<ol>
<li><p>Consider three firms that engage in a first-price auction. Firm <span class="math-container">$i$</span>'s payoff when firm <span class="math-container">$j$</span> wins the auction is <span class="math-container">$S_{i,j}$</span>, which is deterministic and publicly known.</p>
</li>
<li><p>The winning firm <span class="math-container">$i$</span> has to pay its bid and faces transaction costs <span class="math-container">$\tau_i+\epsilon_i$</span>. <span class="math-container">$\tau_i$</span> is stochastic and has p.d.f. <span class="math-container">$g_i(\tau_i)$</span> with <span class="math-container">$E[\tau_i]=\hat{\tau}_i$</span>; <span class="math-container">$\epsilon_i$</span> is stochastic and has p.d.f. <span class="math-container">$h_i(\epsilon_i)$</span> with <span class="math-container">$E[\epsilon_i]=\hat{\epsilon}_i$</span>. (If it helps for tractability, we can assume that each player has the same distributions.)</p>
</li>
<li><p>When going into the auction, each firm <span class="math-container">$i$</span> knows the realization of its own <span class="math-container">$\tau_i$</span>,  the distributions of <span class="math-container">$g_j(\tau_j)$</span>, with <span class="math-container">$i\neq j$</span>, of its competitors, and its own and others <span class="math-container">$h_k(\epsilon_k)$</span>, with <span class="math-container">$k\in \{1,2,3\}$</span>.</p>
</li>
</ol>
<p><strong>Lemma:</strong>
(a) There exists an unique bidding equilibrium <span class="math-container">$(b^*_1,b^*_2,b^*_3)$</span>.</p>
<p>(b) Let <span class="math-container">$\pi_i$</span> be the expected profit of firm <span class="math-container">$i$</span> <strong>prior to learning the realization</strong> of its <span class="math-container">$\tau_i$</span></p>
<p>(i) <span class="math-container">$\frac{\partial \pi_i}{\partial \hat{\tau}_i}\leq 0$</span>  (This is not as trivial as it seems because competing bidders may reduce their equilibrium bid in response.)</p>
<p>(ii) <span class="math-container">$\frac{\partial \pi_i}{\partial \hat{\tau}_{j}}\geq 0$</span>, with <span class="math-container">$j\neq i$</span></p>
<p>Do you know a paper in which something like this treated? How can I prove this Lemma?</p>
<p>Particularly, part (b) seems super intuitive but I do not yet see how to proof it.</p>

CC BY-SA 4.0
2022-05-08T20:41:27.047Z
How is $S_{i,j}$ determined? Is it stochastic?
Walrasian Auctioneer
CC BY-SA 4.0
2022-05-09T08:14:18.703Z
@WalrasianAuctioneer: This is finally deterministic and public information. Thanks for asking.
Paul
<game-theory><auctions>
Public/private knowledge in auctions
Paul
# Public/private knowledge in auctions

(Asked by: username_0 on 2022-05-08)

1. Consider three firms that engage in a first-price auction. Firm $i$'s payoff when firm $j$ wins the auction is $S\_{i,j}$, which is deterministic and publicly known.
2. The winning firm $i$ has to pay its bid and faces transaction costs $\tau\_i+\epsilon\_i$. $\tau\_i$ is stochastic and has p.d.f. $g\_i(\tau\_i)$ with $E[\tau\_i]=\hat{\tau}\_i$; $\epsilon\_i$ is stochastic and has p.d.f. $h\_i(\epsilon\_i)$ with $E[\epsilon\_i]=\hat{\epsilon}\_i$. (If it helps for tractability, we can assume that each player has the same distributions.)
3. When going into the auction, each firm $i$ knows the realization of its own $\tau\_i$, the distributions of $g\_j(\tau\_j)$, with $i\neq j$, of its competitors, and its own and others $h\_k(\epsilon\_k)$, with $k\in \{1,2,3\}$.

**Lemma:**
(a) There exists an unique bidding equilibrium $(b^\*\_1,b^\*\_2,b^\*\_3)$.

(b) Let $\pi\_i$ be the expected profit of firm $i$ **prior to learning the realization** of its $\tau\_i$

(i) $\frac{\partial \pi\_i}{\partial \hat{\tau}\_i}\leq 0$ (This is not as trivial as it seems because competing bidders may reduce their equilibrium bid in response.)

(ii) $\frac{\partial \pi\_i}{\partial \hat{\tau}\_{j}}\geq 0$, with $j\neq i$

Do you know a paper in which something like this treated? How can I prove this Lemma?

Particularly, part (b) seems super intuitive but I do not yet see how to proof it.

**Tags:** game-theory, auctions

> **Comments:**
> **username_1**: How is $S_{i,j}$ determined? Is it stochastic?  
> **username_0**: @username_1: This is finally deterministic and public information. Thanks for asking.  

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