Experiments on Auctions - University College Londonuctpsc0/Teaching/ECON3020/Auction.pdf · An...

Experiments on Auctions

Syngjoo Choi

Spring, 2010

Experimental Economics (ECON3020) Auction Spring, 2010 1 / 25

Auctions

An auction is a process of buying and selling commodities by takingbids and assigning commodities to winning bidder(s).

Auctions have been widely used, ranging from the sale of perishablecommodities like �sh and �owers, to the transfer of assets from publicto private hands, such as timber rights and o¤-shore oil leases, to thesale of rights to use electro-magnetic spectrum for communication.

An interesting historical event is that the Praectorian guard sold o¤the entire Roman Empire by means of an auction in 193 A.D., afterhaving killed the Emperor Pertinax (for more information, go tohttp://en.wikipedia.org/wiki/Auction).


Some Common Auction Forms

The open ascending-price auction (or English auction) is the oldestand perhaps most prevalent auction form. In one variant, the sale isconducted by an auctioneer who begins by calling out a low price andraises it as long as there are at least two interested bidders. Theauction stops if there is only one interested bidder. The object is soldto this bidder at the price at which the second-last bidder droppedout.In the open descending-price auction (or Dutch auction) anauctioneer calls out prices in a descending order until one bidderclaims the item at a given price.In the sealed-bid �rst-price auction, bidders submit bids in sealedenvelopes; the person submitting the highest bid wins the object andpays what he bid.In the sealed-bid second-price auction (or Vickery auction), bidderssubmit bids in sealed envelopes; the person submitting the highest bidwins the object but pays not what he bid, but the second highest bid.


Valuations: Private Value vs. Common Value

An inherent feature of auctions is the uncertainty regarding valuesfacing both sellers and buyers. Auctions are used precisely becausethe seller is unsure of the values that bidders attach to the objectbeing sold (the maximum willingness to pay).

One situation regarding the valuations of the object is that eachbidder knows the value of the object to him-/herself at the time ofbidding. This is called one of private values.

In this situation no bidder knows with certainty the values attached byother bidders and knowledge of other bidders�values would not a¤ecthow much the object is worth to a particular bidder.The assumption of private values is most plausible when the value ofthe object to a bidder is derived from its consumption or use alone.


In another situation values are unknown to each other at the time ofthe auction and may be a¤ected by information available to otherbidder. Such a situation is called one of interdependent values. Aspecial case of this is a situation in which the value, though unknownat the time of bidding, is the same for all bidders, called one of a purecommon value.

A common value model is most appropriate when the value of theobject being auctioned is derived from a market price that is unknownat the time of the auction.


Equivalence of Auctions

Four auction formats (two open auctions and other two sealed-bidauctions) di¤er in the way that they are implemented in the real world.

Open auctions usually require that the bidders collect in the sameplace. In contrast sealed bids may be sumbitted by mail.A bidder may observe the behavior of other bidders in open auctionsbut not in sealed-bid auctions.

However, some of these practical di¤erences are irrelevant for rationaldecision makers.


Observation 1: The open ascending-price auction (or Dutch auction)is strategically equivalent to the sealed-bid �rst-price auction.

A bidder�s strategy in the �rst-price auction maps his privateinformation into a bid.The only relevant information available in the Dutch auction is thatsome bidder has agreed to buy at the current price; but that causes theauction to end.Bidding a certain amount in a sealed-bid �rst-price auction isequivalent to o¤ering to buy at that amount in a Dutch auction.Thus, for every strategy in a �rst-price auction there is an equivalentstrategy in the Dutch auction and vice versa.


Observation 2: When values are private, the open ascending-priceauction (or English auction) is equivalent to the sealed-bidsecond-price auction.

The English auction o¤ers information about when other bidders dropout; and by observing this, it may be possible to infer something abouttheir privately known information.With private values, however, this information is of no use.With interdependent values, seeing some other bidder drop out earlymay bring bad news that may cause a bidder to reduce his ownestimate of the value of the object.With private values, in an English auction, it clearly cannot be optimalto stay in after the price exceeds the value or to drop out before theprice reaches the value.Likewise, in a second-price auction it is best to bid the value.


Revenue Equivalence

The performance of di¤erent auction formats is usually evaluated ontwo grounds: revenue and e¢ ciency.

From the perspective of the seller, a natural criterion in comparingdi¤erent auction forms is the revenue (or expected selling price) thatthey can obtain.

Revenu Equivalence Theorem: If all bidders are risk neutral andhave independent private value for the good, all four auction formats(for the single unit of the good) yield the same expected revenue.

From the perspective of the society as a whole, e¢ ciency (that theobject end up in the hands of the person who values it the most expost) may be more important.


A Classroom Experiment

Go to http://veconlab.econ.virginia.edu/login.htm.Session name is sjc__.


A First-Price Private Value Auction

The auctions you just played is a version of the sealed-bid �rst-priceauction with private values in which (i) there are two bidders in agroup or (ii) there are more than two bidders.

To understand the behavior of bidders, we consider an equilibriumstrategy, as a benchmark, when there are two risk-neutral bidders.

Let Vi denote a value that player i = 1, 2 assigns to the object.Assume that each Vi is independently and identically uniformlydistributed on [0, 10]. Each bidder submits a sealed bid of bi .

Given a realization vi of the value and the choice of bid bi , player i�spayo¤ is given by

ui (bi , bj jvi ) =�vi � bi0

if bi > bjif bi < bj

.


Equilibrium Strategy

Let bidder 1 assume that bidder 2 uses an increasing strategy,b2 (v2) = a� v2, where a > 0. The expected payo¤ for bidder 1 isgiven by

(v1 � b1)� Pr (b1 > b2 (v2))

= (v1 � b1)� Pr�v2 <

b1a

�= (v1 � b1)�

�b1

10� a

�.


Taking the derivative of this and arranging it, we can �nd

b1 =v12.

Thus, an equilibrium strategy for each player i is just bidding half ofhis/her realized value.

In fact, the equilibrium strategy can be easily extended into Nrisk-neutral bidders:

b (v) =(N � 1)N

v .

As the number of bidders become larger (i.e., more competition), theratio, (N�1)N , becomes closer and closer to 1.


Explanations of Overbidding

As we noticed, the bidders in an experiment have a tendency ofoverbidding relative to an equilibrium bidding strategy of arisk-neutral bidder.

One possible explanation of overbidding in a �rst-price auction is riskaversion.

Consider an individual�s utility function has the form of powerfunction:

(Expected payo¤) = (v � b)1�r ��

b10� a

�.


Taking the derivative and rearranging it, we can �nd that

b (v) =v

2� r .

When r = 0 (risk neutrality), we have the same expression as before.As r (< 1) increases from zero (more risk averse), the biddingstrategy becomes more conservative.

An alternative explanation is based on a judgemental error of winningprobability (a systematic underestimation of winning probability).


A Classroom Experiment

Go to http://veconlab.econ.virginia.edu/login.htm.Session name is sjc__.


A First-Price Common Value Auction

You just participated in a �rst-price common value auction in which(i) there are only two bidders in a group or (ii) there are more thantwo bidders.

Two bidders observe separate components of a common prize value,i.e., bidder 1 observes v1 and bidder 2 observes v2.

The monetary value of the prize to the winning bidder is an averageof these two values:

(v1 + v2)2

.

The signals are private in that each bidder only knows his/her ownestimate, but the bidders know that the other bidder�s estimate is theindependent realization of a random variable whose distribution isuniform on the interval, [0, 10].



To understand the behavior of bidders, we consider an equilibriumstrategy, as a benchmark, when there are two risk-neutral bidders.

Let bidder 1 assume that bidder 2 uses an increasing strategy,b2 (v2) = a� v2, where a > 0.Given this, the probability of winning with a bid of b1 is given by

Pr (b1 > b2 (v2)) = Pr�v2 <

b1a

�=

b110� a .

The expected payo¤ conditional on winning with a bid of b1 is

v1 +E�v2jv2 < b1

a

�2

=v1 + b1

2a

2.



The expected payo¤ for bidder 1 is the product of the probability ofwinning and the di¤erence between the conditional expected payo¤and the bid:

Pr (b1 > b2 (v2))�

24v1 +E�v2jv2 < b1

a

�2

� b1

35=

�b1

10� a

��"v1 + b1

2a

2� b1

#.


By taking the derivative and rearranging it, we have

b1 =a

4a� 1v1.

In a symmetric equilibrium, the equilibrium strategy becomes

b (v) =v2.

In fact, risk-averse bidders have the same equilibrium strategy in thissetup (see Holt and Sherman (2000)). It implies that a pattern ofoverbidding can not be explained by risk aversion if players arerational.


Winner�s Curse

We again observe the general pattern of overbidding in this commonvalue auction and the empirical bid/value relationship is signi�cantly�atter than the symmetric Nash equilibrium.

In contrast to the private value auction, the fact that a bidder winsthe auction has a relevant information: having the highest bid meansthat others�value estimates were relatively low. A bidder does notrealize this may bid too high and end up paying more than the prize isworth. This possibility due to judgemental error of overestimating thetrue value of the object is called �Winner�s Curse�.

Rational bidders recognize that winning is informative and makecorrect inferences from the equilibrium bids of others.


One Explanation of Winner�s Curse: Limited-liability e¤ect

One approach to explain the winner�s curse is to investigate whetherlosses are caused by artifacts of the experimental design.Especially, this may occur due to the inability of the experimenter toactually collect losses from laboratory subjects, called alimited-liability e¤ect.In our previous experiment, subjects enjoyed limited liability as theycould not lose more than their starting cash balances.With downside losses eliminated the only constraint on moreaggressive bidding is the opportunity cost of bidding more than isnecessary to win the object. In exchange, higher bids increase theprobability of winning the object and making positive pro�ts.The net e¤ect could be an incentive to bid more than the Nashequilibrium prediction.One possiblity of avoiding this so-called bankruptcy problem iscombining this experiment with another experiment in which subjectsearn some amount of money for certain.


Naive Bidding

The winner�s curse can occur if a bidder does not realize that winningthe auction implies the other bidder�s value estimate is relatively low.

The simplest naive model is obtained by using the unconditionalexpected value of the other bidder�s value estimate, which is 5 whenthe values are uniform on [0, 10]. Since 5 is typically larger than theconditional value, v1/2, this bias will tend to produce higher bids,even with risk neutral bidders, and thus will give rise to a winner�scurse.


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