead of a vector of numbers. If this is the case, then in this mechanism, each Vj is interpreted as a vector of length 1.
  • CPi(k) is called the CM price for the fc-th copy for player i. Notice that Vi(k) is a winning bid, while V 3 (I) is not, therefore CP,(k) ⁇ V t (k).)

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Abstract

Dans les ventes aux enchères traditionnelles d'un ou de plusieurs biens, un joueur fait une offre (basée sur des informations pertinentes) selon sa propre évaluation des biens. Par contre, le système décrit ici traite de nouveaux mécanismes de vente aux enchères où les joueurs peuvent faire des offres fonction de connaissances extérieures c,-à-d. d'informations basées sur les évaluations d'autres joueurs. Ces nouveaux mécanismes incitent les joueurs à faire des offres basées sur leurs connaissances extérieures et à utiliser lesdites offres pour décider de la manière de vendre les biens.
PCT/US2009/003584 2008-06-17 2009-06-16 Mécanismes de vente aux enchères utilisant la connaissance extérieure WO2009154724A1 (fr)

Applications Claiming Priority (14)

Application Number Priority Date Filing Date Title
US13226208P 2008-06-17 2008-06-17
US61/132,262 2008-06-17
US10391108P 2008-10-08 2008-10-08
US61/103,911 2008-10-08
US20056608P 2008-12-01 2008-12-01
US61/200,566 2008-12-01
US20187308P 2008-12-16 2008-12-16
US61/201,873 2008-12-16
US20308608P 2008-12-17 2008-12-17
US61/203,086 2008-12-17
US15780709P 2009-03-05 2009-03-05
US61/157,807 2009-03-05
US21258509P 2009-04-13 2009-04-13
US61/212,585 2009-04-13

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Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20020052828A1 (en) * 2000-05-18 2002-05-02 Ausubel Lawrence M. System and method for an efficient dynamic multi-unit auction
US20030041008A1 (en) * 2001-08-22 2003-02-27 William Grey System and method for facilitating transactions among disparate entities

Patent Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20020052828A1 (en) * 2000-05-18 2002-05-02 Ausubel Lawrence M. System and method for an efficient dynamic multi-unit auction
US20030041008A1 (en) * 2001-08-22 2003-02-27 William Grey System and method for facilitating transactions among disparate entities

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