“On-demand” pricing and capacity management in cloud computing

Abstract

In recent business practice, firms, to fulfill their IT requirements, are using both dedicated “on-premise” capacity infrastructure and “on-demand” capacity requirements provided by companies such as AWS, OpenStack, and VMware. In this research, we analyze the scenario where a business first invests in “on-premise” (or in-house) capacity and also procures the excess demand requirements through the public cloud provider utilizing the pay-as-you-go pricing model. We study the impact of factors such as demand correlation in buyers’ market and demand load profile on the capacity decision. We find various cloud computing strategies and link them with real-life business practices.

Appendix

Proof of Proposition 1

Proof

The expected total profit function may be written as

$$\ \pi (w,m)=wm+P_{0}E_{X}[\min (X,\delta _{s}-m)]-v(m+E_{X}[\min (X,\delta _{s}-m)]).$$

The public cloud provider faces cumulative demand m for elastic capacity from both the buyers and chooses w(m) to maximize $\pi (w,m)$ and hence determines his pricing curve. For any given value of w, the cloud provider would sell $m^{*}$ units of capacity to both buyers combined such it would maximize his expected profit $\pi (w,m)$ with respect to m. Lariviere and Porteus (2001) derive pricing curve in a newsvendor setting using a similar logic. The function $\pi (w,m)$ is uni-modal in m . The first-order condition for optimization may be written as

$$\frac{\partial \pi (w,m)}{\partial m}|_{m=m^{*}}=w-v-(P_{0}-v)(1-F(\delta _{s}-m^{*}))=0.$$

Hence, we get the price-capacity curve of the cloud service provider:

$$\mathbf {w}(m^{*})=v+(P_{0}-v)(1-F(\delta _{s}-m^{*})).$$

(5)

This establishes Proposition 1(a).

• (b) Since $0\le 1-F(\delta _{s}-m)\le 1$, this implies $0\le (P_{0}-v)\left( 1-F(\delta _{s}-m)\right) \le (P_{0}-v)$, $P_{0}>v$, adding v on both sides, we get $v\le \mathbf {w}(m)\le P_{0}$. This establishes Proposition 1(b).

• (c) $\frac{\partial w(m)}{\partial m}=(P_{0}-v)f(\delta _{s}-m^{*})>0$. This establishes Proposition 1(c). $\square$

Proof of Theorem 1

Proof

For (i), we transform the Buyer i’s problem to maximization problem with objective function $\pi _{i}(\mu _{i})=-C_{i}(\mu _{i})$. Further we use the change of variable $\widetilde{\mu _{j}}=-\mu _{j}$. The first-order condition is given by

$$\frac{\partial \pi _{i}(\mu _{i})}{\partial \mu _{i}}=-\left( K_{i}^{'}(\mu _{i})-2\biggl (\frac{P_{0}-v}{b-a}\biggr )E_{D_{i}}\biggl [\Bigl (\max \{D_{i}-\mu _{i},0\}\Bigr )\biggr ]\right) +\left( \biggl (1-G_{i}(\mu _{i})\biggr )\left( v-v_{bi}+(P_{0}-v)\left( \frac{b-\delta _{s}+E_{D_{j}}\biggl [\max \{D_{j}+\widetilde{\mu _{j}},0\})\biggr ]}{b-a}\right) \right) \right).$$

Now, $\frac{\partial ^{2}\pi _{i}}{\partial \mu _{i}\partial \widetilde{\mu _{j}}}=\Bigl (1-G_{i}(\mu _{i})\Bigr )\left(\frac{P_{0}-v}{b-a}\right)(1-G_{j}(-\widetilde{\mu _{j}}))\ge 0,$and hence, $\pi _{i}$ is supermodular in $(\mu _{i},\widetilde{\mu _{j}})$. Topkis (1979, Theorem 1.2 ) immediately establishes Theorem 1. $\square$

Derivation of expression for optimal private cloud capacity $\mu _{m}^{*}$ (benchmark case)

Proof

The expected cost of single buyer if she installs $\mu$ units of capacity is given by

$$C_{mb}({\mu })=\int \left( K(\mu )+\left( \mathbf {w_{u}}\left( \max \{D-\mu ,0\}\right) \right) \Bigl (\max \{D-\mu ,0\}\Bigr )+\,v_{b}\min \{D,\mu \}\right) g(D)\mathrm{d}D.$$

Now, using (1), we get

$$C_{mb}({\mu })= K(\mu )+\biggl (v+(P_{0}-v)\biggl(\frac{b-\delta _{s}}{b-a}\biggr)\biggr )\left( E_{D}\biggl [\max \{D-\mu ,0\}\biggr ]\right) +(P_{0}-v)\left( \frac{E_{D}\biggl [(\max \{D-\mu ,0\})^{2}\biggr ]}{b-a}\right) +v_{b}E_{D}\biggl [\min \{D,\mu \}\biggr ].$$

Now, we solve the first-order conditions and on simplification, we get

$$\frac{\partial C_{mb}({\mu })}{\partial \mu }= K^{'}(\mu )+\biggl (v+(P_{0}-v)\left( \frac{b-\delta _{s}}{b-a}\right) \biggr )\biggl (-\Bigl (1-G(\mu )\Bigr )\biggr )-\biggl (\frac{P_{0}-v}{b-a}\biggr )2E_{D}\biggl [\Bigl (\max \{D-\mu ,0\}\Bigr )\biggr ]+v_{b}\biggl (1-G(\mu )\biggr ).$$

Now, from $\frac{\partial C_{mb}({\mu })}{\partial \mu }|_{\mu =\mu _{m}^{*}}=0$, we get

$$K^{'}(\mu _{m}^{*})-2\biggl (\frac{P_{0}-v}{b-a}\biggr )E_{D}\biggl [\Bigl (\max \{D-\mu _{m}^{*},0\}\Bigr )\biggr ]=\biggl (1-G(\mu _{m}^{*})\biggr )\biggl ((v-v_{bi})+(P_{0}-v)\left( \frac{b-\delta _{s}}{b-a}\right) \biggr ).$$

$\square$

Proof of Proposition 2

Proof

Let $\mu _{m}$ be the solution for the case of single buyer. Now, we evaluate the slope of $C_{i}(\mu _{i})$ at $\mu _{i}=\mu _{m}$, we have

$$\frac{\partial C_{i}(\mu _{i})}{\partial \mu _{i}}\mid _{\mu _{i}=\mu _{m}}=K^{'}(\mu _{m})-2\biggl (\frac{P_{0}-v}{b-a}\biggr )E_{D}\biggl [\Bigl (\max \{D-\mu _{m},0\}\Bigr )\biggr] -\biggl (1-G(\mu _{m})\biggr )\left( v-v_{bi}+(P_{0}-v)\left( \frac{b-\delta _{s}+E_{D_{j}}\biggl [\max \{D_{j}-\mu _{j},0\})\biggr ]}{b-a}\right) \right).$$

Now by (3), we have $K^{'}(\mu _{m})-2\biggl (\frac{P_{0}-v}{b-a}\biggr )E_{D}\biggl [\Bigl (\max \{D-\mu _{m},0\}\Bigr )\biggr ]=\biggl (1-G(\mu _{m})\biggr )\biggl ((v-v_{bi})+(P_{0}-v)\left( \frac{b-\delta _{s}}{b-a}\right) \biggr ),$substituting this in above equation we get

$$\frac{\partial C_{i}(\mu _{i})}{\partial \mu _{i}}\mid _{\mu _{i}=\mu _{m}}=-\left( \biggl (1-G(\mu _{m})\biggr )\left( (P_{0}-v)\left( \frac{E_{D_{j}}\biggl [\max \{D_{j}-\mu _{j},0\})\biggr ]}{b-a}\right) \right) \right) \le 0.$$

Now since the slope is negative at $\mu _{m}$ , this implies that $\mu _{m}$ is less than buyer i’s strategy under multiple buyer case. $\square$

Proof of Proposition 3

Proof

• (a) The expected cost of the buyer i if she installs $\mu _{i}$ units of capacity is given by

  $$C_{i}(\mu _{i}) = \int \int \Bigl (K_{i}(\mu _{i})+\left( \mathbf {w_{u}}\left( \max \{D_{i}-\mu _{i},0\}+\max \{D_{j}-\mu _{j},0\}\right) \right) \Bigl (\max \{D_{i}-\mu _{i},0\}\Bigr )+v_{bi}\min \{D_{i},\mu _{i}\}\Bigl )g_{i}(D_{i})g_{j}(D_{j})\mathrm{d}D_{i}\mathrm{d}D_{j}.$$

  Now, we solve the first order condition, we have

  $$\frac{\partial C_{i}(\mu _{i})}{\partial \mu _{i}}=K_{i}^{'}(\mu _{i})-2\biggl (\frac{P_{0}-v}{b-a}\biggr )E_{D_{i}}\biggl [\Bigl (\max \{D_{i}-\mu _{i},0\}\Bigr )\biggr ]-\biggl (1-G_{i}(\mu _{i})\biggr )\left( v-v_{bi}+(P_{0}-v)\left( \frac{b-\delta _{s}+E_{D_{j}}\biggl [\max \{D_{j}-\mu _{j},0\})\biggr ]}{b-a}\right) \right).$$

  Now, from $\frac{\partial C_{i}(\mu _{i})}{\partial \mu _{i}}=0$ and for symmetrical player we have $\mu _{i}=\mu _{j}=\mu _{d}^{*}$, we get

  $$K^{'}(\mu _{d}^{*})-\biggl (\frac{P_{0}-v}{b-a}\biggr )2\left( E_{D}\biggl [\max \{D-\mu _{d}^{*},0\})\biggr ]\right) =\biggl (1-G(\mu _{d}^{*})\biggr )\left( v+(P_{0}-v)\left( \frac{b-\delta _{s}+E_{D}\biggl [\max \{D-\mu _{d}^{*},0\})\biggr ]}{b-a}\right) -v_{b}\right).$$

  This establishes Proposition 3(a).

• (b) From Theorem 1, we have

  $$\frac{\partial \pi _{i}(\mu _{i})}{\partial \mu _{i}}\mid _{\mu _{i}=\mu _{d}^{*}}=-\left( K^{'}(\mu _{d}^{*})-2\biggl (\frac{P_{0}-v}{b-a}\biggr )E_{D_{i}}\biggl [\Bigl (\max \{D-\mu _{d}^{*},0\}\Bigr )\biggr ]\right) +\left( \biggl (1-G(\mu _{d}^{*})\biggr )\left( v-v_{bi}+(P_{0}-v)\left( \frac{b-\delta _{s}+E_{D}\biggl [\max \{D+\mu _{d}^{*},0\})\biggr ]}{b-a}\right) \right) \right).$$

  Further, we have

  $$\frac{\partial ^{2}\pi _{i}(\mu _{d}^{*})}{\partial \mu _{i}P_{0}}=2\biggl (\frac{1}{b-a}\biggr )E_{D}\biggl [\Bigl (\max \{D-\mu _{d}^{*},0\}\Bigr )\biggr ]+\biggl (1-G(\mu _{d}^{*})\biggr )\left( \frac{b-\delta _{s}+E_{D}\biggl [\max \{D+\mu _{d}^{*}0\})\biggr ]}{b-a}\right) >0.$$

  Now since $\frac{\partial ^{2}\pi _{i}(\mu _{i})}{\partial \mu _{i}P_{0}}>0$, this establishes Proposition 3(b). $\square$

Proof of Proposition 4

Proof

• (a) The expected cost of the buyer i if she installs $\mu _{i}$ units of capacity is given by

  $$C_{i}^{e}(\mu _{i})=\int _{\alpha _{i}}^{\beta _{i}}\int _{\alpha _{j}}^{\beta _{j}}\Bigl (K_{i}(\mu _{i})+\left( \mathbf {w_{u}}\left( \max \{D_{i}-\mu _{i},0\}+\max \{D_{j}-\mu _{j},0\}\right) \right) \Bigl (\max \{D_{i}-\mu _{i},0\}\Bigr )+\psi _{i}(\mu _{i})\min \{D_{i},\mu _{i}\}\Bigl )g_{i}(D_{i})g_{j}(D_{j})\mathrm{d}D_{i}\mathrm{d}D_{j},$$

  which may be further written as

  $$C_{i}^{e}(\mu _{i})=K_{i}(\mu _{i})+\biggl (\phi (\delta _{s})+(P_{0}-\phi (\delta _{s}))\left( \frac{b-\delta _{s}}{b-a}\right) \biggr )\left( E_{D_{i}}\biggl [\max \{D_{i}-\mu _{i},0\}\biggr ]\right) +\psi _{i}(\mu _{i})E_{D_{i}}\biggl [\min \{D_{i},\mu _{i}\}\biggr ]+(P_{0}-\phi (\delta _{s}))\left( \frac{E_{D_{i}}\biggl [(\max \{D_{i}-\mu _{i},0\})^{2}\biggr ]}{b-a}\right) +(P_{0}-\phi (\delta _{s}))\left( \frac{E_{D_{i},D_{j}}\biggl [(\max \{D_{j}-\mu _{j},0\}\max \{D_{i}-\mu _{i},0\})\biggr ]}{b-a}\right).$$

  Now, we solve the first-order conditions:

  $$\frac{\partial C_{i}^{e}(\mu _{i})}{\partial \mu _{i}} = K_{i}^{'}(\mu _{i})-2\biggl (\frac{P_{0}-\phi (\delta _{s})}{b-a}\biggr )E_{D_{i}}\biggl [\Bigl (\max \{D_{i}-\mu _{i},0\}\Bigr )\biggr ]+\psi _{i}^{'}(\mu _{i})E_{D_{i}}\biggl [\min \{\mu _{i},D_{i}\}\biggr ]-\biggl (1-G_{i}(\mu _{i})\biggr )\left( \phi (\delta _{s})-\psi _{i}(\mu _{i})+(P_{0}-\phi (\delta _{s}))\left( \frac{b-\delta _{s}+E_{D_{j}}\biggl [\max \{D_{j}-\mu _{j},0\})\biggr ]}{b-a}\right) \right).$$

  Now, from $\frac{\partial C_{i}^{e}(\mu _{i})}{\partial \mu _{i}}=0$ and for symmetrical player we have $\mu _{i}=\mu _{j}=\mu _{de}^{*}$, we get

  $$K^{'}(\mu _{de}^{*})-\left( \phi (\delta _{s})-\psi (\mu _{de}^{*})+(P_{0}-\phi (\delta _{s}))\left( \frac{b-\delta _{s}+U}{b-a}\right) \right) \biggl (1-G_{i}(\mu _{de}^{*})\biggr )+\psi '(\mu _{de}^{*})Y=\biggl (\frac{P_{0}-\phi (\delta _{s})}{b-a}\biggr )2U,$$

  where $U=E_{D}\biggl [\max \{D-\mu _{de}^{*},0\})\biggr ],$$Y=E_{D}\biggl [\min \{\mu _{de}^{*},D\}\biggr ]$ and $D\sim G(.)$. This establishes Proposition 4(a).

• (b) Let $\mu _{d}$ be the strategy of the buyer under absence server-scale economies effect. Now we evaluate the slope of $C_{i}^{e}(\mu _{i})$ at $\mu _{i}=\mu _{d}$

  $$\frac{\partial C_{i}^{e}(\mu _{i})}{\partial \mu _{i}}\mid _{\mu _{i}=\mu _{d}}=K_{i}^{'}(\mu _{d})-\left( \phi (\delta _{s})+(P_{0}-\phi (\delta _{s}))\left( \frac{b-\delta _{s}+E_{D_{j}}\biggl [\max \{D_{j}-\mu _{j},0\})\biggr ]}{b-a}\right) \right) \biggl (1-G_{i}(\mu _{d})\biggr )- \biggl (\frac{P_{0}-\phi (\delta _{s})}{b-a}\biggr )2E_{D_{i}}\biggl [\Bigl (\max \{D_{i}-\mu _{d},0\}\Bigr )\biggr ]+\psi (\mu _{d})\biggl (1-G_{i}(\mu _{d})\biggr )+\psi '(\mu _{d})E_{D_{i}}\biggl [\min \{\mu _{d},D_{i}\}\biggr ].$$

  Further we have $K_{i}^{'}(\mu _{d})-\left( \phi (\delta _{s})-v_{b}+(P_{0}-\phi (\delta _{s}))\left( \frac{b-\delta _{s}+E_{D_{j}}\biggl [\max \{D_{j}-\mu _{j},0\})\biggr ]}{b-a}\right) \right) \biggl (1-G_{i}(\mu _{d})\biggr )=\biggl (\frac{P_{0}-\phi (\delta _{s})}{b-a}\biggr )2E_{D_{i}}\biggl [\Bigl (\max \{D_{i}-\mu _{d},0\}\Bigr )\biggr ],$substituting this in above equation we get

  $$\frac{\partial C_{i}^{e}(\mu _{i})}{\partial \mu _{i}}\mid _{\mu _{i}=\mu _{d}}=-\left( v_{b}-\psi (\mu _{d})\right) \biggl (1-G_{i}(\mu _{d})\biggr )+\psi '(\mu _{d})E_{D_{i}}\biggl [\min \{\mu _{d},D_{i}\}\biggr ]<0.$$

  Now since the slope is negative at $\mu _{d}$ , this implies $\mu _{d}$ is less than buyer i’s strategy under scale economies. This establishes Proposition 4(b).

• (c) Let $\mu _{d}$ be the strategy of the buyer when public cloud provider doesn’t enjoy the scale economies effect. Similar to proof of Proposition 5(b) on evaluating slope of $C_{i}^{e}(\mu _{i})$ at $\mu _{i}=\mu _{d}$ we have

  $$\frac{\partial C_{i}^{e}(\mu _{i})}{\partial \mu _{i}}\mid _{\mu _{i}=\mu _{d}}=(v-\phi (\delta _{s}))\left( \left( \frac{-a+\delta _{s}-E_{D_{j}}\biggl [\max \{D_{j}-\mu _{d},0\})\biggr ]}{b-a}\right) \right) \biggl (1-G_{i}(\mu _{d})\biggr )>0.$$

  Now since the slope is positive at $\mu _{d}$ , this implies $\mu _{d}$ is less than Buyer i’s strategy when the cloud provider enjoys the scale economies effect. This establishes Proposition 4(c).

$\square$

Proof of Theorem 2

Proof

For (i), we transform the Buyer i’s problem to maximization problem with objective function $\pi _{i}(\mu _{i})=-C_{i}(\mu _{i})$. Further we use the change of variable $\widetilde{\mu _{j}}=-\mu _{j}$

$$\frac{\partial \pi _{i}(\mu _{i})}{\partial \mu _{i}} = -\left( K_{i}^{'}(\mu _{i})-2\biggl (\frac{P_{0}-v}{b-a}\biggr )E_{D_{i}}\biggl [\Bigl (\max \{D_{i}-\mu _{i},0\}\Bigr )\biggr ]\right) +\left( \biggl (1-G_{i}(\mu _{i})\biggr )\left( v-v_{bi}+(P_{0}-v)\left( \frac{b-\delta _{s}}{b-a}\right) \right) \right) +\biggl (\frac{P_{0}-v}{b-a}\biggr )\left( \int _{-\widetilde{\mu _{j}}-\eta \mu _{i}}^{\infty }\int _{\mu _{i}}^{\infty }\left( \eta D_{i}+\epsilon +\widetilde{\mu _{j}}\right) g(D_{i})h(\epsilon )\mathrm{d}D_{i}\mathrm{d}\epsilon \right) +\biggl (\frac{P_{0}-v}{b-a}\biggr )\left( \int _{0}^{-\widetilde{\mu _{j}}-\eta \mu _{i}}\int _{\frac{-\widetilde{\mu _{j}}-\epsilon }{\eta }}^{\infty }\left( \eta D_{i}+\epsilon +\widetilde{\mu _{j}}\right) g(D_{i})h(\epsilon )\mathrm{d}D_{i}\mathrm{d}\epsilon \right).$$

Now we have

$$\frac{\partial ^{2}\pi _{i}}{\partial \mu _{i}\partial \widetilde{\mu _{j}}}=\biggl (\frac{P_{0}-v}{b-a}\biggr )\left( \int _{-\widetilde{\mu _{j}}-\eta \mu _{i}}^{\infty }\int _{\mu _{i}}^{\infty }g(D_{i})h(\epsilon )\mathrm{d}D_{i}\mathrm{d}\epsilon +\int _{0}^{-\widetilde{\mu _{j}}-\eta \mu _{i}}\int _{\frac{-\widetilde{\mu _{j}}-\epsilon }{\eta }}^{\infty }g(D_{i})h(\epsilon )\mathrm{d}D_{i}\mathrm{d}\epsilon \right) \ge 0.$$

Hence, we have $\pi _{i}$ is supermodular in $(\mu _{i},\widetilde{\mu _{j}})$. Topkis (1979, Theorem 1.2) immediately establishes Theorem 2. $\square$

Proof of Proposition 5

Proof

$$\frac{\partial \mu _{1}}{\partial \eta }=\frac{\frac{\partial ^{2}C_{1}(\mu _{1})}{\partial \mu _{1}\partial \eta }}{-\frac{\partial ^{2}C_{1}(\mu _{1})}{\partial \mu _{1}^{2}}}=\frac{\int _{\mu _{2}-\eta \mu _{1}}^{\infty }\int _{\mu _{1}}^{\infty }D_{1}g(D_{1})h(\epsilon )\mathrm{d}D_{1}\mathrm{d}\epsilon +\int _{0}^{\mu _{2}-\eta \mu _{1}}\int _{\frac{\mu _{2}-\epsilon }{\eta }}^{\infty }D_{1}g(D_{1})h(\epsilon )\mathrm{d}D_{1}\mathrm{d}\epsilon }{\frac{\partial ^{2}C_{1}(\mu _{1})}{\partial \mu _{1}^{2}}}>0$$

$$\frac{\partial \mu _{2}}{\partial \eta }=\frac{\frac{\partial ^{2}C_{2}(\mu _{2})}{\partial \mu _{2}\partial \eta }}{-\frac{\partial ^{2}C_{2}(\mu _{2})}{\partial \mu _{2}^{2}}}=\frac{\int _{\mu _{2}-\eta \mu _{1}}^{\infty }\int _{\mu _{1}}^{\infty }D_{1}g(D_{1})h(\epsilon )\mathrm{d}D_{1}\mathrm{d}\epsilon +\int _{0}^{\mu _{2}-\eta \mu _{1}}\int _{\frac{\mu _{2}-\epsilon }{\eta }}^{\infty }D_{1}g(D_{1})h(\epsilon )\mathrm{d}D_{1}\mathrm{d}\epsilon }{\frac{\partial ^{2}C_{2}(\mu _{2})}{\partial \mu _{2}^{2}}}>0$$

Hence, as $\eta$ increases, the private cloud investments by the firms increase. This establishes Proposition 5. $\square$

Proof of Proposition 6

Proof

Let $\mu _{m}$ be the solution for the case of single buyer. Now, we evaluate the slope of $C_{i}(\mu _{i})$ (objective function under demand correlation) at $\mu _{i}=\mu _{m}$.

$$\frac{\partial C_{i}(\mu _{i})}{\partial \mu _{i}}\mid _{\mu _{i}=\mu _{m}} = K^{'}(\mu _{m})-2\biggl (\frac{P_{0}-v}{b-a}\biggr )E_{D}\biggl [\Bigl (\max \{D-\mu _{m},0\}\Bigr )\biggr ]-\biggl (1-G(\mu _{m})\biggr )\biggl (v-v_{bi}+(P_{0}-v)\left( \frac{b-\delta _{s}}{b-a}\right) \biggr )-\biggl (\frac{P_{0}-v}{b-a}\biggr )\left( \int _{\mu _{2}-\eta \mu _{1}}^{\infty }\int _{\mu _{1}}^{\infty }\left( \eta D_{1}+\epsilon -\mu _{2}\right) g(D_{1})h(\epsilon )\mathrm{d}D_{1}\mathrm{d}\epsilon \right) -\biggl (\frac{P_{0}-v}{b-a}\biggr )\left( \int _{0}^{\mu _{2}-\eta \mu _{1}}\int _{\frac{\mu _{2}-\epsilon }{\eta }}^{\infty }\left( \eta D_{1}+\epsilon -\mu _{2}\right) g(D_{1})h(\epsilon )\mathrm{d}D_{1}\mathrm{d}\epsilon \right).$$

Now by (3), we have $K^{'}(\mu _{m})-2\biggl (\frac{P_{0}-v}{b-a}\biggr )E_{D}\biggl [\Bigl (\max \{D-\mu _{m},0\}\Bigr )\biggr ]=\biggl (1-G(\mu _{m})\biggr )\biggl ((v-v_{bi})+(P_{0}-v)\left( \frac{b-\delta _{s}}{b-a}\right) \biggr ),$substituting this in above equation, we get

$$\frac{\partial C_{i}(\mu _{i})}{\partial \mu _{i}}\mid _{\mu _{i}=\mu _{m}} = -\biggl (\frac{P_{0}-v}{b-a}\biggr )\left( \int _{0}^{\mu _{2}-\eta \mu _{m}}\int _{\frac{\mu _{2}-\epsilon }{\eta }}^{\infty }\left( \eta D_{1}+\epsilon -\mu _{2}\right) g(D_{1})h(\epsilon )\mathrm{d}D_{1}\mathrm{d}\epsilon \right) -\biggl (\frac{P_{0}-v}{b-a}\biggr )\left( \int _{\mu _{2}-\eta \mu _{m}}^{\infty }\int _{\mu _{m}}^{\infty }\left( \eta D_{1}+\epsilon -\mu _{2}\right) g(D_{1})h(\epsilon )\mathrm{d}D_{1}\mathrm{d}\epsilon \right) \le 0.$$

Now since the slope is negative at $\mu _{m}$ , this implies that $\mu _{m}$ is less than buyer i’s strategy under multiple-buyer case. This establishes Proposition 6. $\square$