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.


1 Answer 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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