<p>Yes, it is essentially the same idea as with just one unit.
For example, see the text leading to Proposition 14.1 in the book &quot;Auction Theory&quot; by Vijay Krishna.</p>
<p>Let <span class="math-container">$x$</span> be the vector of willingness-to-pay.
Define <span class="math-container">$U(x) = \max_z \{ q(z)x - m(z)\}$</span> as the equilibrium utility of type <span class="math-container">$x$</span>. The incentive compatibility (I am quoting Krishna now) &quot;implies for all <span class="math-container">$x$</span>,
the probability vector <span class="math-container">$q(x)$</span> is a subgradient of the payoff function <span class="math-container">$U$</span>, which
is convex, at the point <span class="math-container">$x$</span>. In other words, the vector <span class="math-container">$q(x)$</span> is perpendicular to
the hyperplane that supports the function <span class="math-container">$U$</span> at <span class="math-container">$x$</span> — the graph of the function <span class="math-container">$U$</span>
lies above the hyperplane.&quot; He goes on to show that
<span class="math-container">$$U(x) = U(0) + \int_0^1 q (t \cdot x) x d t,$$</span>
where instead of directly integrating over the type <span class="math-container">$x$</span> you integrate  over <span class="math-container">$t\in (0,1)$</span>.</p>

CC BY-SA 4.0
2020-09-09T20:05:46.077Z
2020-09-09T20:05:46.077Z
Bayesian
<p>In <a href="https://www.eecs.harvard.edu/~parkes/cs286r/spring07/papers/myerson.pdf" rel="nofollow noreferrer">this</a> paper by Myerson on optimal single item auctions, the incentive compatibility condition is shown to be equivalent to the following simpler conditions on the auction - (i) the allocation probability being monotonic in bid (ii) the form of the payment rule.</p>

<p>Is there a similar characterization for multi-item auctions? Any reference regarding this would be helpful.</p>

<reference-request><auctions><mechanism-design><incentives>
Incentive compatibilty conditions for multi-item auctions
Aniruddha Acharya
# Incentive compatibilty conditions for multi-item auctions

(Asked by: username_0 on 2020-01-02)

In [this](https://www.eecs.harvard.edu/~parkes/cs286r/spring07/papers/myerson.pdf) paper by Myerson on optimal single item auctions, the incentive compatibility condition is shown to be equivalent to the following simpler conditions on the auction - (i) the allocation probability being monotonic in bid (ii) the form of the payment rule.

Is there a similar characterization for multi-item auctions? Any reference regarding this would be helpful.

**Tags:** reference-request, auctions, mechanism-design, incentives

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## 1 answers

### Answer by username_1 on 2020-09-09. Score: 1

Yes, it is essentially the same idea as with just one unit.
For example, see the text leading to Proposition 14.1 in the book "Auction Theory" by Vijay Krishna.

Let $x$ be the vector of willingness-to-pay.
Define $U(x) = \max\_z \{ q(z)x - m(z)\}$ as the equilibrium utility of type $x$. The incentive compatibility (I am quoting Krishna now) "implies for all $x$,
the probability vector $q(x)$ is a subgradient of the payoff function $U$, which
is convex, at the point $x$. In other words, the vector $q(x)$ is perpendicular to
the hyperplane that supports the function $U$ at $x$ — the graph of the function $U$
lies above the hyperplane." He goes on to show that
$$U(x) = U(0) + \int\_0^1 q (t \cdot x) x d t,$$
where instead of directly integrating over the type $x$ you integrate over $t\in (0,1)$.


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