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Can the tools of mechanism design be applied when the decision space $D$ depends on the types $\theta$? For example, when $d \in D$ represents the total funding allocated toward some good and $\theta_i$ represents the funds $i$ has available, $D = \left\{ d \middle| 0 < d < \sum\limits_i\theta_i\right\}$. If not, is there some other approach that analyzes this sort of scenario?

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Yes, they can, because the utility function typically depends on the type. A particular action unavailable to a type can be modelled as that action yielding so low utility that it is never chosen no matter what other players do. Then apply standard mechanism design to the appropriately modified utility function.

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