**Proof of Proposition 1:** *Consider an equilibrium $\{\mathrm{\gamma},c,\mathrm{\pi}\}$ of a competing mechanism game with $\mathrm{\Gamma}$. Suppose that every seller offers a mechanism $\stackrel{\u02c6}{\mathrm{\phi}}$ satisfying properties (i)-(iv) in the proof of Lemma 1, with
$\left[{p}_{1},\dots ,{p}_{n}\right]=\left[{p}_{1}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{m-1},c),\dots ,{p}_{n}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{m-1},c)\right]$as the price vector that specifies prices that the seller charges given the participating buyers’ truthful consistent reporting when there are no deviations by other sellers (i. e., $t=0$). Let each buyer choose each seller with probability $\stackrel{\u02c6}{\mathrm{\pi}}({\stackrel{\u02c6}{\mathrm{\phi}}}^{m})=\mathrm{\pi}({\mathrm{\gamma}}^{m})=1/m$ by reporting $0.$ Then, each seller’s expected price that a buyer faces upon participation is
$\begin{array}{rl}\mathbb{E}\left[{\stackrel{\u02c6}{\mathrm{\phi}}}_{k}({0}^{k})\right]& =\sum _{k=1}^{n}\left(\begin{array}{c}n-1\\ k-1\end{array}\right)\stackrel{\u02c6}{\mathrm{\pi}}({\stackrel{\u02c6}{\mathrm{\phi}}}^{m}{)}^{k-1}{\left(1-\stackrel{\u02c6}{\mathrm{\pi}}({\stackrel{\u02c6}{\mathrm{\phi}}}^{m})\right)}^{n-k}{\stackrel{\u02c6}{\mathrm{\phi}}}_{k}({0}^{k})\\ & =\sum _{k=1}^{n}\left(\begin{array}{c}n-1\\ k-1\end{array}\right)\mathrm{\pi}({\mathrm{\gamma}}^{m}{)}^{k-1}{\left(1-\mathrm{\pi}({\mathrm{\gamma}}^{m})\right)}^{n-k}{p}_{k}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{m-1},c)\\ & =\mathbb{E}\left[{p}_{k}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{m-1},c)\right]\end{array}$*

because $\stackrel{\u02c6}{\mathrm{\pi}}({\stackrel{\u02c6}{\mathrm{\phi}}}^{m})=\mathrm{\pi}({\mathrm{\gamma}}^{m})$ and ${\stackrel{\u02c6}{\mathrm{\phi}}}_{k}({0}^{k})={p}_{k}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{m-1},c)$ for all $k.$ Therefore, it is an incentive consistent truth telling continuation equilibrium that each buyer chooses each seller with the probability $\stackrel{\u02c6}{\mathrm{\pi}}({\stackrel{\u02c6}{\mathrm{\phi}}}^{m})$ and reports $t=0$ upon participation. Because each seller’s expected price is preserved, the buyer’s expected payoff is preserved:
$1-\mathbb{E}[{\widehat{\phi}}_{k}{(0}^{k})]=1-\mathbb{E}[{p}_{k}^{\gamma}({\gamma}^{m-1}\mathrm{,}c)].$Given truthful continuation equilibrium, each seller’s expected payoff satisfies
$\mathrm{\Phi}(\stackrel{\u02c6}{\mathrm{\phi}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},\stackrel{\u02c6}{\mathrm{\pi}})=\mathrm{\Phi}(\mathrm{\gamma},{\mathrm{\gamma}}^{m-1},c,\mathrm{\pi})$[15]because $\stackrel{\u02c6}{\mathrm{\pi}}({\stackrel{\u02c6}{\mathrm{\phi}}}^{m})=\mathrm{\pi}({\mathrm{\gamma}}^{m})$ and ${\stackrel{\u02c6}{\mathrm{\phi}}}_{k}({0}^{k})={p}_{k}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{m-1},c)$ for all $k$.

Now suppose that one seller deviates to $p$. Pick a constant mechanism ${\mathrm{\gamma}}^{{}^{\prime}}$ that always assigns $p$ regardless of the participating buyers’ reports. In the original competing mechanism game $\mathrm{\Gamma}$, the deviating seller’s price is always $p$, but the expected price that a buyer faces upon participating in each non-deviating seller’s mechanism $\stackrel{\u02c6}{\mathrm{\phi}}$ is
$\begin{array}{rl}& \mathbb{E}[{p}_{k}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-2},c)]=\\ & \sum _{k=1}^{n}\left(\begin{array}{c}n-1\\ k-1\end{array}\right){\left(\frac{1-\mathrm{\pi}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-1})}{m-1}\right)}^{k-1}{\left(1-\frac{1-\mathrm{\pi}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-1})}{m-1}\right)}^{n-k}{p}_{k}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-2},c).\end{array}$[16]In the auxiliary competing mechanism game, suppose that a buyer chooses the deviating seller with probability $\stackrel{\u02c6}{\mathrm{\pi}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})=\mathrm{\pi}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-1})$ upon his deviation to $p$ and each non-deviating seller with equal probability. The expected price that a buyer faces upon selecting a non-deviating seller is
$\begin{array}{rl}& \mathbb{E}\left[{\stackrel{\u02c6}{\mathrm{\phi}}}_{k}({1}^{k})\right]\\ & =\sum _{k=1}^{n}\left(\begin{array}{c}n-1\\ k-1\end{array}\right){\left(\frac{1-\stackrel{\u02c6}{\mathrm{\pi}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})}{m-1}\right)}^{k-1}{\left(1-\frac{1-\stackrel{\u02c6}{\mathrm{\pi}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})}{m-1}\right)}^{n-k}{\stackrel{\u02c6}{\mathrm{\phi}}}_{k}({1}^{k})\end{array}$[17]Note that ${\stackrel{\u02c6}{\mathrm{\phi}}}_{1}(1)={p}_{1}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-2},c)$ because ${p}_{1}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-2},c)={p}_{1}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{{}^{\prime \prime}},{\mathrm{\gamma}}^{m-2},c)$ for all ${\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{{}^{{}^{\prime \prime}}}\in \mathrm{\Gamma}$ due to a property of an equilibrium $\{\mathrm{\gamma},c,\mathrm{\pi}\}$. (See item 3 of Definition 1.) Furthermore, ${\stackrel{\u02c6}{\mathrm{\phi}}}_{k}({1}^{k})=0\le {p}_{k}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-2},c)$ for all $k\ge 2.$ Therefore, given the initial set-up of $\stackrel{\u02c6}{\mathrm{\pi}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})=\mathrm{\pi}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-1})$, from eqs [16] and [17], we can infer that
$\mathbb{E}\left[{\stackrel{\u02c6}{\mathrm{\phi}}}_{k}({1}^{k})\right]\le \mathbb{E}\left[{p}_{1}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-2},c)\right].$[18]Equivalently, the buyer’s expected payoff upon selecting a non-deviating seller who offers $\stackrel{\u02c6}{\mathrm{\phi}}$ is
$1-\mathbb{E}\left[{\stackrel{\u02c6}{\mathrm{\phi}}}_{k}({1}^{k})\right]\ge 1-\mathbb{E}\left[{p}_{1}^{\mathrm{\gamma}}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-2},c)\right].$[19]The buyer’s expected payoff, $1-\mathbb{E}\left[{\stackrel{\u02c6}{\mathrm{\phi}}}_{k}({1}^{k})\right]$, is decreasing in $\stackrel{\u02c6}{\mathrm{\pi}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})$ because a non-deviating seller’s expected price $\mathbb{E}\left[{\stackrel{\u02c6}{\mathrm{\phi}}}_{k}({1}^{k})\right]$ is increasing in $\stackrel{\u02c6}{\mathrm{\pi}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})$, as shown on the last line in eq. [17]. Because the buyer’s expected payoff upon selecting the deviating seller is the same whether he offers $p$ in the auxiliary competing mechanism game or he offers ${\mathrm{\gamma}}^{{}^{\prime}}$ in the original competing mechanism game, eq. [19] implies that there exists an incentive consistent truth telling continuation equilibrium in which we have
$\stackrel{\u02c6}{\mathrm{\pi}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})\le \mathrm{\pi}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-1})$[20]upon a seller’s deviation to $p.$ Because eq. [20] is satisfied and ${\mathrm{\gamma}}^{{}^{\prime}}$ is a constant mechanism that always charges $p$, the deviating seller’s expected payoff satisfies
$\mathrm{\Phi}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},\stackrel{\u02c6}{\mathrm{\pi}})=n\stackrel{\u02c6}{\mathrm{\pi}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})p\le n\mathrm{\pi}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-1})p=\mathrm{\Phi}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-1},c,\mathrm{\pi}).$[21]Because $\mathrm{\Phi}(\mathrm{\gamma},{\mathrm{\gamma}}^{m-1},c,\mathrm{\pi})\ge \mathrm{\Phi}({\mathrm{\gamma}}^{{}^{\prime}},{\mathrm{\gamma}}^{m-1},c,\mathrm{\pi})$ for all $p,$ eqs [15] and [21] imply that, for all $p,$
$\mathrm{\Phi}(\stackrel{\u02c6}{\mathrm{\phi}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},\stackrel{\u02c6}{\mathrm{\pi}})\ge \mathrm{\Phi}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},\stackrel{\u02c6}{\mathrm{\pi}})$and hence, no seller has an incentive to deviate to a single fixed price in the auxiliary competing mechanism game. This completes the proof.

**Proof of Theorem 1::** *Consider an equilibrium $\{\stackrel{\u02c6}{\mathrm{\phi}},\stackrel{\u02c6}{\mathrm{\pi}}\}$ of an auxiliary competing mechanism game where each seller either offers a mechanism $\stackrel{\u02c6}{\mathrm{\phi}}$ with the binary report and the worst punishment or deviates to a single fixed price. Suppose that a seller deviates to an arbitrary mechanism in $\mathrm{\Gamma}$ instead of deviating to a single fixed price. Given Lemma 1, we fix the buyer’s truth telling $t=1$ to each non-deviating seller upon participation. Let buyers play an incentive consistent truth telling continuation equilibrium in which each buyer sends a message to the deviating seller according to $c({\mathrm{\gamma}}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})\in \mathrm{\Delta}(C)$ upon selecting him and selects the deviating seller with probability $\mathrm{\pi}({\mathrm{\gamma}}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})\}$ and each non-deviating seller with equal probability. The deviating seller’s expected payoff upon deviating to ${\mathrm{\gamma}}^{{}^{\prime}}$ is $\mathrm{\Phi}({\mathrm{\gamma}}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},c,\mathrm{\pi})=n\mathrm{\pi}({\mathrm{\gamma}}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})\mathbb{E}\left[{p}_{k}^{{\mathrm{\gamma}}^{{}^{\prime}}}({\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1};c)\right],$ where $\mathbb{E}\left[{p}_{k}^{{\mathrm{\gamma}}^{{}^{\prime}}}({\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1};c)\right]$ is the expected price upon selecting the deviating seller:
$\begin{array}{rl}& E\left[{p}_{k}^{{\mathrm{\gamma}}^{{}^{\prime}}}({\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1};c)\right]=\\ & \sum _{k=1}^{n}\left(\begin{array}{c}n-1\\ k-1\end{array}\right)\mathrm{\pi}({\mathrm{\gamma}}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1}{)}^{k-1}(1-\mathrm{\pi}({\mathrm{\gamma}}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1}){)}^{n-k}{p}_{k}^{{\mathrm{\gamma}}^{{}^{\prime}}}({\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1};c).\end{array}$Suppose that the deviating seller offers ${p}^{{}^{\prime}}=\mathbb{E}\left[{p}_{k}^{{\mathrm{\gamma}}^{{}^{\prime}}}({\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1};c)\right]$ instead of ${\mathrm{\gamma}}^{{}^{\prime}}.$ Then it is also an incentive consistent truth telling continuation equilibrium that each buyer chooses the deviating seller with probability ${\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}({p}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})=\mathrm{\pi}({\mathrm{\gamma}}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})$ and each non-deviating seller with equal probability. In this incentive consistent truth telling continuation equilibrium, the deviating seller’s expected payoff satisfies
$\begin{array}{rl}& \mathrm{\Phi}({p}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}})=n{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}({p}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1}){p}^{{}^{\prime}}=\\ & n\mathrm{\pi}({\mathrm{\gamma}}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})\mathbb{E}\left[{p}_{k}^{{\mathrm{\gamma}}^{{}^{\prime}}}({\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1};c)\right]=\mathrm{\Phi}({\mathrm{\gamma}}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},c,\mathrm{\pi})\end{array}$Because $\mathrm{\Phi}({p}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}})=\mathrm{\Phi}({\mathrm{\gamma}}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},c,\mathrm{\pi}),$ eq. [3] ensures
$\mathrm{\Phi}(\stackrel{\u02c6}{\mathrm{\phi}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},\stackrel{\u02c6}{\mathrm{\pi}})\ge \mathrm{\Phi}({\mathrm{\gamma}}^{{}^{\prime}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},c,\mathrm{\pi}).$Therefore, it is not profitable for a seller to deviate to any mechanism ${\mathrm{\gamma}}^{{}^{\prime}}$. This proves the “if” part. Because the set of single fixed prices is also one possible set of (constant) mechanisms, the proof of “only if” part is clear. ■*

**Proof of Proposition 2:** *Consider an equilibrium $\{\stackrel{\u02c6}{\mathrm{\phi}},\stackrel{\u02c6}{\mathrm{\pi}}\}$ where each seller either offers a mechanism $\stackrel{\u02c6}{\mathrm{\phi}}$ with the binary report and the worst punishment or deviates to a single fixed price. For all $k,$ let ${\stackrel{\u02c6}{p}}_{k}={\stackrel{\u02c6}{\mathrm{\phi}}}_{k}({0}^{k})$ be the seller’s equilibrium price conditional on $k$ participating buyers when no seller deviates and each participating buyer reports $t=0.$ We fix a buyer’s truth-telling to each non-deviating seller.*

When no seller deviates from $\stackrel{\u02c6}{\mathrm{\phi}}$, a buyer selects each seller with equal probability $\stackrel{\u02c6}{\mathrm{\pi}}({\stackrel{\u02c6}{\mathrm{\phi}}}^{m})=1/m$. Subsequently, each seller’s expected payoff is
$\mathrm{\Phi}(\stackrel{\u02c6}{\mathrm{\phi}},{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},\stackrel{\u02c6}{\mathrm{\pi}})=\sum _{k=1}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){\left(\frac{1}{m}\right)}^{k}{\left(1-\frac{1}{m}\right)}^{n-k}k{\stackrel{\u02c6}{p}}_{k}.$

In an incentive consistent truth-telling continuation equilibrium, upon a seller’s deviation to a fixed price $p,$ a buyer selects the deviating seller with probability ${\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})$ and each non-deviating seller with equal probability $\frac{1-{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})}{m-1}$. Because the non-deviating seller lowers his price down to zero upon all multiple participating buyers’ truthful reports $t=1,$ the buyer’s expected payoff associated with choosing the non-deviating seller is
$1-{\left(1-\frac{1-{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})}{m-1}\right)}^{n-1}{\stackrel{\u02c6}{p}}_{1}.$By choosing the deviating seller, the buyer’s expected payoff is $1-p.$ If ${\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}=\stackrel{\u02c6}{\mathrm{\pi}},$ then buyers play the original incentive consistent truth-telling continuation equilibrium in $\{\stackrel{\u02c6}{\mathrm{\phi}},\stackrel{\u02c6}{\mathrm{\pi}}\}$. If not, buyers play an alternative incentive consistent truth-telling continuation equilibrium upon a seller’s deviation.

By comparing the buyer’s expected payoffs, $1-{\left(1-\frac{1-{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})}{m-1}\right)}^{n-1}{\stackrel{\u02c6}{p}}_{1}$ and $1-p,$ we can characterize all possible incentive consistent truth-telling continuation equilibria upon a seller’s deviation to $p$ below.
$\begin{array}{rl}{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})& =\\ & \{\begin{array}{cc}1& \mathrm{i}\mathrm{f}p\le {\left(1-\frac{1}{m-1}\right)}^{n-1}{\stackrel{\u02c6}{p}}_{1}\\ 1-(m-1)\left(1-{\left(\frac{p}{{\stackrel{\u02c6}{p}}_{1}}\right)}^{\frac{1}{n-1}}\right)\mathrm{o}\mathrm{r}\phantom{\rule{thinmathspace}{0ex}}1& \mathrm{i}\mathrm{f}{\left(1-\frac{1}{m-1}\right)}^{n-1}{\stackrel{\u02c6}{p}}_{1}<p<{\stackrel{\u02c6}{p}}_{1}\\ 1\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{r}\phantom{\rule{thinmathspace}{0ex}}0& \mathrm{i}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}{\stackrel{\u02c6}{p}}_{1}=p\\ 0& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}(\mathrm{i}.\mathrm{e}.,{\stackrel{\u02c6}{p}}_{1}<p)\end{array}\end{array}$When each buyer selects the deviating seller with probability ${\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})$ upon his deviation to $p,$ the deviating seller’s expected payoff is
$\mathrm{\Phi}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}})=n{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})p.$[22]Let us consider the supremum of the deviating seller’s payoff upon deviation.

*(Case 1)* ${\left(1-\frac{1}{m-1}\right)}^{n-1}{\stackrel{\u02c6}{p}}_{1}\ge p.$ There exists a unique incentive consistent truth-telling continuation equilibrium in which every buyer buys the good from the deviating seller with probability one. In this case, the deviating seller’s expected payoff becomes $\mathrm{\Phi}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}})=np,$ which is monotonically increasing and converges to $n{\left(1-\frac{1}{m-1}\right)}^{n-1}{\stackrel{\u02c6}{p}}_{1}$ as $p$ approaches ${\left(1-\frac{1}{m-1}\right)}^{n-1}{\stackrel{\u02c6}{p}}_{1}.$ Therefore, we have that
$\underset{p}{sup}\mathrm{\Phi}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}})=n{\left(1-\frac{1}{m-1}\right)}^{n-1}{\stackrel{\u02c6}{p}}_{1},\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\text{\hspace{0.17em}}\mathrm{t}\mathrm{h}\mathrm{e}\text{\hspace{0.17em}}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{e}\text{\hspace{0.17em}}{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1}).$[23]

*(Case 2)* ${\left(1-\frac{1}{m-1}\right)}^{n-1}{\stackrel{\u02c6}{p}}_{1}<p<{\stackrel{\u02c6}{p}}_{1}.$ We have two incentive consistent truth-telling continuation equilibria: one mixed-strategy continuation equilibrium and one pure-strategy continuation equilibrium. Whether ${\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})$ equals $1-(m-1)\left(1-{\left(\frac{p}{{\stackrel{\u02c6}{p}}_{1}}\right)}^{\frac{1}{n-1}}\right)$ or $1$, eq. [22] is monotonically increasing and converges to $n{\stackrel{\u02c6}{p}}_{1}$ as $p$ approaches ${\stackrel{\u02c6}{p}}_{1}.$ Therefore, regardless of the incentive consistent truth-telling continuation equilibrium that buyers play, the supremum of the deviating seller’s payoff is
$\underset{p}{sup}\mathrm{\Phi}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}})=n{\stackrel{\u02c6}{p}}_{1}\text{\hspace{0.17em}}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\text{\hspace{0.17em}}\mathrm{a}\mathrm{n}\mathrm{y}\text{\hspace{0.17em}}{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1}).$[24]

*(Case 3)* $p={\stackrel{\u02c6}{p}}_{1}.$ If ${\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})=0,$ then the deviating seller has zero profit. If ${\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})=1,$ then the deviating seller’s payoff is $n{\stackrel{\u02c6}{p}}_{1}.$ Therefore, the supremum of the deviating seller’s payoff is
$\underset{p}{sup}\mathrm{\Phi}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}})=n{\stackrel{\u02c6}{p}}_{1}\text{\hspace{0.17em}}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\text{\hspace{0.17em}}{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})=1.$[25]

*(Case 4)* $p>{\stackrel{\u02c6}{p}}_{1}$. In this case, the deviating seller’s expected profit is zero because $\stackrel{\u02c6}{\mathrm{\pi}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1})=0.$ That is,
$\underset{p}{sup}\mathrm{\Phi}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}})=0\text{\hspace{0.17em}}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\text{\hspace{0.17em}}\mathrm{t}\mathrm{h}\mathrm{e}\text{\hspace{0.17em}}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{e}\text{\hspace{0.17em}}\stackrel{\u02c6}{\mathrm{\pi}}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1}).$[26]

If a seller deviates to a single fixed price $p$ that belongs to Case 2, then the supremum of the deviating seller’s payoff is always, regardless of the incentive consistent truth-telling continuation equilibrium that buyers play. Furthermore, eq. [23], [25], and [26] show that the deviating seller’s expected profit does not exceed $n{\stackrel{\u02c6}{p}}_{1}$, regardless of the truth-telling continuation equilibrium that results upon a seller’s deviation to a single fixed price that belongs to Case 1, Case 3, or Case 4. Therefore, we can conclude that the supremum of the deviating seller’s payoff in *every incentive consistent truth-telling continuation equilibrium* is $n{\stackrel{\u02c6}{p}}_{1}$; specifically, for every incentive consistent truth-telling continuation equilibrium ${\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}}$,
$\underset{p\in [0,1]}{sup}\mathrm{\Phi}(p,{\stackrel{\u02c6}{\mathrm{\phi}}}^{m-1},{\stackrel{\u02c6}{\mathrm{\pi}}}^{{}^{\prime}})=n{\stackrel{\u02c6}{p}}_{1}.$[27]Thus, the necessary and sufficient condition [3] in Theorem 1 is satisfied with equality and hence, every equilibrium $\{\stackrel{\u02c6}{\mathrm{\phi}},\stackrel{\u02c6}{\mathrm{\pi}}\}$ of an auxiliary competing mechanism game is robust to any $\mathrm{\Gamma}$. ■

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