Auctions for Digital Goods Review

Transcription

1 Auctions for Digital Goods Review Rattapon Limprasittiporn Computer Science Department University of California Davis Abstract The emergence of the Internet brings a large change in the economic trend; one such change is the negligible cost of duplicating digital goods An auction is one of the dynamic selling mechanisms that yields a good result dealing with the case when the consumer utility values are dramatically vary such that it is infeasible to do the market research to determine the best fixed price In such scenario, an auction which performs probably well under any market conditions is necessary We review the state-of-the-art auction model from five papers related to auctions for digital goods and categorized them in to sections Then we illustrate the model evolution and compare these models in order to better understand the auction technique proposed in today s market 1 Introduction One certain kind of the economic efficiency is to transfer resources to those who can make the most use of these resources Auctions attempt to achieve this by giving the bidders the opportunity to declare how much they are willing to pay Of course, from the seller s perspective, what matters is not the relative usefulness of the item for sale, but rather the revenue generated by the sale In general, it may not be possible to simultaneously maximize efficiency and revenue, but in many natural situations, the seller can extract reasonable revenue while maximizing efficiency However, a combination of recent economic and computational trends, such the negligible cost of duplicating digital goods and, most importantly, the emergence of the Internet as one of the most important arenas for resource sharing between parties with diverse and selfish interests, has created a number of new and interesting dynamic pricing problems including the profit optimization for the seller in an auctions We consider the design of profit-maximizing auctions for digital, or information, goods such as electronic books, software, and digital copies of music or movies 2 Preliminaries and Notion We consider single-round, sealed-bid auctions for a set of identical items available in unlimited supply A single-round sealed-bid auction is one where each bidder submits a bid, representing the maximum amount they are willing to pay for an item We denote by b the vector of all submitted bids The i th component of b is b i, the bid submitted by bidder i We denote by n the number of bidders Given the vector b, the auctioneer computes a result allocation If bidder i receives the item under the price p i < b i, we say that bidder i wins; otherwise, bidder i loses, or is rejected The price p i is what bidder i pays the auctioneer The profit of the auction is the sum of the revenue that the auctioneer obtains from the bidders Each bidder has a private utility value, representing the true maximum price they are willing to pay for an item We denote by u i the utility value of bidder i Each bidder bids so as to maximize their profit, u i x i p i Bidders bid with full knowledge of the auctioneer's strategy and we assume that bidders do not collude We say that an auction is truthful if, for each bidder i and any choice of bid values for all other bidders, the profit of bidder i is maximized by bidding their utility value, ie, by setting b i = u i Truthful auctions encourage rational bidders to bid at utility value 21 Optimal Single Price Omniscient Auctions A key question is how to evaluate the performance of auctions with respect to the goal of profit optimization An objective would be to design auctions that, on all bid vectors b, achieve a profit close to that of the optimal single-price omniscient auction F The optimal single price omniscient auction F is defined as follows: Let b be a bid vector, and let v i be the i th largest bid in b Auction F on input b determines the value k 2 such that (k)(v k ) is maximized All bidders with b i v k win at price v k ; all remaining bidders lose The profit of F on input b is thus

2 F(b) = max iv 2 i n The value of k must be greater than or equal to 2 If we allow k to be 1, then it is impossible to be competitive with F; consider the case when the highest bid h is much greater than all other bids combined, it is impossible to truthfully extract a constant fraction of h in profit from the high bidder 22 Competitiveness We now formalize the notion of a competitive auction Let A be a truthful auction We say that A is β- competitive against F if for all bid vectors b, the expected profit of A on b satisfies E[A(b)] i F (b) β To gain insight into competitive auctions, consider the k-item Vickrey auction V k This auction is a singleprice auction that sells k items to the k highest bidders at the price equal to the k+1 st highest bid When an unlimited supply of identical items is available, taking k = f(n) yields a truthful auction for any function f with 1 f(n) < n that depends on n only (and not on the bid values) Unfortunately, the resulting auction V f(n) is not competitive For example, consider an input b that has k bids at some very high value h and n k bids at 1 In this case V f(n) (b) = k, whereas F(b) (h)(k) One can attempt to fix this problem by choosing for each bid vector b the value of k that maximizes V k (b) However, the resulting auction is not truthful Consider three bidders with utilities 10, 30, and 40 If the bidders bid their utilities, then optimal choice of k is 1, in which case only the third bidder wins and pays 30 However, if the second bidder changes their bid to 11, then the optimal value of k is 2, the second and the third bidders win, and pay 10 each Therefore bidding utility is not a dominant strategy for the second bidder These examples show that competitive auctions must depend on the bid values in a way that keeps the auction truthful 23 Bid-Independent Auctions This concept is introduced by [1] and is used widely The basic idea is that when we compute the result of bidder i, we will not take into account of his bid value It is shown in [1] that any deterministic bidindependent auction is truthful and any truthful deterministic auction is bid-independent As a result, an auction is truthful if and only if it is equivalent to a bid-independent auction 23 Randomized Auctions We say the auction is randomized if the procedure by which the auctioneer computes the allocation and prices is randomized Note that if the auction is randomized, the profit of the auction, the output prices, and the allocation are random variables Otherwise, the auction is deterministic It is shown in [1] that no truthful deterministic auction is competitive; hence, the studies on the auction for digital goods concentrate on the randomized auction; otherwise, the constraints are relaxed 3 Competitive Auctions The work in [1] presented two auction models that achieve both the truthfulness and competitiveness: DSOT and SCS 31 Dual-Price Sampling Optimal Threshold The idea underlying the first competitive auction introduced in this paper is to use a random sample to get an estimate of the optimal fixed price and then use this price as a threshold for the bids that were not sampled; each bid below the threshold is rejected, and each bid at or above the threshold wins at the threshold price The bids are partitioned into two sets uniformly at random, compute the optimal fixed price for each set, and then use that threshold for the other set We call the resulting family of auctions the dual-price sampling optimal threshold auctions or DSOT r where r is a parameter used by the auction The paper introduced the notion of opt r (b) to be the optimal fixed price for the bid vector b that sells at least r items, ie, opt r (b) is the value v k such that k r and kv k = max iv r i n The algorithm partitions the set of bids uniformly at random into two sets This consists of putting each bid into one set or the other independently with probability 1/2 The full description is described below 1 Partition bids b uniformly at random into two sets, resulting in bid vectors b and b 2 Let p = opt r (b ) and p = opt r (b ), the optimal fixed price thresholds that sell at least r items for b and b, respectively i

3 3 Use p as a threshold for all bids in b (ie, all bids in b of value below p are rejected; all remaining bids win at price p ) 4 Use p as a threshold for all bids in b When r = 1, it refers to this auction simply as DSOT Intuitively, the parameter r enables the auction to avoid outcomes with a small number of winners By this algorithm, DSOT is a dual-priced auction If it is important to have a single-price auction, the paper suggests one can simply skip the 4 th step and reject all the bids in b, at the cost of half the expected profit Since DSOT computes the result of bidder i without taking into account of his bid value, it can be implemented as a bid-independent auction Thus, DSOT is truthful The paper also proved that there is a constant β such that DSOT is β-competitive This upper bound analysis yields result that DSOT is not better than 4- competitive 32 Sampling Cost-Sharing This paper also presents another competitive truthful auction that is simple, easy to analyze, and achieves a good competitive ratio The goal of this technique is, given bids b and cost C, to find a subset of the bidders to share the cost C More precisely the cost-sharing and mechanism is defined as follows Given bids b, the auctioneer finds the largest k such that the highest k bidders can equally share the cost C, and then charges each C/k CostShare C works as follows It maintains a current set X, initialized to the set of all bids At each iteration, bids of value below C/ X are deleted from X When X stops changing, CostShare C declares X to be the set of winning bids and C/ X to be the sale price for every winner The Sampling Cost-Sharing (SCS) auction is described below 1 Partition bids b uniformly at random into two sets, resulting in bid vectors b and b 2 Compute F = F(b ) and F = F(b ), the optimal fixed price profits for b and b, respectively 3 Compute the auction results by running CostShare F on b and CostShare F on b SCS can be implemented in a bid-independent fashion: For each bid, simulate the cost-sharing method assuming that bid is infinite to determine the output of the bid-independent function Thus SCS is truthful The profit of an auction is min(f, F ) One of the CostShare F on b and CostShare F on b will reject all bids The paper proved that SCS is 4-competitive, and this bound is tight 33 Bounded Supply Up to this point, the paper has studied the unlimited supply case, motivated by the digital goods market In addition, it also considers the case where the number of items available for sale is bounded This case is typical for physical goods markets It denotes the number of items available by k As ate before, the seller wishes to maximize profit and is not required to sell all the items The definition of truthful and competitive auctions, which we stated for the unlimited supply case, also applies to the bounded supply case To reduce the bounded supply case to the unlimited supply case, it just simply ignores (rejects) all but the highest k bidders This reduction trivially extends all results presented in the paper to the bounded supply case 4 Envy-Free Auctions for Digital Goods This paper introduces a new concept of envy-free auction After the auction is run, no bidder would be happier with the outcome of another bidder For unlimited supply auctions, this means that there is a single sale price and goods are allocated to all bidders willing to pay this price The main result in this paper is to show that no constant-competitive truthful auction is envy-free Two relaxations of these requirements are considered, allowing the auction to be untruthful with vanishingly small probability, and allowing the auction to give nonenvy-free outcomes with vanishingly small probability Under both of these relaxations we get competitive auctions This paper introduces the notion of β(m)- competitive where the auctioneer sells at least m items It proves that there is no truthful envy-free auction with β(m)-competitive where n log β ( m) O( m ) loglog n However, the paper also proves that there exists a truthful envy-free auction that is Θ(log n)-competitive

4 Nonetheless, both truthfulness and envy-free outcomes are important As a result, the paper presents two auctions that come close to being competitive, truthful, and envy-free at the same time The first auction is truthful, but is only envy-free with high probability; the second auction is envy-free, but only truthful with high probability 41 Almost Envy-Free In this section, the condition that the auction outcome is always envy-free is relaxed and instead it considers auctions that are truthful but only envy-free with high probability One such auction is the Consensus Revenue Estimate (CORE) auction from [5] We review the mass-market version of the CORE auction and discuss its properties The CORE auction combines two general ideas that have proven to be successful for designing competitive mechanisms The first is that of a profit extractor A profit extractor is a truthful parameterized mechanism that, given a target profit, extracts that profit from the bidders if it is possible For basic auctions the maximum extractable profit is the same as the profit of the optimal omniscient auction F The profit extraction mechanism for basic auctions is the following cost sharing mechanism To determine CostShare R of the given bids b, find the largest k such that the highest k bidders can equally share the cost R Charge each of these bidders R/k If no k exists, no bidders win This mechanism is truthful, envy-free, and yields profit R if R F(b) The second technique for the design of competitive mechanisms is that of using a bid-independent consensus estimate The truthfulness of CostShare R implies that it is implemented by some bid-independent function, cs R That is, the bid-independent auction defined by f(b i ) = cs R (b i ) is exactly CostShare R Consider the auction, A, parameterized by function r( ) that is defined by bid-independent function f(b i ) = cs r( ) ( b i ) i b Note that if r is consensus, ie, r(b i ) = R for all i, then A is identically CostShare R Recall that CostShare R gives profit R in the case that R F The consensus estimate technique is to construct an r( ) that gives an estimate of F and is constant on b i for all i For the case that F(b i ) is a constant fraction of F(b), ie, 1 F ( b) F ( b i ) < F ( b), ρ it is possible to pick an r( ) from a distribution of functions that are good estimates of F such that with high probability that r is a consensus Parameterized by constant c with c > ρ, we choose r( ) as follow For u be uniformly distributed on [0, 1], define r(b) is F(b) rounded down to nearest c i+u for integer i The auction is envy-free if r is consensus The paper proves that this probability is 1 log C ρ 42 Almost Truthful Another approach to deal with the non-existence of truthful competitive auctions that always have envyfree outcomes is to relax the requirement that the auction always be truthful Next we define a probabilistically truthful mechanism Let m be the number of winners in the optimal auction on a given set of bids We will be looking for a mechanism with good probabilistically truthful properties in terms of m Here are some definitions defined by the paper An auction is truthful with probability 1 ε if the probability that any bidder can benefit from an untruthful bid is at most ε An auction is truthful with high probability if ε 0 as m, where m is the number of winners in F The paper modifies CORE a little bit as follow 1 Pick u uniformly at random from [0, 1] 2 Let function r( ) be F( ) rounded to nearest c i+u for integer i 3 Run CostShare r(b) on b The paper also proves that this auction is truthful with probability m 1 logc = 1 Θ(1/ m) m 1 And the expected revenue from this auction is F ( b) 1 (1 ) ln c c 5 Online Auctions This paper in [3] presents the concept of online auction in which the bidders know their result immediately The motivation of this study is that most digital goods, for instance, books, music, or software, are sold continuously, without any point at which sales end Further more, since digital goods can be delivered

5 immediately, buyers would presumably expect a fairly rapid, if not immediate, response to a bid for digital goods Even in the case of a digital broadcast, where the delivery is not immediate, buyers want to know quickly whether or not their offers have been accepted In auctioning a pay-per-view broadcast, for instance, we might want to allow bids to come in mere minutes before the broadcast, or maybe even after the broadcast has begun It would be clearly unacceptable to delay all decisions about who receives the broadcast until such a point As a result, it is important to design auctions which work in as online model The auctions must be truthful which decide whether to satisfy a bid and at what price before the next bid arrives The goal of the model is to achieve the good revenue relative to the optimal single price omniscient auction F The model considers auctions for goods available in unlimited supply Each bidder can win at most one item of the good Let n be the number of bidders In the truthful auction, each bidder has his private utility value v i which equal to his bid By normalizing the bids (divide all bids by the smallest bid), we can assume that each v i is a real number in the interval [1, h] Notice that h is the ratio between the highest and lowest bid values The model of an online auction receives bids from the n bidders one by one in a sequence b = [b 1, b 2,, b n ] The auctioneer, when presented with the bid b i of bidder i, determines whether to sell the good to bidder i and if so, at what price This determination must be made before any further bids are presented The price p i charged to bidder i is never more than b i An online auction A can be viewed as a collection of n pricing functions s 1, s 2,, s n, where s i (b 1,, b i ) is a function of the bids received up to step i The interpretation of these functions is that s i sets a selling price for bidder i If s i b i, bidder i wins the good and pay s i ; otherwise, bidder i does not win the good and pay nothing The total revenue of an auction is the sum of the payments received An online auction is truthful is the pricing function for bidder i depends only on the previous bids, b 1,, b i 1, and in the other word, for any two choices for the i th bid, b i and b i, the expected price s i is the same This definition of bid-independent is more general than one given in the previous section Since no auction does well compare to F when there is a single high bidder, it is natural to restrict the attention to the bid sequence for which F αh, for some α 1 Another notion introduced in this study is the bid buckets The bid sequence of range [1, h] is partitioned into l = log h + 1 sub-intervals, I 0,, I l 1, where I k = [2 k, 2 k+1 ) Given a bid sequence b, we define the k th bucket of b to be the collection of bids that fall in the k th interval The weight of the k th bucket is the sum of the bids in the bucket The paper also proves that there is no deterministic auction that performs well in the online auction The proof is different than that in the offline auction presented in [1] 51 Simple Randomized Auction The first idea is an extremely simple truthful randomized online auction that achieves an optimal competitive ratio of O(log h) relative to the total revenue This auction model assumes that we have some way to know the upper bound of the value of h on the range of bids This auction calculates the value s i of bidder i by picking a random k {0,, log h }, and set s i = 2 k Since this auction does not take b i into account, it is truthful The paper presents a proof that this simple online auction is Θ(log h)-competitive relative to the total revenue and relative to the optimal fixed price revenue 52 Weighted Buckets Auction The paper presents an online auction W d which uses the bid buckets previously defined as well as a parameter d 1, which will be tune later The idea is that for the i th bidder, the bid buckets from b 1,, b i 1 are constructed In this context, b i is defined to be the bid values from b 1,, b i 1 Then one bucket is picked at random, with the probabilities weighted according to the weight of the buckets raise to the power d The chosen bucket is essentially the guess for the next bid, and the selling price is set to be the lower limit of that bucket In the other word, d k ( wk ( b i )) Pr[ si = 2 ] = l 1 d r = 0 ( wr ( b i )) Since W d is bid-independent, this weighted buckets auction is truthful The paper illustrates that, by picking d to be log log h and restricting the bid sequences such that F(b) 4h, the weighted buckets auction W d is

6 O(exp log log h) -competitive relative to the optimal fixed price revenue 53 Online Learning in Online Auction The paper in [4] presents some modification to the technique described in section 52 by applying the online learning technique to yield the better decision The key insight connecting the online auction problem to online learning is that setting a single fixed price can be thought of as following the advice of a single expert who predicts that fixed price for every bidder Performing well relative to the optimal fixed price is then equivalent to performing well relative to the best of these experts The model uses the variant of Littlestone and Warmuth s weighted majority (WM) algorithm [6] From the previous section, we have l buckets The selling price is chosen from the lower bound of one random bucket When applying the online learning technique to this context, the candidate prices is chosen from X = {x 1,, x l } Each of this can be viewed as a candidate fixed price corresponding to a set of experts Let r k (b) be the revenue obtained by setting the fixed price x k for the bid sequence b Given a parameter α (0, 1], define weights w ( i) α k r ( v 1,, v ) / h = (1 + ) k i Clearly, the weights can be easily maintained using a multiplicative update Then, for bidder i, the auction chooses s i X with probability wk ( i 1) Pr[ si = xk ] = l j = 1 w j ( i 1) The paper also proves that, restricting the bid 18h 4 sequence such that F ( b) (lnln ln( )) ε 2 h +, the ε auction Weight Majority (WM) is (1+ε)-competitive relative to the optimal fixed price revenue 6 Comparison This section compares and contrast several techniques reviewed in this paper We mention the constraint and situation in which it is feasible to apply the each auction model 61 Offline Model Comparisons The competitive auction is compared with the envyfree technique The CORE technique is suitable for mass-market sale of goods Since the envy-free property is very important to the practical auctions, the CORE technique is preferable to DSPT or SCS Regarding to the revenue, the CORE technique gain some advantages from using the bigger size of bid input In DSOT or SCS, the bids are partition into half, thus lose some profit in computing F 62 Online Model Comparisons For any moderately large auction, the performance guarantee of the weighted majority auction mechanism is dramatically better than that of the weighted buckets auction As a comparison, Bar-Yossef et al show that their weighted buckets auction is O(exp log log h) - competitive [3] However, in that case, the competitive ratio is achieved for the bid sequences with F(b) 4h It is proved in [4] that weighted majority auction fails on such small bid sequences, and indeed, the paper provide a fairly tight lower bound on the sequences for which WM succeeds We can also combine the weight buckets and weighted majority by simply assigning probability ½ to each to achieve performance which is within a factor of two of the better auction 63 Online vs Offline Auctions Comparison The offline auction seems to have some advantage from knowing all the bids before the auction run This implies that the online auction may perform very poor in the very beginning of the auction However, the online auctions have some advantage in some aspect First, it appropriates to some type of digital goods where there is no explicit end time; most digital goods are sold continuously Furthermore, it is impractical for the pay-per-view in which bidders can submit their bids even after the movies start Besides, the online auctions gain benefit from the envy-free problem avoidance Since the price can have different prices at different time, the online auction may be attractive to the auctioneers 7 Open Problem There are some open problems in each of the paper we review The envy-free problem in [1] is addressed by the work in [2] However, in [2], it may also desirable to have one more property in which it prevents the

7 bidders from collude This can happen in the practical and thus can be a problem The online auction may have some trouble as well Since at the beginning of the auction the auctioneer will not have enough information to compute the result, this may require some statistic mechanism to deal with this problem The online auction may have the envy-free problem as well if two bidders submit their bids at almost the same time This may be fixed by running the auction after some short period of time instead of one by one bid References [1] AV Goldberg, JD Hartline, A Karlin, M Saks, AWright Competitive Auctions and Digital Goods, Games and Economic Behavior, 2002 [2] A V Goldberg, JD Hartline Envy-Free Auctions for Digital Goods, 2003 [3] Ziv Bar-Yossef, Kirsten Hildrum, Felix Wu Incentive-Compatible Online Auctions for Digital Goods, in Proceedings of the 13 th Annual ACM-SIAM Symposium on Discrete Algorithm (SODA), pages , 2002 [4] Avrim Blum, Vijay Kumar, Atri Ruda, Felix Wu Online Learning in Online Auctions, 2003 [5] A V Goldberg, JD Hartline Competitiveness via Consensus In Proc 14 th Symp on Discrete Alg ACM/SIAM, 2003 [6] Nick Littlestone and Manfred K Warmuth The weighted majority algorithm Information and computation, 108: , 1994