# The influence of restricted type space on incentive compatible mechanism design?

When we are to design an IC mechanism for an auction, we usually assume that each agent $$i$$'s type $$\theta_i$$ is draw from its type space $$\Theta_i$$, which contains all the possible types of $$i$$. For instance in a single item auction, we may assume that $$\theta_i \in \mathcal{R}^+$$, i.e., $$\Theta_i = \mathcal{R}^+$$.

In some scenario, prior information about the type space can be acquired by the auctioneer by some means. For example, the auctioneer may confirm that $$\theta_i \le M$$ before designing any mechanism. That is, the maximum money each agent can pay is no more than $$M$$. The auctioneer may obtain the information from some statistic query in data market, if we have to explain how she gets it. Then we can validly assume that $$\Theta_i$$ is $$[0, M]$$ instead of $$\mathcal{R}^+$$. Suppose some agent can only benifit himself by misreporting some $$\theta_i^{\prime} > M$$. Then we have no worry about truthfulness because now $$\Theta_i = [0, M]$$. Note that agents would not report higher than $$M$$ because they are informed of the auctioneer's access to the statistics(privacy issue is not considered here).

The above example may not be well constructed, however I'm considering similar setting where some constraints on agents' type space may make truthful mechanisms easier to get, or even make nontruthful mechanism truthful. I failed to find out any literature on this topic. I appreciate if someone can provide guidance on where to look for such research and whether this is valid as a research point.

The type space is the support of the belief about the types. It seems a bit weird to allow a mechanism designer to arbitrarily restrict their belief support. However, you are right: if the designer can find out more about the private types by acquiring information, this would intuitively allow them to reduce information rent, because it makes the type "less private." Of course it would be best to fully unveil types such that no incentive for revelation has to be provided and full surplus extraction becomes possible. In your constructed example it may not be helpful to exclude very high types $$>M$$ as more profitable deviations are often found in closer vicinity to the real type. That depends on the problem at hand of course.