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Two problems are studied in this paper. (1) How much external or internalinformation cost is required to compute a Boolean-valued function with an errorat most $1/2-\epsilon$ for a small $\epsilon$? It is shown that informationcost of order $\epsilon^2$ is necessary and of order $\epsilon$ is sufficient.(2) How much external information cost can be saved to compute a function witha small error $\epsilon>0$ comparing to the case when no error is allowed? Itis shown that information cost of order at least $\epsilon$ and at most$h(\sqrt{\epsilon})$ can be saved. Except the $O(h(\sqrt{\epsilon}))$ upperbound, the other three bounds are tight. For distribution $\mu$ that is equallydistributed on $(0,0)$ and $(1,1)$, it is shown that $IC^{ext}_\mu(XOR,\epsilon)=1-2\epsilon$ where XOR is the two-bit xor function. This equalityseems to be the first example of exact information complexity when an error isallowed.

Trading information complexity for error II: the case of a large error and external information complexity