The theory and practice of auctions

Posted on

In 2008, the authorities of India announced that wireless spectrum for 3G nomadic services would be auctioned in 2009. The 2.1 GHz spectrum is to be sold as paired spectrum ( 2×5 MHz blocks ) in each of 22 parts covering India. There are one to four tonss available in each part and each bidder can obtain at most one batch per part. The authorities ‘s stated aim for the auction emphasizes gross maximization instead than efficiency. In December 2008, the authorities announced the chosen auction design: a distinct clock auction with lowest-accepted command ( LAB ) pricing and probationary victors ( Telecommunications & A ; Information Technology, 2008 ) . Here we examine the equilibrium belongingss of such an auction in a simplified scene. As a comparing, we analyze a distinct clock auction with highest-rejected command ( HRB ) pricing and issue commands. This format is frequently used in high-stake auctions in pattern ( Ausubel & A ; Cramton, 2004 ) .

Interestingly, with net income maximising bidders, we find the HRB auction dominates the LAB auction in both efficiency and grosss. The HRB with issue commands is to the full efficient, since it is a dominant scheme to offer up to one ‘s rating. In contrast, LAB with probationary victors has differential shading, since a probationary victor shades her command, whereas a probationary also-ran does non. This differential shading creates an inefficiency and reduces grosss. Given this strong theoretical consequence, it may look odd that India chose the LAB format.

One possible account comes from behavioral economic sciences. If bidders anticipate the sorrow of losing at a profitable monetary value, they may be loath to shadow commands as a probationary victor. This fright of losing has been shown to explicate overbidding in first-price sealed-bid auctions ( Filiz-Ozbay & A ; Ozbay, 2007 ; Engelbrecht-Wiggans & A ; Katok, 2007, 2008 ) . Risk antipathy is an alternate account for overbidding, but has small empirical support ( Engelbrecht-Wiggans & A ; Katok, 2009 ) . Delgado et Al. ( 2008 ) supply a neurological foundation that fright of losing, non joy of winning, is the beginning of overbidding in first-price auctions. Loss antipathy with a mention point of winning ( Lange & A ; Ratan, 2010 ) provides an correspondent theory for fright of losing. When we extend the criterion theory to include a fright of losing, we find that if bidders ‘ fright of losing is sufficiently strong, so the LAB auction gross dominates the HRB auction. This consequence provides an account for India ‘s choice of a apparently inferior auction format.

Theory: Pricing Rule in a Clock Auction

Auctions have been preponderantly used by authoritiess to sell a broad scope of goods such as spectrum licences, electricity, exchequer measures and emanation rights. Typically, a authorities would take to apportion the good to those who value it most. Although auctions have been regarded as efficient selling mechanisms in literatures, planing efficient auctions is still an huge challenge. Some design issues have been cleared up but others remain unsolved ( Cramton, 2009 ) . One issue that has received small attending so far is the pick of distinct command degrees.

While most auction theoretical accounts have assumed that bidders can subject any command chosen from a uninterrupted interval, command sums are usually restricted to discrete command degrees in pattern. For illustration, FCC ‘s AWS and 700 MHz required that a valid command sum must transcend a designated minimal acceptable bid-one command increase plus the standing high command. In the India 3G spectrum auction, a command increase is a fraction of the old clock monetary value. Ausubel and Cramton ( 2004 ) stated that the motive for distinct command degrees, as opposed to a uninterrupted increase, is their hardiness to communicating failure and facilitation of the monetary value find. Discrete offering unit of ammunitions give bidders clip to repair communicating issues and integrated information revealed throughout the auction into their rating theoretical accounts and command schemes.

Despite the practicality of distinct command, full efficiency is by and large unachievable. Efficiency of the traditional auction theoretical account flexible joints on its ability to screen bidders by rating. Since a command map is a one-to-one monotone function from rating, any standard auction mechanism aa‚¬ ” an auction that awards the point to the bidder who submitted the highest command aa‚¬ ” is to the full efficient. On the other manus, a distinct auction can be perceived as a merchandising mechanism whose message infinite is a finite set. The type and message infinites are no longer isomorphous. A group of bidders with different ratings may subject the same message. The mechanism can non rank these bidders and thereby delegate them the same chance of trade and expected payment. At worst, the mechanism may misassign the point to another bidder who submits the same message as the highest-valuation bidder.

The efficient divider regulation can be implemented with command degrees in many distinct auction formats. In the sealed-bid _first-price auction, efficient command degrees are purely increasing whereas in the second-price auction, command sums may be nonunique.

The efficient command degrees in a clock auction with demand revelation and highest-rejected command can be either increasing or diminishing. One of our cardinal consequences is that uncovering demand improves efficiency even in the independent private rating scene in which monetary value find is irrelevant.

Several documents analyzed offering behavior in distinct auctions. Chwe ( 1989 ) and Mathews and Sengupta ( 2008 ) examined the first-price and second-price sealed command distinct auctions, severally. Cramton and Sujarittanonta ( 2010 ) compared public presentations of assorted pricing regulations in go uping clock auctions with distinct command increases. They besides studied clock auctions in which intraround command is permitted. Rothkopf and Harstad ( 1994 ) determined a set of optimum distinct command degrees in an English auction. David et Al. ( 2007 ) extended the theoretical account of Rothkopf and Harstad and found that diminishing command increases maximise gross. A few paper studied a mechanism design with a limitation on action infinite. Blumrosen and Feldman ( 2006 ) characterized sufficient conditions for dominant-strategy implementability in a mechanism correspondent. Blumrosen, Nisan, and Segal ( 2007 ) studied an optimum mechanism with delimited action infinite.

Most theoretical documents on clock auctions assume a uninterrupted clock for convenience. In pattern, clock auctions use a distinct monetary value clock, since these auctions typically are conducted on the cyberspace, and communicating is non sufficiently dependable to adhere bidders to higher monetary values with the uninterrupted transition of clip. With distinct command degrees, the two prevailing pricing regulations, lowest-accepted command and highest-rejected command, are distinguishable, and the auction interior decorator must choose a pricing regulation every bit good as other elements of the design. Presently, there is small literature for the auction interior decorator to turn to for aid with this issue.

There are a limited figure of documents look intoing auctions with distinct command degrees. These documents focus on explicating offering behavior. Chwe ( 1989 ) studied the first-price auction with distinct command degrees and showed that the expected gross is less than its uninterrupted opposite number. ( Mathews & A ; Sengupta, 2008 ) analyzed a sealed-bid second-price auction with distinct commands.

More closely related is the work that considers picks of command degrees in go uping auctions. Rothkopf and Harstad ( 1994 ) is an of import early part, finding optimum command degrees that maximize expected gross in an unwritten auction. The paper besides introduces the tradeoff between auction continuance and command increases. David et Al. ( 2007 ) extend the theoretical account of Rothkopf and Harstad and happen that diminishing command increases are optimum. Although the pricing regulation in our paper is the same as in Rothkopf and Harstad, the auction formats have of import differences, which result in significantly different offering behavior.

We consider a discrete clock auction with two pricing regulations: highest-rejected command and lowest-accepted command. Bidders have independent private values and unit demands. We foremost analyze offering behavior in an HRB auction. This is our benchmark for comparing public presentation with the LAB auction. The HRB auction is a utile benchmark because of its simpleness, its desirable belongingss ( efficiency and truth laterality ) , and its usage in pattern. In contrast, the LAB auction forces bidders to prosecute in hard trade-offs. We are merely able to work out for equilibrium command behavior in a simplified scene. However, we show that an LAB auction is by and large inefficient. Despite this inefficiency, the LAB auction can give higher grosss if bidders anticipate the sorrow of losing at profitable monetary values, and hence engage in less command shadowing than a bidder focused entirely on net income maximization.

Discrete Clock Auctions

A common method to auction wireless spectrum, electricity, gas, and other merchandises is the distinct clock auction. The auctioneer names a monetary value and each bidder responds with her coveted measure. If there is extra demand, the auctioneer so names a higher monetary value. The procedure continues until there is no extra demand. Discrete unit of ammunitions are used in pattern to simplify communicating, make the procedure robust to communicating failures, and extenuate silent collusion ( Ausubel & A ; Cramton, 2004 ) . An deduction of distinct unit of ammunitions is that the pricing regulation affairs. The two most common pricing regulations are lowest-accepted command and highest-rejected command. Another issue is whether the bidder can stipulate an issue bid-a monetary value less than the current monetary value at which the bidder desires to cut down measure. In the bound as the size of the command increase goes to zero, the differentiation between pricing regulations is irrelevant and exit commands are unneeded. However, in practical auctions where the figure of unit of ammunitions frequently ranges from 4 to 10, distinctness affairs.

In this subdivision, we examine offering behavior under three versions of a distinct clock auction. In each version, after each unit of ammunition the bidders learn the aggregative demand. To forestall bid-sniping, an activity regulation requires that a bidder ‘s measure demanded can non increase at higher monetary values. Bidders can merely keep or cut down measure as the monetary value rises. The three versions differ in the pricing regulation and whether bidders make issue commands to show the monetary value at which a measure decrease is desired.

Highest-rejected command ( HRB ) . If the bidder reduces measure in a unit of ammunition, the bidder names a monetary value for each measure decrease. The monetary value of each decrease must be greater than the anterior monetary value and less than or equal to the current monetary value. Each issue monetary value is interpreted as the monetary value at which the bidder is apathetic between the higher measure and the lower measure. If there is no extra demand at the current monetary value, the supply is awarded to the highest bidders, and each victor pays the highest jilted command for the measure won. The glade monetary value is the lowest monetary value consistent with market clearing-the monetary value at which supply peers demand.

Lowest-accepted command ( LAB ) . This is the same as HRB, except that the victors pay the lowest-accepted command for the measure won. The glade monetary value is the highest monetary value consistent with market glade. Lowest-accepted command with probationary victors ( LABpw ) . This is the same as LAB, except there are no issue commands. Alternatively after each unit of ammunition, probationary victors are determined. Those with the highest monetary value command are selected foremost, and, in the event of a tie, the staying probationary victors are selected at random.