scholarly journals A First Price Auction with an Arbitrary Number of Asymmetric Bidders

2018 ◽  
Vol 19 (2) ◽  
Author(s):  
Nicola Doni ◽  
Domenico Menicucci
Keyword(s):  
Closed Form ◽  
Arbitrary Number ◽  
Equilibrium Strategies ◽  
Private Values ◽  
Price Auction ◽  
Second Price Auction

Abstract We examine an auction setting with private values and $n\geq 2$ bidders, which differ in the probability to have a low (high) value. We prove that for the first price auction, the equilibrium strategies can be written in closed form and there are gaps in the equilibrium bid distribution of each bidder except the strongest two. Our equilibrium characterization allows to prove that in this setting the seller prefers the second price auction to the first price auction for each $n\geq 2$.

Soft Floors in Auctions

2019 ◽  
Vol 65 (9) ◽  
pp. 4204-4221 ◽  
Author(s):  
Robert Zeithammer
Keyword(s):  
The Other ◽  
Symmetric Equilibrium ◽  
Revenue Equivalence ◽  
Bidding Strategies ◽  
Reserve Prices ◽  
Private Values ◽  
Price Auction ◽  
Winning Bidder ◽  
Second Price Auction ◽  
Independent Private Values

Several of the auction-driven exchanges that facilitate programmatic buying of internet display advertising have recently introduced “soft floors” in addition to standard reserve prices (called “hard floors” in the industry). A soft floor is a bid level below which a winning bidder pays his own bid instead of paying the second-highest bid as in a second-price auction most ad exchanges use by default. This paper characterizes soft floors’ revenue-generating potential as a function of the distribution of bidder independent private values. When bidders are symmetric (identically distributed), soft floors have no effect on revenue, because a symmetric equilibrium always exists in strictly monotonic bidding strategies, and standard revenue-equivalence arguments thus apply. The industry often motivates soft floors as tools for extracting additional expected revenue from an occasional high bidder, for example a bidder retargeting the consumer making the impression. Such asymmetries in the distribution of bidder preferences do not automatically make soft floors profitable. This paper presents two examples of tractable modeling assumptions about such occasional high bidders, with one example implying low soft floors always hurt revenues because of strategic bid-shading by the regular bidders, and the other example implying high soft floors can increase revenues by making the regular bidders bid more aggressively. This paper was accepted by Juanjuan Zhang, marketing.

scholarly journals Asymmetric Auctions with Resale

2008 ◽  
Vol 98 (1) ◽  
pp. 87-112 ◽  
Author(s):  
Isa Hafalir ◽  
Vijay Krishna
Keyword(s):  
Asymmetric Auctions ◽  
Monopoly Pricing ◽  
General Revenue ◽  
Private Values ◽  
Price Auction ◽  
Second Price Auction ◽  
Expected Revenue ◽  
Second Price Auctions ◽  
Independent Private Values

We study first- and second-price auctions with resale in a model with independent private values. With asymmetric bidders, the resulting inefficiencies create a motive for post-auction trade which, in our model, takes place via monopoly pricing—the winner makes a take-it-or-leave-it offer to the loser. We show (a) a first-price auction with resale has a unique monotonic equilibrium; and (b) with resale, the expected revenue from a first-price auction exceeds that from a second-price auction. The inclusion of resale possibilities thus permits a general revenue ranking of the two auctions that is not available when these are excluded. (JEL D44)

scholarly journals Entry Deterrence in Dynamic Second-Price Auctions

2016 ◽  
Vol 8 (2) ◽  
pp. 168-201
Author(s):  
Xiaogang Che ◽  
Tilman Klumpp
Keyword(s):  
Social Welfare ◽  
Entry Deterrence ◽  
Private Values ◽  
Price Auction ◽  
Second Price Auction ◽  
Second Price Auctions ◽  
Independent Private Values

We examine a dynamic second-price auction with independent private values and sequential costly entry. We show that delayed revelation equilibria exist in which some buyers place coordinated low early bids. These buyers revise their bids to reflect their true valuations just prior to the end of the auction. Compared to the benchmark immediate revelation equilibrium, in which buyers bid their valuations immediately after entry, fewer high-value bidders enter on expectation in the delayed revelation equilibria. Delayed revelation of buyer values decreases social welfare, but is necessary for bidders to have a strict participation incentive. (JEL D44, D82, D83)

scholarly journals Obviousness around the clock

2021 ◽  
Author(s):  
Yves Breitmoser ◽  
Sebastian Schweighofer-Kodritsch
Keyword(s):  
Mechanism Design ◽  
Behavioral Effect ◽  
Dominant Strategy ◽  
Drop Out ◽  
Behavioral Differences ◽  
Feedback Effects ◽  
Private Values ◽  
Price Auction ◽  
Second Price Auction ◽  
Auction Formats

AbstractLi (Am Econ Rev 107(11):3257–3287, 2017) introduces a theoretical notion of obviousness of a dominant strategy, to be used as a refinement in mechanism design. This notion is supported by experimental evidence that bidding is closer to dominance in the dynamic ascending-clock auction than the static second-price auction (private values), noting that dominance is theoretically obvious in the former but not the latter. We replicate his experimental study and add three intermediate auction formats that decompose the designs’ differences to quantify the cumulative effects of (1) simply seeing an ascending-price clock (after bid submission), (2) bidding dynamically on the clock, and (3) getting (theoretically irrelevant) drop-out information about other bidders. The theory predicts dominance to become obvious through (2), dynamic bidding. We find no significant behavioral effect of (2), however, while the feedback effects (1) and (3) are highly significant. We conclude that behavioral differences between second-price and ascending-clock auctions offer rather limited support for the theory of obviousness and that framing has surprisingly large potential in mechanism design.

scholarly journals Multi-bidder First Price Auction with Beliefs

2019 ◽  
Vol 7 (1) ◽  
pp. 140-160
Author(s):  
Hyeon Park
Keyword(s):  
Closed Form ◽  
Arbitrary Number ◽  
Value Of Information ◽  
Jel Classification ◽  
The Other ◽  
Type Space ◽  
Closed Form Solutions ◽  
Price Auction ◽  
Equilibrium Bidding

This paper characterizes the set of equilibria in the first price auction with multiple bidders—specifically three bidders, each of whose type space is multi-dimensional, incorporating a bidder’s beliefs about others’ valuations. In this auction, each bidder independently and privately learns the other two opponents’ valuations with some probability. This paper derives closed form solutions for equilibrium bidding behaviours parameterized by the degree of information when the bidder has homogeneous beliefs regarding each opponent. This paper demonstrates how much the level of information affects the bidding behaviours of the informed bidders. In addition, this paper extends the model into [Formula: see text] bidders when a high value bidder has fixed beliefs that all other bidders are identical types, and show how the value of information changes as the number of participants increases. Finally, this paper speculates on the possible changes in the efficiency of the model from increasing the valuation space to an arbitrary number. JEL Classification: D44, D82, D83

scholarly journals Online Second Price Auction with Semi-Bandit Feedback under the Non-Stationary Setting

2020 ◽  
Vol 34 (04) ◽  
pp. 6893-6900
Author(s):  
Zhao Haoyu ◽  
Chen Wei
Keyword(s):  
Time Horizon ◽  
Joint Distribution ◽  
Reserve Price ◽  
Total Revenue ◽  
Dynamic Case ◽  
Private Values ◽  
Price Auction ◽  
Second Price Auction ◽  
Private Value ◽  
Over Time

In this paper, we study the non-stationary online second price auction problem. We assume that the seller is selling the same type of items in T rounds by the second price auction, and she can set the reserve price in each round. In each round, the bidders draw their private values from a joint distribution unknown to the seller. Then, the seller announced the reserve price in this round. Next, bidders with private values higher than the announced reserve price in that round will report their values to the seller as their bids. The bidder with the highest bid larger than the reserved price would win the item and she will pay to the seller the price equal to the second-highest bid or the reserve price, whichever is larger. The seller wants to maximize her total revenue during the time horizon T while learning the distribution of private values over time. The problem is more challenging than the standard online learning scenario since the private value distribution is non-stationary, meaning that the distribution of bidders' private values may change over time, and we need to use the non-stationary regret to measure the performance of our algorithm. To our knowledge, this paper is the first to study the repeated auction in the non-stationary setting theoretically. Our algorithm achieves the non-stationary regret upper bound Õ(min{√S T, V¯⅓T⅔), where S is the number of switches in the distribution, and V¯ is the sum of total variation, and S and V¯ are not needed to be known by the algorithm. We also prove regret lower bounds Ω(√S T) in the switching case and Ω(V¯⅓T⅔) in the dynamic case, showing that our algorithm has nearly optimal non-stationary regret.

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