  Replicator equation -- a paradigm equation in evolutionary game dynamics --
mathematizes the frequency dependent selection of competing strategies vying to
enhance their fitness (quantified by the average payoffs) with respect to the
average fitnesses of the evolving population under consideration. In this
paper, we deal with two discrete versions of the replicator equation employed
to study evolution in a population where any two players, interaction is
modeled by a two-strategy symmetric normal-form game. There are twelve distinct
classes of such games, each typified by a particular ordinal relationship among
the elements of the corresponding payoff matrix. Here we find the sufficient
conditions for the existence of asymptotic solutions of the replicator
equations such that the solutions -- fixed points, periodic orbits, and chaotic
trajectories -- are all strictly physical, meaning that the frequency of any
strategy lies inside the closed interval zero to one at all times. Thus, we
elaborate which of the twelve types of games are capable of showing meaningful
physical solutions and for which of the two types of replicator equation.
Subsequently, we introduce the concept of the weight of fitness deviation that
is the scaling factor in a positive affine transformation connecting two payoff
matrices such that the corresponding one-shot games have exactly same Nash
equilibria and evolutionary stable states. The weight also quantifies how much
the excess of fitness of a strategy over the average fitness of the population
affects the per capita change in the frequency of the strategy. Intriguingly,
the weight's variation is capable of making the Nash equilibria and the
evolutionary stable states useless by introducing strict physical chaos in the
replicator dynamics based on the normal-form game.
