Someone who wants everything to be even

I'm looking for a word that would describe someone's behaviour or personality which says that they enjoy it when things are fair. So, for example, when they win money as part of a team, they would be happiest if every member got the same amount, or if they were splitting a cake they would be happiest if everyone got the same size.

I've thought of altruist, communist, etc. but these don't quite describe what I'm looking for.

Egalitarian:

adjective 1. of, relating to, or upholding the doctrine of the equality of mankind and the desirability of political, social, and economic equality

noun 2. an adherent of egalitarian principles

... her strength is the direct product of egalitarian Scandinavian society, where gender roles are merged rather than sharply defined.

(Collins Dictionary)

Well, seeing as a synonym of "even" is "level", perhaps you could use the word leveller:

1. a person or thing that levels

2. a person who works for the abolition of inequalities

^{Collins English Dictionary}

Note that the use of this word in your mentioned situations isn't established usage. As Michael Seifart points out in a comment, definition number 2 stems from a 17th-century English social movement by that name.