Oliver Hinz. Il-Horn Hann

Transcription

1 REEARCH ARTICLE PRICE DICRIMINATION IN E-COMMERCE? AN EXAMINATION OF DYNAMIC PRICING IN NAME-YOUR-OWN PRICE MARKET Oliver Hinz Faculty of Economics and usiness Administration, Goethe-University of Frankfurt, Grüneburglatz, 633 Frankfurt am Main GERMANY Il-Horn Hann Robert H. mith chool of usiness, University of Maryland, 433R Van Munching Hall, College Park, MD 74 U..A. Martin ann Munich chool of Management, Ludwig-Maximilians-University Munich, Geschwister-choll-Platz, 8539 Munich GERMANY Aendix A Otimal Offers for Two-Offer Case for Market with Fixed Threshold Price A buyer who can make a maximum of two offers maximizes the exected consumer surlus: max EC ( WTP L + ( WTP, U L ( δ (. (A The (unrestricted otimization of equation (A for the two-offer model results in the following equations for the otimal first and second offers: dec dec WTP + L δ ( WTP WTP + L δ ( WTP WTP + δ ( WTP + Mutual insertion yields for the otimal first and second offer: WTP δ + L δ, WTP ( 3 δ ( 4 δ + L MI Quarterly Vol. 35 No. Aendices/March A

2 The model can be generalized to the n-offer case where the offers to n are given by the following equations: ( L δ WTP WTP ( δ + δ + + n WTP. n + for <<n It can also be shown that the model converges since monotonically decreases and n monotonically increases with increasing n and and n are bounded. Aendix eller s Decision in Terms of Offer Accetance We want to show that any seller with valuation lower than s strictly refers to accet the offer now instead of waiting for better offers in the future if a seller with valuation s is willing to accet the offer. Define V(s to be the equilibrium ayoff at time + of a seller with valuation s and let Q(s be the discounted robability of a trade for the seller s. uose a seller with valuation s chooses to accet a offer, then s $ δ V(s must hold. Now consider a seller with a valuation s' < s. We want to show that a seller with valuation s' is also better off to accet the offer now and thus s' $ δ V(s'. The seller with valuation s imitates s' only if the exected ayoff is at least as high as the ayoff he exects when he is following his own equilibrium behavior. Thus V( s V(' s Q('( s s s' and since Q(s' # V( s V( s' Q( s' ( s s' V( s' s+ s' s δ ( V( s' s+ s' δ V( s' + s( δ + δ s' since s' # s( δ + δ s' δ V( s' + s' s' δ V( s' q.e.d. Intuitively, it is clear that a seller with valuation s' (remember s' < s loses more money due to the discount factor than a seller with valuation s and thus the seller with valuation s' will always accet an offer if a seller with valuation s is willing to accet the offer. A MI Quarterly Vol. 35 No. Aendices/March

3 Aendix C ehavior in Markets with N and Infinite Offers and One-ided Uncertainty { } The equilibrium is described as a collection of functions (, (, (, (, where is a robability distribution rere β μ μ ( senting the seller s conectures about the buyer s valuation for the events when the buyer s offer is not in the range of the equilibrium offer schedule. The solution strategy for this bargaining model is to reduce the two-sided uncertainty roblem (seller s valuation, buyer s WTP to a case with one-sided uncertainty where the buyer s WTP is known. ubsequently, the two-sided uncertainty case is solved. For the one-sided case the seller knows the buyer s valuation WTP, but the buyer only knows that the seller s valuation is uniformly distributed on [L, U]. The buyer chooses an offer that maximizes the resent value of current and future surlus, given her knowledge of the seller s valuation and subect to the constraint that the seller will accet the offer only if his valuation is sufficiently low so that the seller is better off acceting now than waiting for higher offers in the future. An offer that is reected rovides learning; - is then used as a new lower bound of the distribution assumtion alying ayes rule. With i eriods remaining in the n-stage bargaining game, define to be n + i, so the buyers chooses to maximize her exected gain u(wtp, - given the seller s valuation is uniformly distributed on [ -, U]: such that u ( WTP, max ( WTP ( + δ ( U u ( WTP, + U b δ ( s + where is the indifference valuation in., This yields the following function for the buyer s otimal offering behavior and the seller s cutoff valuations: ( δ + δc + ( WTP, ( WTP ( δ + δc+ δc+ ( δ + δc + ( WTP ( δ + δc+ δc+ ( δ + δ c + cn ; for i > : c ( δ + δ c δ c with. + + The roof is by induction on n. With one eriod remaining, the buyer wishes to choose according to the following rogram: A layer could leave the equilibrium ath with udating of beliefs not being ossible. Cramton (984 shows how such conectures should be constructed to suort a very similar equilibrium. We can safely assume that WTP $ L, since any buyer with L > WTP would not enter negotiations. MI Quarterly Vol. 35 No. Aendices/March A3

4 so WTP n WTP + n n(, n max [( ( n ] max U n U n u WTP WTP dun( WTP, n WTP + WTP + n n WTP + WTP + u ( WTP, ( WTP ( n n n n n U n ( WTP n 4( U n With i eriods remaining, the buyer s exected consumer surlus is given by u ( WTP, max ( WTP ( + δ ( U u ( WTP, + U Assume by the induction hyothesis that with u ( WTP, c ( WTP U ( WTP, c ( WTP ( δ + δ c + cn ; for i > : c ( δ + δ c δ c + + then ( δ + δ ( c ( WTP + ( δ + δ c ( WTP + (C ubstituting yields u ( WTP, max ( WTP ( δ ( ( ( δ c + WTP + δ c + WTP U max ( WTP( + δ ( ( ( c + δ δ + δc + + δ c + WTP U du ( WTP, WTP( + δc+ δ ( δ + δc+ + ( δ + δc+ δ c+ ( WTP d (C A4 MI Quarterly Vol. 35 No. Aendices/March

5 which has an unique maximum when ( δ + δ c δ c WTP( + δ c δ δ c + ( δ + δ c and for strict concavity of u when c ( + δ δ + δ which is clearly satisfied since + < c, δ, δ < WTP( + δc + δ δc + + ( δ + δc + ( δ + δc+ δc+ ( δ + δc + ( WTP ( δ + δc+ δc+ Then by substituting (C3 into (C and (C we get (C3 and ( δ + δc + ( WTP, ( WTP ( δ + δc+ δc+ q.e.d. ( δ + δ c ( WTP ( WTP + (, c U ( U δ + δc+ δc+ u WTP q.e.d. as required by the induction hyothesis. Fudenberg et al. (985 show that the latter-described equilibrium is a unique equilibrium in the infinite-horizon game. The buyer s equilibrium offer, the seller s indifference valuation, and the exected consumer surlus u for eriod in the infinite horizon game is given by where c ( L WTP d ( L WTP d ( L WTP u c U L ( δ + δc c ( δ + δc δc c d ( δ δ δ + δ c MI Quarterly Vol. 35 No. Aendices/March A5

6 Aendix D ehavior in Markets with Infinite Offers and Two-ided Uncertainty In the case of two-sided uncertainty, the assumtion of a known buyer s valuation WTP is relaxed. The seller only assesses the buyer s valuation to be given by the distribution F(WTP with a ositive density f(wtp on [WTP low, WTP high ]. In this case, the buyer must be concerned about the information that her offer reveals to the seller, and the seller must interret this offer as an indication of the buyer s true willingnessto-ay carefully. The searating equilibrium that distinguishes high valuation buyers from low valuation buyers is achieved through discounting over time. The class of high valuation buyers is described by a valuation that is higher than a certain cutoff value β in : WTP > β. To determine the equilibrium, we must follow an iterative rocedure: First, comute the offer sequence and indifference values that result after the buyer s willingness-to-ay has been revealed according to the case with one-sided uncertainty. Determine the offer sequence for the buyer with the highest willingness-to-ay WTP WTP high and her otimal number of offers. econd, stewise decrease WTP from WTP high by some small amount ΔWTP > so that the buyer WTP is indifferent between offering (WTP and (WTP ΔWTP. With decreasing WTP there will come a oint β at which no seller will accet the offer (β. All buyers with WTP < β will thereby offer too low and in this way signal their low willingness-to-ay. For a buyer with willingness-to-ay WTP > β the value of the subsequent offers can easily be calculated since she already revealed her rivate information. All buyers with WTP < β will wait for subsequent rounds to reveal their true willingness-to-ay. For these buyers, we go back to the first ste and determine the offer sequence and the otimal number of offers for a buyer with willingness-to-ay WTP β. This rocess is reeated until the offers of all buyers WTP [WTP low, WTP high ] are determined. y this rocedure the NYOP seller has disunctive classes of buyers that reveal their true willingness-to-ay in the first round (e.g., buyers with WTP (β, WTP high ], in the second round (e.g.,buyers with WTP (β, β ], and so on and so forth. In equilibrium, high valuation buyers reveal their rivate information by submitting an offer that is strictly increasing in WTP. The seller can then infer the buyer s WTP by inverting the offer schedule (WTP by calculating WTP - (. Proof: uose buyer WTP chooses retending to be buyer WTP shade by offering the offer. This means she is trying to imitate the behavior of a low valuation buyer although she actually has a higher valuation of WTP for the roduct offered. Then her exected consumer surlus is determined by the first offer utilizing L as the starting oint lus all other discounted surluses that make use of the belief being udated by a reection in the receding offering round: n u ( WTPshade, ( L( WTP + δ ( ( WTP U L (D where the future offers and indifference valuations are given by with Thus, c ( WTP d shade shade ( WTP d shade shade ( WTPshade δ + δ c shade ( WTP d ( WTP d shade shade shade shade ( WTPshade d ( d A6 MI Quarterly Vol. 35 No. Aendices/March

7 so that WTP WTP c WTP d ( ( shade shade WTP WTP c ( WTP d shade shade ( ( WTP ( WTPshade d ( d ( WTP WTPshade c ( WTPshade d ( WTP ( d ( WTP WTP d c ( WTP ( d shade shade shade (D ubstituting into (D yields u( WTPshade, ( L( WTP + δ ( ( WTP U L + U ( L( WTP δ ( WTPshade ( d ( WTP WTPshade d c ( WTPshade ( d L ( L( WTP + ( WTP U L sha ( ( ( ( de d δ WTP WTPshade d c WTPshade d Performing the summation for the geometric rogression yields (Remember: q for q < q and thus δ ( WTP WTP d c ( WTP ( d shade shade ( WTP WTPshade ( d c ( WTPshade ( d δ δ δ ( WTP WTPshade ( d c ( WTPshade ( d δ δ δ δ ( WTP WTPshade c ( WTPshade δ d d δ u ( WTP, shade ( L( WTP + ( WTPshade ( d δ ( WTP WTPshade c ( WTPshade U L δ d d δ It can be shown that (d /( δ d² -.5 and (d /( δ d -d, so that we can simlify to MI Quarterly Vol. 35 No. Aendices/March A7

8 u ( WTP, shade ( L( WTP + δ c ( WTPshade d ( WTP WTPshade ( WTPshade U L Differentiating the consumer surlus u with resect to yields the first order condition d L + ( WTP + d dwtpshade d dwtpshade dwtpshade δ c ( WTPshade( d ( WTP WTPshade( ( WTPshade At this oint, the buyer can calculate her otimal offer * by substituting WTP shade WTP. This means the buyer has the same incentives to offer * as to shade her offers. This also imlies dwtpshade dwtp dwtp WTP WTP d dwtp + δ + δc w w, and and results in the first order differential equation d d dwtp dwtp L + ( WTP + δ c ( WTP( + d ( WTP dwtp dwtp WTP dwtp WTP dwtp WTP L WTP + + ( WTP + δ c ( ( + d ( w w w w w ( ( dwtp dwtp w L WTP w WTP w w ( WTP c ( WTP ( dwtp δ d w dwtp ( WTP ( dwtp dwtp w L WTP w w w ( WTP c ( WTP ( dwtp d w dwtp δ + ( WTP dwtp ( w L WTP w w w ( WTP c ( WTP ( δ + d w ( WTP dwtp dwtp dwtp ( + ( δ ( δ dwtp + + dwtp dwtp w L WTP WTP w w w c d w which can be solved to yield (WTP, which is tyically lower than her otimal offer in the case where her valuation is known. A8 MI Quarterly Vol. 35 No. Aendices/March

9 Aendix E Exerimental Instructions Information Given to Particiants urf to the following URL: Please do not use the ack-utton on the navigation bar. It is not allowed to oen any other rogram or window during the exerimental session. Once the website is loaded, you can log into the market latform using username and assword from the disensed cards. You are a rosective buyer who is offering for hyothetical roducts. For these roducts, you obtain information about the interval of the seller s costs given by a lower and an uer bound. Moreover, you suffer from bargaining costs in the form of a ercentage discount of your ayoff. The seller faces oortunity costs as well and is therefore interested in a fast agreement. However, the seller will not sell below his costs and thus the information on the distribution of the seller s cost is imortant. Furthermore, you see the number of offers you already submitted on the roduct and the amount of your last offer. For all roducts you have a resale value. Other buyers can have different resale values for the same roduct. Hereby, other buyers can value the same roduct higher or lower than you do. Overall, there are two different resale values for every roduct and you are randomly assigned to one of these. You also obtain information on the second resale value of the given roduct. However, the seller is aware of these two segments by thorough marketing research and tries to maximize his rofit. The sequence of roducts you are trying to buy is different for all buyers. Exlaining the Mechanism The seller alies a mechanism called NYOP. This means that you as buyer make an offer indicating your willingness-to-ay. If your offer surasses a secret threshold rice set by the seller, you buy the roduct for the rice denoted by your offer. If your offer is below the threshold rice, it will be reected. You can lace another offer after a reection. You can reeat offering until you are successful or you are not interested anymore. Note that you do not comete with other buyers; you solely have to surass the secret threshold rice with your offer. How to Earn Money Your ayoff in the game is the difference between your resale value and a successful offer. If you sto offering for a roduct, your ayoff for this roduct is zero. If you offer more than your resale value, you realize negative rofit. The ayoff deends additionally on the number of offers you laced to get the roduct. Deending on the level of a discount factor, your ayoff is discounted for every offer. There are two different discount factors: Either your ayoff is diminished by 5 ercent er offer or by 5 ercent er offer. If you surass the threshold rice with your first offer, your ayoff is always ercent. The seller suffers from bargaining costs as well. The discount factor is the same for both seller and buyer for a given roduct. The closer you offer to the threshold rice, the higher your rofit. The less offers you need, the higher your rofit. Market and Market 3 do not obtain any further information here. Market : For every roduct, the threshold rice is static and already set and thus does not change. Remember that the seller does not sell below his costs. Market 4: The secret threshold rice is not static but changes due to your offering behavior. You actually haggle with the seller. It is therefore ossible that an offer that was reected in early offer rounds gets acceted later because the seller realized that you really have a low willingness-to-ay. The seller does not learn anything across roducts. The game starts anew with a new roduct. Do not forget that the seller does not sell below his costs. Try to maximize your rofit MI Quarterly Vol. 35 No. Aendices/March A9

10 We will draw lots for winners (/3 from the articiants who we will remunerate with their virtual rofit multilied with some factor. The remuneration will take lace in aroximately two weeks. We will inform the winners about time and location of the remuneration. ubsequent to the offering rocess you will have to answer a questionnaire. Please answer in all conscience. References Cramton, P. C argaining with Incomlete Information: An Infinite-Horizon Model with Two-ided Uncertainty, Review of Economic tudies (5:4, Fudenberg, D., Levine, D., and Tirole, J Infinite-Horizon Models of argaining with One-ided Incomlete Information, in Game Theoretic Models of argaining, A. Roth (ed., Cambridge, UK: University of Cambridge Press. A MI Quarterly Vol. 35 No. Aendices/March