In this 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.


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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