  Evolutionary game theory has proved to be a powerful tool to probe the
self-organisation of collective behaviour by considering frequency-dependent
fitness in evolutionary processes. It has shown that the stability of a
strategy depends not only on the payoffs received after each encounter but also
on the population's size. Here, we study $2\times2$ games in well-mixed finite
populations by analysing the fixation probabilities of single mutants as
functions of population size. We proved that 9 out of the 24 possible games
always lead to monotonically decreasing functions, similarly to fixed fitness
scenarios. However, fixation functions showed increasing regions under 12
distinct anti-coordination, coordination, and dominance games. Perhaps
counter-intuitively, this establishes that single-mutant strategies often
benefit from being in larger populations. Fixation functions that increase from
a global minimum to a positive asymptotic value are pervasive but may have been
easily concealed by the weak selection limit. We obtained sufficient conditions
to observe fixation increasing for small populations and three distinct ways
this can occur. Finally, we describe fixation functions with increasing regions
bounded by two extremes under intermediate population sizes. We associate their
occurrence with transitions from having one global extreme to other shapes.
