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The problem of mismatched decoding with an additive metric $q$ for a discretememoryless channel $W$ is addressed. The "product-space" improvement of therandom coding lower bound on the mismatch capacity, $C_q^{(\infty)}(W)$, wasintroduced by Csisz\'ar and Narayan.  We study two kinds of decoders. The {\it $\delta$-margin mismatched decoder}outputs a message whose metric with the channel output exceeds that of all theother codewords by at least $\delta$. The {\it $\tau$-threshold decoder}outputs a single message whose metric with the channel output exceeds athreshold $\tau$. Both decoders declare an error if they fail to find a messagethat meets the requirement. It is assumed that $q$ is bounded.  It is proved that $C_q^{(\infty)}(W)$ is equal to the mismatch capacity witha constant margin decoder. We next consider sequences of $P$-constantcomposition codebooks, whose empirical distribution of the codewords are atleast $o(n^{-1/2})$ close in the $L_1$ distance sense to $P$. Using the CentralLimit Theorem, it is shown that for such sequences of codebooks the supremum ofachievable rates with constant threshold decoding is upper bounded by thesupremum of the achievable rates with a constant margin decoder, and thereforealso by $C_q^{(\infty)}(W)$.  Further, a soft converse is proved stating that if the average probability oferror of a sequence of codebooks converges to zero sufficiently fast, the rateof the code sequence is upper bounded by $C_q^{(\infty)}(W)$. In particular, if$q$ is a bounded rational metric, and the average probability of errorconverges to zero faster than $O(n^{-1})$, then $R\leq C_q^{(\infty)}(W)$.Finally, a max-min multi-letter upper bound on the mismatch capacity that bearssome resemblance to $C_q^{(\infty)}(W)$ is presented.

Converse Theorems for the DMC with Mismatched Decoding