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2007-07-19T15:06:00.000+01:00

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2007-03-04T06:02:00.000Z

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1. Introduction Two of the most common non-price m...

2007-02-04T10:44:00.000Z

1. Introduction Two of the most common non-price mechanisms that allocate
objects to individuals are auctions and lotteries. In auctions the probability
that player i wins depends on the other bids, as well as the size of payments.
In a lottery all agents have the same probability of wining the object, and the
actions of the other players might affect the winning prize (for example, when
there is more than one winner the winning prize will be divided equally) but do
not affect the probability of winning. In this paper we conduct - both
theoretical and empirical - analysis of a selling mechanism that combines
elements of an auction and a lottery. The mechanism studied is used by the
internet portal http://www.BestBidsAuction.com, which also provided data of its
auctions. Before each auction, the auctioneer determines three parameters of the
auction: the highest bidallowed (which is less than 10% of the retail value of
the object), the maximum number of bids allowed before the auction closes, and
the entry fee each bidder needs to pay when submitting his bid. All of these
values are made public before the bidding starts. After the bidders pay the
participation fee, they submit sealed bids, less than or equal to the highest
bid allowed. The winning bid is the highest unique bid (in the sense that no one
else bid exactly the same amount) among all bids received. The winner then pays
his bid price and obtains the object. We call the selling mechanism adopted by
the portal a Gambling Auction, because it has features that make it a
combination of an auction and a lottery. First, the bid and the probability of
winning are not monotonically related, because a lower bid might well win the
auction if many bidders are placing high bids. Consequently there is no obvious
bid that maximizes the probability of winning and, as we show, in equilibriumall
bids provide the same probability of winning. Second, this mechanism is not a
pure lottery either because the winning probability is determined by the action
of the biddersand not by an exogenous randomizing device: the winner is the one
that submits the highest unique bid. Note that, under the symmetric Nash
equilibrium of the game, the equal winning probabilities this auction creates
and the expected payments can be 2

implemented using a lottery and thus the two types of mechanisms are outcome
equivalent if the bidders are risk neutral and follow the symmetric
equilibrium.2The theoretical analysis finds that in a symmetric equilibrium each
bidder chooses his bid using a distribution function over a support that has no
gap. This equilibrium strategy is increasing; namely the probability of placing
a higher bid is not less than that of a lower bid. The intuition is that
otherwise a higher bid would make winning more likely and thus be more
profitable than a lower bid, which would makeeveryone prefer it, destroying the
alleged equilibrium bidding pattern. We test this prediction with a novel data
set collected from the portal http://www.BestBidsAuction.com, which implements auctions
described above. The data confirms that the probability of a higher bid is not
less than a lower bid. We also find that an increase in the number of bidders
increases the number of bids for a given slot, although reduces the probability
that each bidder places his bid at this given slot. This leads to an increase of
the distance between the maximum bid allowed and the actual winning bid. We also
tested the theoretical prediction that each bid has the same probability of
winning by constructed a frequency table (Table 4). This table measures the
frequencywith which the highest bid wins by calculating the number of auctions
in which the highest bid won divided by the number of instances in which a
highest bid was placed. We repeat this exercise at lower bid levels and ask
whether the empirical frequencies are2This Gambling Auction is also interesting,
because it can be used in countries or U.S. states that forbid gambling, because
the rules of the mechanism do not meet the traditional definitions of lottery.
The mechanism might attract people who like participating in gambling
activities, since at a relatively low cost one have the opportunity to win a
sizable prize. The auctioneer will make more money using this mechanism than by
regular auction mechanisms if participants are risk lovers. Empirically, this is
the case since these auctions have a negative expected profit for a bidder. This
mechanism is similar to a rotating saving and credit associations (roscas) in
which group of people save for indivisible good. Each period allthe people
contribute to the rosca and it is given to someone randomly that is able to get
the good. In thenext period it is given to somebody else and so on (see Besley,
Coate and loury (1993)). In our mechanismthe good is also distributed eventually
randomly and each individual pays the participation fees, but theexpected payoff
is negative, since the auctioneer obtains a positive profit and the winner pays
extra amount of money (the winning bid) in order to get the good. 3

Page 4
indeed equal as suggested by the theory. Some formal chi-square tests and
informalanalysis suggest that the theoretical bid distribution is not consistent
with the data. In addition, unlike other studies that estimated the demand for
lottery games and found that consumers respond to the expected returns, we found
that consumer demand for this lottery is not sensitive to the expected payoff
but it is sensitive to the size of theprize. The paper is organized as follows.
In the next section we characterize the equilibrium strategies of the auction
game and provide some comparative static results. Section 3 describes the data,
while Section 4 performs empirical analysis. A final section offers some
concluding remarks. 2. Theoretical considerations We will first describe the
model we consider and then show that in a symmetric equilibrium a higher bid is
chosen with higher probability. There are kbidders3=3who all value the object at
the retail price, v. After paying an entry fee of c each bidder submits a sealed
bid that is less than a maximum value b << v. We assume that each bidder places
only one bid. There is a minimum bid increment, which we normalize to 1. The
winner is the one who placed the highest bid that was not bid by anyone else. If
there is no such bid, then we assume that the seller runs the auctionagain with
the same set of bidders. The internet portal reports that, in the rare event of
no unique bid, the bidders will be notified about the situation and asked to
submit a new bid without additional charge. The winner has to pay an amount
equal to his bid, while the losers only pay the entry fee. In addition we assume
that k, v and b are such that inequilibrium the winning bid is close to b; in
other words, we assume that the bid increment is low compared to the value of
the object, and thus the winning bid is close to 3In the auction at the above
website only the maximum number of bidders is specified, but the number of
actual bidders is usually close to the allowed maximum number of bidders, so one
may assume that the number of bidders is a known constant, k.4

Page 5
the maximum allowed bid b.4Under such conditions we make the simplifying
assumption that each bidder is interested in maximizing his probability of
winning the object, ignoring the payment consequences of his bid.5The entry fee
is already sunk at the bidding stage, so it does not affect bidding strategies.
First, note that the above game has an equilibrium, since after imposing
aminimum bid requirement of 0, the auction becomes a finite game. Moreover,
using Kakutani�s fixed point theorem we may also show that a symmetric (mixed
strategy) equilibrium exists. Claim 1: In any symmetric equilibrium there is no
gap in the support of the equilibrium strategy. Proof: Suppose there was a gap
at b�. Then bidding b� would strictly dominate bidding the next available bid
b�-1, which yields a contradiction in that b�-1 is in the support of the
equilibrium strategy. Note, that the above claim also implies that the high end
of the support is the maximum allowed bid, b. Then a symmetric equilibrium is
characterized by the number of bidsemployed, n, and the probabilities of each of
those bids,)1Pr(+-=ibpiwhere i = 1,�,n. Theorem 1: In a symmetric equilibrium
the probability of a higher bid is not less than a lower bid: i >j implies that
pi= pj. Moreover, pi = pjcan hold only when there are four bidders. In that
case, the unique equilibrium has p1= p2 = 1/2. Proof: See the appendix A. 4On
average, the distance between the winning bids and the maximum allowed bid in
our data is less than14 cents on average, and the maximum distance is less than
$1.5. 5The bidder�s problem is to choose bithat will maximize: P(bi)(V- bi)-C=
P(bi)(V-b+b-bi)-C= P(bi)(V-b)+P(bi)(b-bi)-C, where P(bi) is bidder i probability
of winning the object when placing a bid of bi, V is the object valuation, b is
the highest bid allowed and C is the participation cost. If all bidders follow a
symmetric equilibrium, then the probability of receiving the object is the same
for each bidder. Asmentioned before, the distance between the winning bids and
the maximum allowed bid in our data is lessthan 14 cents on average, and the
maximum distance is less than $1.5. So on average, when one maximizesthe
probability of winning the object and ignores the second part of the objective
function; one ignores a monetary incentive of only a few cents. If we drop this
simplifying assumption then our results do not hold as stated. It is no longer
necessarily true that the equilibrium does not have a gap, since the equilibrium
weidentify in the simplified game is not robust to large deviations, when a
bidder places a bid close to zero.However, since the largest admissible bid is
less than 10% of the value of the object, the incentive for this deviation might
be neglected in a first approach to model this game. This approach is also well
supportedby the data, since winning with a very low bid is very unlikely, as it
will be noted in the next section. 5

Page 6
The intuition behind these results is clear. Suppose, that the other bidders
randomize equally among the bids B = {b1., b2, �, bn}, where b1> b2> � > bn.
Then it is easy to see that if bidder i places the bid b1, then he has a higher
probability of winning then with any other bid that belongs to B. But this
yields a contradiction, because in a symmetric equilibrium bidder i use a mixed
strategy with support on B, and thus he isindifferent between any of the bids
belonging to B. The incentive to bid high iseliminated only if a bidder expects
that there are more bidders who placed a high bid than who placed a lower one.
Thus, in equilibrium each bidder must place a higher bid with higher
probability. Let us consider some examples with a small number of bidders.
First, if there are three bidders, then, in the unique equilibrium all the bids
down to zero are used. With Tpossible bids including 0 it holds that for all 1 <
i < T, pi= 1/ 2T-iand p0=1/ 2T-1is theunique symmetric equilibrium of the game.
If k = 4, it is easy to show that the unique symmetric equilibrium is such that
p1= p2= �. In the case when k = 5 an equilibrium is such that
0.010}.p0.083,p0.197,p0.337,p0.372,{p54321=====We can confirm that it is indeed
equilibrium. A bidder�s utility is his probability of winning plus the
probability of a complete tie divided by five. Suppose that a bidder places the
maximum allowed bid. A bidder wins in this case if no one else placed thisbid,
i.e. with probability .)1(w411p-=A complete tie occurs, if one or two other
bidders placed the highest bid and the other two or three placed the same bid,
or if all others placed the highest bid. This probability is
.)(p})(p)(p)(p){(p)(p6})(p)(p)(p){(p4pt4125242322213534333211++++++++=Since in
equilibrium each bidder obtains a utility of 1/5 we obtain the following
condition:.515w11=+t6

One can compute the corresponding probabilities, wi, tifor i = 2,3,4,5 and write
up thecondition that for all i:6.515wi=+itThen one obtains 5 equations in 5
unknowns (the �s) and this system has a unique real valued solution, the vector
stated above. Finally, one needs to check that by placing a lower bid than bid
5, the achieved utility is not higher than 1/5. By placing such a bid the
deviating bidder wins if and only if the other four bidders tied. Then the
incentive constraint can be written as: ip.51622514=+???=jijiiipppThe proposed
strategy profile satisfies these conditions and thus it is equilibrium.For k = 5
the distribution of the winning bid is
.}011.0,098.0,211.0,325.0,357.0{54321=====pppppFor k = 6, an equilibrium is
0.109},p0.248,p0.309,p0.334,{p4321====and the distribution of the winning bid is
}.122.0,247.0,303.0,329.0{4321====ppppFor k = 7 an equilibrium is
0.078},45p0.296,{54321=0.137=,0.22=,0.26==pppp7and the distribution of the
winning bid is .}084.0,137.0,219.0,272.0,287.0{54321=====pppppIt is apparent
that the size of the support of the equilibrium strategy is not monotonic.
Excluding the case of 3 bidders, which seems non-generic, one conjecture 6The
corresponding probabilities for wiand tiare different for every i. In order to
save space the complete set of equations is not reported here but it is
available upon request from the authors. 7We did not show that the above
equilibria are unique for a given k. For this one would need to show thatif one
considers a different number of bids for a given k than the one considered
above, then no solutionexists to the resulting system of incentive constraints.
We only showed at this point that there are no other equilibria for k=4, 5, 6, 7
when we consider up to 7 possible bids. Our conjecture is that these equilibria
are unique in these cases and moreover, for any k there is a unique equilibrium
of the game.7

that emerges is that the more bidders there are the less concentrated become the
equilibrium strategies. Although there is no monotonicity in the length of the
support with respect the number of bidders, our conjecture is that the expected
distance betweenthe maximum allowed bid and the winning bid (the Gap) increases
with the number ofbidders. Namely, it is more likely that a bid further from the
maximum becomes the winning bid when the number of bidders increases.
Theoretically this is the case when the number of bidders is 4, 5, 6 or 7.83.
The DataThis section describes the data. The data source is
BestBidsAuction.com 9which is the Internet website of Best Bids Auction, a
Arizona company that manages and implements private auctions designed to raise
money for selected charities and member non-profit fundraising organizations.
The internet auction process is a combination of a lottery and an auction.
Before each auction, the auctioneer determines, among other things, the highest
bid allowed and the maximum number of bids that will be accepted for the
auction, and makes this information available for the bidders. In order to
participate in an auction, bidders submit sealed bids, less than or equal to the
highest bid allowed in US dollars and cents and agree to pay a bidding fee for
each submitted bid. The auction is a sealed bid auction in the sense that when a
bidder submits a bid he does not know what the other bids are until the auction
is over. Each auction is closed when it receives the maximum number of bids or
meets the other closing requirements.10After the auction closes, the participant
that submitted the successful (winning) bid is determined. The successful bid
8The expected Gap when k = 4 is 5.04=g, and for the other cases it is,
078.1*4...*055515=++=ppg162.16=gand 458.17=gwhen the number of bidders are 5, 6,
and 7 respectively. 9All the information has been taken from
http://www.BestBidsAuction.com. 10An auction will remain open until either the maximum
number of bids allocated for the auction is reached or the auction reaches
maturation (63 days for auctions requiring less than 200 bids, and 183 days for
auctions requiring 200 or more bids) and has received the minimum number of bids
required to close. If the minimum number of bids has not been reached, the
auction will be extended until the minimum numberof bids is met. At that time, a
closing date of three days will be set and posted on the auction. 8

Page 9
is the highest unique bid out of all bids received in the auction.11Duplicate
bids are used to calculate the number of bids required to close an auction but
are disqualified from being selected as the successful bid. For example, if a
single auction includes the following four bids: $69.42, $69.42, $48.69 and
$65.44, the winner will be the one who submitted $65.44. In the very unlikely
event that an auction closes and there is not a unique bid, all participants
receive an e-mail describing the situation and are asked to submit a new bid
without additional fees. Table 1 gives summary statistics from the different
auctions that took place during 2003 and 2004. The information provided on the
website includes all auctions that havebeen conducted in this period. The
products auctioned were electronic appliances (computers, TV�s, video games etc)
and gift cards (provided by Target, Shell, Wall-Mart, Starbucks etc). The mean
retail value of the items auctioned was $414.169. The most expensive item
auctioned was a Panasonic 42�� Plasma TV with a retail price of $4999, while the
cheapest item was a Nintendo Game Boy with a retail price of $79.99. The Maximum
Allowed Bid was almost always identical to the Maximum Submitted Bid, which
means that in almost all the auctions the highest submitted bid was the highest
allowed bid.12On average, the maximum allowed bid was 7.2% of the retail
price,13and it had a mean of $30.83. The highest Maximum Allowed Bid, $624.38,
occurred in the case of the Panasonic 42�� Plasma TV, while smallest Maximum
Allowed Bid, $2.94, was in the case of a $100 Starbucks gift card. The average
winning bid was $30.70, and it was, on average, 13.69 cents below the Maximum
Allowed Bid (and the maximum submitted bid). We define Gap as thedifference
between the maximum allowed bid and the winning bid. The minimum of this
variable is 0, which mean that the maximum allowed bid was the winner. The
maximum 11A unique bid is a bid that is not a duplicate bid. A "duplicate bid"
is a bid submitted by a participant in anauction where another participant(s)
has submitted a bid(s) for the identical amount.12There are 15 cases out of 310
in which the highest submitted bid is less than the maximum allowed bid. In 10
cases the difference is 1 cent. 13It seems that the auctioneer choose the
Maximum Allowed Bid such that it will be, on average, less than 10% of the
retail price. An OLS regression of the Maximum Allowed Bid on the retail price
yield a coefficient of 0.072 with standard error of 0.0024 (t-value of 29.59)
and R squared of 0.7398. It seems that the Maximum Allowed Bid is also
positively correlated with the Number of Bids per auction. An OLSregression of
the Maximum Allowed Bid on the Number of Bids yield a coefficient of 0.375 with
standarderror of 0.020 (t-value of 18.37) and R squared of 0.5228. An OLS
regression of the Maximum Allowed Bid on both the Retail Price and the Number of
Bids per auction yields coefficients of 0.1765 on the retailprice and -0.6648 on
the Number of Bids, both significant at 1% level. 9

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