ࡱ> xzw7 objbjUU 7|7|ˑ? l6 6 6 .d $ 4 hwG HT 7Bn`j#(#$$%B%&AAAAAAA$C EA- '&%"%'&'&A/ $$yA///'& 8$ $A/'&A/r/%42@h  JA$_H m *@JATB07B@F/FJA/ R u t c o r R e s e a r c h R e p o r t RUTCOR Rutgers Center for Operations Research Rutgers University 640 Bartholomew Road Piscataway, New Jersey 08854-8003 Telephone: 732-445-3804 Telefax: 732-445-5472 Email:  HYPERLINK "mailto:rrr@rutcor.rutgers.edu" rrr@rutcor.rutgers.edu  HYPERLINK "http://rutcor.rutgers.edu/~rrr" http://rutcor.rutgers.edu/~rrr  SLOW DUTCH AUCTIONS Octavian Cararea Michael Rothkopfb  AUTOTEXT "RRR Number" \* MERGEFORMAT RRR ??-2001,  AUTOTEXT "RRR Date" \* MERGEFORMAT JUNE, 2001 Rutcor Research Report  AUTOTEXT "RRR Number" \* MERGEFORMAT RRR ??-2001,  AUTOTEXT "RRR Date" \* MERGEFORMAT JUNE, 2001 SLOW DUTCH AUCTIONS Octavian Carare Michael Rothkopf Abstract. Theorists have long believed that Dutch auctions are strategically equivalent to standard sealed bidding. However, recently in controlled experiments with actual Dutch and sealed bid Internet auctions of collectables, the Dutch auctions produced significantly more revenue. We believe that this happened, in part, because the Internet Dutch auctions are a slow process in which bidders incur incremental transaction costs if they delay bidding. This paper presents models of slow Dutch auctions that include these costs and explain this belief. Acknowledgements: We thank Colin Campbell, Ronald M. Harstad and Martin K. Perry for useful comments. Introduction Dutch auctions are auctions in which the offering price of the item being sold is set high and then lowered until a bidder makes a bid and wins it. They are a long established auction form, used extensively in the Dutch flower markets. They are considered to be a particularly fast auction form with particularly low transaction costs. The Dutch flower auctions use an electronic device to lower the price, and usually sell lots at a rate of 15 per minute [Kambil and van Heck 1998]. Sometimes, slower processes such as the bargain basement of Filenes department store in which the discount on goods is increased over time until the goods are sold are considered Dutch auctions (for example, see Smith 1987). Recently, slow Dutch auctions of collecting cards have been used on the Internet [Lucking-Reiley 1999]. It has long been claimed by auction theorists that Dutch auctions are strategically equivalent to standard sealed bidding [Vickrey 1961, McAfee and McMillian 1987, Lucking-Reiley 1999]. The basic argument for this equivalence is that in each auction form the bidder faces the same trade off between increasing her profit if she wins and increasing her probability of winning. Controlled laboratory experiments comparing Dutch and standard sealed bid auctions have produced lower, not higher, prices [Kagel 1995]. Thus, it was considered surprising when Lucking-Reiley [1999] reported that in controlled experiments with actual Dutch and sealed bid Internet auctions of collectables the Dutch auctions produced significantly more revenue. One of us has long argued in print that the context and the particular details of auctions are important and affect the kind of conclusions that economists like to draw from simple auction models [Rothkopf, Teisberg, and Kahn 1990; Rothkopf 1991; Rothkopf and Harstad 1994; Harstad and Rothkopf 2000]. We believe that Filenes basement, the Internet auctions that Lucking-Reiley experimented with, and other slow Dutch auctions are significantly different from the auctions of the Dutch flower markets. We also believe, as indeed Lucking-Reiley suggests is possible, that this difference accounts or helps account for the unexpected extra revenue that his auctions produced. This paper develops theoretical models of slow Dutch auctions. First, in Section 2 we develop a rather general decision-theoretic model of the decision faced by a bidder in a slow Dutch auction. Then, in Section 3 we build a simple symmetric game-theoretic model. We find and analyze the pure- and mixed-strategy Nash equilibria of the game and provide a revenue comparison result. Finally, in Section 4 we discuss the relevance of these models to Lucking-Reileys data. The basic ideas behind the models are simple. Imagine yourself entering Filenes basement to shop. If you see an item there that you want to buy, you have a choice. You can buy the item at once at its current price or you can return at some future time in the hope that it will still be there and available at a known lower price. Some aspects of this choice are identical to the one you would face in deciding upon a sealed bid on the item. However, some things are clearly different. Coming back in the future involves delay and an extra transaction cost that you would not incur if you bought now. The sealed bid decision has no such incremental cost of bidding less than the maximum price. In addition, the incremental delay involved in taking position of the item might be significant. Our decision-theoretic model deals with both of these effects. The game-theoretic ones concentrate on the transaction costs although a discount factor on future gains could easily be added to account for one kind of time preference. A Decision Theoretic Model of a Slow Dutch Auction This section considers a decision-theoretic model of the decision faced by a bidder in a slow Dutch auction of a single item. We first formulate a rather general model and then specialize it and investigate it. We assume that the auction starts at time zero with a price of P(0) and that the price P(t) thereafter is non-increasing at every point in time t until the item is sold or it reaches a reserve price R at time Tmax. We assume that the bidder has a value for the item being sold, V(t), that depends upon the time t that she wins it. This value allows for any non-sunk transaction costs that would be incurred by an immediate purchase. In addition, she will incur an incremental transaction cost, TC(t), if she decides to bid at time 0 < t < Tmax even if it turns out that the item has already been sold. She assesses a probability p(t) that the item will still be available at time t. The bidders problem is to choose a time t* that maximizes her expected profit from the auction. This profit is zero if she does not bid, V(0) P(0) if she bids at time 0 and [V(t) P(t)] p(t) - TC(t) if she decides to try to bid at time t, 0 < t < Tmax. This is a fairly straightforward one-dimensional optimization problem, which, if not solvable analytically, should present no computational difficulties. Let us consider a simple special case with a constant value, a linearly declining price, a uniform distribution on the time of the items being sold to someone else, and a time-independent incremental cost of bidding after time 0. Thus, we let P(t) be 1 t for 0 < t < 1; we let TC(t) be a constant c; we let p(t) be the right-cumulative of the uniform distribution on [0, 1], 1 - t; and we let the bidder have a value V(t) = v that is independent of the time she wins the item. If v > 1, then a bid at time 0 is potentially profitable. A bid at time t for 0 < t < 1 has expected profit P(t) = (v  1 + t)(1 - t)  c. Setting dP(t)/dt = 0 and solving for t, yields t = 1  v/2, the best value for t in the range 0 < t < 1. It has an expected profit of v2/4  c. If v2/4  c is negative, then no bid is better than a bid in that range. If v 1 is positive and greater than v2/4 c, then an immediate bid at time 0 is the best course. Otherwise, an attempt to buy at time 1 - v/2 is optimal. If c = 0, then this is a standard Dutch auction. For any positive value less than 2, the bidder bids (if the item is still available) in the range between 0 and 1, and her expected profit is v2/4. Suppose, however, that v = 1.4 and c = 0.1. Then the best expected profit if the bidder does not bid at once is 1.96/4 - 0.1 = 0.39. However, an immediate bid will produce a sure profit of 0.4. Hence, the transaction costs will cause the bidder to be more aggressive in the slow Dutch auction and, since the standard Dutch auction without transaction costs gives the same bid as does the sealed bid auction, to be more aggressive than she would be in that auction. Clearly, risk aversion on the part of the bidder would not change this result since the more aggressive action gives a sure profit rather than an uncertain expectation. In this example, a transaction cost of 0.1 will result in a strictly more aggressive optimal bid for any value in the range from 1.36 to 2. Of course, if the transaction costs are large compared with the bidders value, it may cause the bidder to forego bidding entirely. In this example, a transaction cost of c =0.1 will cause a bidder with a value of 0.63 or less to abstain from bidding at all. Next, we investigate the effect of varying the degree of competition. We do that by varying the probability that the item will still be unsold at time t by setting p(t) = 1 at, 0 < t < min (1, 1/a) for a > 0. With this probability, the best time in the range 0 < t < min (1,1/a) to bid is (1+a)/2a v/2, which yields an expected profit of (a/4)[v2 + 2v(1-a)/a + (1/a 1)2] - c. When a = 0, there is no competition, and the bidder with a high enough value to participate at all will wait until time 1 to bid unless c > 1, in which case she will bid immediately. With a = 2, the best bid with value 1.4 is at time .05 if c < .005 yielding a profit of 0.405 c, or at time 0 yielding a profit of 0.4 if c > 0.005. With a = 2, a bidder with a value between 0 and 0.68 should bid if c = 0 but should not bother to bid at all if c = 0.1 A Game Theoretic Model We next investigate a simple game theoretic model in which randomly arriving buyers who face a known decreasing price schedule choose the optimal time when to attempt to purchase the object. We first show that in the case of no cost of return, the pure strategy Nash equilibrium of the game involves high valuation buyers returning to the auction if they arrived when the price was too high. We also compute a mixed strategy equilibrium of the game. Finally, we show that when the cost of return is non-zero the revenue is an increasing function of the cost of return when the cost is in the appropriate range. Pure strategy equilibrium Suppose that there is one object offered for sale in a Dutch auction in which the price declines according to a price rule P(t) that is known in advance by buyers. There are two risk neutral buyers who arrive randomly at the auction site (extension to the case of more than two buyers is possible but tedious.) We assume that the buyers' arrival times are, for simplicity, uniformly distributed on the [0,1] interval. Buyers' valuations assume the values of v > 0 and 0 with probability ( and 1 - (, respectively. Arrival times are independent of valuations, that is, a high value buyer need not arrive earlier during the auction than a low type buyer. A buyer's private information is summarized by his bi-dimensional type, a pair (ta, V) consisting of ta; the buyer's arrival time and V ( {0, v}, his valuation. A symmetric pure strategy equilibrium specifies a pair of functions (pL(ta), pH(ta)) at which low (zero) and high (v) valuation buyers having arrival time ta wish to purchase the object. In such an equilibrium, a buyer with valuation 0 (the low type' buyer) never gets the object when he competes with a high valuation buyer. To see this, note that, if a low valuation buyer decides to purchase the object before t = 1, his expected surplus is negative. He can, however, choose to purchase the object at time t = 1, when his expected surplus is zero. As it turns out, his decision to return at time t = 1 doesn't influence the equilibrium behavior of buyers with a high valuation who, with probability 1, purchase the object before t = 1. In other words, the tie-breaking rule is irrelevant. We can therefore set pL(ta) ( 1 and focus on the equilibrium behavior of the high valuation buyers. In what follows, we assume for simplicity that the price rule is given by P(t) = v(1 - t) and that arrival times are uniformly distributed on [0, 1].  The pure strategy symmetric equilibrium of the game has a different structure depending on whether ( is greater or smaller then 0.5. We first focus on the equilibrium when ( = 0.5. Consider the following proposed equilibrium in this game. Assume that there exists some  EMBED Equation.3 ( (0, 1) such that, for any arrival time ta >  EMBED Equation.3 , a high type buyer finds it optimal to purchase the object right away (instead of returning at a later time.) To find  EMBED Equation.3 , we set the surplus of a high type who arrives at  EMBED Equation.3  equal to the expected surplus from returning at t = 1, from which it follows that  EMBED Equation.3  = 1 - (. Notice that  EMBED Equation.3 , the point where a buyer is indifferent between purchasing upon arrival or returning at the end of the auction, does not depend on the distribution of arrival times. A high type's equilibrium behavior is characterized by a function pH(() defined as follows  EMBED Equation.3  The purchase function of high valuation buyers pH(t) is discontinuous at t =  EMBED Equation.3 . Buyers who have arrival time close (but less than) t will return to purchase the later in the auction, whereas buyers who have arrival time close (but higher than) t will purchase the object right away. The function r(() : [0, t] ( [0; 1] gives the time of purchase when a high valuation buyer's arrival time is less than t: Suppose that r(() is a strictly increasing function. To find r((), consider the problem faced by a high type who arrives at t < t and chooses to return at time ( > t: Given the assumed equilibrium behavior, the expected payoff to a high valuation buyer who chooses to return at time ( is given by  EMBED Equation.3  The high type wins the object at time ( if the opponent has low valuation, or if a high valuation opponent wishes to purchase the object at a time greater than (. It is useful to note that a player's expected payoff from choosing to purchase the object at time ( does not depend on his arrival time. The expected surplus to a high valuation buyer is his valuation net of the price at time ( multiplied by the probability that the object is unsold at (. The object remains unsold at ( if the opponent has valuation 0 (an event with probability 1 - (), or if a high valuation buyer either shows up between r-1(() and  EMBED Equation.3  (which, given the uniform distribution of arrival times on [0, 1] has probability r -1(() -  EMBED Equation.3 ) or after ( -- an event with probability ( (1 - (). Plugging in the price rule P (() = v(1 - (), it follows that v - P(() = v ( and expected surplus can be expressed as  EMBED Equation.3  Differentiating the above expression with respect to ( and setting the result equal to zero implies  EMBED Equation.3  In a symmetric equilibrium, ( = r(t). Hence, r(t) solves the following initial value problem:  EMBED Equation.3   EMBED Equation.3  (1) The following proposition characterizes the equilibrium behavior of buyers in a slow Dutch auction. Proposition 1 If the probability ( that a buyer has high valuation is less than 1/2, the symmetric pure strategy equilibrium of the slow Dutch auction game is given by:  EMBED Equation.3  (2) Proof. It turns out that it is more convenient to express the differential equation that characterizes r() in terms of its inverse. Letting r-1(x) = R(x), solving the initial value problem is equivalent to finding the solution of the following initial value problem:  EMBED Equation.3  The unique increasing (for ( < ) solution is given by  EMBED Equation.3  The inverse of R() is readily found to be r() given in (2). To show that r() is the maximizer of expected revenue, suppose that a high type buyer with arrival time ta <  EMBED Equation.3  chooses to purchase the object at time (. Since r(ta) solves the differential equation in (1), (dE ((() / d() | (=r(ta) = 0. We want to show that a high valuation buyer's expected payoff is not decreasing (not increasing) in ( for ( < (>) r(ta). We have several cases to consider. First, suppose that the high type buyer chooses to purchase the object at time  EMBED Equation.3  < ( = r(t), with t different from ta. His expected surplus is then  EMBED Equation.3  Clearly, since expected surplus is not a function of a bidder's arrival time, it is equal to vt regardless of the value of t ( [ EMBED Equation.3 , 1]. Suppose next that a high valuation buyer with arrival time ta <  EMBED Equation.3  chooses to purchase the object at a time ( satisfying ta < ( <  EMBED Equation.3 . Since in the proposed equilibrium no buyer purchases the object before  EMBED Equation.3 , the high valuation buyer wins the object with probability 1 and pays v(1 - ( ); his surplus is therefore equal to v ( . Note that since ( <  EMBED Equation.3 , as assumed, a high valuation buyer would never choose to purchase the object before  EMBED Equation.3 . Let us now consider the best reply of a high valuation buyer who arrives after  EMBED Equation.3 . Let ta >  EMBED Equation.3  be his arrival time. If he purchases the object right upon arrival, his surplus is v ta. If he returns at ( > ta, he gets the object at price v ( if he is either facing a low valuation buyer, or if a high valuation opponent arrives later than ( or between r-1 (() and  EMBED Equation.3 . His expected surplus is therefore v( [1-((+(( EMBED Equation.3 -r-1(())]. By the same argument as above this is equal to v(1 -(), which is strictly less than vta. Hence, a high valuation buyer who arrives after t would purchase the object upon arrival. ( Proposition 1 above describes the equilibrium behavior of buyers if the probability that a buyer has high valuation is less than 1/2. We next turn to discuss a mixed strategy equilibrium of the game. Mixed strategy equilibrium In addition to the pure strategy equilibrium described above, an equilibrium involving mixing by high valuation buyers exists in which the distribution of purchase times is identical to the distribution of purchase times in the pure strategy equilibrium (implying that the expected revenue is the same as expected revenue in the pure strategy equilibrium above.) To see this, suppose that a high type buyer chooses a cutoff price p, so that if he arrives at the auction site when price is greater than p, he returns at the auction site when price is p (i.e., at time t = P-1(p)) to purchase the object. Conversely, if he arrives when price is less than the proposed return price p, a high type buyer purchases the object right away. While less intuitive, in what follows it is convenient to find the equilibrium in terms of a return time (the time at which price equals the cutoff price). Finding the mixed equilibrium amounts to finding the distribution of the return times used in equilibrium by the high types. If a high type buyer chooses return time equal to 1, his expected surplus is v(1 - (), that is, he wins the object in the event that the opponent is of low type. If he chooses return time t < 1, a high type buyer wins the object in the event that his opponent is of low type (an event with probability 1 - (), if a high valuation opponent arrives after t (an event with probability ( (1 - t)), or if a high type who arrived before t chose a return time greater than t -- an event with probability ( t (1 - F (t)). Therefore, a high type buyer's expected surplus at t is vt (1 - ( t F(t)), where F( ) is the cumulative distribution of return times chosen in equilibrium by the high type buyers. Any strategy that is part of the mixed equilibrium should yield the same payoff to the high type buyer, so in a symmetric equilibrium  EMBED Equation.3  (3) with support on [ EMBED Equation.3 , 1]. Since the distribution of return times is continuous, return time t = 1 is chosen with probability zero in equilibrium, therefore a high type buyer will never lose the object to a low type. In order for F to characterize the mixed strategy equilibrium, it needs to be increasing. It is easily checked that, for ( < , F( ) is strictly increasing. Denote by tp ( [ EMBED Equation.3 , 1] the time at which a high type buyer wishes to purchase the object (for brevity we shall refer to it as the purchase time, although it is not really the time at which a purchase is made.) Given the way in which the equilibrium is constructed, the high type buyers' purchase time is the maximum between their arrival and return times, therefore G( ), the c.d.f. of tp is given by:  EMBED Equation.3  The distribution of purchase times in this symmetric mixed strategy equilibrium, as the next section shows, is the same as the distribution of purchase times in the symmetric pure strategy equilibrium. More importantly, notice that the distribution of purchase times does not depend on the distribution of arrival times, as long as its support remains [0, 1]. To see why, suppose that the distribution of arrival times is T(). The distribution of return times chosen in equilibrium by the high valuation buyers becomes EMBED Equation.3 ; therefore the distribution of purchase times is the same as the one found in (14). We investigate in the next section the properties of the equilibrium in the more interesting case when ( > 0:5. The case of ( > The equilibrium presented in Proposition 1, as well as the mixed strategy equilibrium discussed above are well defined only if ( is less than or equal to 1/2. Both the equilibrium characterized by Proposition 1 and the mixed strategy equilibrium presented above are payoff-equivalent and induce the same distribution of purchase times. It is possible to characterize the symmetric equilibrium in the case ( > 1/2 using both the pure strategy and the mixed strategy equilibrium; for ease of exposition in what follows we focus on the derivation of a symmetric mixed strategy equilibrium. We assume that all the data of the problem remain the same as above, except that the probability that a buyer has high valuation is greater than 1/2. Suppose that there exist  EMBED Equation.3  and  EMBED Equation.3  satisfying 0 <  EMBED Equation.3  <  EMBED Equation.3  < 1 such that the distribution of return times chosen in equilibrium by the high valuation buyers has support on [ EMBED Equation.3 ,  EMBED Equation.3 ]. If a high type buyer chooses return time t, his expected surplus is given by E((t) = vt[1-( EMBED Equation.3 ]. The quantity in square brackets is the probability that the object is not sold at time  EMBED Equation.3 . The following proposition characterizes the distribution of return times in a mixed strategy equilibrium. Proposition 2 If ( > 1/2 the distribution of equilibrium return times is given by EMBED Equation.3 , with support on [ EMBED Equation.3 ]. Proof. Since any strategy that is part of the mixed strategy equilibrium yields the same expected surplus to a buyer, the distribution of equilibrium return times chosen by the high valuation buyers satisfies  EMBED Equation.3  Solving for F ( ) from the above equation yields  EMBED Equation.3  (4) The lower boundary of the support of return times satisfies  EMBED Equation.3  =  EMBED Equation.3 (1-( EMBED Equation.3 ) and clearly 0 <  EMBED Equation.3  <  EMBED Equation.3 . It is easily checked that F ( ) defined above is strictly increasing. We can recover the equilibrium distribution of return times by plugging in  EMBED Equation.3  = 1 in (4). While when ( < 1/2 the support of the distribution of return times doesn't change when the distribution of arrival times changes, when ( > 1/2 the support of the distribution of arrival times does change with the distribution of arrival times. The mixed (as well as the pure strategy) equilibria in the two cases (( greater or less than 0.5) have similar properties. High valuation buyers choose the upper boundary of the support of return times so that their expected surplus is maximized. The value of t that maximizes expected revenue is t =  EMBED Equation.3 , which for ( > 1/2 is strictly less than 1. It follows that the lower boundary of the support of F( ) is  EMBED Equation.3  = EMBED Equation.3 .( Notice that if ( < 1/2 the high type buyers would wish to choose  EMBED Equation.3  = 1/2( > 1 to maximize expected surplus; since the support of arrival times is [0, 1] the high valuation buyers would optimally set  EMBED Equation.3  = 1 (because expected surplus is increasing on [0, 1] whenever ( < 1/2.) In addition, as ( increases from 1/2 to 1,  EMBED Equation.3  decreases from 1 to 1/2 and  EMBED Equation.3  decreases from 1/2 to 1/4. Proposition 3 When  EMBED Equation.3  > 1/2, the symmetric pure strategy equilibrium for the high valuation buyers is characterized by  EMBED Equation.3  The proof is similar to the proof of Proposition 1. Useful in the computation of expected revenue is the inverse of pH(ta) for ta  EMBED Equation.3 , given by  EMBED Equation.3  Having computed the equilibrium for any (, in the next section we focus our attention on expected revenue. Expected revenue To compute expected revenue, we first find the distribution of purchase times. Recall that not all high type buyers choose to return to the auction site; in particular, only high type buyers with arrival time less than t choose to actually return. If a high valuation buyer arrives after t, he purchases the object at the time of arrival. Letting t denote a high type's arrival time, he wishes to purchase the object at time tp, given by  EMBED Equation.3  Therefore, the distribution of his intended purchase times is given by:  EMBED Equation.3  EMBED Equation.3  (5) It is useful to note that when ( < 1/2 (i.e., when  EMBED Equation.3  = 1), G(tp) is equal to F(tp)tp (and therefore it is the same as the distribution of purchase times in the mixed strategy equilibrium above.) Expected revenue can be computed using  EMBED Equation.3 , where t2:2 denotes the minimum of two i.i.d. draws from a distribution with c.d.f. given by (5). If a high and a low type show up, expected revenue is the same as the expectation of the price at the time of purchase. If two high types show up at the auction, expected revenue is the price evaluated at the expected value of the minimum of two draws from a distribution with cumulative G. Letting G1(t) = 1 - (1 - G(t))2 be the c.d.f. of the minimum of two purchase times, expected revenue may then be computed by evaluating  EMBED Equation.3  (7) If ( ( 1/2 the above expression simplifies to ER((; v) = (2v. Note that expected revenue is linear in v, as expected, and that it is increasing at an increasing rate in (. Hence, the auctioneer would prefer to increase the proportion of consumers who value the object at v instead of increasing (at the same cost perhaps) the high type buyers' value. If ( > 1/2 the two integrals in (7) need to be broken down on [ EMBED Equation.3 ,  EMBED Equation.3 ] and [ EMBED Equation.3 , 1] and expected revenue simplifies to ER((, v) = v(( - (2/3 12 (). We assumed above, mainly for ease of computation that the price rule is given by P(t) = v(1 - t). In the next subsection, we investigate the effect on the equilibrium behavior of high valuation buyers of a different price rule. Alternative Price Rules In what follows, we assume that ( ( 1/2. Suppose that instead of the price rule P(t) = v (1 - t) that we previously specified, the auctioneer sets a price rule of the form P(t) = b(1 - t). When the initial price b is less than v the intuition about the expected revenue comparison result below is clear in light of the mixed strategy equilibrium. The proposition below characterizes the effect on expected revenue of such a change. Proposition 4 Changing the price rule to P (t) = b(1 - t), with b satisfying v( < b, has no effect on expected revenue. Proof. Note first that earliest possible time of purchase becomes  EMBED Equation.3  = 1 - v(/b. Following the steps involved in the computation of (11), we have R(x) = 2 - v(/b x b(1-x)(1-()/(((v-b(1-x)). The distribution of purchase times becomes G(tp) =1-b(1-t)(1-()/(((v-b(1-t)). While this is not the same as (5), expected revenue remains the same as expected revenue when the price rule is P(t) = v(1 - t). ( The intuition about the above result is clear in light of the mixed equilibrium outlined earlier. Essentially, in that equilibrium a buyer chooses a cutoff price p, so that if he arrives at the auction site when price is greater than p he returns when price is p, whereas if the price at the time of arrival is less than the cutoff, he purchases the object right away. The fact that the price rule changes (within limits that make any cutoff price p possible) does not alter the behavior of the high type buyers. That is, a high type buyer will choose the same randomization over the cutoff price p, if there exists some t such that P(t) = p: It is useful to note that a lower expected time of a sale implies a higher volume sold per unit of time. Given the revenue equivalence between any linear price rule of the form P(t) = P(0)(1 - t) for P (0) ( (v, a seller wishing to maximize revenue per unit of time would optimally set the starting price P(0) at (v in order to induce buyers to purchase the object upon arrival. To compute the expected time of purchase, note first that expected revenue is the same as the expected revenue computed in above, equal to (2 v. It follows that the expected time of purchase is E [tp] = 1 - (. While expected revenue remains the same, the minimum expected time of purchase corresponds to a starting price in which P(0) = (v. We next turn to investigate the effect of a fixed cost of return on expected revenue. Fixed Cost of Return Suppose that buyers can either purchase the object at arrival, or, by incurring a fixed cost c, they can return at a later time (when price is lower.) This fixed cost is not a cost of waiting; rather, the following interpretation provides a more clear intuition. If the auction goes on for a long period of time, a buyer arriving when the price is relatively large wishes to return at a later time. We assume that staying at the auction site is not an option, therefore if the buyer returns he does so by incurring a fixed cost (perhaps a transportation cost.) We compute below the equilibrium in the game with a fixed cost of return and provide a revenue comparison result. Before computing the equilibrium in the game with a positive cost of return, we investigate the properties of equilibrium in the somewhat trivial case when the cost of return c is greater than v(1 - (). Clearly, no buyer will choose to return at the auction site (if it did, he would have a negative surplus.) Hence, the simple decision rule for the high type buyers in this case is to purchase the object at the time of arrival. Expected revenue may be computed using the formula above to yield Rt = (v(3 - 2()/3. For ( < 3/5 expected revenue is greater than in the case of no cost of return. This example provides the intuition for the revenue comparison result at the end of this section. We now turn our attention to the less trivial case when the cost of return c satisfies c < v(1 - (). The steps involved in the computation of symmetric pure strategy equilibrium with a fixed cost of return are similar to the ones used above for computing r(). The only major difference is that the cutoff point  EMBED Equation.3 , at which a high valuation buyer is indifferent between purchasing the object right away or returning at time t = 1 (and incurring the fixed cost c) is now given by  EMBED Equation.3 c = 1 - ( - c/v <  EMBED Equation.3  (we use the subscript c to emphasize its dependence on the cost of return.) Replicating the steps above, it is easily checked that symmetric pure strategy equilibrium remains the same as the equilibrium computed in section 3.1. It can be easily shown that rc (0) >  EMBED Equation.3 c (the return function remains the same as in section 3.1, except that the cutoff point  EMBED Equation.3  changes; we use the c subscript on both the return function and its inverse to emphasize its dependence on the cost of return.) The distribution of purchase times is  EMBED Equation.3  Observe that no high valuation buyer will purchase the object before  EMBED Equation.3 c (or, alternatively, that all high valuation buyers who arrive between 0 and  EMBED Equation.3 c choose to return and incur the fixed cost c.) The following proposition characterizes the effect that the fixed cost has on expected revenue. Proposition 5 Expected revenue in the slow Dutch auction is an increasing function of the fixed cost of return c. Proof. Let v(1 - () > c1 > c2 > 0. We first show that the distribution of purchase times when the fixed cost is c2 first order stochastically dominates the distribution of purchase times in the game with fixed cost c1. To see this, note that, for t > max{rc1(0), rc2(0)} we have Gc1(t) = Gc2(t) = G(t). Clearly,  EMBED Equation.3 c1 <  EMBED Equation.3 c2. Since, for (< , dG(t)/dt = 1- (/((t2) > 1, the desired result follows. ( It is interesting to see the effect of supposing that the cost c above has the interpretation of a fee charged by the auctioneer whenever a buyer returns to the auction site. In order to maximize revenue, a seller would impose a fee that is equal to v(1 - (), thus inducing high valuation buyers to purchase the object immediately upon their arrival. Discussion of Results Until now, auction theory has contained little discussion of transaction costs. Such discussions as it does contain are focused on the costs of participation in auctions. As far as we know, there is no prior theory on the effect of transaction costs on the relative revenue performance of different auction forms. (Nor, as far as we know, is there any laboratory experimental work that explores this issue.) In this paper, we present both decision theoretic and game theoretic models of the effect of a transaction cost, a cost of bidding later, on the price paid in Dutch auctions. From these models, it is clear that such a cost can raise, not lower, the price received these auctions. Thus, these models offer a potential explanation for Lucking-Reileys unexpected finding in controlled experiments in real Internet auction markets. Transaction costs are not the only potential explanation for Lucking-Reileys results. In particular, if the items he was selling were declining in value to the bidders, the immediacy of purchase available in a Dutch auction could account for his results. Our decision theory model includes this effect, and our game theoretic models can be modified to include an important form of it. Bidding models in which bidders are impatient may also explain some of the revenue results that we provided. However, Lucking-Reileys experiment was not designed to sort out whether the immediacy effect is due to declining value to purchasers or to the ability of an immediate purchase to avoid transaction costs. References Harstad, Ronald M., and Michael H. Rothkopf, 2000, "An 'Alternating Recognition' Model of English Auctions, Management Science 46(1), pp. 1-12. Kagel, John H., 1995, Auctions: A Survey of Experimental Research, in John H. Kagel and Alvin E. Roth, Eds., The Handbood of Experimental Economics, Princeton, NJ, Princeton University Press, Chapter 7, ppl. 501-585; reprinted in Klemperer, Paul H., Ed., 2000, The Economic Theory of Auctions, Vol. II, Elgar, Northampton, MA, pp. 601-686. Lucking-Reiley, David, 1999, Using Field Experiments to Test Equivalence Between Auction Formats: Magic on the Internet, American Economic Review 89(5), pp. 1063-1080. McAfee, R. Preston, and John McMillan, 1987, Auctions and Bidding, Journal of Economic Literature 25(2), pp. 699-738. Rothkopf, Michael H., 1991, "On Auctions with Withdrawable Winning Bids," Marketing Science 10(1), pp. 40-57. Rothkopf, Michael H., and Ronald M. Harstad, 1994, "Modeling Competitive Bidding: A Critical Essay," Management Science 40, pp. 364-384. Rothkopf, Michael H., Thomas J. Teisberg, and Edward M. Kahn, 1991, "Why Are Vickrey Auctions Rare?" Journal of Political Economy 98(1), pp. 94-109. Smith, Vernon L., 1987, Auctions, The New Palgrave Dictionary of Economics, Vol. 1, J. Eatwell, M. Milgate and P. Newman, Eds, pp. 138-144. Vickrey, William, 1961, Counterspeculation, Auctions, and Competitive sealed Tenders, Journal of Finance 16(1), pp. 8-37. a Department of Economics, Rutgers University, 75 Hamilton St., New Brunswick, NJ 08901-1248 b RUTCOR and MSIS Department, Rutgers University, 640 Bartholomew Road, Piscataway, NJ 08854-8003  It is straightforward to modify what follows if the bidder can avoid the incremental transaction cost if the item is already sold.  While in this section we are just considering the bidders problem, it is worth noting that the bid taker will care a great deal more about the effect on bidders with high values than the effect on bidders with low ones.  We deal with more general price rules in the next section.  The domain of R is [ EMBED Equation.3 ,1]. It can be checked that R is increasing on its domain.  Recall that in the mixed strategy equilibrium discussed above a buyer chooses a return time t so that if his arrival time is less than t he returns to purchase the object at t, whereas if his arrival time is greater than or equal to t, he purchases the object upon arrival. 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"B S  ?.; YJ T OLE_LINK2 OLE_LINK3 OLE_LINK1&\\-\\29=C ' 9 B Q W O W Y a r z  X`SW#%P+R+e+g+++++++<,>,....11??_?a?????@ @@@gAiAAA5B7B5D7DDDDDFFMMPP RRRRYYccccZf\ftgvgggggchhhhhiipp;v=v{{Z\`bW_")EM '//7]iȏ)1IQv}#*:=LSewˑNSHOegFH!&vx &4>|####$$g%k%''((g+h+++d,r,..h/j/22x3{33344558899::::W=Y===x>z>>>]?_?@@@@gAiABBDDEEEEG!GII%J(JKKIMJMMMNN\P_PQRRRSS8V;VYYyZZ(\+\]]``ccrgtg=hBhjhohhhiiiijjllllmmoowpypppcqeqssttuvvvzz{{||}}ՃكZ] "/ˑZ\ɕΕ33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333Ying LiuGC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of rrr.asdYing LiuGC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of rrr.asdYing LiuGC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of rrr.asdYing LiuC:\Users\Mike\rrr.dotYing LiuC:\Users\Mike\rrr.dotOctavian CararefE:\Documents and Settings\Octavian Carare\Application Data\Microsoft\Word\AutoRecovery save of rrr.asdOctavian Carare E:\rrr.dotOctavian CarareE:\slow_dutch.dotOctavian CarareE:\slow_dutch.dotMichael RothkofC:\My Documents\slow_dutch.dotO 87oK?ӨPJ!h,kVEp0bG t@F ^`o(.^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.^`o(. \ ^ `\o(..808^8`0o(...808^8`0o(.... ^`o( ..... ^`o( ...... ^`o(....... `^``o(........ `^``o(.........^`o(.^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.@@^@`o( ^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.@@^@`o(0^`0o(.0^`0o(..``^``o(... ^`o( .... ^`o( ..... ^`o( ...... `^``o(....... 00^0`o(........kVEp t7oK?J!hO                            Bn        %%ˑZ@ʑʑpʑʑ(P@P PD@UnknownG:Times New Roman5Symbol3& :ArialAMath1Symbol5& :Tahoma?5 :Courier New#qh@Vf@VfQf7x=!20d 2Q1MS-Word Template for RUTCOR Research Reports v1.0Mikhail NediakMichael Rothkof iD@D Normal$CJ OJPJQJ_HaJ mH sH tH <A@< Default Paragraph FontF6;?@CHGJiY^b fh{lr8"QUWXY[\]_`bcdfgi7 bjbjUU 7|7|lFF?G?G?G?G?G$cGcGoGwGwG HTcGHnkHkHkHkHkHkHkHkHkHmHmHmHmHmHmH$ZJ zLH?GkHkHkHkHkHHkH?G?GkHkHHkHkHkHkH?GkH?GkHkHkHkHkHkHkHkH?G?GiGkHkH_H mcGcGkHkHkHH0HkHzLkHzLkHkHcGcG?G?G?G?G D 00&P1h/ =!"#$x% P0  D 00&P1h/ =!"#$x% P0  jl+.AUL`ak2|3RTVZ^aehkmS@ʑʑpʑʑP@UnknownG:Times New Roman5Symbol3& :ArialAMath1Symbol5& :Tahoma?5 :Courier New"0h0