Optimal selling strategies when buyers name their own prices

Abstract

This paper models a name-your-own-price (NYOP) retailer who allows buyers to initiate their retail interactions by describing a product and submitting a binding bid for it. The buyers have an outside option to buy the same good for a commonly known posted price that also acts as an informative upper bound on the cost the NYOP retailer faces. We conceptualize a selling strategy of such an NYOP retailer to be the probability that a buyer’s bid gets accepted. The selling strategy is a function of only the bid level; it does not depend on the particular realization of the retailer’s procurement cost. Using mechanism-design techniques, we characterize the optimal selling strategy and the equilibrium bidding function that best responds to it. We show that the optimal strategy implements the first-best ex-post optimal mechanism: for every cost realization, the retailer can make as much profit as he would if he could learn his cost first and use the optimal mechanism contingent on it. The complexity involved in credibly communicating an entire bid-acceptance function to buyers can make the first-best strategy impractical in some real-world markets, so we also analyze several simpler NYOP strategies: setting a minimum bid, charging a participation fee, and accepting all bids above cost. We find that under many scenarios, the minimum-bid strategy dominates the other simpler strategies and achieves a majority of the maximal profit improvement available from the first best strategy. However, NYOP retailers in thin markets can do better by charging participation fees than by setting minimum bids.

Appendix: notation table and proofs of propositions

Table 1 Notation

Proof of proposition 1

Let m(x) be the expected payment by a buyer with valuation x. From risk neutrality, the utility of a buyer x who reports type z is U(x) = xq(z) − m(z), and standard incentive-compatibility arguments (see Myerson 1981 for details) imply the expected payment is the following function of q(x):

$$ m(x)=m(0)+xq(x)-{\displaystyle \underset{0}{\overset{x}{\int }}q(t)dt} $$

(IC)

When (IC) does not hold, the buyers do not have the incentive to report their x truthfully.

Consider the direct-revelation retailer who can set an arbitrary bid-acceptance rule π(x, c). Plugging the implied bid-acceptance rule $q(x)={\displaystyle \underset{0}{\overset{p}{\int }}\pi \left(x,c\right)dH(c)}$into (IC) implies that on average over all c, such a retailer receives a payment of

$$ m(x)=m(0)+x{\displaystyle \underset{0}{\overset{p}{\int }}\pi \left(x,c\right)dH(c)}-{\displaystyle \underset{\underline{x}}{\overset{x}{\int }}{\displaystyle \underset{0}{\overset{p}{\int }}\pi \left(t,c\right)dH(c)dt}} $$

(A1)

Note the rule π can use c as an input, so the retailer can set the rule for all possible c levels upfront. However, the buyers do not know c at the time of submitting their bids, so incentive compatibility only restricts the average payment of a given buyer type. Because all buyers with x≥p pay m(p), the expected profit of the retailer is

$$ \begin{array}{l}\varPi \left(\pi \right)= \Pr \left(x<p\right)\left({E}_{x\Big|x<p}\left[m(x)\right]-{E}_{c,x\Big|x\le p}\left[c\pi \left(x,c\right)\right]\right)\\ {}\kern2em +\left[1- \Pr \left(x<p\right)\right]\left[m(p)-{E}_c\left[c\pi \left(p,c\right)\right]\right]\end{array} $$

(A2)

Plugging the m function from (A1) into the profit expression (A2) yields

$\begin{array}{l}\varPi \left(\pi \right)=m(0)+{\displaystyle \underset{\underline{x}}{\overset{p}{\int }}x{\displaystyle \underset{0}{\overset{p}{\int }}\pi \left(x,c\right)dH(c)dF(x)}}-{\displaystyle \underset{\underline{x}}{\overset{p}{\int }}{\displaystyle \underset{0}{\overset{x}{\int }}{\displaystyle \underset{0}{\overset{p}{\int }}\pi \left(t,c\right)dH(c) dtdF(x)}}}-{\displaystyle \underset{\underline{x}}{\overset{p}{\int }}{\displaystyle \underset{0}{\overset{p}{\int }}c\pi \left(x,c\right)dH(c)dF(x)}}\\ {}\kern4em +\left[1-F(p)\right]\left[p{\displaystyle \underset{0}{\overset{p}{\int }}\pi \left(p,c\right)dH(c)-{\displaystyle \underset{0}{\overset{p}{\int }}{\displaystyle \underset{0}{\overset{p}{\int }}\pi \left(t,c\right)dH(c)dt}}}-{\displaystyle \underset{0}{\overset{p}{\int }}c\pi \left(p,c\right)dH(c)}\right]\kern4em (A3)\end{array}$where the second row corresponds to the profit from high buyers (x≥p), and the last term in each row is the expected cost of goods sold.

As in other mechanism-design settings, the term ${\displaystyle \underset{\underline{x}}{\overset{p}{\int }}{\displaystyle \underset{0}{\overset{x}{\int }}{\displaystyle \underset{0}{\overset{p}{\int }}\pi \left(t,c\right)dH(c) dtdF(x)}}}$ in the first row can be simplified by first changing the order of integration from c,t,x to x,t,c, and noting that π(t, c) does not depend on x:

$$ \begin{array}{l}{\displaystyle \underset{\underline{x}}{\overset{p}{\int }}{\displaystyle \underset{0}{\overset{x}{\int }}\pi \left(t,c\right) dtdF(x)}}={\displaystyle \underset{0}{\overset{p}{\int }}\left(\pi \left(t,c\right){\displaystyle \underset{t}{\overset{p}{\int }}dF(x)}\right)dt}=\\ {}={\displaystyle \underset{0}{\overset{p}{\int }}\pi \left(t,c\right)\left[F(p)-F(t)\right]dt}={\displaystyle \underset{\underline{x}}{\overset{p}{\int }}\pi \left(x,c\right)\left(\frac{F(p)-F(x)}{f(x)}\right)dF(x)}\end{array} $$

(A4)

where the last equality simply renames the t variable as x and changes variables. Finally, change the order of integration to be first over x and then over c throughout, and collect terms:

$$ \begin{array}{l}\varPi \left(\pi \right)=m(0)+{\displaystyle \underset{0}{\overset{p}{\int }}{\displaystyle \underset{\underline{x}}{\overset{p}{\int }}\left(x-\frac{F(p)-F(x)}{f(x)}-c\right)\pi \left(x,c\right)dF(x)dH(c)}}+\\ {}+\left[1-F(p)\right]\left[{\displaystyle \underset{0}{\overset{p}{\int }}\left(p-c\right)\pi \left(p,c\right)dH(c)}\right]-\left[1-F(p)\right]{\displaystyle \underset{0}{\overset{p}{\int }}{\displaystyle \underset{\underline{x}}{\overset{p}{\int }}\pi \left(x,c\right) dxdH(c)}}\end{array} $$

(A5)

The last term in (A5) is the expected surplus of the high buyers (x≥p). It obviously depends on the allocation rule for all x≤p, and after rewriting it as ${\displaystyle \underset{0}{\overset{p}{\int }}{\displaystyle \underset{\underline{x}}{\overset{p}{\int }}\pi \left(x,c\right)\left(\frac{1-F(p)}{f(x)}\right)dF(x)dH(c)}}$, we can incorporate it into the first row of (A5) to result in (Standard individual rationality arguments also imply m(0)=0):

$$ \varPi \left(\pi \right)={E}_c\left[{\displaystyle \underset{\underline{x}}{\overset{p}{\int }}\left(x-\frac{1-F(x)}{f(x)}-c\right)\pi \left(x,c\right)dF(x)+\left[1-F(p)\right]\left(p-c\right)\pi \left(p,c\right)}\right] $$

(A6)

Equation (A6) implies the retailer profit is as if all high customers paid p and all low customers delivered the same profit they would in the absence of the posted-price competitor. In other words, the surplus of high-value buyers implied by the IC constraint affects the payments of low-value buyers exactly as it would in the absence of the posted-price competitor.

The optimal allocation rule is obvious, and it maximizes the expected profit pointwise:

$$ \pi \left(x,c\right)=1\iff \left(x<p\ \mathrm{and}\ c<x-\frac{1-F(x)}{f(x)}\right)\ \mathrm{or}\ x=p $$

(A7)

To see the optimality of always selling to high-value buyers with x=p, note that although the term π(p, c)appears in (A6) twice, its impact on profits inside the integral is measure zero, whereas its impact on profits in the [1 − F(p)](p − c)π(p, c) term has positive measure.

To derive the c-contingent profit shown in the proposition, use integration by parts to show that $\left[1-F(p)\right]\left(p-c\right)={\displaystyle \underset{p}{\overset{\overline{x}}{\int }}\left(\psi (x)-c\right)dF(x)}$, and verify that the right-hand expression results when we plug A7 into the expression in the large square brackets in A6. QED Prop 1

Proof of Lemma 1

Because q(x) = U′(x),${\beta}^{\prime }(x)=\frac{U(x){U}^{{\prime\prime} }(x)}{{\left({U}^{\prime }(x)\right)}^2}>0\iff {U}^{{\prime\prime} }(x)>0\iff {q}^{\prime }(x)>0$. From Proposition 1, q′(x) = ψ′(x)h(ψ(x)) > 0 ⇔ ψ′(x) > 0. To derive the optimal bidding function, plug q and U from Proposition 1$q(x)={\displaystyle \underset{0}{\overset{p}{\int }}\pi \left(x,c\right)dH(c)}=H\left(\psi (x)\right)$ and $U(x)={\displaystyle \underset{\underline{x}}{\overset{x}{\int }}q(t)dt}={\displaystyle \underset{\psi^{-1}(0)}{\overset{x}{\int }}H\left(\psi (t)\right)dt}$ into Eq. 2 :$\beta (x)={\displaystyle \underset{\psi^{-1}(0)}{\overset{x}{\int }}t\frac{dH\left(\psi (t)\right)}{H\left(\psi (x)\right)}}={\displaystyle \underset{0}{\overset{\psi (x)}{\int }}{\psi}^{-1}(c)\frac{dH(c)}{H\left(\psi (x)\right)}}=E\left[\left.{\psi}^{-1}\left(\mathrm{cost}\right)\right|{\psi}^{-1}\left(\mathrm{cost}\right)<x\right]$, where the second equality follows from a change in variables c = ψ(z). QED Lemma 1

Proof of Lemma 2

To be incentive compatible, β(p)must satisfy, for every x<p:

$$ \begin{array}{l}p-\beta (p)\ge A\left(\beta (x)\right)\left(p-\beta (x)\right)\kern2em (IC1)\\ {}U(x)\ge x-\beta (p)\kern8em (IC2)\end{array} $$

where the first inequality(IC1) ensures type p bids β(p) and the second inequality (IC2)ensures types x<p do not deviate to β(p). The deviation surplus for type p is

$$ A\left(\beta (x)\right)\left(p-\beta (x)\right)=H\left(\psi (x)\right)\left(p-{\displaystyle \underset{0}{\overset{\psi (x)}{\int }}{\psi}^{-1}(w)\frac{h(w)}{H\left(\psi (x)\right)}dw}\right)={\displaystyle \underset{0}{\overset{\psi (x)}{\int }}\left(p-{\psi}^{-1}(w)\right)dH(w)} $$

which is obviously increasing in x, so the best deviation from bidding p is to bid β ⁻ _p . Therefore, (IC1) reduces to $\beta (p)\le p-{\displaystyle \underset{0}{\overset{\psi (p)}{\int }}\left(p-{\psi}^{-1}(w)\right)dH(w)}=p-{\displaystyle \underset{\psi^{-1}(0)}{\overset{p}{\int }}H\left(\psi (z)\right)dz}$. The LHS of (IC2)is the expected equilibrium surplus of type x, which Proposition 1 pins down as $U(x)={\displaystyle \underset{\psi^{-1}(0)}{\overset{x}{\int }}H\left(\psi (t)\right)dt}$. Therefore, (IC2)is $\beta (p)\ge p-{\displaystyle \underset{\psi^{-1}(0)}{\overset{p}{\int }}H\left(\psi (z)\right)dz}$ because $x-{\displaystyle \underset{\psi^{-1}(0)}{\overset{x}{\int }}H\left(\psi (z)\right)dz}$ is increasing in x. Therefore, the two incentive-compatibility constraints together uniquely determine the bid of type p as$\beta (p)=p-{\displaystyle \underset{\psi^{-1}(0)}{\overset{p}{\int }}H\left(\psi (z)\right)dz}$. It is easy to show that the invertibility constraint β(p) > β ⁻ _p is always satisfied. QED Lemma 2

Proof of Lemma 3

The upper bound on probability acceptance must simply satisfy, for every x,

A(b)(x − b) ≤ H(ψ(x))(x − β(x))

The RHS is maximized by x=p, so using the same arguments as in the proof of Lemma 2, the constraint is thus $\begin{array}{cc}\hfill A(b)\le \left(\frac{1}{p-b}\right){\displaystyle \underset{\psi^{-1}(0)}{\overset{p}{\int }}H\left(\psi (z)\right)dz}\hfill & \hfill QED\ Lemma3\hfill \end{array}$

Proof of Proposition 4

The expected profit of a retailer who uses a participation fee such that buyers with x ≥ x _e enter solves the following problem:

$$ {x}_e=\underset{z}{ \arg \max}\left[1-F(z)\right]\underset{S(z) \equiv \mathrm{expected}\ \mathrm{surplus}\ \mathrm{of}z}{\underbrace{H\left({\beta}_0(z)\right)\left(z-{\beta}_0(z)\right)}}+{\displaystyle \underset{z}{\overset{1}{\int }}{\pi}_0\left( \min \left(x,p\right)\right)dF(x)} $$

where ${\pi}_0(x)={\displaystyle \underset{0}{\overset{\beta_0(x)}{\int }}\left({\beta}_0(x)-c\right)dH(c)}=H\left({\beta}_0(x)\right)\left[{\beta}_0(x)-E\left(\left.c\right|c<{\beta}_0(x)\right)\right]$is the expected profit from a buyer with valuation x. The envelope theorem implies S′(z) = H(β ₀(z)), so the FOC of the retailer’s problem is [1 − F(x _e )]H(β ₀(x _e )) = f(x _e )[S(x _e ) + π ₀(x _e )].

In words, raising the fee increases the payment of everyone above the entry threshold (LHS) while decreasing the number of entrants, which results in a marginal loss of the fee and bidding profit (RHS). In other words, the RHS is the marginal loss of the gains from trade, because the H(β ₀(x _e ))β ₀(x _e ) cancels out in S(x _e ) + π ₀(x _e ):

[1 − F(x _e )]H(β ₀(x _e )) = f(x _e )H(β ₀(x _e ))[x _e  − E(c|c < β ₀(x _e ))]

Both sides of the FOC are weighted by H(β ₀(x _e ))because no trade occurs when c > β ₀(x _e ). Canceling out the weights results in the final FOC equation: ψ(x _e ) = E(c|c < β ₀(x _e )). QED

Proof of Proposition 5

For p<2/3, the relative profit advantage of the first-best strategy over the minimum-bid strategy is ${\varPi}_A(p)-{\varPi}_m(p)=\frac{{\left(2p-1\right)}^3}{12p}\to 0\ \mathrm{a}\mathrm{s}\ p\ \mathrm{a}\mathrm{pproaches}\ \frac{1}{2}.$

• Claim a): Let Δ(p) = Π ^* _m (p) − Π ^* _e (p). There are three regions of p: First, for $p<\frac{4}{7}$, $\varDelta (p)=\frac{p\left(1-p\right)}{8}>0$ and obviously decreasing in p. Second, for $\frac{4}{7}<p<\frac{2}{3}$, Δ(p) is decreasing on the interval because ${\varDelta}^{\prime}\left(\frac{4}{7}\right)=-\frac{1}{56}<0$ and ${\varDelta}^{{\prime\prime} }(p)=-\frac{5}{6}-\frac{8}{147{p}^3}<0$, and Δ(p) > 0 because and $\varDelta \left(\frac{2}{3}\right)=\frac{127}{5292}>0$ and Δ(p) is decreasing. Third, for $p>\frac{2}{3}$, Δ(p) is decreasing on the interval because ${\varDelta}^{\prime }(1)=-\frac{55}{10584}<0$ and ${\varDelta}^{{\prime\prime} }(p)=\frac{1}{6}+\frac{124}{1323{p}^3}>0$, and Δ(p) > 0because $\varDelta (1)=\frac{55}{10584}>0$ and Δ(p) is decreasing.

• Claim b): The equivalence for $p\le \frac{2}{3}$ is obvious from both strategies charging p. The dominance for $p>\frac{2}{3}$ follows from ${\varPi}_m^{*}(p)-{\varPi}_{fix}^{*}(p)=\frac{2}{27p}-{\left(\frac{1-E(c)}{2}\right)}^2=\frac{\left(8-3p\right){\left(3p-2\right)}^2}{432p}>0$.

• Claim c): Let Δ(p) = Π ^* _fix (p) − Π ^* _e (p). Consider $p>\frac{2}{3}$, where $\varDelta (p)=\frac{1}{4}-\frac{4}{147p}-\frac{3p}{8}+\frac{7{p}^2}{48}$. We see that $\varDelta (1)=-\frac{5}{784}<0$ and $\varDelta \left(\frac{2}{3}\right)=\frac{127}{5292}>0$, so there must be at least one p* s.t. Δ(p *) = 0. To prove uniqueness, note Δ(p) is decreasing on the interval because ${\varDelta}^{\prime }(1)=-\frac{11}{196}<0$ and ${\varDelta}^{{\prime\prime} }(p)=\frac{7}{24}-\frac{8}{147{p}^3}>0$.

• Claim d): Given the ranking established in a)-c), it is enough to show the passive strategy is strictly less profitable than the participation-fee strategy for all p and the fixed-price strategy for$p>\frac{2}{3}$. The former follows from ${\varPi}_e^{*}(p)-{\varPi}_0(p)=\left\{\begin{array}{c}\hfill p<\frac{4}{7}:\frac{p\left(6-7p\right)}{24}>0\hfill \\ {}\hfill p\ge \frac{4}{7}:\frac{4}{147p}>0\kern2em \hfill \end{array}\right.$. The latter follows from ${\varPi}_{fix}^{*}(p)-{\varPi}_0(p)=\frac{12-p\left(18-7p\right)}{48}$, which is decreasing for $p>\frac{2}{3}$ and positive at p=1.

• QED Proposition 5.