The most influential egalitarian perspective is undoubtedly Rawls's (1971, 1993), which assigns absolute priority to the least advantaged in society (the difference principle). However, many have claimed that even though an egalitarian perspective should imply some priority to the worst off, the Rawlsian perspective is too demanding. One response to this criticism is to argue in favour of an egalitarian perspective that never assigns absolute priority to the worse off, but which still includes limited priority to those members of society in distributive conflicts. A different response to the demandingness criticism is to agree that the worse off should not always be given absolute priority, but to argue that there are some cases where they should be. In this paper, we elaborate on this view, and look at the possibility of deviating from the leximin approach within this category of egalitarian principles.