Contrary to the previous section, assume now that the number of potential participants $n$ is not common knowledge among all the players. More precisely, the seller still knows $n$ with certainty whereas the buyers do not. This different information structure is certainly more appropriate to capture the features of PRAs that take place over the Internet, where indeed buyers do not know the number of rivals. Notice however that, in order to participate, agents must first register on the seller’s website. As such, while somehow stark (registration does not necessarily imply participation), the assumption that the seller knows $n$ with certainty seems justifiable.

This information asymmetry has important implications for what concerns possible equilibrium outcomes and the profitability of the mechanism. Since the optimal initial price that the seller sets is a function of the number of participants, uncertainty about $n$ leads in fact to uncertainty about ${p}_{0}^{\ast}$. To see this point, assume that, from the point of view of each agent $i\in N$, the number of participants is a random variable $\tilde{N}$. Let $g\left(\cdot \right)$ be the non-degenerate probability mass function of $\tilde{N}$ and ${S}_{\tilde{N}}=\left\{{\tilde{n}}_{min},...,{\tilde{n}}_{max}\right\}$ with ${\tilde{n}}_{min}\ge 1$ its support. Agents’ beliefs about ${p}_{0}^{\ast}$ are then captured by the distribution ${H}_{i,1}\left({p}_{0}\right)$, which is defined over the support ${S}_{{p}_{0}^{\ast}}=\left\{{p}_{0}^{\ast}\left({\tilde{n}}_{min}\right),...,{p}_{0}^{\ast}\left({\tilde{n}}_{max}\right)\right\}$ and assigns the probability mass function ${h}_{i,1}\left({p}_{0}^{\ast}\left(\tilde{n}\right)\right)=g\left(\tilde{n}\right)$ for every $\tilde{n}\in {S}_{\tilde{N}}$, where ${p}_{0}^{\ast}\left(\tilde{n}\right)$ is the optimal initial price the seller would set if the actual number of participants was $n=\tilde{n}$ (indeed, we will show that in equilibrium the seller faces the same problem, and thus adopts the same behavior, as the ones described in Section 3.2). Uncertainty about ${p}_{0}^{\ast}$, in turn, implies uncertainty about the current price ${p}_{t}^{\ast}$ at any $t\ge 1$. In such an environment, agents still play according to the behavioral rule defined in eq. [3], but now action ${a}_{{\stackrel{\u02c6}{i}}_{t},t}^{2}=\mathrm{\gamma}\left(\cdot \right)$ with $\mathrm{\gamma}\left({p}_{t}^{\ast}\right)=0$ can be observed even on the equilibrium path. Indeed, it may well be the case that an agent rationally decides to observe the price, discovers an actual level that would instead lead to a negative payoff, and thus decides not to buy the item. This implies that, in equilibrium, the price may actually fall.

The following proposition formalizes these results:

**Proposition 3:** *In a “profit maximizing trading equilibrium” (PMTE) of a PRA in which* $n$ *is uncertain, agents believe the initial price to be distributed over* ${S}_{{p}_{0}^{\ast}}=\left\{{p}_{0}^{\ast}\left({\tilde{n}}_{min}\right),...,{p}_{0}^{\ast}\left({\tilde{n}}_{max}\right)\right\}$ *according to the distribution* ${H}_{i,1}^{\ast}\left({p}_{0}\right)$ *that assigns probability mass function* ${h}_{i,1}\left({p}_{0}^{\ast}\left(\tilde{n}\right)\right)=g(\tilde{n})$ *for any* $\tilde{n}\in \left\{{\tilde{n}}_{min},...,{\tilde{n}}_{max}\right\}$*. Each* ${p}_{0}^{\ast}\left(\tilde{n}\right)$ *is defined as* ${p}_{0}^{\ast}\left(\tilde{n}\right)=\underset{{p}_{0}\in \left[{v}_{s}+\mathrm{\Delta}-c,\stackrel{\u02c9}{v}+\mathrm{\Delta}-c\right]}{arg\text{\hspace{0.17em}}max}{\mathrm{\pi}}_{s}\left(\tilde{n}\right)$ *where*:
${\mathrm{\pi}}_{s}\left(\tilde{n}\right)=\left(1-{\left[F\left({p}_{0}-\mathrm{\Delta}+c\right)\right]}^{\tilde{n}}\right)({p}_{0}-\mathrm{\Delta}+c-{\mathrm{\upsilon}}_{s})$

*Each agent* $i\in N$ *plays* ${\left({a}_{i,t}^{\ast}\right)}_{t=1}^{{t}_{e}}$ *where* ${a}_{i,t}^{\ast}=\left({a}_{i,t}^{\ast 1},{a}_{i,t}^{\ast 2}\right)$ *is such that*:
${a}_{i,t}^{\ast 1}=\{\begin{array}{c}\mathrm{\varnothing}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thickmathspace}{0ex}}if\text{\hspace{0.17em}}\left({\mathrm{\pi}}_{i}|{H}_{i,t}^{\ast}\left({p}_{t-1}\right)\right)\le 0\\ wto\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}if\text{\hspace{0.17em}}\left({\mathrm{\pi}}_{i}|{H}_{i,t}^{\ast}\left({p}_{t-1}\right)\right)>0\end{array}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}for\text{\hspace{0.17em}}any\text{\hspace{0.17em}}i\in N$
${a}_{i,t}^{\ast 2}=\{\begin{array}{c}\mathrm{\gamma}\phantom{\rule{thickmathspace}{0ex}}\left(\cdot \right)\phantom{\rule{1em}{0ex}}with\text{\hspace{0.17em}}\mathrm{\gamma}\text{\hspace{0.17em}}\left({p}_{t}\right)=0\phantom{\rule{thickmathspace}{0ex}}\text{\hspace{0.17em}}if\text{\hspace{0.17em}}{p}_{t}>{b}^{\ast}({\mathrm{\upsilon}}_{i})\\ \mathrm{\gamma}\phantom{\rule{thickmathspace}{0ex}}\left(\cdot \right)\phantom{\rule{1em}{0ex}}with\text{\hspace{0.17em}}\mathrm{\gamma}\text{\hspace{0.17em}}\left({p}_{t}\right)=1\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}if\text{\hspace{0.17em}}{p}_{t}\le {b}^{\ast}({\mathrm{\upsilon}}_{i})\end{array}\phantom{\rule{1em}{0ex}}for\text{\hspace{0.17em}}i={\stackrel{\u02c6}{i}}_{t}\phantom{\rule{thickmathspace}{0ex}}and\phantom{\rule{thickmathspace}{0ex}}{a}_{i,t}^{\ast 2}=\mathrm{\varnothing}\text{\hspace{0.17em}}for\text{\hspace{0.17em}}any\text{\hspace{0.17em}}i\ne {\stackrel{\u02c6}{i}}_{t}$

*and* $\left({\mathrm{\pi}}_{i}|{H}_{i,t}\left({p}_{t-1}\right)\right)$ *is as in [1], agents’ beliefs* ${H}_{i,t}^{\ast}\left({p}_{t-1}\right)$ *evolve from* ${H}_{i,1}^{\ast}\left({p}_{0}\right)$ *according to Bayes’ rule*, ${b}^{\ast}({v}_{i})={v}_{i}$ *for any* $i\in N$*, and* ${t}_{e}\in \left\{1,...,T\right\}$.

**Proof:** *In the appendix. ■The equilibrium is now characterized by three possible (classes of) outcomes:
*

*(1)*

*An agent observes the price in $t=1$, and immediately buys the item.*

*(2)*

*No agent ever observes the price and the item remains unsold.*

*(3)*

*More than one agent observes the price and the auction ends at ${t}_{e}\in \left\{2,...,T\right\}$.*

The first two outcomes are analogous to the ones that characterize the PMTE of a PRA in which $n$ is common knowledge; the third one is instead specific of a PRA where there is uncertainty about the number of participants. In this third equilibrium outcome, the entry of multiple bidders and repeated observations of the price can occur. The following example illustrates such a possibility:

**Example 3:** *Consider a PRA where* $F\text{\hspace{0.17em}}\sim \text{\hspace{0.17em}}U\left[0,150\right]$, ${v}_{s}=0$, $c=2$*, and* $\mathrm{\Delta}=1$*. Let* ${S}_{\tilde{N}}=\left\{1,...,5\right\}$ *with* $g\left(\tilde{n}\right)=0.2$ *for every* $\tilde{n}\in \left\{1,...,5\right\}$*. The actual number of participants is* $n=4$ *with* ${v}_{1}=50$, ${v}_{2}=88$, ${v}_{3}=93$*, and* ${v}_{4}=130$*. In equilibrium, the game can thus unfold in the following way*:

*–*

*period* $t=1$*: agents’ initial beliefs are described by the distribution* ${H}_{i,1}^{\ast}\left({p}_{0}\right)$*. More precisely, all agents expect the initial price to be distributed over the support*:
${S}_{{p}_{0}^{\ast}}=\left\{74,\phantom{\rule{thickmathspace}{0ex}}85.6,\phantom{\rule{thickmathspace}{0ex}}93.49,\phantom{\rule{thickmathspace}{0ex}}99.31,\phantom{\rule{thickmathspace}{0ex}}103.82\right\}$

*with probabilities* $\left(0.2,0.2,0.2,0.2,0.2\right)$*. As such, the choice of observing the price leads to a positive expected payoff for any* ${v}_{i}>\underset{t=1}{\underset{\_}{v}}=83$ *where* $\underset{t=1}{\underset{\_}{v}}$ *solves* $\left({\mathrm{\pi}}_{i}|{H}_{i,1}^{\ast}\left({p}_{0}\right)\right)=0$*. Buyer* $1$ *thus remains inactive whereas buyers* $2$, $3$*, and* $4$ *are willing to observe the price. With probability* $\frac{1}{3}$*, buyer* $2$ *is chosen. He observes* ${p}_{1}^{\ast}=92.49$*, such that* ${p}_{1}^{\ast}>{v}_{1}$ *and thus does not buy the item*.

*–*

*period* $t=2$*: agent* $1$*’s beliefs do not change (i. e*., ${H}_{1,2}^{\ast}\left({p}_{1}\right)={H}_{1,1}^{\ast}\left({p}_{0}\right)$*) such that the agent remains inactive. Agent* $2$*’s beliefs are instead given by* ${h}_{1,2}^{\ast}\left({p}_{1}^{\ast}=92.49\right)=1$*; the agent thus also plays* ${a}_{1,2}^{\ast}=\left(\mathrm{\varnothing},\mathrm{\varnothing}\right)$*. Agents* $3$ *and* $4$ *instead observe that the auction is still open. Therefore, they update their beliefs to* ${H}_{i,2}^{\ast}\left({p}_{1}\right)$ *with* $i\in \left\{3,4\right\}$ *and expect* ${p}_{1}^{\ast}$ *to be distributed over the support*:
${S}_{{p}_{1}^{\ast}}=\left\{73,\phantom{\rule{thickmathspace}{0ex}}84.6,\phantom{\rule{thickmathspace}{0ex}}92.49,\phantom{\rule{thickmathspace}{0ex}}98.31,\phantom{\rule{thickmathspace}{0ex}}102.82\right\}$

*with probabilities* $\left(0,0.02,0.153,0.3,0.527\right)$.
*Given* ${H}_{i,2}^{\ast}\left({p}_{1}\right)$*, it is worthwhile for a player to observe the price if and only if* ${v}_{i}>{\underset{\_}{v}}_{t=2}=99.07$*, where* ${\underset{\_}{v}}_{t=2}$ *solves* $\left({\mathrm{\pi}}_{i}|{H}_{i,2}^{\ast}\left({p}_{1}\right)\right)=0$*. Therefore, agent 3 now decides to remain inactive whereas agent* $4$ *observes the price, discovers* ${p}_{2}^{\ast}=91.49$ *such that* ${p}_{2}^{\ast}<{v}_{4}$*, and thus buys the item. The auction closes and payoffs are* ${u}_{1}=0$, ${u}_{2}=-2$, ${u}_{3}=0$, ${u}_{4}=36.51$*, and* ${u}_{s}=95.49$.

Example 3 shows which forces drive the Bayesian updating of agents’ beliefs. On the one hand, learning that a rival observed the price and did not buy the item brings good news, as this implies that the price fell by $\mathrm{\Delta}$. As such, the support of possible prices shifts to the left. On the other hand, the fact that the rival who observed the price did not buy the item means that the actual price that he discovered was higher than his valuation, where the latter was certainly above the entry threshold ${\underset{\_}{v}}_{t=1}$. As such, the probability distribution becomes more skewed towards high values. For standard values of the parameters (small $\mathrm{\Delta}$, see footnote 1), the latter effect dominates such that the expected price, as well as the threshold ${\underset{\_}{v}}_{t}$ that makes the choice to observe the price worthwhile, weakly increases over time. Figure 2 below illustrates the effects of these two conflicting forces in the context of Example 3.

Figure 2: Updating of agents’ beliefs.

The way agents update their beliefs about the hidden price implies that, in equilibrium, two further results hold. First, an agent that is willing to observe the price in a given period but does not get the opportunity to do so may decide not to observe the price in subsequent periods, because the threshold ${\underset{\_}{v}}_{t}$ that solves condition $\left({\mathrm{\pi}}_{i}|{H}_{i,t}^{\ast}\left({p}_{t-1}\right)\right)=0$ increases over time. As such, a player may have a valuation ${\underset{\_}{v}}_{t}<{v}_{i}<{\underset{\_}{v}}_{{t}^{{}^{\prime}}}$ with ${t}^{\text{'}}>t$ (this is the case of agent 3 in Example 3).

Second, no agent that is unwilling to observe the price in a given period will decide to observe the price in subsequent periods. The intuition is the following. A player plays ${a}_{i,t}^{\ast 1}=\mathrm{\varnothing}$ if his expected payoff from observing the price is negative, i. e., if ${v}_{i}<{\underset{\_}{v}}_{t}$. Now suppose that in period ${t}^{\text{'}}>t$, agent $i$ observes that the auction is still open. The player realizes that two paths may have occurred: (1) No rival observed the price in any of the previous periods. If this was the case, then $i$‘s beliefs would not change and thus the expected payoff of playing ${a}_{i\mathrm{,}{t}^{\text{'}}}^{\ast 1}\ne \varnothing $ would remain negative as it was in $t$. (2) Some other player observed the price in some of the previous periods. However, if this was the case, the agents that observed the price necessarily refused to buy the item, otherwise the auction would have closed. Moreover, if a player $j\in N$ observed the price in period ${t}^{\text{'}\text{'}}\in \left\{t\mathrm{,...,}{t}^{\text{'}}\right\}$, then it must be the case that ${v}_{j}>{\underset{\_}{\text{v}}}_{{t}^{\text{'}\text{'}}}$, which in turn implies ${v}_{j}>{v}_{i}$ given that ${\underset{\_}{\text{v}}}_{{t}^{\text{'}\text{'}}}\ge {\underset{\_}{\text{v}}}_{t}$. The fact that $j$ did not buy the item thus indicates that, a fortiori, agent $i$ would find the transaction unattractive.

A noticeable feature of the equilibrium is that it remains optimal for the agents to play ${b}^{\ast}({v}_{i})={v}_{i}$. In other words, and despite the fact that, in equilibrium, the price may actually fall, agents’ gross willingness to pay still coincides with their valuation. The reason is that in a PRA, the price falls endogenously in response to agents’ behavior, rather than exogenously, as it does in a standard Dutch auction. Indeed, the probability that the price falls (i. e., the probability that one or more rivals observe the price and then do not buy the item) is too low to justify a strategy ${b}^{\text{'}}({v}_{i})<{b}^{\ast}({v}_{i})={v}_{i}$. We discuss this result in more detail in the proof of Proposition 3; however, as an informative example, consider a hypothetical PRA in which there are $n$ participants but only agent $i$ has a valuation ${v}_{i}>{\underset{\_}{v}}_{t=1}$. Assume moreover that the initial price is such that ${b}^{\text{'}}({v}_{i})-c<{p}_{1}^{\ast}\left(n\right)\le {v}_{i}-c$. If the price was falling exogenously, the current price ${p}_{t}^{\ast}\left(n\right)$ would eventually reach a level ${p}_{t}^{\ast}\left(n\right)\le {b}^{\text{'}}({v}_{i})-c$. Agent $i$ would then enjoy a more substantial surplus by playing ${b}^{\text{'}}({v}_{i})$ rather than ${b}^{\ast}({v}_{i})$. However, in a PRA, the price remains stuck at the initial level and thus reaches ${b}^{\text{'}}({v}_{i})$ with zero probability. Therefore, ${b}^{\ast}({v}_{i})$ dominates ${b}^{\text{'}}({v}_{i})$, as the former leads to ${u}_{i}={v}_{i}-{p}_{1}^{\ast}\left(n\right)-c$ with ${u}_{i}\ge 0$, whereas the latter leads to ${u}_{i}^{\text{'}}=0$.

We conclude the analysis of the mechanism by studying its profitability. The fact that, in equilibrium, multiple entries can occur benefits the seller, as revenues increase by $(c-\mathrm{\Delta})>0$ every time an agent observes the price. Indeed, the following Lemma shows that a PRA with uncertainty about the number of participants raises higher expected revenues than a PRA in which $n$ is common knowledge.

**Lemma 3:** *Consider a PRA with* $n$ *participants. Let* $PRA\left(n\right)$ *denote the case in which* $n$ *is common knowledge and* $PRA\left(\tilde{n}\right)$ *the case in which, from the buyers’ point of view, the number of participants is a random variable with mean* $n$*. Then*, ${\mathrm{\pi}}_{PRA(\tilde{n})}>{\mathrm{\pi}}_{PRA(n)}$*, i. e*., $PRA\left(\tilde{n}\right)$ *yields higher expected revenues than* $PRA\left(n\right)$.

**Proof:** *In $PRA\left(n\right)$ the optimal initial price ${p}_{0}^{\ast}\left(n\right)$ is as defined in Proposition 1. Given ${p}_{0}^{\ast}\left(n\right)$, let $\underset{\_}{v}\left(n\right)$ solve ${u}_{i}\left({p}_{0}^{\ast}\left(n\right)\right)=\underset{\_}{v}\left(n\right)-{p}_{0}^{\ast}\left(n\right)+\mathrm{\Delta}-c=0$. Now consider $PRA\left(\tilde{n}\right)$ where by construction ${\sum}_{{\tilde{n}}_{min}}^{{\tilde{n}}_{max}}\tilde{n}g\left(\tilde{n}\right)=n$. Let ${v}_{i}=\underset{\_}{v}\left(n\right)$ and evaluate $\left({\mathrm{\pi}}_{i}|{H}_{i,1}^{\ast}\left({p}_{0}\right)\right)$ where ${H}_{i,1}^{\ast}\left({p}_{0}\right)$ is as defined in Proposition 3. By Lemma 1, ${p}_{0}^{\ast}\left(\tilde{n}\right)<{p}_{0}^{\ast}\left(n\right)$ for any $\tilde{n}<n$ and ${p}_{0}^{\ast}\left(\tilde{n}\right)>{p}_{0}^{\ast}\left(n\right)$ for any $\tilde{n}>n$. Then, ${u}_{i}\left({p}_{0}^{\ast}\left(\tilde{n}\right)\right)=-c$ whenever $\tilde{n}>n$ while ${u}_{i}\left({p}_{0}^{\ast}\left(\tilde{n}\right)\right)=\underset{\_}{v}\left(n\right)-{p}_{0}^{\ast}\left(\tilde{n}\right)+\mathrm{\Delta}-c>0$ whenever $\tilde{n}<n$. It follows that $\left({\mathrm{\pi}}_{i}|{H}_{i,1}\left({p}_{0}\right)\right)={\sum}_{{\tilde{n}}_{min}}^{{\tilde{n}}_{max}}g\left(\tilde{n}\right){u}_{i}\left({p}_{0}^{\ast}\left(\tilde{n}\right)\right)>0$. Given that $\left({\mathrm{\pi}}_{i}|{H}_{i,1}^{\ast}\left({p}_{0}\right)\right)$ is strictly increasing in ${v}_{i}$, it then must be the case that $\left({\mathrm{\pi}}_{i}|{H}_{i,1}^{\ast}\left({p}_{0}\right)\right)=0$ if and only if agent $i$ has a valuation $\underset{\_}{v}\left(\tilde{n}\right)<\underset{\_}{v}\left(n\right)$. In other words, the valuation that makes an agent indifferent between observing the price or not is lower when the number of participants is uncertain. Now consider any profile of valuations $\left({v}_{1},...,{v}_{n}\right)$ where without loss of generality ${v}_{1}<...<{v}_{n}$. Three situations may occur: (a) ${v}_{n}<\underset{\_}{v}(\tilde{n})<\underset{\_}{v}(n)$ and thus ${u}_{s|PRA\left(\tilde{n}\right)}={u}_{s|PRA\left(n\right)}=0$; (b) $\underset{\_}{v}(\tilde{n})<{v}_{n}<\underset{\_}{v}(n)$ and thus ${u}_{s|PRA\left(\tilde{n}\right)}>0$ and ${u}_{s|PRA\left(n\right)}=0$; (c) $\underset{\_}{v}(\tilde{n})<v(n)<{\underset{\_}{v}}_{n}$ and thus ${u}_{s|PRA\left(\tilde{n}\right)}>0$, ${u}_{s|PRA\left(n\right)}>0$, and ${u}_{s|PRA\left(\tilde{n}\right)}\ge {u}_{s|PRA\left(n\right)}$ as multiple observations can occur in $PRA\left(\tilde{n}\right)$. Therefore, $PRA\left(\tilde{n}\right)$ raises actual revenues that are weakly larger than those that $PRA\left(n\right)$ raises. It follows that ${\mathrm{\pi}}_{PRA(\tilde{n})}>{\mathrm{\pi}}_{PRA(n)}$, i. e., $PRA\left(\tilde{n}\right)$ yields larger expected revenues than $PRA\left(n\right)$.*

The lower bound on the revenues that a PRA with uncertainty about $n$ can raise is thus given by ${u}_{s|PRA\left(n\right)}={p}_{0}^{\ast}\left(n\right)+c-\mathrm{\Delta}$. As for the upper bound, notice that in every round in which tied players update their beliefs, the probability of at least one of the possible prices (namely, the lowest one) drops to zero. Given that the number of possible initial prices coincides with the number of possible realizations of the random variable $\tilde{N}$, it follows that revenues can at most reach the level ${u}_{s|PRA\left(\tilde{n}\right)}={p}_{0}^{\ast}\left(n\right)+\left|{\tilde{N}}_{n}\right|\left(c-\mathrm{\Delta}\right)$, where ${\tilde{N}}_{n}=\left\{{\tilde{n}}_{min},...,n\right\}$ is the set of possible realizations of $\tilde{N}$ that are smaller or equal than the actual number of buyers $n$.

This latter result implies that, in some (highly specific) situations, a PRA with uncertainty about $n$ may revenue dominate standard auction formats. Necessary conditions for this to happen are the following:

(1)

Buyers’ uncertainty about $n$ is highly pronounced (i. e., the distribution of $\tilde{N}$ has a large variance),

(2)

The true number of participants (i. e., the actual realization of $\tilde{N}$) is very large.

Indeed, as the number of participants gets larger, the revenue gap between a PRA where $n$ is common knowledge and a standard auction shrinks (see Section 3.2). At the same time, the possibility that there are multiple players that are simultaneously willing to observe the price increases. However, if the distribution of $\tilde{N}$ has a high variance, so does the distribution that captures agents’ beliefs about the hidden price. It follows that the chances that a player who observes the price refuses to buy the item become non-negligible. As such, the seller may collect multiple fees and the accrual of these fees may more than compensate for the fact that the expected selling price in a PRA is lower than in a standard auction. Conditional on the occurrence of the two above-mentioned necessary conditions, the expected revenues of $PRA\left(\tilde{n}\right)$ may even exceed agents’ maximal valuation $\stackrel{\u02c9}{v}$. The following example illustrates this possibility. The example also highlights how the ranking of the different mechanisms depends on both the specific characteristics of the distribution of $\tilde{N}$ and the actual realization of $n$.

**Example 4:** *Consider two versions of a PRA with* $F\sim U\left[0,150\right]$, ${v}_{s}=0$, $c=2$*, and* $\mathrm{\Delta}=1$.

*–*

*In the first version, agents believe that the random variable* $\tilde{N}$ *is distributed over the support* ${S}_{\tilde{N}}=\left\{1,2,3,4,5\right\}$ *with* $g\left(\tilde{n}\right)=0.2$ *for every* $\tilde{n}\in {S}_{\tilde{N}}$*. The actual number of participants is* $n=3$*. Therefore (see examples 1, 2, and 3)*, ${\mathrm{\pi}}_{PRA(n)}=70.87$, ${\mathrm{\pi}}_{PRA(\tilde{n})}=72.05$*, and* ${\mathrm{\pi}}_{FPA}={\mathrm{\pi}}_{SPA}=81.25$*. It follows that* $\stackrel{\u02c9}{v}>{\mathrm{\pi}}_{FPA}={\mathrm{\pi}}_{SPA}>{\mathrm{\pi}}_{PRA(\tilde{n})}>{\mathrm{\pi}}_{PRA(n)}$.

*–*

*In the second version, agents instead believe that* $\tilde{N}$ *is distributed over* ${S}_{\tilde{N}}=\left\{3,5,10,30,2999\right\}$ *with* $g\left(\tilde{n}\right)=0.2$ *for every* $\tilde{n}\in {S}_{\tilde{N}}$*. The actual number of participants is* $n=2999$*. In such a case*, ${\mathrm{\pi}}_{PRA(n)}=149.55$, ${\mathrm{\pi}}_{PRA(\tilde{n})}=151.26$*, and* ${\mathrm{\pi}}_{FPA}={\mathrm{\pi}}_{SPA}=149.9$*. It follows that* ${\mathrm{\pi}}_{PRA(\tilde{n})}>\stackrel{\u02c9}{v}>{\mathrm{\pi}}_{FPA}={\mathrm{\pi}}_{SPA}>{\mathrm{\pi}}_{PRA(n)}$.

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