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Selling in channels with uncertain price

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Abstract

In this article, we analyze the problem of selling through channels where the sales price and hence revenue is uncertain. Examples of such channels include real estate listings, traditional and online auctions, name-your-own-price channels and price guarantees. More specifically, we consider a seller, with a fixed inventory, who has to decide when to make items available to the market. We assume that there is a holding or depreciation cost associated with the items, which means that the seller would like to make items available for sale as soon as possible. On the other hand, we also assume that the expected selling price is decreasing in the number of items made available to the market. That is, there is a price cannibalization effect among the items made available at the same time, resulting in an incentive to make items available sequentially over time. We formulate the problem as a discrete time Markov Decision Problem and consider two cases: guaranteed successful sales and possibly unsuccessful sales. The reason for considering the two cases is that they require different analyses and give rise to different results. The article focuses on the case when the seller only has two items, which has sufficient complexity to raise interesting insights and challenging questions.

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Notes

  1. www.dellauction.com; cgi3.ebay.com/ws/eBayISAPI.dll?ViewUserPageuserid=dell_financial_services; auction.mlb.com; www.clearance-comet.co.uk; www.shopgoodwill.com/; accessed 5 July 2011.

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Acknowledgements

We would like to thank Mahesh Nagarajan and the anonymous referees for their helpful suggestions. We would also like to acknowledge financial support through NSERC Discovery Grants 5527-201 (Puterman) and 372074-2009 (Odegaard).

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Correspondence to Fredrik Ødegaard.

Appendix

Appendix

Optimality Equations for the Multiple Re-Posting Case

Table A1, Table A2

Table A1 Optimality equations for the multiple re-posting case
Table A2 Resulting optimality equations for the multiple re-posting case

Formal Proof of Results for the Single Posting Case

We first introduce some notation. When both items have been posted, we define E[Xi, τS t =s] to be the conditional expected sales price of item i, i=1, 2, in period t, and state s=([x1, y1; x2, y 2 ], z):

where Pr{ X i,τ = q∣X i, yi = x i, Z t = z,…,Zt+(τ−yi) = z′} is derived using the Chapman-Kolmogorov equations. The main issue regarding the expected sales price is that it only depends on the current price of a posting, and how many postings will be underway for the duration of the posting. Once both items have been posted, we know how many postings there will be for the remainder of each individual posting. As a consequence of the assumptions that postings progress independently and Assumption 1, E[Xi, τS t ] is increasing in x i and independent of x j , for ij.

Corollary A1:

  • If postings progress independently of the price in the other posting, and Assumption 1 holds, then the conditional expected sales price, E[Xi, τS t =([x1, y1; x2, y2], z)], is increasing in x i and independent of x j , i=1, 2, ij.Proof of Corollary A1.The result regarding independence of the price in the other posting is immediate by the assumption that price transitions do not depend on the price in the other posting. Proof by induction on the number of remaining periods n=τy i . Without loss of generality, consider item 1. For n=0, E[X1, τS t =([x1, τ; x2, y2], z)]=x1, which is increasing in x1. Assume that the result holds for n=0, 1,…, l−1, that is for y1=τ, τ−1,…, τ−(l−1). Let n=l then y=τl,

    where inequality holds because of Lemma 4.7.2 in Puterman (1994), the induction assumption and Assumption 1. Although Lemma 4.7.2 in Puterman (1994) is with respect to discrete variables and infinite sequences, it can be adapted to continuous variables and/or finite sequences. Alternatively, the results from Lemma 9.1.1 and Proposition 9.1.2 in Ross (1996) can be applied. □

When both items have been posted, we define R(S t ) to represent the total expected profit over the remainder of the planning horizon in period t, for s=([x1, y1; x2, y2], z),

There is a slight misuse of notation when y i =δ, i=1, 2. In this case, we implicitly define τδ=0, as no holding cost will be incurred. Note that R(S t ) is not necessarily increasing or decreasing in the elapsed sales time of the postings. Although the incurred holding cost will decrease, the expected sales price of the postings will also decrease. It is this trade-off that is the crux of the problem regarding when to post the second item. However, R(S t ) is increasing in x1 and x2.

Corollary A2:

  • If Assumption 1 holds, then R(S t ) is increasing in x1 and x2, for all y1, y2, and z=0, 1, 2.Proof of Corollary A2. Each posting progresses independently of the price in the other posting; the result is therefore immediate by Corollary A1. □

Lemma A3:

  • If we assume a vigilant seller and that postings are guaranteed to be successful, then the value functions for discrete prices are as follows:

    Proof of Lemma A3. Owing to the vigilant seller assumption and that we are considering the case when postings are guaranteed to be successful, we can explicitly write out the value function (6) according to the following table:

illustration

figure a

Note that there are only non-trivial decisions to be made for t<τ and z=1. Consequently, once the second item has been posted, we can evaluate the expected total future reward (=expected sales price for both items – total remaining cost). The implication of this is summarized in the following two lemmas, which will facilitate the book keeping and establish Lemma A3. □

Lemma A4:

  • If we assume a vigilant seller and each posting is guaranteed to be successful, then once item 2 has been posted we can explicitly evaluate the value function, for (1) τtT, z=0, 1, or (2) t<τ, z=2,

Proof of Lemma A4. There are three cases to consider.

(1) For τtT and z=0, proof by backward induction on t. Let t=T then y1=δ and y2=τ or δ, and therefore V t ([x1, y1; x2, y2], z)=−h0+x1+x2=−h(2τδδ)+E[X1, τ∣([x1, δ; x2, δ], 0)]+E[X2, τ∣([x1, δ; x2, δ], 0)], and the result holds. Assume that the result holds for t=l+1, l+2,…, T. Let τt=l, then y1=τ or δ and y2=τ or δ, and therefore V t ([x1, y1; x2, y2], z)=Vt+1([x1, δ; x2, δ], 0)=−h(2τδδ)+E[X1, τ∣([x1, δ; x2, δ], 0)]+E[X2, τ∣([x1, δ; x2, δ], 0)], where the second equality holds because of the induction hypothesis. Therefore, the result holds for all τtT and z=0. As noted earlier, there is a slight abuse of notation when y i =δ, i=1, 2, where we define τδ=0.

(2) For τtT and z=1, proof by backward induction on t. Let t=T−1 and z=1, then t1=δ and t2=τ−1, therefore

And the result holds (note that if t=T, then owing to the vigilant seller assumption all postings are completed and z≠1). Assume that the result holds for t=l+1, l+2,…, T. Let τt=l and z=1, then t1=τ or δ, therefore

where the second equality holds because of the induction hypothesis when z′=1, or case (1) above when z′=0. Therefore, the result holds for all τtT and z=1.

(3) For t<τ and z=2, proof by backward induction on t. Let t=τ−1 and z=2, then y1=τ−1, therefore

where the second equality follows from case (1) above when z′=0, or case (2) above when z′=1, and the third equality follows from that we assume each posting progresses independently of the price of the other posting. Therefore, the result holds for t=τ−1 and z=2. Assume that the result holds for t=τ−(l−1), τ−(l−2), …, τ−1. Let t=τl and z=2, then y1=τl and

where the second equality follows from the induction hypothesis when z′=2, case (1) above when z′=0, or case (2) above when z′=1, and the third equality holds because of the assumption that each posting progresses independently of the price in the other posting. Therefore, the result holds for all t<τ and z=2. □

Lemma A5:

  • If we assume a vigilant seller and each posting is guaranteed to be successful, then for t<τ and z=1,

Proof of Lemma A5. Proof by backward induction on t. Let t=τ−1 and z=1, then y1=τ−1 and

where the first equality holds because of Lemma A4 with z′=1, and the second equality holds because of the assumption that each posting progresses independently of the price in the other posting. Assume that the result holds for t=τ−(l−1), τ−(l−2),…, τ−1. Let t<τ and z=1, then y1=t and,

where the first equality holds because of Lemma A4 with z=1 (if t+1=τ) or with z=2 (if t+1<τ), and the second equality holds because of the assumption that each posting progresses independently of the price in the other auction. □

Owing to Lemmas A4 and A5, and to the fact that we mainly are interested in states sS such that A(s)={0, 1}, we have the value functions listed in Lemma A3.  □

Proof of Proposition 1. By Lemma A3, there are only three cases to consider.

Case (1) If t=T, then by Lemma A3, V t *([x1, y1; x2, y2], z)=x1+x2, and the result is immediate.

Case (2) If t<τ and z=2, or τt<T, then by Lemma A3, V t *([x1, y1; x2, y2], z)=R([x1, y1; x2, y2], z), and the result follows from Corollary A2.

Case (3) For t<τ and z=1, by Lemma A3,

We establish the result using backward induction on t. Let t=τ−1 and hence y1+1=τ, then by Lemma 4.7.2 in Puterman (1994), Case (2) above and Assumption 1, is increasing in x1, and by Corollary A2, R([x1, y1; p, 0], 2) is increasing in x1. As V t *([x1, y1; x2, y2], z) is the maximum of two increasing functions, it is also increasing in x1 and the result holds. Assume that Proposition 1 holds for t=τ−(l−1),…, τ−2, τ−1. Let t=τl and hence y1+1=τ−(l−1), and again by Lemma 4.7.2 in Puterman (1994), the induction assumption and Assumption 1, is increasing in x1, and by Corollary A2, R([x1, y1; p, 0], 2) is increasing in x1. As V t *([x1, y1; x2, y2], z) is the maximum of two increasing functions, it is also increasing in x1 and the result holds. Similar to the proof of Corollary A1, the results from Lemma 9.1.1 and Proposition 9.1.2 in Ross (1996) can be applied. □

Proof of Theorem 2. Let prices be discrete. Sufficient to show that V t *([x1, y1; p, 0], 1)−R([x1, y1; p, 0], 2) is decreasing in x1, for all t<τ. Define g2(S t ) to be the gain in the expected sales price of the second posting by delaying the posting one period, for s=([x1, y1; x2, y2], z), s′=(x1, y1+1; x2, y2], z′),

where z, z′=0, 1, 2 and by definition if y1=τ, δ then y1+1=δ. □

Corollary A6:

  • If postings progress independently of price in other posting and Assumption 2 holds, then g2(S t )⩾0 and independent of x1 and x1.

Proof of Corollary A6. As, by assumption, each posting progresses independently of the price in the other posting, the independence of x1 is immediate. If y1=τ, δ, then y1+1=δ, and E[X2, τSt+1=s′]=E[X2, τS t =s], as price transitions are independent of calender time. Therefore, assume that y1<τ. Proof by induction on y1. If y1=τ−1,

where the inequality holds because of Lemma 4.7.2 in Puterman (1994), Corollary A1 and Assumption 2, and the last equality holds because of the assumption that price transitions are independent of calender time. Assume that the result holds for y1=τ−1, τ−2,…, τl. Let y1=τ−(l+1),

where the inequality holds because of Lemma 4.7.2 in Puterman (1994), Corollary A1 and the induction assumption, and the last equality holds because of the assumption that price transitions are independent of calender time and the price of the other posting. Similar to Corollary A1, the results from Lemma 9.1.1 and Proposition 9.1.2 in Ross (1996) could have been applied.  □

By Corollary A2 and Proposition 1, R([x1, y1; p, 0], 2) and V t *(x1, y1; p, 0], 1) are increasing in x1. We make use of the following relationship,

Proof by backward induction on t. Let t=τ−1, then y1=τ−1 and by Lemma A3 and A6,

by Corollaries A1 and A6, E[X2, τ∣([x1, τ, p, 0], 1)], respectively g2([x1, τ−1; p, 0]) are independent of x1, and as,

define Λ t *(x)=V t *([x, y; p, 0], 1)−R([x, y; p, 0], 2). Let t=τl. Then by Lemma A3 and A6,

First show that is decreasing in x1. By Corollary A2, R([x, y; p, 0], 2) is increasing in x. Define α(x+1)=R([x+1, y; p, 0], 2)−R([x, y; p, 0], 2). Therefore,

where the inequality holds by Assumption 3. Therefore, is decreasing in x1. Next show that is decreasing in x1. By the induction assumption Λt+1(x) is decreasing in x, therefore define β(x+1)=Λt+1*(x)−Λt+1*(x+1), then,

where the inequality holds because of Assumption 1. Therefore, is decreasing in x1. As g2([x1, t; p, 0]) is independent of x1, the result holds for all t<τ.  □

Proof of Corollary 3. For a given decision epoch t, we know that for X1p t * any additional holding cost by deferring to post the second item is not compensated by the gain in expected final price for the two items. Therefore, if h increases and as the expected final sales prices remains the same, then any additional holding cost will still not be compensated (in fact, it is even less compensated), and the result follows. □

Proof of Proposition 4. From equation (3), we can compare various fixed policies and determine when each one dominates another. Let OP(j) and OP(j+m) be the open loop policies of posting the second item j and (j+m) periods, respectively after the first item. We then have, VO(j)VO(j+m) ⇔−(2τ+j)h+2(p+1+(τj)π2)⩾−(2τ+j+m)h+(p+(j+m)π1+(τjm)π2)⇔h⩾2(π1π2). As this condition is independent of j and j+m the result is that simultaneous posting is optimal iff h⩾2(π1π2). By symmetry non-overlapping sequential posting is optimal if h<2(π1π2) and there are no other optimal policies. □

Formal Proofs of Results for the Multiple Re-Posting Case

Proof of Lemma 5. (1) x1>0, y1=τ, δ and z=0, 1, or x1>0, y1, y2<τ, and z=2. There are four cases to consider.

(1a) For x1>0, y1=δ and z=0, 1, R I ([x1, δ; x2, y2], z)=−h(τy2)+E[X2, τ∣([x1, δ; x2, y2], z)]+

Let x1>0, y1=δ, and z=0. If y2=δ then V([x1, y1; x2, y1], z)=0=R I ([x1, δ; x2, δ], 0). If y2=τ then z=0 and V([x1, y1; x2, y1], z)=x2+V(Δ)=−h(ττ)+E[X2, τ∣([x1, δ; x2, τ], 0)]=R I ([x1, δ; x2, τ], 0).

Let x1>0, y1=δ, and z=1, that is y2<τ. Proof by backward induction on y2. Let y2=τ−1 and x2>0 then

where the second equality holds because of the case above with y1=δ, z=0.

Let y2=τ−1 and x2=0 then

where the second equality holds because of the case above with y1=δ and z=0, and that V([x1, δ; 0, 0], 1)=v(0, 0), as the first item has been sold and by the vigilant seller assumption the second item will be continuously re-posted until it is sold.

Assume that the result holds for y2=τ−(l−1), τ−(l−2), …, τ−1. Let y2=τl then

where the second equality holds because of induction hypothesis. Therefore, Lemma 5 holds for the case (1a).

(1b) For x1>0, y1=τ and z=0, 1,

Let x1>0, y1=τ, and z=0, that is y2=τ. Therefore V([x1, y1; x2, y1], z)=x1+x2+V(Δ)=R1([x1, τ; x2, τ], 0).

Let x1>0, y1=τ, and z=1, that is y2<τ. Proof by backward induction on y2. Let y2=τ−1 and x2>0 then

where the second equality holds because of case (1a) above.

Let y2=τ−1 and x2=0 then

where the second equality holds because of case (1a) above and that V([x1, δ; 0, 0], 1)=v(0, 0), which holds as the first item has been sold and by the vigilant seller assumption the second item will be continuously re-posted until it is sold.

Assume that the result holds for y2=τ−(l−1), τ−(l−2),…, τ−1. Let y2=τl then

where the second equality holds because of case (1a) above. Therefore, Lemma 5 holds for the case (1b).

(1c) Let y1=τ−1. Proof by backward induction on y2. Let y2=τ−1 and x2>0, then

where the second equality holds because of case (1b) above, and the third equality holds because of the assumption that each posting progresses independently of the price in the other posting.

Let y2=τ−1 and x2=0, then

where the second equality holds because of case (1b) above, and the fourth equality holds because of (8). Assume that the result holds for y2=τ−(l−1), τ−(l−2), …, τ−1. Let y2=τl, then

where the second equality holds because of the case above with y1=τ, and the third equality holds because of the assumption that each posting progresses independently of the price in the other posting and that πx2, rz=0 for r<x2. Therefore, Lemma 5 holds for the case (1c).

(1d) Let y1<τ−1. Proof by backward induction on y1. Let x1>0, y1=τ−2, and z=2, then

where the second equality holds from case (1c) above, and the third equality holds because of the assumption that each posting progresses independently of the price in the other posting and that πx2, r∣z = 0 for r<x2. Assume that the result holds for y1=τ−(l−1), τ−(l−2),…, τ−2. Let y1=τl, x1>0, and z=2, then

where the second equality holds from the induction hypothesis, and the third equality holds because of the assumption that each posting progresses independently of the price in the other auction and that for r<x2. Therefore, Lemma 5 holds for the case (1d). And consequently, Lemma 5 holds for (1) x1>0,y1=τ, δ and z=0, 1, or x1>0, y1, y2<τ and z=2.

It remains to show that Lemma 5 also holds for (2) x1>0, y1<τ, y2=0, z=1. Proof by backward induction on y1. Let y1=τ−1 and z=1, then

where the first equality holds because of case (1) above, and the second equality holds because of the assumption that each posting progresses independently of the price in the other posting. Assume that the result holds for y1=τ−(l−1), τ−(l−2),…, τ−1. Let y1=τl and z=1, then

where the first equality holds because of case (1) above, and the second equality holds because of the assumption that each posting progresses independently of the price in the other posting. Therefore, Lemma 5 holds for the case (2) x1>0, y1<τ, y2=0 and z=1. □

Proof of Lemma 6. Let y1=τ−1, x1=0, x2>0 and z=2. Proof by backward induction on y2. Let y2=τ−1, then

where the second equality holds because of Lemma 5, the fifth equality holds because and the sixth equality holds because Assume that the result holds for y2=τ−(l−1), τ−(l−2),…, τ−1. Let y2=τl, then

where the second equality holds because of Lemma 5, the fourth equality holds because , and the fifth equality holds because Therefore, Lemma 6 holds for y1=τ−1, x1=0, x2>0, and z=2.

For y1<τ−1, x1=0, x2>0, z=2, proof by backward induction on y1. Let y1=τ−2, then

where the second equality holds because of the case above with y1=τ−1, the fourth equality holds because the expected sales price of the first posting is independent of the price in the second posting and that πq, 0∣2=0 for q>0, the fifth equality holds because πq, 0∣2=0 for q>0 and the second posting progresses independently of the price in the first posting, and the sixth equality holds because of the assumption that and that the second posting progresses independently of the price in the first posting.

Assume that the result holds for y1=τ−(l−1), τ−(l−2),…, τ−2. Let y1=τl, then

where the second equality holds because of the induction hypothesis, and the other equalities because of the same reasoning as above. Therefore, Lemma 6 holds for y1<τ−1, x1=0, x2>0, and z=2. □

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Ødegaard, F., Puterman, M. Selling in channels with uncertain price. J Revenue Pricing Manag 11, 386–420 (2012). https://doi.org/10.1057/rpm.2011.25

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