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Muller's ratchet is a prototype model in mathematical population genetics. In an asexual population of constant size N, individual lineages are assumed to slowly acquire slightly deleterious mutations over the generations. Due to randomness, every once and a while the individuals with the currently smallest number of mutations disappear from the population; this is a click of the ratchet. The classical variant of the model, which assumes so-called proportional selection, so far has resisted against a fully rigorous asymptotic analysis of the clicking rate. In [1] this hurdle has been overcome by considering tournament (instead of proportional) selection, where selective competition within pairs is won by the fitter individual. In our talk we will explain the graphical construction which was used in [1] to obtain a hierarchy of dual processes for the tournament ratchet. We will apply this duality also in the "near-critical" regime. We will reveal the form of the type-frequency profile between clicks of the ratchet, as well as the asymptotic click rates in various subregimes. Finally, we will discuss the mapping which takes (m; s) into the corresponding parameter pair of the "classical" ratchet so that the click rates have similar asymptotics under appropriate approximations, and will illustrate this by simulations. The talk is based on joint work in progress with A. Gonzalez Casanova, Ch. Smadi and A. Wakolbinger.