US 20100161388 A1 Abstract An embodiment in accordance with the present invention provides a method for determining a demand function for an item. For example, the method includes determining a first estimate of the demand function for the item by utilizing a first auction having a first set of auction parameters. Additionally, the method includes determining a second estimate of the demand function for the item by utilizing a second auction using auction data from the first auction. The second auction has a second set of auction parameters based on the first estimate of the demand function.
Claims(13) 1. A computer readable storage medium having readable code embodied therein for causing a computer system to perform operations comprising:
(a) determining a sample size for estimating a demand function for an item; (b) configuring first auction parameters for a first auction to minimize cost of collecting said sample size; (c) determining a first estimate of said demand function for said item utilizing bid data of said first auction; (d) in response to a pre-established criterion not being satisfied, determining a second estimate of said demand function by utilizing a second auction, said second auction having second auction parameters based on said first estimate of said demand function. 2. The computer readable storage medium as described in 3. The computer readable storage medium as described in 4. The computer readable storage medium as described in 5. The computer readable storage medium as described in 6. The computer readable storage medium as described in 7. A computer-implemented automated demand estimator system for estimating demand of a given item, comprising:
a sample size determiner that determines a minimum auction sample size for estimating said demand for a specified accuracy; an auction configurer that configures auction parameters to minimize cost of collecting said sample size for estimating said demand; and a demand function estimator that generates a demand function estimation for said item utilizing bid data of an auction having said auction parameters. 8. The system as described in 9. The system as described in 10. The system as described in 11. The system as described in 12. The system as described in 13. The system as described in Description This application is a divisional of and claims the benefit of co-pending U.S. patent application Ser. No. 10/175,661 filed on Jun. 19, 2002, entitled “Determining a Demand Function for an Item” and is assigned to the assignee of the present invention, and is hereby incorporated by reference in its entirety herein. The present invention relates to the field of determining a demand function for an item and/or a product. A seller or marketing research provider is frequently faced with the challenge of designing an appropriate pricing strategy for a new product or an existing product undergoing change in market conditions. Such a pricing strategy usually requires reliable information on cost and demand. Reliable information on cost is usually readily attainable. Reliable information on the demand function, on the other hand, may not be so readily available. The seller or market researcher may be able to use historical price and volume information, when available, to obtain an estimate of the demand function. However, with new products or existing products undergoing significant changes in the market structure, historical price and volume information is rarely available. In order to design a pricing strategy for a new product or an existing product undergoing significant change, the seller or marketing research provider might decide to employ a survey or a field test-marketing technique. However, surveys are of limited value in that it is difficult to control the pool of participants, and the participants bear no consequences for their responses. Field test-marketing techniques are very costly to employ and, similarly, are of limited value because of limited experimental control. Another option that can be used to determine a viable pricing strategy is an econometric analysis of historical market data. In some cases historical market data may exist in the form of bid data from standard auctions. However, econometric analysis of historical market data is also of limited value because there is likely to be limited applicable data and because strong modeling assumptions are needed to identify the underlying demand function under the current market conditions. One conventional method for estimating the demand function utilizes an analysis of bid data acquired from standard auctions. Attempts to estimate the demand function from bid data acquired from standard auctions have two main disadvantages. First, standard auctions are not necessarily designed and conducted with demand estimation in mind. Therefore, the data from such auctions are rarely ideal for demand estimation. Thus, additional data gathering effort is needed to obtain the information on covariates that are crucial for reliable structural estimation. Also, it is often necessary to make unpalatable behavioral assumptions regarding any inference based on bids from standard auctions. For example, the risk concerns which factor into the bidders' decisions for most standard auctions make it difficult to discern the price that they are truly willing to pay. Thus, there exists a need for an apparatus and a method for extracting the underlying demand function for new products and for existing products that are subject to new market conditions. An embodiment in accordance with the present invention provides a method for determining a demand function for an item. For example, the method includes determining a first estimate of the demand function for the item by utilizing a first auction having a first set of auction parameters. Additionally, the method includes determining a second estimate of the demand function for the item by utilizing a second auction using auction data from the first auction. The second auction has a second set of auction parameters based on the first estimate of the demand function. The accompanying drawings, which are incorporated in and form a part of this specification, illustrate embodiments of the invention and, together with the description, serve to explain the principles of the invention. System It should be understood that the conducted auctions During auctions The invited bidders (e.g., The computer system that embodies the automated demand estimator system The automated demand estimator system The automated demand estimator system The auction configurer The auction configurer The demand function estimator The report generator Network Network At step The total number of actual participants will vary depending on the auction parameters such as number of invited participants in each auction, number of auctions to run, number of items to offer per auction. The goal is to find optimal values for these parameters so that the number of participants is maximized under user-specified constraints such as budget, time, total quantity, and other system constraints such as the maximum number of auctions per period, and the maximum number of bidders per auction. In one embodiment, the optimum number of participants selected to participate in the auction is not known at the time of the first Vickrey auction, so an initial estimate of the number of participants is used. The initial estimate of the number of participants becomes refined through the iterative process of holding additional Vickrey auctions and gathering additional data. At step and the reserve prices and any other parameters of the auctions to be conducted. Auctions can be “self-financing” in that the profit generated from selling the items can be used to run more auctions at a later time. It might be beneficial to offer more than one item per auction so as to extract additional revenue. However, offering more items in an auction will lead to a lower selling price. Therefore, it is desirable to determine the number of items to sell at each auction that provides the maximum profit. In the present embodiment, the auction parameters begin as an estimate and are reconfigured for subsequent auctions based on results and parameters from previous auctions. The number of items to sell that provides the maximum profit is one parameter that is determined through such an iterative process. In step A sealed-bid-second-price auction is the simplest form of a Vickrey auction for a situation where a single item is available for sale. In a sealed-bid-second-price auction the highest bidder wins the item and pays the second highest bid price. In the present embodiment, selected participants form a representative sample of the set of potential buyers. Therefore, their bids reflect an accurate picture of the potential demand function for the product. In a case where the demand for a product is such that the buyers demand different quantities, a multi-unit version of the Vickrey auction, which is a “price equals highest rejected bid” version, may be used. In the multi-unit Vickrey auction, the method of inducing “truth telling” may be to ensure that the price paid by the bidder be equal to the price that would clear the market if this bidder's demand were to be removed from the market. This eliminates the situation in which buyers wishing to buy more than one unit might be tempted to misrepresent their demand if all available units were to be sold at a uniform price equal to the highest rejected bid. As shown by step In step According to one embodiment of the present invention, as shown in step When the “stop rules” are satisfied at step At step Auction parameters may include the following: the auction format to be used; the number of auctions to be conducted; the reserve prices in the auctions to be conducted; the number of units available in each auction to be conducted; the set of participants in each auction to be conducted; the timing and duration of each action to be conducted; and the set of products available in each auction to be conducted. In the present embodiment, the user first specifies the item for which the demand is to be estimated as well as the target market in the form of observable buyer characteristics. The user also provides operational information on terms and conditions, such as payment terms, delivery options, etc. that are common to all of the auctions that are to be conducted. In addition, the user specifies the starting date (Start Date) for the test-marketing program and the user specifies the total duration of the test-marketing program (Time ( In the present embodiment, the user specifies the time frame of demand to be estimated (Time_Frame); e.g., he/she states if the relevant demand is quantity demanded per day, per week, per month, etc. The user also specifies the total duration of the test-marketing program (Time), e.g., a month, three weeks, two months, etc. In the present embodiment the number of time periods (T) is determined by the equation T=Time/Time_Frame. In the present embodiment, the number of time periods is at least two time periods. In each period a number of auctions are conducted simultaneously. The reason for having two or more time periods is threefold. The first reason is to exploit potential savings from sequential sampling. More particularly, sample information from early periods may provide sufficient accuracy for the estimation of the unknowns so that it may not be worth the further costs of getting additional information. Second, the design decisions (e.g., size of the sample, number of auctions and auction parameters) are based on current information. Initial information about the unknowns is coarser than the information in later periods. Thus, a sequential approach makes it possible to refine and improve the design decisions during the course of the data collection process. Finally, there may be random temporal variation in the unknown elements. In this case, sequential sampling allows one to control for such time effects. In the following discussion, time periods are indexed by the variable t. In this embodiment, for periods t=1 to T, the following notation is used to describe the present invention: -
- Revenue
_{t}=Revenue collected in period t from auction sales. - Cost
_{t}=Total cost incurred in period t for conducting auctions and inviting bidders as well as supplying the number of units sold in the auctions. - Budget
_{t}=Budget amount available at period t=Budget_{t-1}+Revenue_{t-1}−Cost_{t-1}. - x
_{t}=total quantity sold in all auctions at period t. - Q
_{t}=total quantity available at period t=Q_{t-1}−x_{t-1}. - q
_{t}=total number of units auctioned at period t. - S
_{t}=target total sample size calculated on the basis of information available up to, and including, period t−1. - s
_{t}=the size of the sample collected at period t. - ê
_{t}=Estimate of auction participation cost, e, based on information available up to, and including, period t−1. - {circumflex over (F)}
_{t}(.)=Estimate of the distribution of valuations, F(.) based on information available up to, and including, period t−1. - a
_{t}={circumflex over (F)}_{t-1}(p_{L}), value of the empirical distribution function based on observations up to period t−1, evaluated at p_{L}, lower end of the relevant price range. - b
_{t}={circumflex over (F)}_{t-1}(p_{H}), value of the empirical distribution function based on observations up to period t−1, evaluated at p_{H}, upper end of the relevant price range.
- Revenue
In the present embodiment, the user provides an initial program budget (Budget (0)) for test-marketing and demand estimation purposes and the user specifies the total number of units available for test-marketing and demand estimation purposes (Quantity (0)). The user also specifies the cost that has to be incurred to set up an auction (Cost_per_Auction). This includes the listing and other transaction fees charged by an auction service provider if a third party service provider conducts the auctions. Moreover, the user specifies the cost incurred to recruit or maintain a subject (Cost_per_Bidder) and the cost of making a unit of the item available for sale (Cost_per_Unit). In addition, the user specifies the range of relevant prices under consideration (Price_Range) such as, for example, an interval [p If the user has an initial estimate of the demand function based on previous marketing studies on the same or similar products, it can optionally be specified (Initial_Estimate). If available this estimate is used to configure the initial stage decisions on the number of auctions, the number of bidders and the number of units in each auction. In the present embodiment, the following constraints are used for the following decision variables (the number of auctions, the number of bidders per auction, the number of units per auction, reserve price in each auction) should satisfy the following user-specified bounds: -
- a. 0<A<=MaxA=maximum number of auctions per period
- b. 0<n<=MaxB=maximum number of bidders per auction
- c. 0<k<=MaxU=maximum number of units per auction
- d. MinR<=r<=MaxR, bounds on the reserve price.
With regard to pricing of items to be sold at auction, in the present embodiment, various price levels are a priori ruled out as irrelevant to potential decisions. For example, if the demand estimates are to be used in a pricing decision and the unit production cost is expected to stay at its current level, then demand at prices below the current unit production cost is obviously irrelevant. In this case, the user may specify the price range as an interval with a lower bound that is no smaller than the current unit cost. In other situations, the relevant price range may include price levels below the current unit production cost. An example of this may be a situation where the unit production cost at the test-marketing phase may be higher than the unit production cost that would be available in the actual marketing phase. For instance, the objective of demand estimation may be to determine whether or not it is worthwhile to invest in a technology that reduces unit production costs from its current level. The user also specifies a target accuracy (Target_Accuracy) for the demand estimate in terms of two parameters, a precision level, δ, and a significance level, α. These parameters represent the condition that the relative discrepancy between the estimated and actual demand functions is less than δ with probability at least 1−α. Alternatively, the user may state a target accuracy for the estimated demand in terms of the size of the sample. It is understood that the target accuracy may be measured by a precision parameter, δ, and a significance level α. In the present embodiment, the initial demand function, is represented by the unknown distribution function F(x) of a random variable X, is represented by the function {circumflex over (F)} In this embodiment, the precision parameter, δ, can be specified in two alternative ways. In the first case, the precision parameter, δ, represents the upper bound on relative deviation of {circumflex over (F)} [{circumflex over (F)} For a sample size s and a subinterval [a, b] in the unit interval, if the precision criterion is given in terms of relative deviation, Pr(s, δ, a, b) represents the probability that the empirical distribution {circumflex over (F)} If the precision criterion is given in terms of absolute deviations, Pr(s, δ, a, b) represents the probability that the empirical distribution {circumflex over (F)} In both of the above formulations, the numbers a and b are related to the user specified range of relevant price levels, [p The user-specified accuracy criterion (δ,α) can be converted to a sample size s(δ, α, a, b) where s(δ, α, a, b) is the minimum sample size that satisfies the relationship Pr(s(δ, α, a, b), δ, α, b)=1−α. For this specification of the accuracy criterion, the task is to minimize the total cost of collecting a sample of size s(δ, α, a, b) by selecting the auction parameters. In an alternative embodiment, when a significance level a, and a subinterval [a, b] in the unit interval are given, the sample size s is selected to minimize δ(s, α, a, b) subject to a budget constraint, where δ(s, α, a, b) is the best precision that can be achieved with a sample of size s with probability 1−α, e.g., the solution to the equation Pr(s, δ, a, b)=1−α. In an alternate embodiment Pr (s, δ, α, b) is maximized by selecting s subject to a user-specified budget constraint. In step In the present embodiment, the initial estimate of the demand function F In step Thus, an auction with n invited bidders provides a sample of size n from the unknown distribution F(.) provided all invited bidders choose to submit a bid. If the auction induces only some of the bidders to submit bids, then the observed bids reveal only partial information about the distribution function to be estimated. A plausible assumption is that, due to costs involved in evaluating the auctioned item and submitting a bid, only those bidders with valuations greater than some particular level choose to participate in the auction. If this is the case, then the observed bids form a sample from the distribution F(.) truncated on the left. The truncation point is expected to depend on the auction parameters such as the number of bidders, the number of units sold, the reserve price, etc. Each bidder has an auction participation cost, e, which he/she has to incur to evaluate the item, and to prepare and submit a bid. A bidder with valuation v and participation cost e participates in an auction if the expected payoff from participation exceeds the cost of participation. Such a bidder's expected payoff from the auction depends on the number of units for sale and the number of potential rivals. For demand estimation purposes, a bidder participates in an auction with n invited bidders and k units for sale if and only if his valuation of the item exceeds a critical valuation level, denoted as v To determine v The critical valuation v
Thus, the critical valuation v At step In step A simple estimate of v If an estimate of the participation cost e, say ê, and an estimate of F(v), say {circumflex over (F)}(v), are available, then one can obtain an alternative estimate of v
for parameter configurations (n, k, r) that do not appear in the sample with sufficient frequency this alternative approach is going to be adopted to estimate v In the present embodiment participation cost is estimated as follows. An upper bound for the auction participation cost, e, can be estimated by using estimates of the unknowns, v
where the index i refers to the i'th auction in the sample. An estimate of an upper bound for e is obtained by taking the average of E
can be used as an estimate for an upper bound for the unknown e. At A representative customer in the target market has observable characteristics denoted by a vector X. These characteristics include what is known or can be learned about the customer, e.g., demographic information, past purchases, income, etc. For the purpose of illustration, a method is presented for estimating the demand for a case where each potential buyer is interested in at most one unit of the product. In the single unit demand case, a customer's maximum willingness to pay, i.e., his valuation, for the object to be sold, conditional on X is denoted by V. That is, a customer with valuation V buys the item if and only if the price of the item, p, is less than V. The seller cannot observe V directly. From the seller's point of view V is a random variable. The distribution function of V is denoted by F(v). The function F(v) is also unknown by the seller. The goal of the test-marketing effort is to gather data to estimate the unknown function F(v). Once an estimate of the function F(v) is determined, the demand function for the item is obtained as follows. A representative customer's demand for the item under consideration depends on the price, p. If the customer's valuation, V, is less than the price, p, the customer does not buy the item, otherwise he/she buys it. Thus, the quantity bought by the customer in response to a price p is a random variable with the following distribution: D (p)=0 with probability F (p), and D (p)=1 with probability 1-F (p). When Q (p) denotes the total quantity sold in the market at a price p, if the number of potential customers in the target market is N, then Q (p) is a Binomial random variable with parameters Nand (1-F (p)). Accordingly, Q (p) is distributed Binomial (N, 1-F (p)). Since the seller's revenue depends on Q (p) and the cost may also depend on Q (p), the seller's profit is also a random variable, the distribution of which is determined by the distribution of Q (p). For the purposes of illustration, the unit cost of production is a constant and it is assumed that production is made-to-order. In this case, the seller's profit corresponding to a price p and unit cost c is π=(p−c)Q(p) which is a random variable. If the distribution of Q (p) is known then the distribution of profit can be obtained by well-known statistical methods. One takes a sample from this distribution of Q (p) and forms a sample of π values by evaluating the function (p−c)Q(p) at the observed values of Q (p). The goal of the test-marketing effort is to gather data to estimate the unknown function F(.) Once an estimate of the function F(.) is determined, the distributions of quantity sold and the associated profits are readily obtained. The following are models of demand that are used to update the demand function where the buyers may wish to buy multiple units. Common characteristics of this class of models is that the randomness in demand is generated by some cumulative distribution function, F(.), and thus the issue of demand estimation is reduced to estimation of a distribution function. In the present multiple unit models for determining demand, a representative customer's demand per period for various quantities of the product corresponding to various price levels is determined by two factors: Z, which represents the maximum quantity per period the customer can possibly buy; and F(.), a cumulative distribution function that determines the customer's marginal valuation for various quantity levels. The two elements (Z, F(.) can be used in several alternative ways to model a situation where customers have random demand schedules. A customer is represented by a random sample, (W In a first model for multiple unit demand, a customer's demand schedule is represented by the order statistics, (W In this model, a customer's willingness to pay for z units, V(z), is the sum of the first z marginal valuations:
In the second model for multiple unit demand, a customer's demand schedule is a random vector, (W
Continuing with step For fixed n, k and r, the observed bids form a random sample from the distribution F(v) truncated at v In the present embodiment, Ĝ(v) is used to denote the empirical cumulative distribution of observed bids. Since observed bids are a random sample from the distribution F(v) truncated at v Using Q (p) to denote the total quantity sold in the market at a price p. If the number of potential customers in the target market is N, then Q (p) is a Binomial random variable with parameters Nand (1-F (p)). For p≧v Therefore, given an estimated demand function, the distribution of any function of Q (p) can be estimated using any of a number of statistical methods. In particular, the distribution of profit can be estimated as follows: a sample is taken from the estimated distribution of Q (p) and forms a sample of profit values by evaluating the function profit=(p−c) Q (p) at the observed values of Q (p). Additional auctions are subsequently held and will use refined values of the auction parameters based on the results from the most recently run auction. At step At step When the demand function does not satisfy the specified accuracy level (step In one embodiment, the auction parameters are updated by first computing the total number of units auctioned in period t. This is done by dividing the total available quantity at period t, Q The target total sample, S The target sample size is then determined. More particularly, in the present embodiment the target sample size is determined for the period t. In the present embodiment, the user specified accuracy parameters (δ, α) and the price range [p In one embodiment, the function s(δ, α, a
then bidders with valuations that exceed p Next, the difference between the target total sample size S The number of auctions, the number of bidders per auction, the number of units per auction and the reserve price are then determined. In the present embodiment, these parameters are determined using the sample size s Number of units may then be determined. If the number of bidders in the auction is n, the number of units offered in the auction, k, is determined to find the least-costly way of inducing p If S is the total sample size (number of bidders in all auctions conducted) and n bidders are invited in each auction then the number of auctions needed is S/n. If the total number of units available is restricted to be at most Q, and k units are offered for sale in each auction, then there is an additional constraint on the number of auctions, that is, A k<q. Thus, in order to ensure that both the sample size constraint and the quantity constraint are satisfied, the number of units per auction may be restricted to the range 1≦k≦(q/s)n. When M (n, k, p It is noted that actual quantity sold in an auction with parameters (n, k, r) is Min {M, k}, which is random since M is random. The price per unit in such an auction is also a random variable. If M≦k, then M units are sold in the auction and the price for each of the M units sold is equal to the reserve price r. If M>k, then k units are sold in the auction and the price for each of the k units sold is equal to the (k+1)st highest bid among the M submitted bids. In this case, the price per unit exceeds the reserve price since p Denote by ê and {circumflex over (F)}(v) the current estimates of the unknowns e and F(v), respectively. Expected net revenue from sales in an auction with Mactual bidders is Exp_Net_Revenue (n, k, r, M|ê, {circumflex over (F)}(.), C The value of k that maximizes is denoted by k(n, p Specifically, starting with an initial value for k: k*=k(0)=largest integer smaller than (n+1)/2 and solving using the following logic: If ExpNR(k(0)+1)≧ExpNR(k(0)) While {k(0)≦k*≦(q/s)n, ExpNR(k*+1)≧ExpNR(k*); k*=k*+1}. Else k (n, p The number of bidders and number of auctions is then updated. In the present embodiment, the variables C the expected net revenue is obtained as a function of the decision variable n and the parameters (p When S is the desired sample size (the total number of bidders from all auctions), then the number of auctions, A, and the number of bidders per auction, n, are related by s=n A, or equivalently, A=s/n. The expected net cost of collecting a sample of size S with A auctions each with n bidders can be expressed in terms of a single decision variable n in the following equation: In the present embodiment, since n can take integer values in a finite range from 2 to MaxB (given by system constraints), the minimization of ENC (n, s, Q, p Additional auctions are then run using the updated auction parameters from step At the time of the first Vickrey auction, little is known about the probability distribution function F(v). In one embodiment, as additional auctions are run, more information becomes available and the people marketing the product are provided better knowledge of the demand function and can make decisions on the parameter of interest, such as number of items to offer, based on the value of F(v). If the value of F(v) is not considered good enough, additional auctions are run until F(v) is considered good enough as shown in step Additionally, there are budgetary concerns that are to be met as shown in step At step At step It is noted that step In summary, embodiments of the present invention provide a method and system thereof for test marketing and determining demand function for new products and existing products under new market conditions. This is achieved by running auctions that are designed for the specific purpose of efficient demand estimation. This method of the present invention avoids strong a priori assumptions about the unknown demand function and about bidding behavior in auctions and uses auctions as part of the data collection process designed specifically for efficient demand estimation. In the foregoing specification, the invention has been described with reference to specific embodiments thereof. It will, however, be evident to those skilled in the art that various modifications and changes may be made thereto without departing from the broader spirit and scope of the invention. The specification and drawings are, accordingly, to be regarded in an illustrative rather than a restrictive sense. Referenced by
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