# Incentive compatibilty conditions for multi-item auctions

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.

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 $$t\in (0,1)$$.