Designing Optimal Pre-Announced Markdowns in the Presence of Rational Customers with Multi-unit Demands - Online Appendix

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1 087/msom Designing Optimal Pre-Announced Markdowns in the Presence of Rational Customers with Multi-unit Demands - Online Appendix Wedad Elmaghraby Altan Gülcü Pınar Keskinocak RH mith chool of Business, University of Maryland, College Park, MD chool of Industrial and ystems Engineering Georgia Institute of Technology, Atlanta, GA 3033 Proof of Theorem Given the bid quantities q i, q i of all customers i j, we want to find the best response optimal bid quantities of customer j If v j < p, customer j bids zero at both price steps If p > v j p, then from Observation customer j bids his entire demand at step If v j p, customer j s expected profit from bidding q j, q j = D j q j can be expressed as follows Π j = v j p q j + v j p E[A j ] where E[A j ] denotes the expected quantity allocated to customer j in step Note that since we consider effective markdowns, customer j is guaranteed to receive all his bid quantity at step, ie, his profit at step is v j p q j Customer j s expected profit from bidding at step depends on the expected quantity that will be allocated to that customer at step Recall that when the total bid quantity exceeds supply at a given step, the seller uses the random allocation rule This is equivalent to choosing a permutation of N customers randomly and satisfying the customers demand in sequence based on their position in this permutation There are N! distinct permutations of N customers and the seller will choose any of these permutations with equal probability /N! Let us denote the set of all permutations by U and let B j π denote the set of customers whose bid quantities are satisfied before customer j in some permutation π U B j π = indicates that customer j s demand is satisfied first E[A j ] = N! π U min{k D i q i q j +, D j q j }, i B j π We will show that the profit function Π j is convex in q j by analyzing its slope Π j = Π j q j = v j p + v j p E[A j ], where E[A j ] = E[A j] q j evaluates to x N! for some positive real x Π j is piece-wise linear and continuous The breakpoints are where the evaluation of the min{} changes from one argument being minimum to the other At these points, the derivative of the min{} either stays the same at -, or goes from - to 0 We use an abuse of notation considering the parts of the function where the derivative exists The higher the value of q j, the higher is the number of permutations for which K i B j π D i i/ B j π q i q j + is equal to zero, ie, the lower is the value x Hence we observe that as q j increases, the slope of the profit function either increases possibly changing from negative to positive or remains the same Based on these observations, we conclude that the profit function is convex, and is maximized at one of the extreme points {0, D j }, implying an all-or-nothing bidding strategy at each step i/ B j π Proof of Proposition 4 ince p = v, using the equilibrium bidding strategies from Table 3 we can write the total profit of the seller as follows: { p D Π p = + v K D if p ˆp CI v, v v K if p > ˆp CI v, v

2 Π is strictly monotonically increasing in p up to ˆp CI v, p, after which it sharply drops to v K Hence, this break-point ˆp CI v, p =p is the optimal price At p, customer is indifferent between buying at step or By assumption A customer bids his entire demand at step Proof of Theorem 5 uppose there exists some partition {, }, which satisfies conditions C and C simultaneously We will demonstrate that the proposed bidding strategies constitute an equilibrium by considering any profitable deviations uppose that all buyers, except buyer k, bid according to E From Theorem, we know that it is always optimal for customer k to bid all of his potential demand in one step For k, if buyer k bids according to E, his profit is given by Π k = D k v k p On the other hand, if customer k deviates from E and bids D k at step, he will compete with the other customers in for the allocation of the remaining units; in this case customer k s expected profit is Π k = v k p E[A k ] where E[A k ] is customer k s expected allocation at step We know that Π k = D k v k p > v k p E[A k ] = Π k by the definition of the partition {, } and condition C Therefore, given all other buyers bid according to E, it is an optimal response for buyer k to do the same imilarly, for k, if buyer k bids according to E, his profit is given by Π k = v k p E[A k ] If buyer k deviates and bids D k in step, his profit is given by Π k = D k v k p From the definition of {, } and condition C, we know that Π k = D k v k p < v k p E[A k ] = Π k Therefore, given all other buyers bid according to E, it is an optimal response for buyer k to do the same Proof of Proposition 6 We can rewrite condition C as follows: v k p v k p E[A k] D k C As we increase D k by one unit, the denominator of E[A k] D k increases by at most one unit, ie, as D k increases, E[A k] D k other hand, v k p v k p is increasing in v k Then we have v j p v j p > v k p v k p > E[A k] D k E[A j] D j increases by one unit, but the numerator decreases or remains the same On the for any j < k since v j > v k and D j D k This implies that if condition C holds for customer k for a given partition {, }, then it should hold for all customers j < k Proof of Theorem 7 Let q jt denote customer j s bid quantity at step t, t {,, m} Note that for any customer j who can afford to bid at p m, we have q jm = D j t<m q jt The expected profit of customer j can be expressed as: Π j q j,, q jm = m v j p t E[A jt ] where E[A jt ] is the expected allocation to customer j at step t Recall that when the total bid quantity exceeds supply at a given step, the seller uses a random allocation rule and this is equivalent t=

3 to choosing a permutation of N customers randomly and satisfying the demands based on their sequence in this permutation The seller will choose each of N! permutations with equal probability N! We denote the set of all permutations with U and define Bj π as the set of customers whose bids are satisfied before customer j in some permutation π U Using this notation, we can define the expected allocation of customer j at step t as follows: where K = K q i, E[A jt ] = N! i Bπ j k t π U K m = K i/ B π j k<t min{k + t, q jt} D i q ik i Bπ j i Bπ j i/ B π j k<m k<m K t = K q ik q ik q jk, m > t > k<t We look at the derivative of Π j with respect to q jk and show that the profit is maximized at the end-points of the range for q jk E[A jt ] q jk Π j q j,, q jm q jk = m t= v j p t E[A jt] q jk evaluates to zero for k > t For k < t, E[A jt] q jk evaluates to x k value of q jk, the higher is the number of permutations for which K t + q jk N for some x t! k The higher the is equal to zero, ie, the lower is the value of x t For k = t, E[A jt] q jt evaluates to wt N t! for some w t Note that K t + is constant with respect to q jt, hence, as q jt increases, the number of permutations for which min{k t +, q jt} = q jt decreases or remains the same, implying that E[A jt] q jt is non-decreasing in q jt In summary, as q jk increases, E[A jt] q jk increases or remains the same From this observation, it follows that the slope of the profit function in the direction of q jk in non-decreasing in q jk implying that customer j s profit is maximized at one of the end-points 0 or D j l<t q jl ince the optimal bid quantity at step t is 0 or D j l<t q jl, customer j submits 0 or D j at step By induction, if the customer submits 0 or D j at steps,, t, t < m, then he will submit 0 or D j at step t + Hence, it is optimal for customer j to submit all-or-nothing bids at any step Proof of Observation 8: Consider a markdown where p > p > > p j > v k > p j+ > where p c [v k, v k ] i ince the total demand is less than K for any price higher than v k, customers will not purchase but simply wait until the markdown reaches p j, which is the lowest price above v k Hence, one can eliminate p, p,, p j from the markdown without affecting the expected profit of the seller ii uppose p > > v k > p l > > p m, ie, p l is the highest price which is smaller than v k We claim that the seller would be better off by eliminating the last price step p m ince D[p i ] > K p i p l, eliminating p m does not decrease the number of units sold We also need to show that the seller is guaranteed to sell all K units at the same or higher prices than with the m-step markdown Customers originally bidding at step m may bid at higher price steps or they may not bid at all in the new markdown with fewer steps If they do not bid in the new mechanism, then the competition at the higher price levels is unaffected On the other hand, if they bid at higher price steps, then the competition can only increase since there would be more customers bidding at a price step As 3

4 a result, at any price step j < m, the competition either increases or remains the same, preventing the seller s revenue from decreasing Hence, the revenue of the seller is not made any worse by eliminating the lowest price step from the markdown Using these arguments repeatedly, we can eliminate all price steps strictly smaller than p l iii By contradiction, suppose p m [ v j+, v j ] for some j < N By setting p = v j, the seller does not decrease the demand at any price step, hence, the competition at each price step remains the same However, due to the increase in p m, some customers who were previously bidding at p m may now choose to bid at higher price steps, potentially increasing the seller s revenues Proof of Observation 0: ince p c [v N, v N ], from Observation 8, there is at most one price, p, exceeding v N By the definition of an INT markdown, only one price can be chosen from [v N, v N ] Proof of Proposition 9: If customer bids D at p, his surplus is Π = v p D Alternatively, if he bids D at p his expected surplus is Π = v p { F p D + [ F p ] D +K D } Hence, customer bids at p if and only if Π Π Rearranging terms, we get Equations 4 and 5 To identify the range of p values for which T or P markdowns are feasible, first we find the p values which satisfy ˆp IV v, p = v the lower bound on allowable prices for p under an INT markdown, and find two solutions, namely, v and p v where p v = v D D + D K v v 6 Note that when customer valuations are uniformly distributed, the threshold step price, ˆp IV v, p, is a quadratic convex function of p To induce some customer types to bid at p, we need p p v In addition, p < v from A4-IV, hence, we need p min{ v, p v } Combining this with the result of Corollary, we get the following condition for a feasible P markdown Observation A feasible P markdown satisfies the following: v p < min{ v, p v } and max{ˆp IV v, p, v } < p ˆp IV v, p 7 imilarly, we can identify conditions for the existence of a T markdown Observation 3 A feasible T markdown satisfies the following: v p < min{ v, p v } and v < p ˆp IV v, p 8 Note that since ˆp IV v, p > ˆp IV v, p, a P markdown exists whenever a T markdown exists Results 4 and 5 spell out conditions 7 and 8 presented in Observations and 3 for uniformly distributed valuations, and are used in the proofs of some of the theorems that follow Result 4 When the valuations are uniformly distributed, there exists a P markdown with price p if and only if one of the following conditions holds: i for p [v, p v ] if v v v v D +D K D < v v v v or equivalently if v p v < v 4

5 ii for all p [v, v if v v v v D +D K D or equivalently if v p v The proof of Result 4 follows immediately by inserting the appropriate values for the uniform distribution into the conditions 7 of Observation We have a similar result for the T markdown Result 5 When the valuations are uniformly distributed, there exists a T markdown with price p if and only if one of the following conditions holds: i for p [v, p v ] if v v v v ii for all p [v, v if v v v v D +D K D < v v v v or equivalently if v p v < v D +D K D or equivalently if v p v In the following, whenever there is a strict inequality a < b as a constraint, this should be perceived as a b ɛ, where ɛ 0 To simplify the notation, we use a < b ɛ can be thought of as the minimum possible price increment Figure 7 shows how the existence of P and T markdowns depends on the scarcity of supply measured by D +D K D the ratio of customer s expected unmet demand to his entire demand, if both customers bid at step : if scarcity is low, it is more likely to see a pooling rather than a separating outcome Figure 7 is based on Results 4 and 5 presented above a Pooling for all p v v v v b Only P exists for some p v v v _ P and T exist for some p v_ v v_ v v_ D _ c v v P exists for all p,t for some p d v v _ P and T exist for all p e D + D K Figure 7: Existence of INT markdowns satisfying A4-IV as a function of market parameters The seller can determine the optimal P markdown by maximizing Π P subject to 7, where: Π P = [ F ˆv p, p ]p D + F ˆv p, p p D + K D [ F p ]p 9 imilarly, the seller can determine the optimal T markdown by maximizing Π T subject to 8, where: Π T = p D + K D [ F p ]p 0 Define P P and P T to be the set of p values for which a P and a T markdown exists, respectively Note that since P T P P, we have P T P P = P T In what follows, we characterize the optimal P and T markdowns Based on those results, we are able to derive sufficient conditions for a P or a T markdown to be optimal Define p M p to be the optimal first step price given a second price step of p under markdown type M = T, P Characterization of the Optimal T Markdown To characterize the optimal T markdown, we find p T p and then demonstrate some properties of the seller s revenue function 5

6 Theorem 6 The optimal T markdown has the following properties: i For a given second step price p, the optimal first price step is ˆp IV v, p ii Π T ˆp IV v, p, p is convex in p if K < D +D /3, and it is concave in p if K D +D /3 iii If Π T iv If Π T ˆp IV v, p, p is concave in p, then the optimal step price is: p T = v if p T v p T if v < p T p v v ɛ min{p v, v } if p v < min{p T, v } if v min{p T, p v } ˆp IV v, p, p is convex in p, then the optimal step price is: { p T D v = if +D K D D v Kv D v v + K D v v p v otherwise where p T = v 3K D D D v D +D Kv 3K 3D D Proof of Theorem 6: D v v v p v p v i The seller s profit is increasing in p since ΠT p = D > 0 Therefore, for a given p, the seller would prefer to set p to its upper bound, which is ˆp IV v, p ii The first derivative of seller s revenue with respect to p after substituting p = ˆp IV v, p and uniform CDF and pdf for F i p i and f i p i is as follows: Π T ˆp IV v, p, p = D p v D + D K v v Taking the second derivative with respect to p yields: Π T ˆp IV v, p, p p = 3D + D 3K v v + v p 3K 3D D v v ince v > v we obtain a sufficient and necessary condition for the function to be concave convex in p as K D + D /3 K < D + D /3 iii When the revenue function is concave in p, its maximizer is p T If pt [v, min{p v, v }], then it is the optimal step price Otherwise, the optimal p is one of the boundary values of the feasible p range iv When the revenue is a convex function of p, one of the boundary points will be the optimal step price First we show that if a T markdown exists for all p [v, v, then p T = v For v to be the optimal step price, revenue at this price should be higher than the revenue at p = v, ie, Π T ˆp IV v, v, v Π T ˆp IV v, v, v Kv + v v D + D K D v D + D K D v Kv v v From Result 5ii, we have v v v v as the condition for a T markdown to exist for all p, hence we conclude that v is always optimal provided that a T markdown exists for all p as a result of the following series of inequalities, D +D K D 6

7 D + D K D v v v v > v v v v > v v v v K D D v v v = D v Kv D v v If a T markdown exists for only p [v, p v ], where p v < v, then either p T = p v or p T = v We compare the corresponding revenues to find conditions under which either one is optimal For p T = v, we need Π T ˆp IV v, v, v Π T ˆp IV v, p v, p v = v, p v, ie, Π T D + D K D v Kv D D v v + K D v p v D v v p v v v From Result 5 we know that a T markdown exists for p [v, p v ] when v v v v v v Hence p T = v if v v v v > D +D K D > v v v v D v Kv D v v + K D Combining the ranges that yield the same p T we get the desired conditions v v < D +D K D < v p v p v D v v v v 7

8 Characterization of the Optimal P Markdown To characterize the optimal P markdown, we first show some properties of the seller s revenue function in and then derive p P p for a given p Theorem 7 The optimal P markdown has the following properties maximizes the uncon- i Π P is concave in p and p P p = p + v p [ F p ] D +D K 4D strained Π P ii The optimal step price is the following: { p P p P p = p if v < v +p max{ˆp IV v, p, v } + ɛ otherwise Proof of Theorem 7: and p < p = v 4D D +D K v v i First we show that the revenue function is concave in p from the second order condition, and find the maximizer from the first order condition The first and second order partial derivatives of p, p in Equation 9 with respect to p are: Π P Π P p = [ F ˆv p, p ]D ˆv p, p p f ˆv p, p p p D Π P p = f ˆv p, p ˆv p, p p D ince ˆv p,p D p = [ F p ] D +D K, and f are both positive and independent of p under uniform distribution, the revenue function is concave in p for all p ubstituting ˆv p, p from Equation 5, ˆv p,p D p = [ F p ] D +D K, and uniform CDF and pdf for F and f in the first derivative and setting equal to zero, we get the first order condition: v p v v D p p v v [ F p ] 4D D + D K = 0 We solve for the p value that satisfies the first order condition and get p P p = p + v p [ F p ] D +D K 4D ii To find out the optimal p for a given p, we need to understand the conditions under which p P p max{ˆp IV v, p, v }, ˆp IV v, p ] as stated in constraint 7 It is easy to see that p P p < ˆp IV v, p always, hence, we only need to check the lower bounds Next, we show that p P p max{ˆp IV v, p, v }, ˆp IV v, p ] if and only if v < v +p and p < p iia p P p > ˆp IV v, p if and only if v < v +p We substitute v = v in Equation 4, and compare with p P p p P p = p + v p [ F p ] D + D K 4D implifying the expression, we get v < v +p as the equivalent condition iib p P p 4D > v if and only if p < p = v D +D K v v > p + v p [ F p ] D + D K D 8

9 In order for p P p to exceed v, we get: p + v p [ F p ] D + D K 4D > v ubstituting uniform CDF for F and rearranging we get p < p as the equivalent condition Plugging p = p P p in Equation 9, now we can determine the optimal step price, p P, corresponding to each case Result 8 When p P p = p P p, the optimal step price for a P markdown is: p P = v if p P v p P if v < p P p v v ɛ < min{ v, p v } if p v p P and p v < v otherwise Proof of Result 8: ubstituting p = p P p into Equation 9, we get: Π P p P v p D + D K p, p = p D + [ F p ] + K D p 8 v v The first derivative with respect to p yields: Π P v p D + D K = D [ F p ] + K D p 4 v v v p D + D K f p + K D p 8 v v The second derivative with respect to p after substituting uniform distribution for F yields: Π P p = v p D + D K v v v v K D v v + v p D + D K 4 v v v v olving for p from the first order condition ΠP p = 0, we get p = A± A BC C, where A = v + v D +D K 8K D v v, B = 8v D v K v v + v v + v D +D K and C = 3D + D K The second order condition required for p to be a maximizer yields p > A C, hence we get pp = as the unique maximizer of the unconstrained revenue function By comparing p P A+ A BC C with boundary points of the feasible p range,[v, min{p v, v },we determine the optimal step price In order not to clutter the presentation in the proofs in this document, we will drop ɛ when substituting arguments with ɛ in other expressions and use lim ɛ 0 in expressions that involve = Result 9 When p P p = v + ɛ, then in a P markdown we have: i Π P v + ɛ, p is convex in p if and only if K D v v < D v v, and it is concave in p otherwise ii If Π P v + ɛ, p is concave in p, then the optimal step price is: 9

10 p P v = v if p P v v p P v if v < p P p v v p v if p v < p P v where p P v be the value of p that solves ΠP v +ɛ,p p = 0 iii If Π P p P v = v + ɛ, p is convex in p, then the optimal step price is: v ɛ if p v K D v v D v v p v if p v D + K D v v v v v and p v > v Π P v + ɛ, v and p v v v otherwise where lim ɛ 0 Π P v v v + ɛ, v = Kv + D v v v v v v D v v D +D K Proof of Result 9: i We substitute p = v + ɛ in the seller s revenue given in 9 and uniform CDF for F i lim ɛ 0 ΠP v p v + ɛ, p = v p + v v D v v v v v v D v v v v D + D K D + D K The first derivative with respect to p is: Π P v + ɛ, p = v D + p D + v p + v v p v v v v v v v v Taking the second derivative with respect to p, we get: Hence the function is concave if K D v v D v v ii Given that Π P p v D v v D [ K D v v v D + p D + K D p v p v v Π P v + ɛ, p p = D K D v v v v D D + D K and concave otherwise D +K D v p v v v + ɛ, p is concave in p, the optimal step price is p P v = v + ], provided that p P v is in the p range for which a P markdown exists v v We first show that p P v is less than v ince Π P v + ɛ, p is concave, the denominator of the second term in p P v expression above is negative, hence p P v < v < v If p P v < v, we conclude that the revenue function is decreasing in p over the entire range and v is optimal If p P v > v, then the revenue function is increasing in p at p = v, and we have to compare p P v with the maximum p for which a P markdown exists in order to find the optimal step price From Observation we know that a P markdown exists for all p < min{p v, v } If p P v < p v, then the markdown with prices v, p P v is optimal If p P v p v, then p v is optimal iii When Π P v + ɛ, p is convex in p, one of the endpoints of the p range will be the optimal step price If p v < v, then the p range for which a P markdown exists is [v, v Otherwise, the p range is [v, p v ] 0

11 In each case, we identify which endpoint maximizes the seller s revenue by evaluating Π P v +ɛ, p at each endpoint and comparing the resulting revenues For p [v, v, we show that p P v = v ɛ if Π P v + ɛ, v Π P v + ɛ, v ɛ By substituting the p and p values in 9, we get the condition for p P v = v ɛ: lim ɛ 0 Π P v + ɛ, v ɛ = v D, hence we require p v D + K D v v v v v D Rearranging terms and simplifying we get, p v K D v v D v v v imilarly, for p [v, p v ], we show that p P v = p v if Π P ɛ, p v lim ɛ 0 Π P v + ɛ, p v = p v D + K D v v v v, and lim ɛ 0 Π P v +ɛ, v = Kv +D v v as the condition for p P v = p v p v D + K D v v v v Kv + D v v v v v v v v v v D D +D K v v v v v v D v v D +D K v + ɛ, v Π P v + Hence, we have the following For both p ranges, the other alternative optimal is to have p at the lower bound, which is v, hence we get the conditions given in the result Comparing P and T Markdowns Proposition 0 When P T, i the optimal markdown is P if v < v +v ii the optimal markdown is T if v v + v Proposition 0i implies that it is optimal for the seller to only partially separate the high types when the range of the high types is large relative to the low valuation range Proposition 0ii implies that the seller is best served by inducing all of the high types to purchase at the first price step when the high and low types are fairly far apart similar to EX in Table?? Proof of Proposition 0: i From Theorem 7 we know that for a given p, the P revenue function is maximized at p P p = p P p if v < v +p ie, p P p > ˆp IV v, p and p < p ie, p P p > v ince Proposition 0 is stated only for the case when T is feasible, from Equation 8, we have v < ˆp IV v, p Hence, the condition v < v +p combined with Equation 8 automatically implies that p P p > v and therefore p P p is optimal, ie, the P markdown s optimal step price is not at the boundaries ince the P revenue is concave and the boundaries approach the T revenue, the P markdown dominates the T markdown ii From Theorem 7, we know that for a given p, the P revenue function is maximized at p P p = max{ˆp IV v, p, v } + ɛ if v v +p As ɛ 0, the markdown converges to a T markdown, and hence the T markdown revenue exceeds the optimal P markdown revenue in this case If v v + v, then v v +p for all p [v, v since the right-hand-side is maximized at p = v Tabulated Results for Numerical Examples ection ection 43 Each table corresponds to one parameter being changed while keeping everything else the same First column of the header identifies the parameter that is altered and the original value Each row corresponds to a different instance and the new value of the parameter is given in the first column in that row Whenever the P markdown converges to a T markdown, ɛ is used to differentiate

12 D = 3 ingle Price T Markdown P Markdown p Revenue Type p T p T Revenue p P p P Revenue P none exists P ɛ P ɛ P ɛ P ɛ P ɛ P ɛ P ɛ P ɛ P ɛ 00 Table 5: Optimal markdown and single price performance for different values of D D = 9 ingle Price T Markdown P Markdown p Revenue Type p T p T Revenue p P p P Revenue P none exists none exists P ɛ P ɛ Table 6: Optimal markdown and single price performance for different values of D step prices of P and T markdowns In this case, the corresponding P markdown revenue is slightly less than the T markdown revenue, hence the corresponding cell has been left blank K = 0 ingle Price T Markdown P Markdown p Revenue Type p T p T Revenue p P p P Revenue P ɛ P ɛ P none exists none exists Table 7: Optimal markdown and single price performance for different values of K

13 v = ingle Price T Markdown P Markdown p Revenue Type p T p T Revenue p P p P Revenue P none exists P ɛ P ɛ P ɛ 300 Table 8: Optimal markdown and single price performance for different values of v v = 5 ingle Price T Markdown P Markdown p Revenue Type p T p T Revenue p P p P Revenue P ɛ P ɛ P ɛ P none exists P none exists P none exists none exists Table 9: Optimal markdown and single price performance for different values of v v = ingle Price T Markdown P Markdown p Revenue Type p T p T Revenue p P p P Revenue P none exists P none exists P none exists P none exists P none exists P P ɛ P ɛ P ɛ P ɛ P ɛ P ɛ 5 Table 0: Optimal markdown and single price performance for different values of v 3

14 v = 8 ingle Price T Markdown P Markdown p Revenue Type p T p T Revenue p P p P Revenue P ɛ P ɛ P ɛ P ɛ P ɛ P ɛ P ɛ P ɛ P ɛ P P P P P P Table : Optimal markdown and single price performance for different values of v 4