Buying from the Babbling Retailer? The Impact of Availability Information on Customer Behavior

Abstract

Provision of real-time information by a firm to its customers has become prevalent in recent years in both the service and retail sectors. In this chapter, we study a retail operations model where customers are strategic in both their actions and in the way they interpret information, while the retailer is strategic in the way it provides information. This chapter focuses on the ability (or the lack thereof) to communicate unverifiable information and influence customers’ actions. We develop a game-theoretic framework to study this type of communication and discuss the equilibrium language emerging between the retailer and its customers. We show that for a single-retailer and homogeneous customer population setting, the equilibrium language that emerges carries no information. In this sense, a single-retailer providing information on its own cannot create any credibility with the customers. We study how the results are impacted due to the heterogeneity of the customers. We provide conditions under which the firm may be able to influence the customer behavior. In particular, we show that the customers’ willingness-to-pay and willingness-to-wait cannot be ranked in an opposite manner. However, even when the firm can influence each customer class separately, the effective demand is not impacted.

This chapter is based on Allon and Bassamboo (2011).

Appendix: Proofs

Proof of Proposition 3. The proof follows along the same line as the proof of Proposition 2. As before the firm’s incentive compatibility condition requires that the firm signals from the set of messages that maximizes

$$\displaystyle{\lambda _{L}y_{L}(m) +\lambda _{H}y_{H}(m).}$$

Thus, given the equilibrium (ν, y _L , y _H ), we have that there exist a constant $\theta$ for all realization of quantity on hand that satisfies the definition of AGNI. This completes the proof.

Proof of Proposition 4. Let $A_{R,i}^{m_{1}}$ be the implied availability in equilibrium for class i customer during the regular season when the firm signals m ₁. Let $A_{S,i}^{m_{1}}$ be the implied availability in equilibrium for class i customer during the sales season when the firm signals m ₁. We know from Lemma 1 that

$$\displaystyle\begin{array}{rcl} \vert A_{R,1}^{m_{1} } - A_{R,2}^{m_{1} }\vert \leq \varDelta ^{R},\quad \vert A_{ S,1}^{m_{1} } - A_{S,2}^{m_{1} }\vert \leq \varDelta ^{S}.& &{}\end{array}$$

(12.16)

For the equilibrium to be CWI, there must be a message such that

$$\displaystyle\begin{array}{rcl} \ \ (A_{R,L}^{m_{1} } - A_{S,L}^{m_{1} })v_{L} + c_{L}& \geq & pA_{R,L}^{m_{1} } - sA_{S,L}^{m_{1} }, \\ (A_{R,H}^{m_{1} } - A_{S,H}^{m_{1} })v_{H} + c_{H}& \leq & pA_{R,H}^{m_{1} } - sA_{S,H}^{m_{1} }.{}\end{array}$$

(12.17)

Using (12.16), we have that

$$\displaystyle{ \vert pA_{R,L}^{m_{1} } - sA_{S,L}^{m_{1} } - (pA_{R,H}^{m_{1} } - sA_{S,H}^{m_{1} })\vert \leq p\varDelta ^{R} + s\varDelta ^{S} \leq v_{ L}(\varDelta ^{R} +\varDelta ^{S}). }$$

(12.18)

Thus, (12.17) implies that

$$\displaystyle\begin{array}{rcl} (A_{R,L}^{m_{1} } - A_{S,L}^{m_{1} })v_{L} + c_{L} \geq (A_{R,H}^{m_{1} } - A_{S,H}^{m_{1} })v_{H} + c_{H} - v_{L}(\varDelta ^{R} +\varDelta ^{S}).& &{}\end{array}$$

(12.19)

Rearranging we obtain

$$\displaystyle\begin{array}{rcl} c_{H} - c_{L}& \leq & (A_{R,L}^{m_{1} } - A_{S,L}^{m_{1} })v_{L} - (A_{R,H}^{m_{1} } - A_{S,H}^{m_{1} })v_{H} + v_{L}(\varDelta ^{R} +\varDelta ^{S}) {}\\ & \leq & (A_{R,L}^{m_{1} } - A_{S,L}^{m_{1} })v_{L} - (A_{R,H}^{m_{1} } - A_{S,H}^{m_{1} })v_{L} + (A_{R,H}^{m_{1} } - A_{S,H}^{m_{1} })(v_{L} - v_{H}) {}\\ & & + v_{L}(\varDelta ^{R} +\varDelta ^{S}) {}\\ & \stackrel{\text{(a)}}{\leq }& (A_{R,L}^{m_{1} } - A_{S,L}^{m_{1} })v_{L} - (A_{R,H}^{m_{1} } - A_{S,H}^{m_{1} })v_{L} + v_{L}(\varDelta ^{R} +\varDelta ^{S}) {}\\ & \stackrel{\text{(b)}}{\leq }& 2v_{L}(\varDelta ^{R} +\varDelta ^{S}), {}\\ \end{array}$$

where (a) follows by noting that v _H  > v _L and $A_{R,H}^{m_{1}} \geq A_{S,H}^{m_{1}}$, and (b) follows from (12.16). This completes the proof.

Lemma 1.

 For any equilibrium, we have that for any message m

$$\displaystyle\begin{array}{rcl} \vert A_{R,L}^{m_{1} } - A_{R,H}^{m_{1} }\vert \leq \varDelta ^{R}&:=& 2e^{-\lambda _{L} } + 2e^{-\lambda _{H} } + \mathbb{E}\left [ \frac{1} {Q_{0}}\right ],{}\end{array}$$

(12.20)

$$\displaystyle\begin{array}{rcl} \vert A_{S,L}^{m_{1} } - A_{S,H}^{m_{1} }\vert \leq \varDelta ^{S}&:=& 2e^{-\lambda _{L} } + 2e^{-\lambda _{H} } + \mathbb{E}\left [ \frac{1} {D_{2}}\right ].{}\end{array}$$

(12.21)

Proof of Lemma 1. Let y be the equilibrium behavior for the customers. Thus, for the message m ₁ customer of class-L and class-H purchase during the regular season with probability y _L (m ₁) and y _H (m ₁). Thus, we can represent the

$$\displaystyle\begin{array}{rcl} A_{R,L}^{m_{1} }& =& \mathbb{E}{\Biggl [ \frac{Q_{0}} {\sum _{i=2}^{N_{L}}\mathbb{I}_{\{\omega _{ i}^{L}\leq y_{L}\}} + 1 +\sum _{ j=1}^{N_{H}}\mathbb{I}_{\{\omega _{ j}^{H}\leq y_{H}\}}} \wedge 1\Biggm |N_{L} \geq 1\Biggr ]},{}\end{array}$$

(12.22)

$$\displaystyle\begin{array}{rcl} A_{R,H}^{m_{1} }& =& \mathbb{E}{\Biggl [ \frac{Q_{0}} {\sum _{i=1}^{N_{L}}\mathbb{I}_{\{\omega _{ i}^{L}\leq y_{L}\}} + 1 +\sum _{ j=2}^{N_{H}}\mathbb{I}_{\{\omega _{ j}^{H}\leq y_{H}\}}} \wedge 1\Biggm |N_{H} \geq 1\Biggr ]},{}\end{array}$$

(12.23)

where ω _j ^H and ω _j ^L are iid sequences of uniform random variables on [0, 1]. Let us define

$$\displaystyle\begin{array}{rcl} X_{L} ={\Biggl [ \frac{Q_{0}} {\sum _{i=2}^{N_{L}}\mathbb{I}_{\{\omega _{ i}^{L}\leq y_{L}\}} + 1 +\sum _{ j=1}^{N_{H}}\mathbb{I}_{\{\omega _{ j}^{H}\leq y_{H}\}}} \wedge 1\Biggr ]},& & {}\\ X_{H} ={\Biggl [ \frac{Q_{0}} {\sum _{i=1}^{N_{L}}\mathbb{I}_{\{\omega _{ i}^{L}\leq y_{L}\}} + 1 +\sum _{ j=2}^{N_{H}}\mathbb{I}_{\{\omega _{ j}^{H}\leq y_{H}\}}} \wedge 1\Biggr ]}.& & {}\\ \end{array}$$

Since,

$$\displaystyle\begin{array}{rcl} & & {\biggl |{\biggl [\sum _{i=2}^{N_{L} }\mathbb{I}_{\{\omega _{i}^{L}\leq y_{L}\}} + 1 +\sum _{ j=1}^{N_{H} }\mathbb{I}_{\{\omega _{j}^{H}\leq y_{H}\}}\biggr ]} -{\biggl [\sum _{i=1}^{N_{L} }\mathbb{I}_{\{\omega _{i}^{L}\leq y_{L}\}} + 1 +\sum _{ j=2}^{N_{H} }\mathbb{I}_{\{\omega _{j}^{H}\leq y_{H}\}}\biggr ]}\biggr |} {}\\ & & \qquad ={\bigl | \mathbb{I}_{\{\omega _{1}^{H}\leq y_{H}\}} - \mathbb{I}_{\{\omega _{1}^{L}\leq y_{L}\}}\bigr |} \leq 1, {}\\ \end{array}$$

we obtain that

$$\displaystyle\begin{array}{rcl} \vert X_{L} - X_{H}\vert \leq \frac{1} {Q_{0}}.& &{}\end{array}$$

(12.24)

Using the definitions of X _L and X _H , we have

$$\displaystyle\begin{array}{rcl} \vert A_{R,L}^{m_{1} } - A_{R,H}^{m_{1} }\vert & = & \vert \mathbb{E}[X_{L}\vert N_{L} \geq 1] - \mathbb{E}[X_{H}\vert N_{H} \geq 1]\vert {}\\ & = & {\biggl |\frac{\mathbb{E}[X_{L}\mathbb{I}_{\{N_{L}\geq 1\}}]} {\mathbb{P}(N_{L} \geq 1)} -\frac{\mathbb{E}[X_{H}\mathbb{I}_{\{N_{H}\geq 1\}}]} {\mathbb{P}(N_{H} \geq 1)} \biggr |} {}\\ & \leq &{\biggl |\frac{\mathbb{E}[X_{L}\mathbb{I}_{\{N_{L}\geq 1,\ N_{H}\geq 1\}}]} {\mathbb{P}(N_{L} \geq 1)} -\frac{\mathbb{E}[X_{H}\mathbb{I}_{\{N_{H}\geq 1\,N_{L}\geq 1\}}]} {\mathbb{P}(N_{H} \geq 1)} \biggr |} {}\\ & & +{\biggl | \frac{\mathbb{E}[X_{L}\mathbb{I}_{\{N_{L}\geq 1,\ N_{H}=0\}}]} {\mathbb{P}(N_{L} \geq 1)} \biggr |} +{\biggl | \frac{\mathbb{E}[X_{H}\mathbb{I}_{\{N_{H}\geq 1,\ N_{L}=0\}}]} {\mathbb{P}(N_{H} \geq 1)} \biggr |} {}\\ & \stackrel{\text{(a)}}{\leq }& {\biggl |\frac{\mathbb{E}[X_{L}\mathbb{I}_{\{N_{L}\geq 1,\ N_{H}\geq 1\}}]} {\mathbb{P}(N_{L} \geq 1)} -\frac{\mathbb{E}[X_{H}\mathbb{I}_{\{N_{H}\geq 1\,N_{L}\geq 1\}}]} {\mathbb{P}(N_{H} \geq 1)} \biggr |} + e^{-\lambda _{L} } + e^{-\lambda _{H} } {}\\ & = & {\biggl |\frac{\mathbb{E}[X_{L}\mathbb{P}(N_{H} \geq 1)\mathbb{I}_{\{N_{L}\geq 1,\ N_{H}\geq 1\}}] - \mathbb{E}[X_{H}\mathbb{P}(N_{L} \geq 1)\mathbb{I}_{\{N_{H}\geq 1\,N_{L}\geq 1\}}]} {\mathbb{P}(N_{L} \geq 1)\mathbb{P}(N_{H} \geq 1)} \biggr |} {}\\ & & + e^{-\lambda _{L} } + e^{-\lambda _{H} } {}\\ & = & {\biggl |\frac{\mathbb{E}[(X_{L}\mathbb{P}(N_{H} \geq 1) - X_{H}\mathbb{P}(N_{L} \geq 1))\mathbb{I}_{\{N_{L}\geq 1,\ N_{H}\geq 1\}}]} {\mathbb{P}(N_{L} \geq 1)\mathbb{P}(N_{H} \geq 1)} \biggr |} + e^{-\lambda _{L} } + e^{-\lambda _{H} } {}\\ & = & {\bigl |\mathbb{E}[(X_{L}\mathbb{P}(N_{H} \geq 1) - X_{H}\mathbb{P}(N_{L} \geq 1))\vert N_{L} \geq 1,\ N_{H} \geq 1]\bigr |} {}\\ & & + e^{-\lambda _{L} } + e^{-\lambda _{H} } {}\\ & \stackrel{\text{(b)}}{\leq }& {\bigl |\mathbb{E}[(X_{L} - X_{H})\vert N_{L} \geq 1,\ N_{H} \geq 1]\bigr |} + 2(e^{-\lambda _{L} } + e^{-\lambda _{H} }) {}\\ & \stackrel{\text{(c)}}{\leq }& \mathbb{E}\left [ \frac{1} {Q_{0}}\right ] + 2(e^{-\lambda _{L} } + e^{-\lambda _{H} }). {}\\ \end{array}$$

Here (a) and (b) follows by noting that X _L and X _H are bounded by 1, $\mathbb{P}(N_{L} = 0) = e^{-\lambda _{L}}$ and $\mathbb{P}(N_{H} = 0) = e^{-\lambda _{H}}$. The inequality in (c) follows by (12.24) and mutual independence of random variables Q ₀, N _L and N _H . This completes the proof for (12.20). The proof of (12.21) follows along the same line and using the definitions:

$$\displaystyle\begin{array}{rcl} A_{S,L}^{m_{1} }& =& \mathbb{E}{\Biggl [ \frac{Q_{0} -\sum _{i=2}^{N_{L}}\mathbb{I}_{\{\omega _{ i}^{L}\leq y_{L}\}} -\sum _{j=1}^{N_{H}}\mathbb{I}_{\{\omega _{ j}^{H}\leq y_{H}\}}} {N_{B} + 1 +\sum _{ i=2}^{N_{L}}\mathbb{I}_{\{\omega _{ i}^{L}>y_{L}\}} +\sum _{ j=1}^{N_{H}}\mathbb{I}_{\{\omega _{ j}^{H}>y_{H}\}}} \wedge 1\Biggm |N_{L} \geq 1\Biggr ]}{}\end{array}$$

(12.25)

$$\displaystyle\begin{array}{rcl} A_{S,H}^{m_{1} }& =& \mathbb{E}{\Biggl [ \frac{Q_{0} -\sum _{i=1}^{N_{L}}\mathbb{I}_{\{\omega _{ i}^{L}\leq y_{L}\}} -\sum _{j=2}^{N_{H}}\mathbb{I}_{\{\omega _{ j}^{H}\leq y_{H}\}}} {N_{B} + 1 +\sum _{ i=1}^{N_{L}}\mathbb{I}_{\{\omega _{ i}^{L}>y_{L}\}} +\sum _{ j=2}^{N_{H}}\mathbb{I}_{\{\omega _{ j}^{H}>y_{H}\}}} \wedge 1\Biggm |N_{H} \geq 1\Biggr ]}.\qquad {}\end{array}$$

(12.26)

Proof of Theorem 1. As in the proof of Proposition 4, we let $A_{R,i}^{m_{1}}$ be the implied availability in equilibrium for class i customer during the regular season when the firm signals m ₁, and $A_{S,i}^{m_{1}}$ be the implied availability in equilibrium for class i customer during the regular season when the firm signals m ₁. We know from Lemma 1 that

$$\displaystyle\begin{array}{rcl} \vert A_{R,1}^{m_{1} } - A_{R,2}^{m_{1} }\vert \leq \varDelta ^{R},\quad \vert A_{ S,1}^{m_{1} } - A_{S,2}^{m_{1} }\vert \leq \varDelta ^{S}.& &{}\end{array}$$

(12.27)

For the equilibrium to be CWI, there must be a message m ₁ and m ₂ such that

$$\displaystyle{ \begin{array}{rl} (A_{R,L}^{m_{1}} - A_{S,L}^{m_{1}})v_{L} + c_{L}& \geq pA_{R,L}^{m_{1}} - sA_{S,L}^{m_{1}}, \\ (A_{R,H}^{m_{1}} - A_{S,H}^{m_{1}})v_{H} + c_{H}& \leq pA_{R,H}^{m_{1}} - sA_{S,H}^{m_{1}}; \\ (A_{R,L}^{m_{2}} - A_{S,L}^{m_{2}})v_{L} + c_{L}& \leq pA_{R,L}^{m_{2}} - sA_{S,L}^{m_{2}}, \\ (A_{R,H}^{m_{2}} - A_{S,H}^{m_{2}})v_{H} + c_{H}& \geq pA_{R,H}^{m_{2}} - sA_{S,H}^{m_{2}}. \end{array} }$$

(12.28)

Further, using (12.18), we have

$$\displaystyle\begin{array}{rcl} c_{H} - c_{L}& \leq & (A_{R,L}^{m_{1} } - A_{S,L}^{m_{1} })v_{L} - (A_{R,H}^{m_{1} } - A_{S,H}^{m_{1} })v_{H} + v_{L}(\varDelta ^{R} +\varDelta ^{S}) {}\\ & \leq & (A_{R,L}^{m_{1} } - A_{S,L}^{m_{1} })v_{L} - (A_{R,H}^{m_{1} } - A_{S,H}^{m_{1} })v_{L} + (A_{R,H}^{m_{1} } - A_{S,H}^{m_{1} })(v_{L} - v_{H}) {}\\ & & + v_{L}(\varDelta ^{R} +\varDelta ^{S}) {}\\ & \stackrel{}{\leq }& (A_{R,L}^{m_{1} } - A_{S,L}^{m_{1} })v_{L} - (A_{R,H}^{m_{1} } - A_{S,H}^{m_{1} })v_{L} + v_{L}(\varDelta ^{R} +\varDelta ^{S}) {}\\ & & + (A_{R,H}^{m_{1} } - A_{S,H}^{m_{1} })(v_{L} - v_{H}) {}\\ & \stackrel{}{\leq }& 2v_{L}(\varDelta ^{R} +\varDelta ^{S}) + (A_{ R,H}^{m_{1} } - A_{S,H}^{m_{1} })(v_{L} - v_{H}) {}\\ \end{array}$$

We thus obtain:

$$\displaystyle\begin{array}{rcl} & \frac{c_{L}-c_{H}} {v_{H}-v_{L}} \geq -2(\varDelta ^{R} +\varDelta ^{S}) \frac{v_{L}} {v_{H}-v_{L}} + (A_{R,H}^{m_{1}} - A_{S,H}^{m_{1}}),& {}\\ & \frac{c_{L}-c_{H}} {v_{H}-v_{L}} + 2(\varDelta ^{R} +\varDelta ^{S}) \frac{v_{L}} {v_{H}-v_{L}} \geq (A_{R,H}^{m_{1}} - A_{S,H}^{m_{1}}). & {}\\ \end{array}$$

The other inequalities follows in the same manner. This completes the proof.