## Optimization in the Private Value Model: Competitive Analysis Applied to Auction Design (2003)

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Citations: | 4 - 0 self |

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@TECHREPORT{Hartline03optimizationin,

author = {Jason D. Hartline},

title = {Optimization in the Private Value Model: Competitive Analysis Applied to Auction Design},

institution = {},

year = {2003}

}

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### Abstract

We consider the study of a class of optimization problems with applications towards profit maximization. One feature of the classical treatment of optimization problems is that the space over which the optimization is being performed, i.e., the input description of the problem, is assumed to be publicly known to the optimizer. This assumption does not always accurately represent the situation in practical applications. Recently, with the advent of the Internet as one of the most important arenas for resource sharing between parties with diverse and selfish interests, this distinction has become more readily apparent. The inputs to many optimizations being performed are not publicly known in advance. Instead they must be solicited from companies, computerized agents, individuals, etc. that may act selfishly to promote their own self-interests. In particular, they may lie about their values or may not adhere to specified protocols if it benefits them. An auction is

### Citations

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Citation Context ..., we say that B j is c-good if #cj# # n # (B j ) # j - #cj#. Otherwise, B j is c-bad. We prove that B j is likely to be c-good using the following version of the Cherno# bound: Theorem 4.24 (see e.g. =-=[36]-=-, page 70) Let X i , 1 # i # n be independent Bernoulli trials such that for all i, Pr[X i = 1] = 1/2. Then for X = P 1#i#n X i , and 0s# 1, Pr h Xs(1 - #) n 2 is- # 2 n 4 . Since the partition of b i... |

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Citation Context ...ecause we can always use lexicographic tie-breaking to achieve it. Our competitive multi-item auctions are randomized. We use the following lemma, which is a variation of the Chernoff bound (see e.g. =-=[12, 36]-=-), as the main tool in our analysis. Lemma 9.1 Consider a set A and its subset B ⊂ A. Suppose we pick an integer k such 91s92 that 0 < k < |A| and a random subset (sample) S ⊂ A of size k. Then for 0 ... |

729 |
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Citation Context ...and approximation of the common welfare maximization objective for the combinatorial auction problem (See, e.g., [40, 32, 2]). The fundamental observation made is that the Vickrey-Clarke-Groves (VCG) =-=[51, 13, 27]-=- mechanism that implements common welfare maximization, when applied to the combinatorial auction problem, is required to solve an NP-hard optimization problem. Further, use of an approximately optima... |

648 | Worst-Case Equilibria
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Citation Context ...as first proposed by Nisan and Ronen [39]. Other notable work in the intersection of Computer Science, Game Theory, and Economics includes the study of the price of anarchy [31] and market equilibria =-=[14]-=-. The price of anarchy considers the di#erence between the social welfare of an optimal allocation of resources and the social welfare of a allocation obtained when a Nash equilibrium is found by self... |

599 |
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Citation Context ...and approximation of the common welfare maximization objective for the combinatorial auction problem (See, e.g., [40, 32, 2]). The fundamental observation made is that the Vickrey-Clarke-Groves (VCG) =-=[51, 13, 27]-=- mechanism that implements common welfare maximization, when applied to the combinatorial auction problem, is required to solve an NP-hard optimization problem. Further, use of an approximately optima... |

568 | Algorithmic mechanism design
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Citation Context ...pricing problem both with the objective of budget-balanced, i.e., zero profit and zero deficit, and the objective of common welfare maximization. Finally, looking beyond these traditional objectives, =-=[39, 3]-=- consider the private value scheduling problem where the objective is to minimize the makespan (the completion time of the last job). One interesting observation about the objective of common welfare ... |

524 | How bad is selfish routing
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Citation Context ...ion obtained when a Nash equilibrium is found by selfish agents competing for the resources. Along these lines, Roughgardin et al. have studied the problem of routing tra#c in a communication network =-=[46, 45]-=-. The market equilibrium question looks at taking a market with agents and goods and computing prices for the goods such that they can be reallocated among the agents in an optimal way such that there... |

394 |
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Citation Context ...ve pricing is unlikely to be optimal for the seller. We investigate adaptive pricing via auctions. In the scarce supply case, multiple item auctions have been studied by Shapley and Shubik [49]. (See =-=[44]-=- for a survey of the area.) Results for the scarce case, however, do not directly apply to the unlimited supply case. Consider the case where each item for sale is unique -- for example the real estat... |

372 | Auction Theory: A Guide to the Literature
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Citation Context ...of the double auction problem in which all sellers are willing to sell their items for price zero. 1.2 Related Work There is an extensive amount of Economics literature on auctions. For a survey, see =-=[30]-=-. Unlike the approach taken in this thesis, the standard approach to profit maximization is to look for the Bayesian optimal mechanism. This approach assumes that the private values come from some kno... |

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Citation Context ...ng enough for one movie to be seen. The airline wants to price movies to maximize its profit. Another generalization of the basic auction problem is the multicast pricing problem of Feigenbaum et al. =-=[18]-=-. In this problem a content provider is attempting to sell and distribute a digital good via multicast over a computer network such as the Internet. It is assumed that the consumers are located at var... |

190 | Truthful mechanisms for one-parameter agents
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- 2001
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Citation Context ...pricing problem both with the objective of budget-balanced, i.e., zero profit and zero deficit, and the objective of common welfare maximization. Finally, looking beyond these traditional objectives, =-=[39, 3]-=- consider the private value scheduling problem where the objective is to minimize the makespan (the completion time of the last job). One interesting observation about the objective of common welfare ... |

189 | Computationally feasible VCG mechanisms
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- 2000
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Citation Context ...In the first direction, the most notable example is work looking at the intractability and approximation of the common welfare maximization objective for the combinatorial auction problem (See, e.g., =-=[40, 32, 2]-=-). The fundamental observation made is that the Vickrey-Clarke-Groves (VCG) [51, 13, 27] mechanism that implements common welfare maximization, when applied to the combinatorial auction problem, is re... |

189 | Truth revelation in approximately efficient combinatorial auctions
- Lehmann, O’Callaghan, et al.
(Show Context)
Citation Context ...In the first direction, the most notable example is work looking at the intractability and approximation of the common welfare maximization objective for the combinatorial auction problem (See, e.g., =-=[40, 32, 2]-=-). The fundamental observation made is that the Vickrey-Clarke-Groves (VCG) [51, 13, 27] mechanism that implements common welfare maximization, when applied to the combinatorial auction problem, is re... |

165 |
The assignment game I: the core
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Citation Context ... non-adaptive pricing is unlikely to be optimal for the seller. We investigate adaptive pricing via auctions. In the scarce supply case, multiple item auctions have been studied by Shapley and Shubik =-=[49]-=-. (See [44] for a survey of the area.) Results for the scarce case, however, do not directly apply to the unlimited supply case. Consider the case where each item for sale is unique -- for example the... |

160 | S.: Strategyproof sharing of submodular costs: budget balance versus efficiency. Economic Theory 18
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- 2001
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Citation Context ...neer. We assume that 0 # p i # b i for all winning bidders and that p i = 0 for all losing bidders (these are the standard assumptions of no positive transfers and voluntary participation. See, e.g., =-=[37]-=-). 3. The profit of the auction (or auctioneer) is A(b) = P i p i . 15 An auction is deterministic if the allocation and prices are a deterministic function of the bid vector. An auction is randomized... |

129 | Competitive Auctions and Digital Goods
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- 2001
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Citation Context ...yon, Mike Saks, and Andrew Wright. Many of the results presented have been published with subsets of the above as coauthors [26, 22, 20, 15, 23, 24] and some are in manuscript form as of this writing =-=[25, 8]-=-. One main contribution of this work is the introduction of competitive analysis to economic mechanisms [26, 20, 24]. This type of analysis allows for design and analysis of profit maximizing mechanis... |

127 | The Simple Economics of Optimal Auctions
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- 1989
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Citation Context ...ty distribution. The goal then is to design the mechanism that obtains the highest expected profit on said input distribution. For i.i.d. prior distribution's the Bayesian optimal auction is given in =-=[38, 11, 42]-=-. We will discuss these results more in Chapter 2. For arbitrarily correlated prior distributions, Ronen considers the computational issues of implementing the Bayesian optimal auction for selling a s... |

116 |
An Exploration in the Theory of Optimal Income Taxation
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Citation Context ...uctions. Related formulations to the one we give here have appeared in numerous places in recent literature (e.g., [3, 48, 25, 32]). To the best of our knowledge, the earliest dates back to the 1970s =-=[35]-=-. Definition 2.6 Let b-i denote the vector of bids b with b i removed, i.e., b-i = (b 1 , . . . , b i-1 , ?, b i+1 , . . . , b n ). We call such a vector masked. Definition 2.7 (Bid-independent Auctio... |

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Citation Context ...fare maximization, the mechanism profit is always non-positive. Archer and Tardos have studied the private value shortest path problem where the goal is to design a mechanism to buy the cheapest path =-=[4]-=-. They note that for 9 this problem the VCG mechanism, the mechanism that maximizes the common welfare, may end up spending a lot more than the cost of the cheapest path. It turns out that all mechani... |

109 | E.: An approximate truthful mechanism for combinatorial auctions with single parameter agents - Archer, Papadimitriou, et al. - 2003 |

89 | Competitive Generalized Auctions
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Citation Context ...i Azar, Kostub Deshmukh, Amos Fiat, Andrew Goldberg, Anna Karlin, Claire Kenyon, Mike Saks, and Andrew Wright. Many of the results presented have been published with subsets of the above as coauthors =-=[26, 22, 20, 15, 23, 24]-=- and some are in manuscript form as of this writing [25, 8]. One main contribution of this work is the introduction of competitive analysis to economic mechanisms [26, 20, 24]. This type of analysis a... |

86 | Online interval scheduling
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Citation Context ...lues, h, is not known, but the lower bound is of 1 is, Yossi Azar has observed that we can still get close to log h-competitive [7]. This result is obtained by using the classify and select technique =-=[33]-=- (and a good description can be found in [6]). 32 Profit Maximization via k-Vickrey Now we consider obtaining a profit maximizing auction via the k-Vickrey mechanism. Again, for any fixed k that is no... |

77 |
Competitive Sealed Tenders
- “Counterspeculation
(Show Context)
Citation Context ...and approximation of the common welfare maximization objective for the combinatorial auction problem (See, e.g., [40, 32, 2]). The fundamental observation made is that the Vickrey-Clarke-Groves (VCG) =-=[51, 13, 27]-=- mechanism that implements common welfare maximization, when applied to the combinatorial auction problem, is required to solve an NP-hard optimization problem. Further, use of an approximately optima... |

71 | The prize collecting steiner tree problem: theory and practice
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- 2000
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Citation Context ...uction mechanism to maximize the auctioneer's profit. We assume that each buyer only desires one item and that each seller 1 this is a private value version of a prize collecting Steiner tree problem =-=[28]-=-. The prizes at nodes in the Steiner tree problem correspond to private values of bidders in the multicast problem. 7 only has one item to sell. Naturally, the basic auction is a special case of the d... |

60 | Online learning in online auctions
- Blum, Kumar, et al.
- 2003
(Show Context)
Citation Context ... have developed in this thesis was originally extended to this problem by Bar-Yossef et al. [9]. An improved solution to the online auction problem, based on online learning, was given by Blum et al. =-=[10]-=-. This improvement also includes analysis of the online posted price problem where the mechanism must o#er each consumer a price and the consumer may choose to accept or reject the o#ered price. The m... |

56 |
A measure of the asymptotic eciency for tests of a hypothesis based on the sum of observations
- Cherno
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(Show Context)
Citation Context ...because we can always use lexicographic tie-breaking to achieve it. Our competitive multi-item auctions are randomized. We use the following lemma, which is a variation of the Cherno# bound (see e.g. =-=[12, 36]-=-), as the main tool in our analysis. Lemma 9.1 Consider a set A and its subset B # A. Suppose we pick an integer k such 92 that 0s|A| and a random subset (sample) S # A of size k. Then for 0s# 1 we ha... |

52 | Optimal Pricing Mechanisms with Unknown Demand
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- 2003
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Citation Context ...optimal auction was extended to the multicast pricing problem that we discuss in Chapter 10. Recently, Segal has considered the uninformed auction problem in a setting similar to the Bayesian setting =-=[48]-=-. He shows that auction mechanisms related to the ones presented later in this thesis are asymptotically optimal when the bidders valuations are drawn independently from an unknown or partially unknow... |

48 |
Collusive Behavior in Noncooperative Epsilon-Equilibria of Oligopolies with Long but Finite Lives
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- 1980
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Citation Context ...ds. We will be looking for a mechanism with good probabilistically truthful properties in terms of m. For other solution concepts related to approximate or probabilistic truthfulness, see for example =-=[41, 47]-=-. 68 Definition 6.2 [2] An auction is truthful with probability 1 - # if the probability that any bidder can benefit from an untruthful bid is at most #. An auction is truthful with high probability i... |

45 | Incentive-compatible online auctions for digital goods
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- 2002
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Citation Context ...he auction must compute a price for them before seeing the next bidder's bid. The analysis framework that we have developed in this thesis was originally extended to this problem by Bar-Yossef et al. =-=[9]-=-. An improved solution to the online auction problem, based on online learning, was given by Blum et al. [10]. This improvement also includes analysis of the online posted price problem where the mech... |

44 |
The value of knowing a demand curve: Bounds on regret for on-line posted-price auctions
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Citation Context ...price. The mechanism does not learn anything about the bidder's value except whether it was above or below the price the mechanism o#ered. 8 This problem was further studied by Klienberg and Leighton =-=[29]-=-; however, additive loss is considered instead of a multiplicative competitive ratio. Finally, Awerbuch et al. consider a general approach for adding incentive properties to online algorithms. They sh... |

42 |
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Citation Context ...rtest path share this problem [4, 17]. In contrast, for matriod optimization problems, e.g., minimum spanning tree, the VCG mechanism does find a solution at close to the cost of the optimal solution =-=[50]-=-. We note that using the VCG mechanism for the shortest path and minimum spanning tree problems was first proposed by Nisan and Ronen [39]. Other notable work in the intersection of Computer Science, ... |

39 | Envy-free auctions for digital goods - Goldberg, Hartline - 2003 |

35 |
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- Ronen
- 2001
(Show Context)
Citation Context ...al auction for selling a single item. Ronen shows that it is computational infeasible (i.e., NP-hard) to implement the optimal auction but instead an approximately optimal solution can be implemented =-=[43]-=-. In [34], the Bayesian approach to the optimal auction was extended to the multicast pricing problem that we discuss in Chapter 10. Recently, Segal has considered the uninformed auction problem in a ... |

33 | Reducing truthtelling online mechanisms to online optimization
- Awerbuch, Azar, et al.
- 2003
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Citation Context ...proach for adding incentive properties to online algorithms. They show how to convert an online algorithm into a truthful online mechanism with only logarithmic additive loss to the competitive ratio =-=[6]-=-. There has been substantial recent interest private value problems in Computer Science. This work has been primarily focused in two directions, (a) looking at the computational complexity and communi... |

32 | Competitive Auctions for Multiple Digital Goods
- Goldberg, Hartline
- 2001
(Show Context)
Citation Context ...i Azar, Kostub Deshmukh, Amos Fiat, Andrew Goldberg, Anna Karlin, Claire Kenyon, Mike Saks, and Andrew Wright. Many of the results presented have been published with subsets of the above as coauthors =-=[26, 22, 20, 15, 23, 24]-=- and some are in manuscript form as of this writing [25, 8]. One main contribution of this work is the introduction of competitive analysis to economic mechanisms [26, 20, 24]. This type of analysis a... |

25 | Truthful and Competitive Double Auctions
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- 2002
(Show Context)
Citation Context ...i Azar, Kostub Deshmukh, Amos Fiat, Andrew Goldberg, Anna Karlin, Claire Kenyon, Mike Saks, and Andrew Wright. Many of the results presented have been published with subsets of the above as coauthors =-=[26, 22, 20, 15, 23, 24]-=- and some are in manuscript form as of this writing [25, 8]. One main contribution of this work is the introduction of competitive analysis to economic mechanisms [26, 20, 24]. This type of analysis a... |

19 |
Beweis eines Ausdrucks von welchem die Binomial-Formel ein einzelner Fall ist
- Abel
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Citation Context ...v. Thus if g c u is a consensus for v then it is a consensus estimate with value within a factor of c from v. We define G c as a distribution of functions of the form g c U with U chosen uniformly on =-=[0, 1]-=-. We repeatedly make use of the following result. Lemma 4.10 For g from G c , g(v) is distributed identically to c U # v/c for U # uniform on [0, 1]. Proof: Consider a random variable Y = log c g(v) a... |

19 | Private communication - Fiat |

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- 2003
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Citation Context |

14 | Almost-dominant strategy implementation
- Schummer
- 1999
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Citation Context ...ds. We will be looking for a mechanism with good probabilistically truthful properties in terms of m. For other solution concepts related to approximate or probabilistic truthfulness, see for example =-=[41, 47]-=-. 68 Definition 6.2 [2] An auction is truthful with probability 1 - # if the probability that any bidder can benefit from an untruthful bid is at most #. An auction is truthful with high probability i... |

6 |
Market equilibria via a primal-dual-type algorithm
- Devanur, Papadimitriou, et al.
- 2002
(Show Context)
Citation Context ...g a market with agents and goods and computing prices for the goods such that they can be reallocated among the agents in an optimal way such that there is neither a surplus or a shortage of any good =-=[14, 16]-=-. 1.3 Contributions The work presented in this thesis is the result of collaborations between the author and Yossi Azar, Kostub Deshmukh, Amos Fiat, Andrew Goldberg, Anna Karlin, Claire Kenyon, Mike S... |

4 |
personal communication
- Azar
- 1995
(Show Context)
Citation Context ... , and F (2) ). These bounds are tight. If the upper bound on the bid values, h, is not known, but the lower bound is of 1 is, Yossi Azar has observed that we can still get close to log h-competitive =-=[7]-=-. This result is obtained by using the classify and select technique [33] (and a good description can be found in [6]). 32 Profit Maximization via k-Vickrey Now we consider obtaining a profit maximizi... |

4 |
Truth Revelation in Approximately E#cient Combinatorial Auctions
- Lehmann, O'Callaghan, et al.
- 1999
(Show Context)
Citation Context ...In the first direction, the most notable example is work looking at the intractability and approximation of the common welfare maximization objective for the combinatorial auction problem (See, e.g., =-=[40, 32, 2]-=-). The fundamental observation made is that the Vickrey-Clarke-Groves (VCG) [51, 13, 27] mechanism that implements common welfare maximization, when applied to the combinatorial auction problem, is re... |

4 | Profit-Maximizing Multicast Pricing by Approximating Fixed Points
- Mehta, Shenker, et al.
- 2003
(Show Context)
Citation Context ...n for selling a single item. Ronen shows that it is computational infeasible (i.e., NP-hard) to implement the optimal auction but instead an approximately optimal solution can be implemented [43]. In =-=[34]-=-, the Bayesian approach to the optimal auction was extended to the multicast pricing problem that we discuss in Chapter 10. Recently, Segal has considered the uninformed auction problem in a setting s... |

3 |
Amazon may spell end for `dynamic' pricing
- Press
- 2000
(Show Context)
Citation Context ... competitive assuming that bidders bid their true values. Auctions that do not have the envy-free property are impractical selling mechanisms for markets where consumers object to di#erential pricing =-=[5]-=-. For these applications, auctions that are truthful and almost envy-free or envy-free and almost truthful may be acceptable. 6.1 Definitions We now formalize our notion of an envy-free auction. Defin... |

1 |
The Price of Truth: Frugality in Truthful Mechanisms
- Koutsoupias, Papadimitriou
- 1999
(Show Context)
Citation Context ...um spanning tree problems was first proposed by Nisan and Ronen [39]. Other notable work in the intersection of Computer Science, Game Theory, and Economics includes the study of the price of anarchy =-=[31]-=- and market equilibria [14]. The price of anarchy considers the di#erence between the social welfare of an optimal allocation of resources and the social welfare of a allocation obtained when a Nash e... |