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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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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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