Competitive reasons for the Name-Your-Own-Price channel

Abstract

This paper shows that the Name-Your-Own-Price (NYOP) business model can help soften competition. When consumers differ in their frictional costs (i.e., the shopping hassle) they experience when bidding at an NYOP retailer, the NYOP format can be a mechanism for differentiating a retailer from a posted-price rival. Beyond providing a motivation for using an NYOP mechanism, competition also has important implications for the optimal structure of the NYOP format. For example, this paper shows that prohibiting rebidding may benefit an NYOP firm by reducing price rivalry.

Appendix

1.1 Derivation of consumers’ optimal bids when retailer B uses an NYOP format

Retailer B’s loyal consumers can either bid at B’s NYOP site or they can forego bidding. A bid of b is accepted with a probability of b, in which case the consumer gets a net surplus of (1 − b). Thus, the consumer’s expected value from bidding b is:

$${\text{EV}}_{\text{L}} \left( b \right) = \left( {1 - b} \right)b - c_i $$

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The first-order condition (FOC) of this maximization problem is: $\frac{{\partial {\text{EV}}_{\text{L}} \left( b \right)}}{{\partial b}} = 1 - 2b \equiv 0$. This FOC is satisfied when $b = \frac{1}{2} \equiv \hat b$ and ${\text{EV}}_{\text{L}} \left( {\hat b} \right)$ is non-negative (so as to induce bidding rather than abstaining) iff $c_i \leqslant \frac{1}{4} \equiv \hat c$.

Suppose retailer A sells at a posted price of P _A . A switcher who purchases at the posted price earns a surplus of (1 − P _A ). A switcher can bid at B’s NYOP site if she so chooses. The expected value from placing a bid of b and then purchasing at the posted price if that bid is rejected is:

$${\text{EV}}_{{\text{S1}}} \left( b \right) = \left( {1 - b} \right)b - c_i + \left( {1 - b} \right)\left( {1 - P_A } \right)$$

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Using the FOC, this expected value is maximized when $b = \hat b_1$and weakly exceeds the surplus from buying at the posted price without bidding iff $c_i \leqslant \hat c_1$, where $\hat b_1$and $\hat c_1$ are defined in Eq. 2.

Now suppose retailer A also uses an NYOP format. If a switcher were to not bid on either site, she would earn a surplus of 0. Another alternative bidding strategy would be to place a bid of b at one of the two sites and then not bid at the other site if this bid were rejected. This strategy generates an expected value given by Eq. 8, which is maximized when $b = \hat b$, yielding an expected value of $\frac{1}{4} - c_i$. Finally, a switcher could place a bid of b ₁ at one of the two sites and then place a bid of b ₂ at the other site if the first bid is rejected. This strategy yields an expected value of:

$${\text{EV}}_{{\text{S2}}} \left( {b_1 {\text{,}}b_2 } \right) = \left( {1 - b_1 } \right)b_1 - c_i + \left( {1 - b_1 } \right)\left[ {\left( {1 - b_2 } \right)b_2 - c_i } \right]$$

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Using the resulting FOCs, the consumer’s expected value is maximized when $b_1 = \hat b_{11}$ and $b_2 = \hat b$ where $\hat b_{11}$ and $\hat b$ are defined in Eqs. 3 and 1, respectively. This dual-bidding strategy yields a higher expected surplus than the single-bidding strategy $\forall c_i \leqslant \hat c$. This dual-bidding strategy is weakly superior to not bidding at all iff $c_i \leqslant \hat c$.

1.2 Derivation of equilibrium prices and profit for each subgame

In accordance with Proposition 1, the remainder of the derivations in the “Appendix” proceeds under the assumptions $\frac{1}{4} \leqslant \bar c \leqslant \frac{1}{2}$ and $\lambda <\frac{1}{9}$.

1.2.1 Both firms use posted prices

The loyal-switcher model whereby both firms sell at a posted price is a slightly simplified version of a model which has previously been analyzed by Varian (1980; using the terms “informed” and “uninformed” to describe the two segments). Rather than replicating that entire analysis, below, I simply report the results.

There is no pure-strategy equilibrium to this game. Instead, there is a mixed-strategy equilibrium in prices in which each retailer charges price P with a probability f(P). f(P) = 0 if $P <\hat P$ or P > 1, where $\hat P = \frac{\lambda }{{1 - \lambda }}$. On the interval $\left[ {\hat P{\text{,}}1} \right]$, ${\text{f}}\left( P \right) = \frac{\lambda }{{\left( {1 - 2\lambda } \right)P^2 }}$. Each firm’s expected profit is $\Pi _A^{{\text{PP,PP}}} = \Pi _B^{{\text{PP,PP}}} = \lambda$.

1.2.2 Firm A uses posted prices; firm B uses an NYOP format

Retailer A chooses its posted price in order to maximize the expected profit given in Eq. 6 (with the profit function for retailer B, $\Pi _B^{{\text{PP,NYOP}}}$, given by Eq. 5). At the interior solution of $P_A^{{\text{IS}}} = \sqrt[{\text{3}}]{{\frac{{{\text{2}}\overline {\text{c}} \left( {{\text{1}} - \lambda } \right)}}{{{\text{1}} - {\text{2}}\lambda }}}}$, profit for firm A (who uses a posted price) is $\Pi _A^{{\text{PP,NYOP}}\left( {{\text{IS}}} \right)} = \frac{{3\left( {{\text{1}} - \lambda } \right)}}{2}\sqrt[{\text{3}}]{{\frac{{\overline {\text{c}} \left( {{\text{1}} - \lambda } \right)}}{{4\left( {{\text{1}} - {\text{2}}\lambda } \right)}}}}$. The profit for firm B is $\Pi _B^{{\text{PP,NYOP}}\left( {{\text{IS}}} \right)} = \frac{\lambda }{{16\overline {\text{c}} }} + \frac{{1 - {\text{2}}\lambda }}{4}\sqrt[{\text{3}}]{{\frac{{\overline {\text{c}} ^{\text{3}} \left( {{\text{1}} - \lambda } \right)^4 }}{{4\left( {{\text{1}} - {\text{2}}\lambda } \right)^4 }}}}$. There is also a potential corner solution at $P_A^{{\text{CS}}} = 1$. Here, firm A’s profit is $\Pi _A^{{\text{PP,NYOP(CS)}}} = 1 - \lambda - \frac{{{\text{1}} - {\text{2}}\lambda }}{{{\text{8}}\overline {\text{c}} }}$ and firm B’s profit is $\Pi _B^{{\text{PP,NYOP}}\left( {{\text{CS}}} \right)} = \frac{{1 - \lambda }}{{{\text{16}}\overline {\text{c}} }}$. Firm A chooses the posted price that yields the highest profit:

$$\Pi _A^{{\text{PP,NYOP}}} = {\text{Max}}\left[ {\Pi _A^{{\text{PP,NYOP}}\left( {{\text{IS}}} \right)} {\text{,}}\Pi _A^{{\text{PP,NYOP}}\left( {{\text{CS}}} \right)} } \right]{\text{ = }}\left\{ {\matrix {\frac{{3\left( {{\text{1 - }}\lambda } \right)}}{2}\sqrt[{\text{3}}]{{\frac{{\overline {\text{c}} \left( {{\text{1}} - \lambda } \right)}}{{4\left( {{\text{1}} - {\text{2}}\lambda } \right)}}}}} \hfill {{\text{if }}\overline {\text{c}} \leqslant \frac{{{\text{1}} - 2\lambda }}{{{\text{2}}\left( {{\text{1}} - \lambda } \right)}}} \hfill \\ {1 - \lambda - \frac{{{\text{1}} - {\text{2}}\lambda }}{{{\text{8}}\overline {\text{c}} }}} \hfill {{\text{if }}\overline {\text{c}} > \frac{{{\text{1}} - 2\lambda }}{{{\text{2}}\left( {{\text{1}} - \lambda } \right)}}} \hfill \ } \right.$$

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1.2.3 Both firms use a NYOP format

When both retailers use an NYOP format, profit is given by Eq. 7. Taking these integrals,

$$\Pi _A^{{\text{NYOP,NYOP}}} = \Pi _B^{{\text{NYOP,NYOP}}} = \frac{{2 - \lambda }}{{48\overline c }}$$

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1.3 Derivation of the optimal market format—proposition 1

For (PP, NYOP) to be a subgame perfect Nash equilibrium, then firm B must earn greater profit by choosing the NYOP format rather than the posted-price format. One can calculate the difference in profit under these two formats as:

$$\Pi _B^{{\text{PP,NYOP}}} - \Pi _B^{{\text{PP,PP}}} {\text{ = }}\left\{ {\matrix {\frac{{1 - {\text{2}}\lambda }}{4}\sqrt[{\text{3}}]{{\frac{{\overline {\text{c}} ^{\text{3}} \left( {{\text{1}} - \lambda } \right)^4 }}{{4\left( {{\text{1}} - {\text{2}}\lambda } \right)^4 }}}} - \lambda + \frac{\lambda }{{16\overline c }}} \hfill {{\text{if }}\overline {\text{c}} \leqslant \frac{{{\text{1}} - 2\lambda }}{{{\text{2}}\left( {{\text{1}} - \lambda } \right)}}} \hfill \\ {\frac{{1 - \lambda }}{{16\overline c }} - \lambda } \hfill {{\text{if }}\overline c > \frac{{{\text{1}} - 2\lambda }}{{{\text{2}}\left( {{\text{1}} - \lambda } \right)}}} \hfill \ } \right.$$

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This difference in the bottom term is positive as long as $1 > \lambda \left( {1 + {\text{16}}\overline {\text{c}} } \right)$ or $\lambda < \frac{1}{{1 + {\text{16}}\overline {\text{c}} }}$. Note that this condition will be met when $\lambda <\frac{1}{9}$ (since $\overline {\text{c}} < \frac{1}{2}$). Furthermore, the top term is decreasing in $\overline {\text{c}}$. Thus, on the range, $\overline c \in \left[ {\frac{{\text{1}}}{{\text{4}}}{\text{,}}\frac{{{\text{1}} - 2\lambda }}{{{\text{2}}\left( {{\text{1}} - \lambda } \right)}}} \right]$, the difference in profit $\left( {\Pi _B^{{\text{PP,NYOP}}} - \Pi _B^{{\text{PP,PP}}} } \right)$is minimized at $\overline c = \frac{{{\text{1}} - 2\lambda }}{{{\text{2}}\left( {{\text{1}} - \lambda } \right)}}$. When $\overline c = \frac{{{\text{1}} - 2\lambda }}{{{\text{2}}\left( {{\text{1}} - \lambda } \right)}}$, $\Pi _B^{{\text{PP,NYOP}}} - \Pi _B^{{\text{PP,PP}}} = \frac{{\left( {1 - \lambda } \right)^2 \left( {1 - 5\lambda } \right)}}{{12\left( {1 - 2\lambda } \right)^2 }}$, which is positive for all $\lambda <\frac{1}{5}$. Therefore, $\lambda <\frac{1}{9}$ is a sufficient condition for retailer B to prefer the NYOP format over a posted-price format.

In addition, for (PP, NYOP) to be a subgame perfect Nash equilibrium, firm A must earn greater profit by choosing the posted-price format rather than the NYOP format. The preceding analysis shows that if firm A chooses the posted-price format, then firm B will select the NYOP format. To determine firm A’s profit from selecting the NYOP format instead, one needs to anticipate whether firm B would best respond by selecting the posted-price format or the NYOP format. Specifically, one can calculate:

$$\Pi _B^{{\text{NYOP,PP}}} - \Pi _B^{{\text{NYOP,NYOP}}} {\text{ = = }}\left\{ {\matrix {\frac{{3\left( {{\text{1}} - \lambda } \right)}}{2}\sqrt[{\text{3}}]{{\frac{{\overline {\text{c}} \left( {{\text{1}} - \lambda } \right)}}{{4\left( {{\text{1}} - {\text{2}}\lambda } \right)}}}} - \frac{{2 - \lambda }}{{48\overline {\text{c}} }}} \hfill {{\text{if }}\overline {\text{c}} \leqslant \frac{{{\text{1}} - 2\lambda }}{{{\text{2}}\left( {{\text{1}} - \lambda } \right)}}} \hfill \\ {1 - \lambda - \frac{{{\text{8}} - {\text{13}}\lambda }}{{{\text{48}}\overline {\text{c}} }}} \hfill {{\text{if }}\overline c > \frac{{{\text{1}} - 2\lambda }}{{{\text{2}}\left( {{\text{1}} - \lambda } \right)}}} \hfill \ } \right.$$

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Numerical calculations verify that this difference is positive over the entire relevant parameter range. Therefore, firm B will choose the posted-price format if firm A has chosen the NYOP format. For (PP, NYOP) to be a subgame perfect Nash equilibrium, we must have $\Pi _A^{{\text{PP,NYOP}}} \geqslant \Pi _A^{{\text{NYOP,PP}}}$ We can show

$$\Pi _A^{{\text{PP,NYOP}}} - \Pi _A^{{\text{NYOP,PP}}} {\text{ = }}\left\{ {\matrix {\frac{{10\overline {\text{c}} \left( {{\text{1}} - \lambda } \right)\sqrt[{\text{3}}]{{2\overline {\text{c}} \left( {1 - \lambda } \right)}} - \sqrt[{\text{3}}]{{1 - 2\lambda }}}}{{{\text{16}}\overline {\text{c}} \sqrt[{\text{3}}]{{1 - 2\lambda }}}}} \hfill {{\text{if }}\overline {\text{c}} \leqslant \frac{{{\text{1}} - 2\lambda }}{{{\text{2}}\left( {{\text{1}} - \lambda } \right)}}} \hfill \\ {1 - \lambda - \frac{{{\text{3}} - {\text{5}}\lambda }}{{{\text{16}}\overline {\text{c}} }}} \hfill {{\text{if }}\overline {\text{c}} > \frac{{{\text{1}} - 2\lambda }}{{{\text{2}}\left( {{\text{1}} - \lambda } \right)}}} \hfill \ } \right.$$

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For both the upper and lower expressions of Eq. 15, the difference, $\Pi _A^{{\text{PP,NYOP}}} - \Pi _A^{{\text{NYOP,PP}}}$, is increasing in $\overline c$. Thus, the minima over the two ranges are reached at $\overline c = \frac{1}{4}$ and $\overline c = \frac{{{\text{1}} - 2\lambda }}{{{\text{2}}\left( {{\text{1}} - \lambda } \right)}}$, respectively. At these minimizing points, the differences, $\Pi _A^{{\text{PP,NYOP}}} - \Pi _A^{{\text{NYOP,PP}}}$, are $\frac{{5\left( {{\text{1}} - \lambda } \right)\sqrt[{\text{3}}]{{4\left( {1 - \lambda } \right)}} - 4\lambda \sqrt[{\text{3}}]{{1 - 2\lambda }}}}{{{\text{16}}\sqrt[{\text{3}}]{{1 - 2\lambda }}}}$ and $\frac{{\left( {1 - \lambda } \right)\left( {5 - 11\lambda } \right)}}{{8\left( {1 - 2\lambda } \right)}}$, respectively. These values are positive for all $\lambda <\frac{1}{9}$.

1.4 Incentive to allow rebids

Customers who are loyal to retailer B (who uses the NYOP format) can place up to two bids. The optimal bidding strategy is:

$$\left\{ {\matrix {{\text{bid }}b_{12} {\text{, then bid }}b_{22} {\text{ if }}b_{12} {\text{ is rejected}}} \hfill {{\text{if 0}} \leqslant c_i \leqslant c_2 } \hfill {} \hfill \\ {{\text{place a single bid of }}\hat b} \hfill {{\text{if }}c_2 <c_i \leqslant \hat c} \hfill {{\text{where}}b_{12} = \frac{{1 + 2c_i }}{3}{\text{,}}b_{22} = \frac{{2 + c_i }}{3}{\text{,}}c_2 = \frac{{2 - \sqrt 3 }}{2}} \hfill \\ {{\text{do not bid at all}}} \hfill {{\text{if }}c_i >\hat c} \hfill {} \hfill \ } \right.$$

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For the switchers, the optimal bidding strategy is:

$$\left\{ {\matrix {{\text{bid }}\hat b_{12} {\text{, then bid }}\hat b_{22} {\text{ if }}\hat b_{12} {\text{ is rejected; purchase at }}P_A {\text{ if }}\hat b_{12} {\text{ is rejected}}} \hfill {{\text{if 0}} \leqslant c_i \leqslant \hat c_2 } \hfill {} \hfill \\ {{\text{place a single bid of }}\hat b_1 {\text{; purchase at }}P_A {\text{ if }}\hat b_1 {\text{ is rejected}}} \hfill {{\text{if }}\hat c_2 <c_i \leqslant \hat c_1 } \hfill {{\text{where }}\hat b_{12} = \frac{{P_A + 2c_i }}{3},\hat b_{22} = \frac{{2P_A + c_i }}{3},\hat c_2 = \frac{{3 - P_A - \sqrt {9 - 6P_A } }}{2}} \hfill \\ {{\text{purchase at }}P_A {\text{ without bidding at all}}} \hfill {{\text{if }}c_i >\hat c_1 } \hfill {} \hfill \ } \right.$$

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The following analysis proceeds under the assumption that ${\text{ }}\overline c \geqslant \frac{{1 - 2\lambda }}{{2\left( {1 - \lambda } \right)}}$ and $\lambda \leqslant \frac{1}{9}$. These restrictions ensure that, when retailer B (i.e., the NYOP firm) restricts consumers to a single bid, retailer A selects a posted price P _A  = 1. The equilibrium expected profit for each firm is:

$$\Pi _A^{{\text{PP,NYOP}}\left( {1{\text{ bid}}} \right)} = 1 - \lambda - \frac{{1 - 2\lambda }}{{8\overline c }}$$

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$$\Pi _B^{{\text{PP,NYOP}}\left( {1{\text{ bid}}} \right)} = \frac{{1 - \lambda }}{{16\overline c }}$$

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If retailer A allows consumers to rebid if their initial bid is rejected, then the profits for the two firms are given by:

$$\Pi _A^{{\text{PP,NYOP}}\left( {{\text{2 bids}}} \right)} = P_A \left\lceil {1 - \lambda - \left( {1 - 2\lambda } \right)\left( {\int\limits_{c_i = 0}^{\hat c_2 } {\int\limits_{P_{{\text{NYOP}}} = 0}^{\hat b_{22} } {{\text{f}}\left( {c_i } \right){\text{d}}P_{{\text{NYOP}}} {\text{ d}}c_i } } + \int\limits_{c_i = \hat c_2 }^{\hat c_1 } {\int\limits_{P_{{\text{NYOP}}} = 0}^{\hat b_1 } {{\text{f}}\left( {c_i } \right){\text{d}}P_{{\text{NYOP}}} {\text{ d}}c_i } } } \right)} \right\rceil $$

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$$\begin{aligned} \Pi _B^{{\text{PP,NYOP}}\left( {{\text{2 bids}}} \right)} = \lambda \left[ {\int\limits_{c_i = 0}^{c_2 } {\left( {\int\limits_{P_{{\text{NYOP}}} = 0}^{b_{12} } {b_{12} {\text{f}}\left( {c_i } \right){\text{d}}P_{{\text{NYOP}}} } + \int\limits_{P_{{\text{NYOP}}} = b_{12} }^{b_{22} } {b_{22} {\text{f}}\left( {c_i } \right){\text{d}}P_{{\text{NYOP}}} } } \right){\text{d}}c_i + } \int\limits_{c_i = c_2 }^{\hat c} {\int\limits_{P_{NYOP} = 0}^{\hat b} {\hat b{\text{f}}\left( {c_i } \right){\text{d}}P_{{\text{NYOP}}} } {\text{d}}c_i } } \right] \\ + \left( {1 - 2\lambda } \right)\left[ {\int\limits_{c_i = 0}^{\hat c_2 } {\left( {\int\limits_{P_{{\text{NYOP}}} = 0}^{\hat b_{12} } {\hat b_{12} {\text{f}}\left( {c_i } \right){\text{d}}P_{{\text{NYOP}}} } + \int\limits_{P_{{\text{NYOP}}} = \hat b_{12} }^{\hat b_{22} } {\hat b_{22} {\text{f}}\left( {c_i } \right){\text{d}}P_{{\text{NYOP}}} } } \right){\text{d}}c_i + } \int\limits_{c_i = \hat c_2 }^{\hat c_1 } {\int\limits_{P_{{\text{NYOP}}} = 0}^{\hat b_1 } {\hat b_1 {\text{f}}\left( {c_i } \right){\text{d}}P_{{\text{NYOP}}} } {\text{d}}c_i } } \right] \\ \end{aligned} $$

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Retailer A chooses P _A to maximize Eq. 20. The closed-form solution is extremely complex. But Fig. 2 provides an example that verifies the claim in Proposition 2. In particular, when λ = 0.05, $\Pi _B^{{\text{PP,NYOP}}\left( {1{\text{ bid}}} \right)} >\Pi _B^{{\text{PP,NYOP}}\left( {2{\text{ bids}}} \right)}$ for $\overline c \in \left[ {\frac{9}{{19}},0.476} \right]$.

Furthermore, one can verify that in the absence of this strategic effect on prices (i.e., that P _A is lower when rebidding is allowed), the NYOP firm is always more profitable when rebidding is allowed. In particular,

$$\left( {\left. {\Pi _B^{{\text{PP,NYOP}}\left( {{\text{2 bids}}} \right)} } \right|P_A = 1} \right) - \left( {\left. {\Pi _B^{{\text{PP,NYOP}}\left( {{\text{1 bid}}} \right)} } \right|P_A = 1} \right) = \frac{{1,605 - 882\sqrt 3 + \left( {214 - 156\sqrt 3 } \right)\lambda }}{{6,912\overline c }} \approx \frac{{77.3 - 56.2\lambda }}{{6{\text{,}}912\overline c }} > 0$$

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