<p>Consider bidder <span class="math-container">$i$</span>. Let <span class="math-container">$p = \max_{(2)}\{b_{-i}\}$</span>.</p>
<p>By bidding <span class="math-container">$v_i$</span>, she wins if <span class="math-container">$v_i &gt; p$</span> and not if <span class="math-container">$v_i &lt; p$</span> (and indifferent if <span class="math-container">$x_i = p$</span>).</p>
<p>Suppose, however, she bids <span class="math-container">$z_i &lt; v_i$</span>.</p>
<p>(i) If <span class="math-container">$v_i &gt; z_i &gt; p$</span>, she still wins the auction and still gets <span class="math-container">$v_i - p$</span>.</p>
<p>(ii) If <span class="math-container">$p &gt; v_i &gt; z_i$</span>, she still loses the auction and still gets <span class="math-container">$0$</span>.</p>
<p>(iii) However, if <span class="math-container">$v_i &gt; p &gt; z_i$</span>, she now loses instead of winning.</p>
<p>Thus, bidding less than <span class="math-container">$v_i$</span> can never improve <span class="math-container">$i$</span>'s payoffs.</p>
<p>Can you finish the argument for the case <span class="math-container">$z_i &gt; v_i$</span>?</p>

CC BY-SA 4.0
2021-11-02T08:44:37.887Z
Thank you! It makes good sense. 

However, do you happen to know if there is a way to solve this as an equation? Or at least write up the expected utility given that each bidder bids her true value, sinse I am also interested in comparing bidder's expected utility in a third-price auction with her expected utility in a second-price auction?
Sarah Plaug
CC BY-SA 4.0
2021-11-02T11:45:27.807Z
I dont see the benefit of attempting to write this as an equation? This is enough to show this is indeed the equilibrium. As for expected utilities, I would advice you to start a new question instead asking for that!
Walrasian Auctioneer
CC BY-SA 4.0
2021-11-02T01:05:35.010Z
2021-11-02T01:05:35.010Z
Walrasian Auctioneer
<p>I am interest in finding the optimal strategy for a single bidder (the BNE), when I have a third-price auction with N bidders and two identical goods. The bidders have iid valuations: U(0,1)</p>
<p>I have been told by my teacher that it is a weakly dominant strategy for the bidders to bid their valuations, but I cannot see why this would be? <br />
(and I know from the literature that it is optimal to bid slightly above your true valuation if only one item is auctioned off, i.e. to bid <span class="math-container">$b_i=\frac{n-1}{n-2}v_i$</span>)</p>
<p>The utility function for bidder <em>i</em> looks like this: <br />
<span class="math-container">$ u_i(b_i, b_{-i}) =  \left\{
\begin{array}{lr} 
v_i-\max_{(2)}\{b_{-i}\} &amp; \text{if } b_i&gt;\max_{(1)}\{b_{-i}\}&gt;\max_{(2)}\{b_{-i}\} \\
v_i-\max_{(2)}\{b_{-i}\} &amp; \text{if } \max_{(1)}\{b_{-i}\}&gt;b_i&gt;\max_{(2)}\{b_{-i}\} \\
0 &amp; \text{if } b_i&lt;\max_{(2)}\{b_{-i}\}
\end{array}
\right. $</span> <br />
since in this auction, the highest and second highest bidder wins a good each, and both pay the third highest bid, because the bidders gain equal value from either good.</p>
<p>If anyone knows why it supposedly is a weakly dominant strategy to bid your true value in this type of auction, or where I might find a good explanation, I would really appreciate any help I can get!</p>

CC BY-SA 4.0
2021-11-02T09:49:42.820Z
Should we assume that $n \geq 3$?

Also: I guess that, if this is indeed dominant, then it should be dominant for any value distribution (not just uniform)?
afreelunch
CC BY-SA 4.0
2021-11-02T10:57:40.463Z
Yes, sorry for not clarifying that I assume $n\geq 3$.
And indeed, I would think that it should hold for any continuous distribution with increasing probability. The reason why I asked specifically for case with a uniform value distribution is because that is the distribution I assumed in my thesis.
Sarah Plaug
<game-theory><auctions><bayesian-game>
Is it a weakly dominant strategy to bid your true value in a third-price auction with two identical goods and N uniform distributed bidders?
Sarah Plaug
# Is it a weakly dominant strategy to bid your true value in a third-price auction with two identical goods and N uniform distributed bidders?

(Asked by: username_0 on 2021-11-01)

I am interest in finding the optimal strategy for a single bidder (the BNE), when I have a third-price auction with N bidders and two identical goods. The bidders have iid valuations: U(0,1)

I have been told by my teacher that it is a weakly dominant strategy for the bidders to bid their valuations, but I cannot see why this would be?   
(and I know from the literature that it is optimal to bid slightly above your true valuation if only one item is auctioned off, i.e. to bid $b\_i=\frac{n-1}{n-2}v\_i$)

The utility function for bidder *i* looks like this:   
$ u\_i(b\_i, b\_{-i}) = \left\{
\begin{array}{lr}
v\_i-\max\_{(2)}\{b\_{-i}\} & \text{if } b\_i>\max\_{(1)}\{b\_{-i}\}>\max\_{(2)}\{b\_{-i}\} \\
v\_i-\max\_{(2)}\{b\_{-i}\} & \text{if } \max\_{(1)}\{b\_{-i}\}>b\_i>\max\_{(2)}\{b\_{-i}\} \\
0 & \text{if } b\_i<\max\_{(2)}\{b\_{-i}\}
\end{array}
\right. $   
since in this auction, the highest and second highest bidder wins a good each, and both pay the third highest bid, because the bidders gain equal value from either good.

If anyone knows why it supposedly is a weakly dominant strategy to bid your true value in this type of auction, or where I might find a good explanation, I would really appreciate any help I can get!

**Tags:** game-theory, auctions, bayesian-game
> **Comments:**
> **username_1**: Should we assume that $n \geq 3$?  Also: I guess that, if this is indeed dominant, then it should be dominant for any value distribution (not just uniform)?  
> **username_0**: Yes, sorry for not clarifying that I assume $n\geq 3$. And indeed, I would think that it should hold for any continuous distribution with increasing probability. The reason why I asked specifically for case with a uniform value distribution is because that is the distribution I assumed in my thesis.  

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## 1 answers

### Answer by username_2 on 2021-11-02. Score: 3 | Accepted answer

Consider bidder $i$. Let $p = \max\_{(2)}\{b\_{-i}\}$.

By bidding $v\_i$, she wins if $v\_i > p$ and not if $v\_i < p$ (and indifferent if $x\_i = p$).

Suppose, however, she bids $z\_i < v\_i$.

(i) If $v\_i > z\_i > p$, she still wins the auction and still gets $v\_i - p$.

(ii) If $p > v\_i > z\_i$, she still loses the auction and still gets $0$.

(iii) However, if $v\_i > p > z\_i$, she now loses instead of winning.

Thus, bidding less than $v\_i$ can never improve $i$'s payoffs.

Can you finish the argument for the case $z\_i > v\_i$?

> **Comments:**
> **username_0**: Thank you! It makes good sense.   However, do you happen to know if there is a way to solve this as an equation? Or at least write up the expected utility given that each bidder bids her true value, sinse I am also interested in comparing bidder's expected utility in a third-price auction with her expected utility in a second-price auction?  
> **username_2**: I dont see the benefit of attempting to write this as an equation? This is enough to show this is indeed the equilibrium. As for expected utilities, I would advice you to start a new question instead asking for that!  

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