<p>This is not quite the same as the usual generalized-second-price-auction setting because you are assuming the items are identical, whereas in a GSP setting there is an order on the items (slot A is better than slot B is better than ...).</p>

<p>But anyway, your auction is not dominant-strategy-incentive compatible. Suppose the bidders' true valuations are $v_1 &gt; v_2 &gt; \dots &gt; v_k &gt; \dots &gt; v_n$, and suppose every bidder reports her valuation truthfully.</p>

<p>The first bidder wins an item and pays $v_2$. If she had bid $v_k + \epsilon$, she would have still won an item, but only paid $v_k$. So she would have preferred to lie and bid $v_k + \epsilon$.</p>

CC BY-SA 3.0
2013-05-29T15:53:50.960Z
2013-05-29T15:53:50.960Z
usul
<p>Consider multiunit auction (as it is defined in Introduction to Mechanism
Design by Noam Nisan) , where $k$ identical units of some good are sold in an auction (where $k &lt; n$). In the simple case each bidder is interested in only a single unit. In this case $A = \{S–wins|S ⊂ I, |S| = k\}$, and a bidder’s valuation $v_i$ gives some fixed value $v^∗$ if $i$ gets an item, i.e. $v_i(S) = v^∗$ if $i ∈ S$ and $v_i(S) = 0$ otherwise. </p>

<p>Let consider possible solutions:</p>

<p>The first incentive compatible solutions seems like generalization of Vickrey auction. Maximizing social welfare means allocating the items to the $k$ highest bidders, and in the VCG mechanism with the pivot rule, each of them should pay the $k + 1$’st highest offered price. It's obvious incentive compatible just by generalization of Vickrey auction.</p>

<p>But what if, the bidder with the highest bid pays the price equal to the second highest bid and the bidder with the second highest bid pays the price equal to the third highest bid ans so on. Obviously their valuation will be much less as in the first solution, but more interesting question is whether the second solution incentive compatible?</p>

CC BY-SA 3.0
2013-05-29T10:42:03.893Z
What you are describing is the generalized second price auction, which is not, in general, incentive compatible. The equilibria of such auctions have been studied extensively in recent years. E.g., 

http://faculty-gsb.stanford.edu/ostrovsky/papers/gsp.pdf
JoelO
<gt.game-theory>
Multiunit Auction
com
# Multiunit Auction

(Asked by: username_0 on 2013-05-29)

Consider multiunit auction (as it is defined in Introduction to Mechanism
Design by Noam Nisan) , where $k$ identical units of some good are sold in an auction (where $k < n$). In the simple case each bidder is interested in only a single unit. In this case $A = \{S–wins|S ⊂ I, |S| = k\}$, and a bidder’s valuation $v\_i$ gives some fixed value $v^∗$ if $i$ gets an item, i.e. $v\_i(S) = v^∗$ if $i ∈ S$ and $v\_i(S) = 0$ otherwise.

Let consider possible solutions:

The first incentive username_0patible solutions seems like generalization of Vickrey auction. Maximizing social welfare means allocating the items to the $k$ highest bidders, and in the VCG mechanism with the pivot rule, each of them should pay the $k + 1$’st highest offered price. It's obvious incentive username_0patible just by generalization of Vickrey auction.

But what if, the bidder with the highest bid pays the price equal to the second highest bid and the bidder with the second highest bid pays the price equal to the third highest bid ans so on. Obviously their valuation will be much less as in the first solution, but more interesting question is whether the second solution incentive username_0patible?

**Tags:** gt.game-theory

> **Comments:**
> **username_1**: What you are describing is the generalized second price auction, which is not, in general, incentive username_0patible. The equilibria of such auctions have been studied extensively in recent years. E.g.,   http://faculty-gsb.stanford.edu/ostrovsky/papers/gsp.pdf  

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## 1 answers

### Answer by username_2 on 2013-05-29. Score: 1 | Accepted answer

This is not quite the same as the usual generalized-second-price-auction setting because you are assuming the items are identical, whereas in a GSP setting there is an order on the items (slot A is better than slot B is better than ...).

But anyway, your auction is not dominant-strategy-incentive username_0patible. Suppose the bidders' true valuations are $v\_1 > v\_2 > \dots > v\_k > \dots > v\_n$, and suppose every bidder reports her valuation truthfully.

The first bidder wins an item and pays $v\_2$. If she had bid $v\_k + \epsilon$, she would have still won an item, but only paid $v\_k$. So she would have preferred to lie and bid $v\_k + \epsilon$.


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